FLOW IN A DEFORMABLE POROUS MEDIUM
A T H ES IS
SUBM ITTED TO THE DEPARTMENT OF MATH EMATICS
UN I VERSITY COLLEGE
UNIVERSITY OF N EW SO UTH WALES
AUSTRALIAN DEFENCE FORCE A C ADEMY
FOR THE DEG REE OF
DOCTOR OF PH I LOSOPHY
By Steven I an Barry December 1 990 © Copyright 1990 by Steven Ian Barry
11 Certificate of Originality
I hereby declare that this submission is my own work and that, to the best of my knowl edge and belief, it contains no material previously published or written by another person nor material which to a substantial extent has been accepted for the award of a ny other degree or diploma of a university or other institute of higher learning, except wh ere due acknowledgement is made in the text.
I also hereby declare that this thesis is written in accordance with the University's Policy with respect to the Use of Project Reports and Higher Degree Theses.
S. I. Barry
11 l Acknowledgements
This thesis was completed while on a Teaching Fellowship at the Mathematics Department, University College, UNSW and while on a British Council Bursary to the Physiological Flow Studies Unit (PFSU), Imperial College, London. This financial support is gratefully acknowl edged.
I would like to acknowledge the guidance and supervision of Geoff Aldis who has been both an excellent supervisor and friend. Also Kim Parker of the PFSU who supervised me during my stay in London. I would also like to thank Peter Winlove, John Lever, Phil Broadbridge, Jim Hill and Neville de Mestre for some interesting discussions. I would also like to thank Glenn Fulford and Geoff Aldis for their instructive comments while writing this thesis.
The assistance of all my associates in the Mathematics Department (ADFA) deserves par ticular mention, especially the administrative support given me by Colin Pask and invaluable computer support by Ian French, Patrick Tang, Ted Catchpole and Rowland Sammut.
I would like to thank my friends who helped me avoid insanity in the last few years. My good friends Rainer Ignetik and Neil Viney for wasting so much of my time discussing philosophy, women, politics and anything else we could think of to kill a Friday afternoon. Rod Weber for being such a clown. Ted Catchpole for introducing me to the masochism of running and actually encouraging me to do triathlons. Lucy Sargent who saw me through the early years. My close friends Diana Lucic and Remco Marcelis and some of my other friends who helped, Mark Jay, Wen Wang, Rhonda Perriman, Liz Cole and Lindy Spindler.
The ability and enthusiasm of my former lecturers and teachers should also be recognised, especially Chris Packer, Br. Brendon O'Hearn, Robyn Denton, Tony Roberts, Ernie Tuck, Bill Henderson and Tony Thomas.
I would like to thank my parents, Jack and Jenny Barry, for being so supportive, always listening and putting up with me during my undergraduate years.
IV This thesis is dedicated to my parents Jack and Jenny Barry
V Abstract
This thesis is a mathematical investigation of the flow of fluid through elastic porous materials. The passage of fluid through the porous medium deforms the material causing a coupling between the solid deformation and the fluid flow.
A summary of the theory of flow through deformable porous media and an overview of the applications of such flows to biological tissue deformation are given. To gain a better understanding of the fundamental dynamics of the solid-fluid interaction various practical applications of the theory are studied: The one dimensional flow-induced compression of a porous material, flow of fluid in a channel lined with a porous layer, flow radially out through cylindrical and spherical shells and high speed flow into a rigid porous material.
These geometries were considered as they can be used to model biological problems such as the permeation testing of biological tissues, flow in an artery lined with a thin porous gel, the permeability of the artery wall and flow in a placenta.
Solutions are found to the governing nonlinear diffusion equations using various analyti cal and numerical techniques. Various models for the flow-induced deformation of a porous material are examined using published experimental data, leading to a nonlinear finite defor mation model being proposed. The effects of the nonlinearities on unsteady one dimensional deformation are considered. By developing and solving the full coupled equations for flow over a thin porous layer a new boundary condition for the fluid at the porous layer surface is proposed. A general set of equations and solutions for radially directed flow in either cylin drical or spherical geometries are given. A comparison between equations governing low and high speed flow in a porous material are also considered.
VI Contents
Certificate of Originality Ill
Acknowledgements IV
Abstract VI
1 Introduction 1
1.1 Historical Perspective 1
1.2 Thesis Outline .... 5
2 Deformable Porous Media: A Review 8
2.1 Assumptions ...... 8
2.2 Mixture Theory for a Biphasic Medium 9
2.2.1 Continuity Equation . 10
2.2.2 Momentum Equations 10
2.3 Permeability a Function of the Strain 12
2.4 Stress Relation ...... 14
2.4.1 Infinitesimal Deformations . 14
2.4.2 Finite Deformations .... 15
2.5 Boundary Conditions at Porous Medium-Fluid Interface 17
2.6 Viscoelastici ty ...... 19
2.6.1 Stress Relaxation 20
2.6.2 Creep . . . 25
2. 7 Steady Permeation 27
2.8 Radial Flows . . . 28
2.9 Further Modeling of Deformable Tissues 29
vu 3 Steady Flow and Deformation 30 3.1 Flow Geometry and Governing Equations 30 3.2 Infinitesimal Models .... 32
3.3 Finite Deformation Models 34 3.4 Results ... 36 3.5 Discussion . 40
4 Unsteady Flow-Induced Deformation 42
4.1 Mixture Theory for Unsteady Flow 42
4.2 Nonlinear Diffusion Equation 45
4.3 Infinitesimal Deformations .. 46 4.3.1 Exact Solution for given v(t) 47 4.3.2 Exact Solution for given t:i..P(t) 47 4.3.3 Small Time Solution for given t:i..P(t) . 48 4.4 Perturbation Analysis for Slowly Varying Pressure 48
4.4.1 Solution for given v(t) ...... 48 4.4.2 Small Time Solution for given v( t) 49
4.4.3 Solution for given t:i..P(t) ...... 50
4.5 Singular Perturbation Analysis for Rapidly Varying Pressue 52 4.5.1 Solution for given v(t) .. 52
4.5.2 Solution for given t:i..P( t) . 53
4.6 Numerical Solution for Finite Deformations 53 4. 7 Results and Discussion 54
4.8 Conclusions ...... 55
5 Fluid Flow Over a Thin Deformable Porous Layer 59
5.1 Mathematical Model ...... 60
5.2 Flow Geometry and Governing Equations 61
5.3 Steady State Deformation 63 5.4 Rigid Porous Layer .... 64
5.5 Deformable Porous Layer 65 5.6 Results and Discussion 66
5. 7 Conclusions ...... 71
VIII 6 Radial Flow Through Shells 72 6.1 Geometry ...... 73
6.2 Governing Equations 74
6.3 Boundary Conditions . 75
6.3.l Constrained Flow. 75 6.3.2 Unconstrained Flow 76
6.4 Steady Radial Flow . . . . 77 6.4.l General Solutions . 78 6.4.2 Solution for Constrained Flow. 78 6.4.3 Solution for Unconstrained Flow 79
6.5 Perturbation Solution for Infinitesimal Displacements . 80
G.6 Small Time Solutions in Unsteady Flow 81 6.6.1 Cylindrical Geometry ...... 81 6. 7 Perturbation Solutions for Slow Compression Rates . 82
6.8 Results and Discussion 83 6.9 Conclusions ...... 87
7 High Speed Flow in a Rigid Porous Medium 88
7.1 Introduction ...... 88 7. l. l Inertial Effects 89 7.1.2 Forcheimer's Law . 89 7.1.3 Theoretical Backing for Forcheimer's Law 90 7.1.4 Gradient Form for Forcheimer's Law 91
7.1.5 Darcy Flow versus Nonlinear Flow 91
7.1.6 Flow into Boxes 92 7.2 Method of Solution ... 93
7 .3 Results and Discussion . 96
8 Summary and Discussion 99 8.1 Summary .... 99
8. 2 Future Research 102
IX Appendices
A Mathematical Relations 105 A.l Modified Bessel Functions Iv, Kv 105
A.2 Laplace Transforms . 105
A.3 Error Function ... 106
A.4 Exponential Integral 106
B Notation 107
C Theoretical Relations 110
C.1 Proof of k = 7/K . 110 C.2 Relation between apparent permeability and intrinsic permeability 111
C.3 Relation between the porosity and strain in one dimension . 112
C.4 Relation between the porosity and strain in two dimensions 113
C.5 Relation between initial and final radii in cylindrical deformations 114
D Algebraic Results 116
D.l Integrals from chapter 3 116
D.2 Integrals from chapter 6 on steady radial flow 117
D.2.1 Constrained flow, k = 1, n = 1 .... 117
D.2.2 Constrained flow, k = exp(m), n = 1 118
D.2.3 Constrained flow, k = 1/(1 - m), n = 1. 118
D.2.4 Constrained flow, k = 1, n = 2 .... 119
D.2.5 Constrained flow, k = exp(m), n = 2 119 D.2.6 Constrained flow, k = 1/(1 - m), n = 2. 119 D.3 Constants from equation (6.55) ...... 120
D.4 Solution for unsteady radial flow, equation (6.57) 120
D.5 Constants from equation (6.68) 122
E Computational Scheme 123
E. l Euler-Cauchy Numerical Scheme 123
Bibliography 124
X Chapter 1
Introduction
This thesis is a mathematical investigation of the flow of fluid through porous materials. I consider the situation in which the porous medium deforms due to the passage of fluid, leading to a coupling between the solid deformation and the fluid pressure. I will provide a detailed summary of the theory of flow in deformable porous media and by considering various practical applications of such flows, attempt to gain a better understanding of the fundamental dynamics involved. Such problems as one dimensional flow-induced compression, flow of a fluid past a porous layer and radially directed flow will be considered. A brief chapter concerning the high speed flow of fluid in a rigid porous medium is also included.
Throughout the thesis the biological applications of these situations will be discussed although no direct biological modeling will be done. Biological tissues provide interesting and relevant examples of this work and are the original motivation for many of the problems discussed. Such applications as flow in placental subunits, permeation of fluid through the artery wall, flow within an artery lined with a thin porous gel and permeation experiments on general soft tissue will be considered. It is not my intention to accurately model these biological tissues, more to use them as motivation for understanding the solid-fluid interaction in a variety of applicable geometries.
In this introductory chapter I will give a brief overview of the historical development of flow through porous materials and then give a brief description of the problems covered in this thesis.
1.1 Historical Perspective
The flow of fluid into a porous medium has been studied in a number of diverse fields such as soil mechanics, engineering, chemistry, physics and biology. In hydrology, the physics of fluid flow through the soil relates to irrigation, dam construction, soil degradation and
1 1. Introduction 2
salinity problems. The consolidation ( compaction) of soils under buildings as fluid moves out from the soil is of prime concern to civil engineers as are the fluid-solid interactions in earthquakes. In engineering, the design of filters and separation columns requires a good knowledge of porous media mechanics. Biology provides an interesting source of examples of porous materials. The flow of synovial fluid through articular cartilage, passage of proteins in the arterial wall, formation of decubitus ulcers (pressure or bed sores), blood flow in placental subunits, swelling of tissues and most soft tissue mechanics are all examples where an understanding of the mechanism of fluid flow through a porous solid matrix is essential.
Henry Darcy (also written d'Arcy) in 1856 was the first to study the passage of fluid through porous materials. As an engineer engaged in the modification of the water works in the French town of Dijon, Darcy was concerned with the amount of sand required to filter a given flux of water. As an appendix to his report, Darcy described an experiment he performed for flow through a cylindrical column of sand. The relationship he found between the applied pressure difference and the flux of fluid can be written in the form,
Q = k~p ( Ll) I where Q is the fluid flux, ~p is the difference in pressure over a length I of porous medium and k is the permeability which is normally a constant dependent on the type of porous material. The permeability represents the ease with which fluid can flow for a given pressure gradient. The above equation can be generalised to 1 Vp = --v, ( 1.2) k where p is the pressure and v is the velocity vector for the macroscopic motion of the fluid. This is known as Darcy's law.
Darcy's law forms the basis for most models of flow through porous media. Further modifications to this law have since been established to include such effects as high speed flow ( chapter seven) and more recently flow in deformable materials ( chapter two).
When fluid moves through a porous material the pressure gradient forcing this flow may be sufficient to deform the porous material. Alternatively if a porous material (or sponge) is compressed, the internal fluid will be forced to move within and out of the sponge. During this compression the properties of the material will change, in particular the permeability. Compression reduces the pore size in the material increasing the resistance to the passage of fluid. Hence there is a complex interaction between the deformation of the solid and the movement of the fluid.
The first studies to be done on this coupled process were by Terzaghi ( 192.S) and Biot ( 1941) on the consolidation of soils under loading culminating in the work of Biot ( 1956a,b ). Essentially these works considered the flow from a compressed region of sand. Biot extended his theory to include anisotropy (Biot 1955), compressible materials and wave propagation 1. Introduction 3
(Biot 1962). These equations have been verified by Burridge and Keller (1981) using an averaging technique with Berryman (1980) and Plona (1980) confirming the wave propagation results of Biot experimentally.
The application of these theories to biological tissues essentially began with the work of Kenyon (1976b,c) who developed a theory for solid-fluid (biphasic) mixtures which was then applied to the radial flux of fluid through the walls of a porous cylinder, a model of flow through arterial tissue (Kenyon 1976a, 1979). The theory was then developed further by Mow, Lai and co-workers (Holmes, Hou, Kwan, Mak) in a series of papers from 1977. Their main application was the lubrication properties of diarthrodial joints, the knee being a classic example. The main premise of their modeling was that articular cartilage acts as a porous sponge saturated with synovial fluid. As the cartilage is compressed the fluid moves through the pores to form a surface lubricating layer. The biomechanical modeling of cartilage is a difficult problem but excellent reviews are by Mow and Lai (1979, 1980), Mow et al. ( 198ci) and Mow et al. ( 1990).
Mow and co-workers used a mixture theory based on the work of Bowen ( 1980), Truesdell and Toupin (1960) and Truesdell and Knoll (1965). In developing their theory, which results in similar equations to Biot's, they have considered the difference between apparent and intrinsic permeabilities (Holmes 1985) and found an appropriate form for the dependence of the permeability on the local strain (Mow and Mansour 1977, Lai and Mow 1980). They also considered a nonlinear stress-strain relation for the tissue by using finite deformation theory (Mow et al. 1985, Holmes 1986, Kwan et al. 1990) and have recently considered a poro-viscoelastic theory where the porous solid is modeled as being intrinsically viscoelastic rather than just elastic (Mak 1986, Setton et al. 1990). The appropriate boundary condition between the fluid and the porous medium has been considered by Hou et al. (1989) which improves on the work of Taylor (1971), Richardson (1971) and Saffman (1971). Further applications to the theory of swelling tissues has also been considered by Lai et al. ( 1990) with an extension to a tri-phasic theory of cartilage.
These theoretical developments have been coupled with a study of the viscoelastic prop erties of the cartilage, specifically those of creep and stress relaxation 1, due to the internal movement of the fluid within the tissue (Eisenfeld et al. 1978, Mow et al. 1980, Lai et al. 1981, Holmes 1983, Holmes et al. 1985, Mow et al. 1985, Holmes 1986, Spilker et al. 1990). Other problems concerned with indentation (Mak et al. 1987), unconfined compression (Armstrong et al. 1984) and lubrication of the cartilage (Hou et al. 1990) have been considered by this group.
Other theoretical developments have been made by Parker et al. ( 1987) who considered the flow through a polyurethane sponge and Lanir ( 1987) who considered a model for swelling
1 After application of a constant force a tissue may creep towards its final displacement. After applicatiou of a constant displacement the force required to hold this displacement decreases or the stress relaxes. 1. Introduction 4
tissues. In chapter three I consider the experiment of Parker et al. ( 1987) on polyurethane sponges to compare various models for one-dimensional flow-induced compression. Because the polyurethane sponge is much simpler in structure than biological tissue the essential components of solid and fluid interaction can be more readily ascertained.
When the fluid flux is large the drag force between the solid and the fluid becomes nonlinear. The effect of nonlinear drag laws on flow through deformable porous material has been considered experimentally by Beavers et al. (1975) and Beavers et al. (1981a,b) who used nonlinear quadratic velocity terms to represent the effect of high flow rates on the drag within the medium. The effect of inertia has also been considered by Madhav and Basak ( 1977) who used a power law for the velocity to account for inertial effects and Basak and Madhav ( 1978) who include the time dependent inertial term with only the linear Darcy drag term being included. The effect of nonlinear drag laws on the motion of a fluid in a rigid porous medium will be considered in chapter seven.
The 'poro-elastic'2 theory developed for articular cartilage has been adapted for use in a number of different biological applications. Skin deformation has been modeled by Oomens et al. (1987a,b) who developed finite element algorithms for two dimensional deformation. The indentation of skin has also been modeled by Lanir et al. (1990). The artery wall has been modeled by Kenyon (1976a, 1979), Jayaraman (1983), Klanchar and Tarbell (1987) and Jain and Jayaraman (1987) giving rise to new approximate forms for the permeability and developing the theory to include cylindrically orientated flows. I have illustrated a general theory and solution method for radial flow through cylindrical and spherical shells in chapter
SIX.
Other biological systems studied have been the intervertebral disk (Simon et al. 1985), developing chick heart (Yang and Taber 1990) and lung tissue (Sachs et al. 1990, Ford et al. 1990).
The applicability of poroelastic modeling has been noted for myocardium (heart muscle) (Tsaturyan et al. 1984, Sorek and Sideman 1986), the formation of decubitus ulcers (Reddy et al. 1981), corneal stroma swelling (Friedman 1971), injection into brain tissue (Nicholson 1985), blood flow in placental subunits (see chapter seven, Erian 1977, Aifantis 1978) and general flow through the interstitium (Levick 1987). In chapter seven I consider flow in a crude model of a placental subunit by solving the nonlinear flow through a rigid porous medium.
It is also worth noting some of the work of Parker and Winlove ( 1988) who have looked into the structure of biological tissue in terms of its component parts, particularly looking at the permeability of the collagen and proteoglycans. They have proposed in Winlove and Parker {1990) that the elastin fibres are porous and fluid may move within these pores.
2 The term poroelastic refers to a theory or model that incorporates the elasticity of a porous material. 1. Introduction 5
Other work on the flow through deformable porous materials has been by Kubik ( 1986a,b) who has incorporated pore structure into his models and Auriault (1980) who also has incor porated information at the pore scale. Rice and Cleary ( 1975) also derive the basic diffusion equations for the stress in their model of para-elasticity. Micro-inertial effects have been considered by Thigpen and Berryman (1985) in their mixture model of multiphase flow. These were designed to model the flow in soils. Silberberg (1989) also developed a model for transport of a macromolecular solute through deformable matrices. Lew and Fung ( 1970) considered a model of dense networks of tubules in their model of deformation of a porous material and Palciauskas and Domenico ( 1989) considered a model of pressure in deforming rocks.
1.2 Thesis Outline
The general theory used in this thesis, based on the work of Mow and co-workers will be outlined in chapter two. Here I will use mixture theory to develop the governing equations and discuss some of the various forms for these governing equations. This is not original work since papers by Mow et al. (1990), Mow et al. (1985) and Holmes (1986) cover much the same ground. In the absence of a book covering deformable porous materials I have rewritten this theory in a form that I feel contains all the necessary information but without discussing the underlying mixture theory in too much depth. A more detailed review of some of the more useful results from the literature will also be included in chapter two.
Chapter three contains an analysis of one dimensional steady flow-induced compression using the data of Parker et al. (1987) to examine various permeability and stress relations. The suitability of using biological based models to examine polyurethane sponge deformation is considered. The aim of this chapter is to find the most suitable theory applicable to a variety of situations. As a result, a finite-deformation model with permeability dependent exponentially on strain is proposed. This chapter has been published in slightly modified form in Barry and Aldis (1990a).
Unsteady deformation is important since many biological applications involve oscillatory flow and experiments on tissues often involve a step change in pressure. In chapter four the unsteady flow of fluid through a one-dimensional porous medium is considered. The deformation profile for various given time-dependent velocities will be used to gain a better understanding of the interaction between the fluid and the solid. Both analytical and nu merical results will be given which show that the nonlinear effects of deformation can lead to some unexpected results, such as a possible increase in mean flow rate through a tissue by varying the pressure around a mean value. This chapter forms the basis of the paper Barry and Aldis (19906). 1. Introduction 6
When the deformation is extended to two or three dimensions, shearing effects must be considered. In chapter five I will consider the problem of the flow of fluid through a channel that is lined with a thin deformable porous material, a situation occuring in arteries. Using a linear theory, the deformation and fluid flux through the porous layer will be given for both steady and unsteady pressure gradients. This will include a novel boundary condition for the flow in the channel, at the porous media interface. This replaces the conventional no slip condition for cases in which the channel wall is layered with a thin porous material.
In biology many problems may be modeled by spherical or cylindrical geometries. In chapter six general equations and solutions are derived for radial flow through both cylin drical and spherical shells. Exact integral solutions are found for steady flow. Small time solutions for unsteady flow are found and perturbation solutions for slow compression rates are illustrated. Detailed results for the flow through a cylindrical shell constrained at the outer boundary by a rigid mesh will be given.
With high speed flow, there exist a number of models used extensively in the literature, the most appropriate beiug first postulated by Forcheimer in 1901. This is a one dimensional extension of Darcy's law involving the addition of a quadratic velocity term which is written here in vector form ( 1.3) where c is a constant, PJ is the fluid density and"'= k/µJ is another measure of the permeability, common in the soil literature. It was found that this equation was the most versatile of the existing high speed models with significant experimental verification and theoretical backing. This is discussed in detail in the seventh chapter along with a comparison of other models and a review of the literature on high speed flow through a porous medium. Equation (1.3) will be used to analyse flow in a two-dimensional box geometry. By using the stream function, the difference between Darcy flow (flow calculated using Darcy's law) and high speed flow (using equation 1.3) will be analysed for two box geometries that model blood flow in the placenta and industrial filters. It is shown that the use of the high speed model predicts moderate changes in the streamline patterns with more fluid moving into the corners of the box.
In appendix (A) I give a list of some of the nontrivial mathematical identities used in this thesis. These are adapted from Abramowitz and Stegun (1972). The second appendix (B) contains a list of the notation that is common throughout the text although notation is described as it arises in the text. Some useful theoretical results regarding the development of poroelastic theory are given in appendix (C). Appendix (D) contains some tedious algebra removed from the main text. Appendix (E) describes a numerical scheme used in the text.
As a notational guide all boldface characters are assumed to be either vectors or tensors. In referencing material the name of the author(s) will be given, followed by the date of 1. Introduction 7
publication and in the case of large publications the appropriate page numbers. The symbol K will be used for the permeability in chapter seven and the symbol k = K/ µ 1 will be used in the other chapters on deformable porous materials but will still be referred to as the 'permeability'. This is to conform to the notation which varies between the soil and biological literature. Chapter 2
Deformable Porous Media: A Review
This chapter deals with the background theory and general developments in the 'poro-elastic' modeling of flow through deformable porous materials. I will first give a general description of porn-elastic theory using 'biphasic' mixture theory (mixture theory applied to single solid and fluid phases). This development will not emphasize the governing equations for unsteady flow, radial or shear flows which are dealt with in more depth in later chapters. Some of the more relevant applications of this theory are also described. Limited reviews (Mow et al. 1984, Mow et al. 1990) have appeared on poroelasticity but as yet no book summarising this work has appeared. Therefore I have attempted in this chapter to gather the major results and methods together in a common notation. I have not included a detailed analysis of mixture theory as this has been done already (Bowen 1980, Bedford and Drumheller 1983) and would distract from the poroelastic modeling considered in this thesis.
2.1 Assumptions
The assumptions that I will hold implicit to this derivation and the rest of the thesis are:
1. There exist only two components, a viscous fluid and an elastic solid matrix.
2. Each of the components are intrinsically incompressible.
3. There exist no external body forces such as gravity.
4. The elastic solid matrix is not intrinsically viscoelastic.
5. No osmotic pressures are present.
6. The medium is initially isotropic and homogeneous. 8 2. Deformable Porous Media - A Review 9
7. The porous medium has a structure of randomly interconnected pores.
8. All variables are assumed continuous at every point in the medium by using a standard continuum mechanical assumption based on a volume average that contains a large number of pores.
The second assumption is crucial to the theory considered in this thesis. The bulk com pression of the medium occurs due to a change in the relative volume fractions of the solid and fluid phases, not due to any compression of the constituents themselves. Thus a squeezing out of fluid leads to a reduction in porosity and hence a bulk compression of the mixture.
In this chapter it is also assumed that viscous shear in the fluid is negligible and that inertial effects can be ignored. These assumptions are relaxed in later chapters and hence discussed in more detail there. Many of these assumptions are not valid for a variety of phys iological tissues which are anisotropic (for example muscle), possess strong osmotic pressure (for example intervertebral disks) and often have a very fine and intricate structure. The applicability of these assumptions to polyurethane sponges (Parker et al. 1987) will be shown in chapter three. The poroelastic theory based on these assumptions has also been successful in modeling a variety of biological tissues. Thus I feel that these assumptions incorporate the essential elements for a poroelastic theory and analysis of the resulting governing equations will lead to more insight into the fundamental mechanisms of porous media mechanics.
2.2 Mixture Theory for a Biphasic Medium
The following derivation follows the notation of Mow et al. ( 1985) and Holmes ( 1986) and represents the current 'basic' biphasic model for flow in elastic sponges.
The theory of mixtures is based on the idea that the individual components of the mixture (in this case the solid matrix and the fluid) can be "modeled as superimposing continua so that each point in the mixture is occupied simultaneously by a material point of each con stituent" (l3edfor 2.2.1 Continuity Equation The volumes of the two constituents are denoted v.a where /3 = s, J denotes either the solid or the fluid phase. The apparent densities of the constituents are given as dm.6 p.6 = lim --, (2.1) dV-'-+O dV where dm.6 is the mass of the /3 phase in the small volume dV. The true ( or intrinsic) density of each phase is ,a . dm.6 Pr= hm dV,,, (2.2) dVtl-+O ,_, where dV.il is the small volume occupied by the /3 phase. The relative porosities of each phase are ,a _ . dV.il 1 p PS+ pf. (2.7) The continuity equation for each phase can then be derived from simple continuum me chanics ( Mow et al. 1985) as (2.8) where v.6 represents the velocity of the /3 phase. This is derived from the conservation of mass of each phase within an elemental volume dV that contained both phases. Since the the constituents are intrinsically incompressible, p~ is constant and use of equation (2.4) in equation (2.8) yields (2.9) Adding the solid and fluid forms of this equation and using equation (2.6) leads to (2.10) 2.2.2 Momentum Equations The momentum equation for each phase can be written as (2.11) 2. Deformable Porous Media - A Review 11 where T/3 is the stress tensor for the (3 phase, b/3 is the resultant external body force and 1r/3 is a drag force between the constituents representing the internal forces due to frictional interaction between the two phases. For small velocities and deformation rates the inertial terms can be assumed negligible. A derivation of the appropriate equations with inertial terms will be given in chapter four. Excluding the effect of body forces, equation (2.11) becomes (2.12) where Newton's third law implies that 1r 8 = -1rl; the force on the solid by the fluid is opposite to the force on the fluid by the solid. The stress tensors can be modeled as - 8 K(v - v 1) - pVq/, (2.14) where a 8 represents a solid stress relation, known as the 'contact stress' (Kenyon 19766), as a function of the strain, /( is the drag coefficient of relative motion and I is the identity tensor. For the fluid phase (2.15) represents the viscous stresses acting within the fluid. The term µa is the apparent viscosity of the fluid in the porous material. Lundgren ( 1972) calculated µa for flow through an array of spheres to find µa:::::: µ1/f( 8 ) where µl is the intrinsic viscosity of the fluid and the function J( For one dimensional and radial flows the viscous stress in the fluid can be taken as zero. Hence is is assumed for the rest of this derivation that al = 0 and as = a. In chapter five al will be reintroduced and analysed in detail. Substituting equation (2.13) into (2.12) and adding both phase equations to eliminate T 3 gives Vp=V·a. (2.16) Substituting the interaction term (2.14) into equation (2.12) gives - V ( 8 p) + V · a K(v 8 - vl)- pV -V( 4>1 p) -K(vs - v 1 ) + pV Making use of 8 = 1 - 4>f leads to "v·a K(vs - vf) + s"'ilp, (2.19) 0 -K(vs - vf) + f"'ilp. (2.20) Elimination of p from these equations yields [(( s f) "'il·a= 4>f V -V . (2.21) The continuity equation can be rewritten as V · v = 0, where (2.22) is the macroscopic fluid velocity vector. Using this in equation (2.21) gives "'ilp = 'V. a=.!. (au -v) (2.23) k at ' where k = (1)2/K is the permeability, u is the displacement of the solid and au/at= v 8 • A derivation of this relationship between k and [( is given in Lai and Mow (1980) with a similar derivation given in appendix C.l. Equation (2.23) can be easily explained in physical terms. Taking Darcy's law and ex pressing it relative to the movement of the solid it is easy to obtain "'ilp = ¼(t - v). Considering the stress in the solid matrix as being governed by the standard equilibrium equation of elasticity gives V ·a= "'ilp where the gradient of pressure is acting as an internal body force on the solid matrix. 2.3 Permeability a Function of the Strain It has been noted earlier that a medium's properties change as the medium is deformed. In particular the porosity of the medium decreases with compression. Since this results in smaller pathways for the passage of fluid, the permeability would be expected to decrease. The relationship between the permeability function and the strain of the tissue is then required for substitution into the governing equations (Mow and Mansour 1977). This will involve first finding a relationship between the porosity and the strain in the medium and then analysing a simple experiment to find the functional form for k. For a one dimensional deformation the porosity is related to the strain au/ az by f f Sau (2.24) = Assuming that the permeability is dependent on the porosity, then to a first approxima tion, k = k(&u/&z). At this stage it is useful to describe the difference between the apparent and intrinsic permeabilities, ka and k. The apparent permeability is readily found from Darcy's law written as ka = vh/ b.P for applied pressure gradient b.P over a length h of medium with a fluid velocity v. In measuring this value, however, the drag of the fluid on the solid has imposed a nonuniform strain on the medium. This apparent permeability does not account for the deformation in the tissue or the possibility that some sections of the medium may be more compressed than other sections. The intrinsic permeability is the permeability of an infinitesimally small slice of tissue if it were to be magically separated from the rest of the medium without change. The apparent permeability is then some average of the intrinsic permeability. Unfortunately only the apparent permeability can be measured and the intrinsic permeability must be inferred. Consider then an experiment by Lai and Mow ( 1980) shown in schematic form in figure 2.1. A section of tissue is compressed by a given distance and allowed to settle to produce a uniform strain. A pressure difference is then applied. The resulting velocity gives the apparent permeability for the uniform strain plus that strain induced by the fluid flow. By using numerous values for the pressure difference and extrapolating down to zero pressure gradient the apparent permeability at a given strain can be found. In the limit of b.P _. 0 the medium has only the uniform strain imposed by the initial compression. Repeating this for numerous compressive strains allows determination of the permeability as a function of the strain. A simple form which fitted the experimental data was k = koexp (m ;:)- (2.25) The constant ko represents the permeability in the absence of any strain and m is a fitting constant typically between 1 and 10. As the strain increases, a limit 4>1 _. 0 is reached when the fluid has no available pathway through the medium, forcing the permeability to become zero. Thus an exponential decrease in permeability will model this limiting process over most of the strain range but will not give a zero permeability when &u/&z = 1 - 1/ It can be shown (Holmes 1985, appendix C.2) that the relationship between apparent and intrinsic permeability is 1 l(p ka = - k(-r)dr, (2.26) fp 0 where fp = b.P /Ha· This shows that ka is found from the integral of the intrinsic permeability over the range of the strains in the tissue. A corresponding result for finite deformations is also given in Holmes ( 1986 ). Inverting this expression shows that given a functional dependence for the apparent permeability the intrinsic permeability may be found. Equation (2.25) proposed by Lai and Mow ( 1980) for articular cartilage seems to be valid 2. Deformable Porous Media - A Review 14 Impermeable boundary Z=O Z=h Z=L Figure 2.1: Schematic diagram of a permeation experiment with imposed strain (Lai and Mow 1980). By compressing the porous medium between two rigid filters and applying a pressure gradient the dependence of the permeability on the strain can be found for a variety of tissues (Oomens 1987, Yang and Tarber 1990). Various other forms for t he dependence of the permeability on the strain have been used. Holmes (1986) considered a form similar to the exponential dependence but with an added factor accounting for the limit of zero porosity. Klanchar and TarbeU (1987) used k :::::: 1/(1 - mau/az) as a first order approximation to the exponential form. Also Parker et al. (1987) posed a form where the permeability is dependent exponentially on the square root of the strain. A comparison of these forms against the data of Parker et al. (1987) for a simple permeation experiment is given in chapter three and in Barry and Aldis (1990a). 2.4 Stress Relation 2.4.1 Infinitesimal Deformations The stress relation wiU be a function of the local strain in the tissue. The most common assumption is that the stress is linearly related to the strain. This is standard infinitesim al deformation theory. The stress tensor is then a 3 = >.eI + 2µe, (2 .27 ) with e = 1(vu+ (Vuf), ( 2.28) Lame constants >. , J.L and e = trace(e). 2. Deformable Porous Media - A Review 15 This form for the stress tensor leads, in one dimension, to the nonlinear diffusion equation EJ2u 1 (au ) Ha 8z2 = k (t) 8t - v(t) ' (2.29) obtained by substituting equations (2.27) and (2.28) into (2.23). In one dimension the stress is given the notation a, the displacement u and the velocity v since these variables are no longer vectors or tensors. 2.4.2 Finite Deformations For large deformations the stress can no longer be expected to be a linear function of the strain and hence a finite deformation theory is used. Applications of finite deformation to fluid flow in a deformable porous medium have been published by Mow et al. (1985), Holmes (1986) and Kwan et al. (1990). These papers form the basis of the finite deformation model shown here. Detailed developments of finite deformation are given in Bedford and Drumheller ( 1983), Bowen (1980), Fung ( 1965 ), Truesdell and Noll (1965) and Wang and Truesdell ( 1973). The Eulerian coordinates Zi, with displacements ui(z) are related to Lagrangian coordi nates (Xi) and displacements Ui(X) by Zi = X; + Ui(X), (2.30) so that (2.31) and 8 DXm 8 ----- (2.32) {)zi Oz; 8Xm This latter result converts Eulerian derivatives to Lagrangian derivatives. The l\ronecker delta is given the notation 6; 1 . The conventional way to define a nonlinear stress relation is in terms of the Helmholtz free energy function, A, which is a measure of the potential energy in a system. Kwan et al. (1990) and Mow et al. ( 1985) consider the free energy for both the solid component As and fluid component Ai and combine these using the density weighted average pA = p5 As+ pi Ai to obtain the mixture energy function A. They then assume that As = Ai = A which simplifies the resulting equations. The resulting equations then become equivalent to those of Holmes ( 1986) who considered the fluid inviscid so that Ai is independent of the deformation. The work of Kenyon ( 1976b,c) and Wijesinghe ( 1978) who derive the governing equations for fluid flow through an elastic sponge using similar methods is also noted. The finite deformation model requires a deformation gradient defined as oz; Fi1 = ax , (2.33) J 2. Deformable Porous Media - A Review 16 and the Left Cauchy-Green deformation tensor B = F(Ff. (2.34) The assumption of isotropy implies that the strain energy function can be written as a function A= A(B) = A(J1 ,h,h) for invariants trace(B) = b11 + b22 + b33, (2.35) ½((traceB) 2 - trace(B2 )), (2.36) The contact stress tensor for the solid phase can then be written in any of the forms (Mow et al. 1985), a (2.38) (2.39) (2.40) So, for example, if the flow is one dimensional then the deformation tensor is oz 8U 0 0 l+- 0 0 8X 8X F= 0 1 0 0 1 0 (2.41) 0 0 1 0 0 1 This gives the Left Cauchy-Green deformation tensor as (2.42) where y = 1 + 8U /8X. The invariants are J1 = y2 + 2, h = 2y 2 + 1, h = y2 . (2.43) The form for the free energy function used by Mow et al. ( 1985) is A= (2.44) i )3 {(3,\ + 2µ)J1 + (,\ + 2µ)Jl - (9,\ + lOµ)h}. In one dimension this leads to the stress relation a= HM [l + d1(Y - l)] (y 2 _ _!_) = HMF(y). (2.45) 4 y y2 2. Deformable Porous Media - A Review 17 Holmes ( 1986) uses a form ,e/J(J1 -3) A---- (2.46) - (h - 5)1' where 1 , n, /J are constants. This leads to the one dimensional stress (1 _ 2)n+l a(y) = HM O a*(y), (2.47) 2 where (2.48) n = /J(I-5), /J is a constant and The permeability function has been shown to be a function of the strain so in one dimen sion k = k( t ). When this is converted to Lagrangian coordinates k = k( -wt;) = k( 1 - 1/ y) To simplify the notation it is assumed that k(8u/8z) =k(y) =k(8U /8X). Thus the notation k always refers to the permeability although the precise form of this function will naturally change slightly when different coordinate systems are considered. The effect of inclusion of a finite deformation theory becomes apparent by looking at the term V · a in the one dimensional governing equation. Letting the nonlinear stress function be a(y) = H M F(y) it is easy to show H OCT = H F(y) oy = _l_ ( au - ) (2.49) Moz M y ax k(y) 8t v ' where 8/oz = (l/y)8/8X. If a new function is defined as y ((y) = HMF(y)k(y) (2.50) then the appropriate equation for the displacement is a2 u (au ) ax2 = ((y) at - v (2.51) so the effect of the finite deformation is to replace 1/k(y) by y/(k(y)F'(y)). Thus some of the analysis of equation (2.29) can easily be extended to finite deformations. The pressure variation with strain will, however, be modified by the nonlinear variation of stress with strain. 2.5 Boundary Conditions at Porous Medium-Fluid Interface The boundary conditions for the solid and fluid at solid boundaries are of zero tangential or normal velocities for the fluid and zero normal component for the solid displacement. For most of the applications considered here the solid matrix is assumed to be able to slide 2. Deformable Porous Media - A Review 18 without friction, tangentially along the rigid boundary. The boundary condition between the fluid and solid in the porous medium and a fluid flowing over this medium are nontrivial and have been discussed by a number of authors. The boundary condition between a rigid porous medium and a viscous fluid has been stud ied experimentally and theoretically by Beavers and Joseph (1967), Taylor (1971), Richardson (1971) and Saffman (1971). The result is a boundary condition for the fluid which matches the flow in the fluid to the Darcy fluid flow in the porous medium, -oq] = c(qo - v) (2.52) oz O for fluid velocity q in the channel, fluid flux v in the porous layer (given by Darcy's law), c is a constant and the subscript 'O' denotes evaluation at the boundary. This assumes, however, that a fully developed Darcy flow exists within the porous medium, which is not true for thin porous layers. An equation first proposed by Brinkman ( 194 7) for flow through a sparse array of spheres 1 2 'vp = -kV+ µa 'v V, (2.53) has also been used to model the interface between a fluid and a porous medium. This equation, proposed initially as a simple combination of Stokes flow and Darcy's law, has been theoretically verified for arrays of spheres by Tam (1969) and Lundgren (1972). It has the advantage over Darcy's law that it is of the correct order to match the fluid velocity governed by the N avier-Stokes equations in the purely fluid region. It has been shown by Kim and Russell ( 1985) that application of equation (2.53) gives the same boundary condition proposed by Taylor ( 1971 ). The theory of mixtures predicts a similar governing equation. The boundary conditions between a fluid and a deformable porous medium have been considered by Hou et al. (1989) who use biphasic mixture theory to derive a set of boundary conditions based on conservation laws. These conditions were then applied to the problem of steady shear flow over a rigid porous layer with examples given for different physical parameters. In chapter six I extend this work to consider unsteady flow over a porous deformable layer and an appropriate boundary condition for the free fluid at the interface. At the fluid-porous medium interface the boundary conditions derived by Hou et al. (1989) are [ [Ts + Ti - PJVf(vf - vs)]. n 0 (2.55) where [·] is the jump going from the purely fluid phase to the porous medium, n is the normal unit vector at the interface pointing into the purely fluid phase. These two matching conditions represent the conservation of mass and momentum. 2. Deformable Porous Media - A Review 19 For the case of purely normal flow to the boundary these conditions reduce to a continuity of velocity equation and a zero contact stress condition on the free surface, (2.56) (7 0, (2.57) where [vf]+ denotes the velocity of the fluid in the purely fluid region and [vfJ- the velocity in the porous medium. This is derived in Hou et al. (1989). For the one dimensional cases considered here the zero contact stress condition reduces to the condition OU= O oz (2.58) on the boundary. This has been supported by the results of Parker et al. (1987). For the case of flow tangential to the boundary (shear flow), the nondimensional boundary conditions are (2.59) (2.60) (2.61) These represent conservation of mass and momentum across the interface between fluid and porous medium. The boundary conditions for tangential flow are discussed again in more details in chapter SlX. 2.6 Viscoelasticity Viscoelasticity is a phenomenon associated with many biological tissues. Two viscoelastic phenomena, stress relaxation and creep, are considered here. Stress relaxation describes the process where a constant displacement is imposed on a medium and the force required to hold this displacement decreases. If a constant force is applied to the medium the tissue will creep towards its equilibrium displacement. Classical methods for these phenomena are described in Fung (1981) and use convolution integrals of the form (2.62) where G(t) is the normalized reduced relaxation function and T is the elastic response due to a continuous stretch ye. Using poroelastic theory, however, it has been shown that viscoelastic behaviour of a tissue can arise from the interaction of the fluid and the porous solid matrix. This has been noted 2. Deformable Porous Media - A Review 20 Impermeable boundary Compressive Stress Displacement=U(t) filter - Z=h Figure 2.2: Schematic diagram for stress relaxation. A displacement of U(t) is applied using the rigid filter. The force required to hold this displacement then decreases in time. by Tsaturyan et al. (1984) for passive myocardium and McCutchen (1982) also commented on this property for cartilage prefering to call it poroelasticity instead of viscoelasticity. In a separate theory for an elastic solid with voids Cowin (1985c) noted the viscoelastic behaviou r of a porous sponge. In mixture theory these processes can be explained by the finite time it takes the fluid to move within the tissue. If a displacement is imposed on the boundary then fluid is squeezed out of the region close to the boundary, compressing the solid matrix. This will produce a large stress whi ch is reduced as fluid moves within the tissue to balance the appli ed stress. 2.6.1 Stress Relaxation An analysis of stress relaxation has been done by Eisenfeld et al. ( 1978) for an arbitrary permeability fun ction. The geometry for this experiment is shown in figure 2.2. 2. Deformable Porous Media - A Review 21 The equations for stress relaxation are of the form H &2 u _ 1 au (2.63) a Oz 2 - k ( t) at ' where the infinitesimal stress <7 = Hal is used with constant Ha. The velocity v = 0, since there is no flow through the bottom boundary, z = h. The boundary and initial conditions are u(h, t) = 0, u(z,0) = 0, (2.64) where h is the final length of the medium. They took the displacement of the medium to be of the form 0 < t < T compression, u(O,t) = { U(t) (2.65) U(T) t ?:. T relaxation, Using a scaled stress <7o(t) = -fz(O, t), a suitable relaxation time is T(/) = _ [ lim ln (<7o(t) - <7o)]-l (2.66) t-+oo t It was then shown that (2.67) This shows that for a medium that has a very low permeability the relaxation time is large since it takes longer for the fluid to move within the medium. The time dependent behaviour during stress relaxation and creep has been analysed in Holmes (1983) and Holmes et al. (1985) with geometry shown in figure 2.3. Following their work the variables are nondimensionalised using u = uuc, z = zh, k = kko, t = it0, where Uc is the final compression, h the length of the medium, k0 the permeability of the medium in the absence of strain and to the time for the initial compression to take place. This gives (2.68) where b = uc/ h and (2.69) The term ,/koHato is a diffusive length scale representing the distance a disturbance has reached into the cartilage at time t 0 • The hats denoting nondimensional variables are now dropped for the remainder of this section as all variables are nondimensional. The boundary conditions are generalised to those of u(0, t) = g(t), u(l,t)=0, (2.70) for some smooth function g with g(O) = 0. 2. Deformable Porous Media - A Review 22 Impermeable Boundary Compressive Stress l!lllil!l!illifillllllll 1lll!llliiiillllillll!illilliifflll111111:111:11111111:1::::1:1:1:111111111111:11111111:1 Rigid Displacement=U(t) Boundary Rigid Z=h Filter Figure 2.3: Schematic diagram for stress relaxation. The displacem ent is obtained using the top impermeable boundary. The fluid then flows out through the bottom rigid filt er allowing the applied stress at the top boundary to relax. 2. Defonnable Porous Media - A Review 23 If R » 1 for a given medium this means that to is large so that the compression proceeds slowly enough to approximate uniform compression. The effect of the compression is felt throughout the medium practically simultaneously. If R ~ 1, fast compression, then the disturbance has not had time to reach the lower boundary by the time the compression phase is over, so during compression the medium is effectively infinite in length. For slow compression a perturbation expansion can be found using the small quantity 1/ R2 . The solution is then - 1 1 . g'( t) u(z,t)- g(t)(l - z)- R26 z(z - l)(z - 2) k(-t5g(t)) (2.71) which is valid fort » 1/ R2 • This restriction on t is due to the singular nature of the diffusion equation as R - oo. This solution clearly is not valid for t ~ 0 as g'( t) -:/ 0 in general and hence a solution for small times must be obtained. By rescaling the time variable as i = R2 t, the governing equation becomes a2 u 1 au (2.72) az 2 - k(t5 au) al" Oz Expanding as a perturbation series in 1/ R2 - - 1 - u(z, t) = uo(z, t) + R2 u1(z, t) + ... , (2. 73) it is easy to show that the zeroth order solution satisfies the Ii near form of the governing equation (2.63), equivalent to the constant permeability case. This should be expected since for small times the deformation should not be large enough for the permeability to become nonlinear. Taking the Laplace transform with respect to time and allowing the Laplace transform variables---. oo gives the solution for u(z,i) as i---. 0. If the function g(t) = t then the stress is 2 /T 1 ao(t)~ RV;, t < R2. (2. 7'1) The relaxation phase of the analysis is also treated as a singular perturbation problem for T = t - 1 ~ 1/ R2 in the same fashion to obtain 1 2~ (2. ,5) ao(r) - ao(O) ~ RV 1rk(-l)' 0 ~ T ~ R2. These results can be summarized in figure 2.4. The initial rise in stress corresponds to the small time solution. The second phase (b) corresponds to the linear. zeroth order, part of equation (2.71). The deformation is not large enough for nonlinear permeability effects to dominate. The third region (c) shows the effect of the nonlinear terms illustrated by the first order terms in equation (2.71). The final phase (d) shows the relaxatiou of the tissue when the displacement is held constant. I have gone into a little detail on this since the analysis is similar to the unsteady perme ation problem considered in chapter four. 2. Deformable Porous Media - A Review 24 Surface Stress 1.0 Time Figure 2.4: Stress relaxation due to imposed linear surface displacement u(O,t) = t, t < l. A small time solution due to the singular nature of the equation is found followed by a linear steady solution and nonlinear deviation effects. The tissue then relaxes for t > 1, (Holmes et al. 1985). If the displacement is applied rapidly, R ~ 1, then a rescaling of the spatial variable must be used. Setting z = z/R gives 1 au (2.76) k(li¥fi) at' where 8 = b / R. The boundary conditions are then u(0,t) = g(t), _lim u(z,t) = 0. (2.77) z-oo Expanding the permeability function in a power series in 8 gives 1 (·OU) · au ·t =( b-a_ :::::1+6('(0)8_. (2.78) k(b ~) z z z The displacement is also expanded in the form u(z, t) = uo(z, t) + 8u1 (z, t) + ... (2. 79) The zeroth order equation is then auo ouo (2.80) az2 - Tt· This is simply the linear version of the governing equation (2.68) on an infinite domain. Thus for small times the solution will correspond to this infinite domain solution since for fast compression rates the disturbance due to the compression has not yet arrived at the far boundary, hence the medium will behave as if it is infinite in extent. The imposed 2. Deformable Porous Media - A Review 25 displacement function g( t) is now assumed to be g( t) t. By using a similarity variable, T/ = z/-/i,, the solution is written as uo(z, t) = tfo(T/), (2.81) which gives the ordinary differential equation !~' (r,) + ½r, JM r,) - JM r,) = o, (2.82) with f(O) = 1 and J( oo) = 0. This has solution 2 2 fo(T/) = ( 1 + ~ ) erfc (~) - ft exp (- : ). (2.83) The stress at the boundary can then be written as ao(t)~ ( .,fit2 1/2) R.1 (2.84) Stress relaxation has also been considered by Holmes ( 1984) in a more general fashion to that described above. Lai et al. ( 1981) also looked at nonlinear effects in stress relaxation by solving the diffusion equations for the exponential form of the permeability function. Mow et al. (1980) have also done some early work on this problem. Mak (1986) also considered the stress relaxation and creep problems but for an intrinsically viscoelastic solid matrix as well as a viscous fluid. Similar work has been done by Oomens et al. ( 1987) for a mixture model of skin deformations. 2.6.2 Creep Coupled with stress relaxation is the process of creep. If a constant stress is applied to the same system described previously then the surface displacement will creep gradually towards equilibrium. This is due to the fluid slowly moving through the medium and through the porous boundary. The following analysis by Holmes ( 1984) shows some interesting behaviour for creep. The equation for creep is H {J2u _ 1 ou (2.85) a 8z2 - k ( fz) ot ' with boundary conditions u(h, t) = 0, ou(O t) = _ Fo (2.86) oz ' Ha' Fo being the applied stress. Nondimensionalising this equation using z = hz, t = ( Hakoh2 ) i, u -_ (hHF.ao) u-, k = kok,- (2.87) 2. Deformable Porous Media - A Review 26 gives 1 au 0 < z < l, {2.88) k (ft) ai' with {2.89) The parameter f = Fo/ Ha. The hats, denoting nondimensional variables, are now dropped for convenience. An interesting result (Holmes 1984) using the maximum principle of partial differential equations is that 0 ~ uf(z, t) ~ u(z, t) ~ uo(z, t) ~ 1 - z, 0 ~ z ~ l, t 2'.'. o. (2.90) where uf, u, uo are solutions to the equations a2uf 1 auf (2.91) az2 k(f) at ' 82u 1 au (2.92) 2 at' 8z k (ft) {J2uo auo (2.93) {)z2 at These represent the equations of motion with limiting values for the permeability function. That is when the strain is zero, the actual strain as a function of time and the final value for the strain. Hence this states that the actual solution is sandwiched between the solutions to the equations that have the extremes of strain imposed on the permeability function. This is intuitively obvious. If for a time interval O < t < T the function w( z, t) is defined to be the solution to the diffusion equation defined on O < z < oo instead of O < z < h then 0 ~ w(z,t)- u(z,t) ~ w(l,T), 0 ~ z ~ 1, 0 ~ t ~ T. (2.94) Since for fast compression the information about the compression has not had time to diffuse to the right hand boundary then the medium is effectively like an infinite medium and so it is expected that the displacement should be close to that of the solution on the infinite domain for small times. This corresponds well with the similar observation for stress relaxation remarked on earlier. The solution on the infinite domain is expressed in the form z w(z, t) (TJ), (2.9-5) = JtF TJ = Jt' for some function F, where the similarity variable indicates the lack of length scale for the infinite medium problem. The creep displacement then has the form u(O, t) ~ F(O)Vt (2.96) 2. Deformable Porous Media - A Review 27 in the initial moments of the compression. This has been verified experimentally (Mow et al. 1984). After this initial rise the displacement decays exponentially to the equilibrium solution. Creep has also been dealt with for finite deformations by Mow et al. (1985) and Holmes ( 1986) who showed that the extent of creep is over-estimated when infinitesimal deformation theory is used. Some tissues have viscoelastic behaviour without fluid movement which has led Mak (1986) to consider a tissue which comprises a viscoelastic solid matrix saturated with a viscous fluid, a poro-viscoelastic model. The viscoelastic behaviour then occurs on two scales, the macroscopic fluid movement and the microscopic structure of the solid matrix. It has been noted by Winlove and Parker ( 1990) that elastin fibres possess a porous structure on a number of scales. The elastin is comprised of numerous fibres, the gaps between the fibres form one 'macroscopic' scale. The fibres themselves can also be shown to have a porous structure with fluid movement within these microscopic pores leading to a viscoelastic behaviour on the microscopic scale. Hence it is possible that a para-elastic model on many scales is needed. 2. 7 Steady Permeation The simplest problem in deformable porous media is that of the steady permeation experiment where the fluid pressure is allowed to compress the medium against a rigid yet permeable filter. Most of the work on this problem has been done with an added strain imposed as discussed in the evaluation of the functional form for the intrinsic permeability ( Holmes 1985, Lai and Mow 1980). Experimental results for this problem have been published by Beavers et al. (198la,b), Beavers et al. (1975a,b) and Parker et al. (1987). Theoretical results for constant permeability and exponential permeability were published by Parker et al. ( 1987). An analysis of various permeability models and stress relations using the data of Parker et al. (1987) is given in Barry and Aldis (1990a) and in chapter three. In their experiments Beavers et al. ( 1981) considered a nonlinear extension to Darcy's law to account for the increased drag at high velocities. The results from these experiments then differ from those of the rest of the deformable porous media field in that they deal with high Reynolds' number flows. It is unclear which effects are due to the deformation and which are due to the high speed flow and hence I have not used this data. Once the solid-fluid interaction has been correctly considered this high speed flow can be incorporated. In chapter seven the nonlinear equation for high speed flow is discussed. 2. Deformable Porous Media - A Review 28 2.8 Radial Flows Cylindrical geometries for flow in deformable media have been predominantly used to model the flow through an arterial wall. The flow of proteins through the artery wall has been linked (Schettler et al. 1983, Nerem and Cornhill 1980) to the formation of atherosclerotic plaques ( a form of arteriosclerosis). The deformation of the tissue may then be a factor in this process. For a cylindrical geometry with purely radial flow and infinitesimal deformations the governing equations become op o ( 1 o ) 1 ( ou v( t) ) or= Ha8r ;ar(ru) = k() &t - -r- ' (2.97) where u( r, t) is the radial displacement, v( t) is the radial component of velocity and q; is the change in porosity given by 1 a (r,t)=--0 (ru). (2.98) r r The boundary conditions for the problems associated with flow through an artery wall are of free boundaries at both the inner and outer wall so that OU u Ha-+,\-= 0 ( 2.99) or r on each boundary. Kenyon (1976, 1979) has considered this problem for steady flow with constant perme ability and for a time dependent model with a unit impulse of pressure. Klanchar and Tarbell ( 1987) also solved the steady flow problem with the linear form for the permeability k _ ko (2.100) - 1 - m' which is an approximation to the exponential permeability function used by Lai and ivlow ( 1980 ). They also considered the case in which there is a small membrane resistance on the inside wall of the artery. This is a model of the decreased permeability associated with the endothelial cell layer lining the arteries. J ayaraman ( 1983) looked at the oscillatory flow for a constant permeability and Jain and Jayaraman (1987) also looked at the same problem but with two layers in the wall each with different permeabilities. The general equations and solutions for cylindrical flow are given in chapter six. I know of no published solutions in the spherical geometry which would serve as a model for injection phenomena. On this note Nicholson (1985) states that during an injection of fluid into brain tissue there occur two separate phenomena. Firstly where the fluid is simply allowed to diffuse from the point source of injection and secondly where the pressure of the injection causes a cavity of fluid to form. From this he looked at the absorption of the injected chemical from the cavity. His model does not account for the fluid diffusion that might result 2. Deformable Porous Media - A Review 29 during the action of the formation of the cavity. It would then be a useful analysis to extend his work to the situation where the radius of the cavity can be predicted and hence the effect of this cavity upon absorption rates. I have considered some aspects of the spherical problem in chapter six. 2.9 Further Modeling of Deformable Tissues Other applications and modeling using poroelastic theory have been to the unconfined com pression of articular cartilage by Armstrong et al. ( 1984 ). They considered a cylindrical section of tissue compressed between two rigid impermeable plates with fluid allowed to flow through the side walls of the tissue. This analysis was done using a constant permeabil ity function and linear deformation theory. The indentation of a porous material with a porous indenter was dealt with by Mak et al. (1987) using an axisymmetric geometry. The indentation of a porous material by a solid indenter is considered in Hou et al. (1990). Lanir (1987) has incorporated osmotic effects into a mixture theory to model the swelling of biological tissues as have Lai et al. (1990) and Snijders et al. (1990). Bogen (1987a, b) has looked at the strain energy description of swelling in terms of a fluid compartment model. This work can be of use for defining the appropriate strain energy function for a finite deformation analysis of fluid-solid interaction in swelling tissues. A non-mixture theory for the swelling of the corneal stroma has been proposed by Friedman ( 1971 ). Other models of swelling have been put forward by Cowin (1985a,b) using his theory of an elastic medium with voids (Cowin 1983a,b ). Extensions to two dimensions are difficult and therefore finite element modeling has re cently become prevalent as a method of solving the poroelastic equations. Spilker et al. ( 1990) indicate a useful method for implementing a finite element routine and give a useful overview of this field. Originally poroelastic finite element modeling was used in soil con solidation (Zienkiewicz and Shioma 1984, Simon et al. 1986a,b) with recent modeling of the intervertebral disk (Simon et al. 1985), the artery wall (Simon and Gaballa 1988) and skin deformation (Oomens et al. 1987). Chapter 3 Steady Flow and Deformation In this chapter I consider steady one-dimensional flow-induced compression, experimentally equivalent to the flow through a slab of material confined to axial movement by frictionless rigid walls. The aim is to investigate combinations of existing theories to find a simple model which describes the experimental data of Parker et al. ( 1987) for large deformations of a polyurethane sponge. The theory for this case has been summarised in chapter two so only the governing equations will be restated in the next section. The deformation models will be solved with different functional dependences of the permeability on strain. The models predict the height of the solid matrix, velocity of the fluid against applied pressure and the displacement against position in the medium. These models will be compared in the following results and discus sion sections to illustrate the usefulness of assuming particular forms for the stress-strain and permeability relations. Both infinitesimal and finite deformation stress relations will be considered. 3.1 Flow Geometry and Governing Equations A pressure difference is applied to the ends of a porous medium producing a fluid flow which deforms the solid matrix. The upstream end of the solid matrix, z = h, is a free boundary and the downstream end, z = 0, is fixed by a rigid porous mesh which supports the solid material but offers no resistance to the flow of fluid (figure 3.1). The initial length of the medium is L, the final length h and the pressure at the downstream end is zero, without loss of generality. The direction of flow is in the negative z direction with z = 0 being the downstream end. The displacement of the medium is u(z). The boundary conditions are no displacement at z = 0 and zero contact stress at z = h. 30 3. Steady Flow 31 Impermeable boundary J ~: Rigid Filter D 1 ~ ~:P=~P ~ ~:u·(z)=O ~ ~I D~======------~ Z=O Z=h Figure 3.1: Schematic diagram of uni-axial compression ofa deformable porous medium by an applied pressure difference. The medium is constrained by a rigid porous mesh and side walls. The flow and boundary conditions are indicated. The general governing equation rewritten here from equation (2.23) is "vp="v·a=k1 (8uDt-v(t ).) (3.1) For one dimensional steady flow this becomes dp da -v (3.2) dz dz k(u'(z))' where the prime denotes differentiation with respect to the argument. In one dimension the contact stress is given the notation a, the velocity v and the displacement u. The variables are nondimensionalised by setting u = uL, z = iL, h = hL, k = kk0 , (3 .3 ) v = (3.4) where ko is the permeability in the absence of deformation. Also HM = 8a(l)/8y is the fini te deformational analogue of the aggregate elastic modulus ( Holmes 1987), where y is the axial stretch. The one dimensional steady state equation then becomes df> da _1 - = - = v- (3 .5) di di k' where k is a function of the strain du/di. From now on the hats, denoting nondimensional variables, will be dropped as all variables are assumed nondimensional unless otherwise stated. Various forms of the dependence of the permeability k on strain have been given in the li t erature. The simplest is constant permeability, k = 1 (IDCON for Infinitesimal D eformation with CONstant permeability, FDCON for Finite D eformation CONstant permeability). 3. Steady Flow 32 The form k = exp(mu'(z)), with constant m has been developed for articular cartilage by Lai and Mow (1980) (the exponential permeability models IDEXP and FDEXP). A form sug gested by Klanchar and Tarbell (1987) for arterial tissue is k = 1/(1- mu'(z)), as an approxi mation to the exponential form (the approximate permeability models IDAPP and FDAPP). Parker et al. (1987) also postulated a square root form of k = exp(-mJ-u'(z)/(1 + u'(z)) as a fit to experimental data (the square root models IDSQU and FDSQU). 3.2 Infinitesimal Models The infinitesimal deformation theory outlined in chapter two is used here with stress tensor a= >.eI + 2JLe, e is the infinitesimal strain tensor, e = trace(e) and >.,Jl are the Lame stress constants. The aggregate elastic modulus is then Ha = >. + 2/L, and Ha replaces HM in the nondimensionalisations. In the one-dimensional infinitesimal case the governing equation is "( ) V u z = k(u'(z)) ( 3.6) The boundary conditions are expressed as u(0) 0, u'(h) = 0, p(0) 0, p(h) = b.P, (3. 7) where 6.P is the applied pressure difference. There is also a compatibility condition h = 1 + u(h), (3.8) to find the height h of the matrix. Integration of equation (3.5) with the substitution of a yields p(z) = u'(z) - u'(0). (3.9) This then gives 6P = -u'(0) which implies that the infinitesimal models become unphysical if 6.P exceeds 1 because of the resulting elimination of all pore space at z = 0. For the permeability set ask = 1 (IDCON model), the solution after first solving equation (3.6) is V 6.P (1 +bit)' u(z) V (; - hz), (3.10) 1 h 1 +6.P. 2 3. Steady Flow 33 Letting the permeability function be k = exp (mu'(z)), (IDEXP model) the solution is 1 11P exp(-m/1P))-I h = ( 1+-+------m exp(-m/1P) - 1 1 - exp(-m/1P) (3.11) V = mh 1 + vm( z - h) z 1 - vmh u(z) -----log(l + vm(z - h)) --- log(l - vmh). vm 2 m vm 2 Following Klanchar and Tarbell (1987) where k = 1/(1 - mu'(z)), (IDAPP model) the solution by integration is h = (1-_!_+ 11P )-1 m log(l + m!1P) log( 1 + m/1P) V = (3.12) mh z 1 u(z) = - + - 2 (exp[vm(h - z)] - exp(vmh)). m vm The form postulated by Parker et al. (1987) is unsuitable for infinitesimal deformations due to a singularity ink when u'(z) = -1. As an approximation the form k = exp ( -mJ-u'(z)) , (3.13) is used (IDSQU model). Solving equation (3.6) for this model by the substitution of 17(z) = -mJ-u'(z) yields ) vm2(z - h) eri(z [1 - 77(z)] = ---- + 1, (3.14) 2 so u( z) can be found from 1 1z 1 111(z) dz u(z) = -- 172 (s)ds = -- 172 (z)-d17. (3.15) m 2 o m 2 ri(O) d17 Using equation (3.14) to define dz/d17 2 1ri(z) u(z) = -- 173e11 d17, vm4 11(0) = ~ [( 773 - 3772 + 677 - 6)eT/) TJ(z), (3.16) vm TJ(O) where 17(z) is found numerically from equation (3.14). Since 17(h) = 0 and 77(0) = -m..Ji;? the height is 6 - ( w3 3w2+ 6w - 6)ew ( - )-l (3.17) h = 1 + m 2 ((w - l)exp(w) + 1) (3.18) v = h: 2 (1+(w-l)ew), where w = -m,,Jt;l5. 3. Steady Flow 34 3.3 Finite Deformation Models Since the stress-strain relation cannot be expected to be linear over the whole range of strains a nonlinear finite deformation model should be employed. Using the form for the nondimensional contact stress (Mow et al. 1985), a=~ [1 + d1(Y - 1)] (y2 __..!:__) = F(y), (3.19) 4 y y2 where d1 is a constant and y(X) = 1 + U'(X), equation (3.5) becomes (1 + U'(X))-1d~ ((1 + d1 U'(X)) [(l + U'(X)) - (1 + u\x))3]) = k(l + ;'(X)) · (3.20) The displacement is denoted by U(X) in the Lagrangian coordinate system. The Lagrangian boundary condition at the upstream boundary is U'(l) = 0. Equation (3.20) is conveniently written in the form dF(y) 1 (3.21) ~ = vy k(y). Integrating equation (3.21) by separation of variables gives j k~) dF = j k~) F'(y) dy = g(y) = vX + c, (3.22) where c = g(l) - vis a constant. The value of the displacement is found from x J,y(X) dX U(X) = 1 (y - 1) dX = (y - 1)-d dy. (3.23) 0 y(O) Y Separation of the integral into two parts gives y(X) dX U(X) = J, y-dy-X y(O) dy l J,y(X) = - k(y)F'(y) dy - X. (3.24) V y(O) The pressure p is denoted P(X) in the Lagrangian framework and by integrating equation (3.5) is P(X) = F(y(X)) - F(y(O)) (3.25) giving 6.P = -F(y(O)). Now consider each model in turn. Letting the function k(y) = 1 (FDCON model) for all y, the integrals above become straightforward with 3 1 2 1 g(y) = (1 - di) ln(y) - -(1 - di)-+ 2d1y - -di - (3.26) 4 y4 3 y3 and the displacement as 1 U(X) = -X + - [F(y(X)) - F(y(O))]. (3.27) V 3. Steady Flow 35 It was found that setting the permeability as k = exp(mdU /dX) gave better results than the form k = exp(mdu/dz) = exp(m - m/y) used in the IDEXP model. Hence the permeability function k(y) = exp(my - m) is used (FDEXP model) and the integrals for g(y) and U(X) can be evaluated in terms of the Exponential integral Ei(z) (Abramowitz and Stegun 1972, appendix A.4). For the permeability function k(y) = 1/(1 - my+ m), (FDAPP model) the integrals are done using numerous partial fractions and, as with the FDEXP model, were done analytically using the REDUCE algebraic manipulation package and are listed in the appendix D.l. A different choice of the contact stress function has been used by Holmes ( 1986) (FDHOLMES model) as ( ) ( 1 - k(y) = (1 _\o)2k*(y), (:3.30) where (3.31) Defining the deformation gradient as w = dU / dX = y - l the deformation equation (3.2) becomes dw dX = ((w), (3.32) where Ay ((w) (3.33) k*(y)oa* joy' 2v(l - o) 2 A ( 3.34) ( 1 - 5 )n+ 1 . The pressure takes the form of P(X) = (l - ~5)n+i [a*(y(X)) - a*(y(O))]. ( 3 .35) The solution of equation (3.32) was found using a simple step by step third or A summary of the models investigated is given in table 3.1 showing the form of the permeability and stress function used and where the solution can be found. 3. Steady Flow 36 Model Permeability Stress-strain Solution IDCON k = 1 er= dU /dx (3.10) IDEXP k = exp(mdU /dx) er= dU /dx (3.11) IDAPP k = 1/ ( 1 - mdU / dx) er= dU /dx (3.12) IDSQRT k = exp(-mj-dU /dx) cr=dU/dx (3.17) FDCON k = l (3.19) (3.27) FDEXP k = exp(mdU /dX) (3.19) Fig. 3.3-3.6 FDAPP k = 1/(1 - mdU /dX) (3.19) not shown FDHOLMES (3.31) (3.28) (numerical) Fig. 3.3-3.6 Table 3.1: Summary of the types of models investigated for uni-axial flow in a de formable porous medium. The equation numbers for each solution are indicated. The source of each model is given in the text. 3.4 Results To allow for the different stress-strain relations between biological tissue and polyurethane sponges the stress axis is translated by 1.lkPa as shown in figure 3.2. In figure 3.3, for example, this corresponds to a nondimensional pressure translation of 1.6. For consistency the same pressure translation is used in figure 3.4 although the velocity is still significant for pressures less than 1.6. Note that both the nonlinear stress relations used are almost identical over the entire range of the strain. The displacement of the free boundary (1-height) is given in figure 3.3 where the single parameter m has been adjusted to give the best fit to the data. In the case of the infinitesimal models only the initial data points were used since it was not possible to get a good overall fit. In calculating the velocity shown in figure 3.4, no free parameters remain since the parameter m has already been fixed for each model. The velocity data of Parker et al. (1987) were only found to agree with all the models when scaled with a factor of 0.035. This factor is almost certainly due to the differences in nondimensionalisation. The scaling does not affect the fitting of the displacement data and the shape of the velocity curves remains unchanged. Comparing the infinitesimal models, figure 3.3 indicates that most of these were accurate for displacements less than 0.2 with the IDSQRT model accurate up to 0.3. A better fit can be obtained for the displacement from the IDAPP model with m ~ 700 but this choice leads to an impossibly small velocity. Alternatively for m ~ 20 a good fit can be obtained in the velocity but with a correspondingly poor fit with the displacement. The finite deformation models FDEXP and FDHOLMES are shown in figures 3.3 and 3.4 to give good agreement with the data. The FDAPP model gives a very poor fit and FDCON 3. Steady Flow 37 8 • 6 ...... (/) (D 4 en en •• •• 0 0.2 0.4 0.6 0.8 1.0 !strain I Figure 3.2: Compressive stress-strain relations for a saturated porous medium: •, ex perimental data from Parker et al. (1987) for polyurethane foam;---- , equation (3.19) with HM = 0.55; - - - - , equation (3.28) with HM = 0.55, /3 = 0.4, d1 = 1.0077, 'Po = 0.03. The theoretical lines appropriate for biological tissues have been displaced vertically to allow for the initial rapid rise shown in the foam data. 3. Steady Flow 38 o_s _, .,., ___ FDEXP, m=6.8 _, ., o_ .... _.. IDEXP, m=3.5 _ __ . _ FDHOLMES. m=5.0 /./ ---·---·-- IDAPP, m=11.6 0 ___ • IDSOU, m=4.0 (/) 0_3 -0 l -, ------PJ ,'" ------/ / . ,-- (') - , CD , , , 3 0_2 ~-----·-·------····------·------·-----·------CD ::J .--+ 0_ 1 0 2 4 6 8 ~p Figure 3.3: Displacement of the free boundary (1-height) of the medium versus pressure compared with, •. experimental data from Parker et al. {1987). Both finite deformation models give good agreement with the experimental data. overestimated both the velocity and displacement. The latter two are not included on the graphs for clarity. The relation between the displacement and position in the medium is shown graphically in figure 3_5 where the pressure is chosen to give a height of 0.5 in each model. The IDCO N model shown in figure 3.5b, typical of the infinitesimal models, shows an impossible deformation near X = 0 with no pore space remaining. Figures 3-5c and 3.5d (FDCON, FDHOLMES) illustrate a deformation pattern which is too uniform, while figure 3.5e (FDEXP model) gives a more plausible result with the medium highly consolidated downstream and relatively unconsolidated upstream. Continuing the analysis of the FDEXP model briefly, a graph of displacement versus position is given in figure 3.6 form= 0 (equivalent to FDCON) and m = 6.8. The pressures have been chosen to give matrix heights of 0.25, 0.5 and 0.75. This illustrates that for high pressures the displacement profile forms a small 'boundary layer' near X = 1 where dU/dX :::::: 0 followed by an almost linear section to the boundary condition of U(0) = 0. As the pressure increases the boundary layer becomes smaller. The 'boundary layer' is significantly smaller for m = 0 than for m = 6.8. 3. Steady Flow 39 0.5 ___ FDEXP, m=6.8 •...••. IDEXP, m=3.5 • -·-·- FDHOLMES, m=5.0 • .•.•.•.•• IDAPP,m=11.6 .,.,.,·· ,. . , .,.,·""' ___ • IDSQU, m=4.0 .. ············-;,,·:,,,,,,..·· ...... --· :;.;.:,,, ...... 0.3 · ,,.,.,· ... ,,. ,. ,·,. ,.,. ,·,. V ,.,. ,.,· ,· ,.,· ,,,_,.,· -, .. -·-·············· ,,.,l' ,,.,,. 0.1 ,,. 0 2 4 6 8 ~p Figure 3.4: Fluid flow velocity versus pressure. •, experimental data from Parker et al. (1987); Although no additional parameters have been used to fit the data, the FDEXP model still represents the data well. ------a b C d e Figure 3.5: Illustration of the solid matrix deformed to 50% of original height. a) Undeformed flow, b) IDCON model, c) FDSQU model, d) FDHOLMES model, e) FDEXP model. Only the FDEXP model gives a realistic deformation pattern consistent with Parker et al. (1987). 3. Steady Flow 40 1.0 ___ m=6.8 ____ m=O 0.75 C o.5 ..__.,x 0.25 0 0.25 0.5 0.75 1.0 X Figure 3.6: Displacement versus position for the exponential permeability finite defor mation model (FDEXP). Note the apparent formation of a boundary layer near X = 1 at high pressures corresponding to the high displacements. 3.5 Discussion Typical stress-strain relations for biological tissues (Holmes 1986, Mow et al. 1985, Chuong and Fung 1984) do not, in general, show the initial rapid rise that is exhibited by polyurethane sponge (Parker et al. 1987). The data of Parker et al. (1987) is still useful, however, in examining the models of deformation of soft biological tissue in compression by assuming a translation of the stress axis. The stress-strain relation is translated by l.lkPa in an attempt to account for differences in the structure of biological tissue and polyurethane sponge. The stress-strain relations, (3.19) and (3.28), developed for biological tissues (Holmes 1986, Mow et al. 1985) then closely fitted the data of Parker et al. ( 1987). The structure of open celled polyurethane sponges are of interconnecting cavities. These cavities resist the initial application of stress until a stress is reached at which cavities first collapse (buckle). After the onset of buckling the stress-strain relation shows typical characteristics of biological tissue. Thus as the motivation of this work is the study of biological based models for stress compression the stress relations used here are translated to fit the data of the polyurethane sponge. In modeling the nonlinear behaviour of the stress relation, Parker et al. ( 1987) assumed a dimensional form of a= HM(U'(X))U'(X), obtaining the function HM from the parameter :J. Steady Flow 41 fitting of three quadratic pieces. Instead I have assumed that HM is a constant and fitted the stress-strain data with equations (3.19) and (3.28). The dimensional form of equation (3.19) has only one fitting constant HM and equation {3.28) uses the same value of HM with the fitting constant (J, since the initial solidity Features of biological tissues that are not present in polyurethane sponges include osmotic pressure and anisotropy. These effects can be important in biological applications including cartilage (Lanir 1987a,b) and in swelling problems. While I believe that the models inves tigated here will be useful in describing soft tissue deformation I recognise that they were compared using data from a polyurethane sponge. However they are simpler than models which include the effects of osmosis and anisotropy, and would be preferred in situations where these effects are relatively unimportant. The internal deformation of the sponge given in figure 3.5e agrees qualitatively with the photograph published in Parker et al. (1987). The nonlinear pattern of deformation, with high consolidation at the downstream boundary and a relatively undeformed medium upstream, is well reproduced. The most successful model investigated (FDEXP) used a nonlinear stress-strain relation (Mow et al. 198.'5) with an exponential dependence of the permeability on the strain (Lai au Unsteady Flow-Induced Deformation This chapter will consider the one-dimensional deformation of a porous medium by an applied fluid pressure gradient. Mixture theory will be used to derive equations for the solid-fluid interaction including inertial effects. An upper bound on the frequency of pressure oscillation is found below which the effects of inertia can be ignored. Solutions are then found for the inertia-free regime. Exact solutions for the linearized theory are found in terms of a specified velocity or pressure. When a nonlinear constitutive equation and nonlinear permeability relation are used, a perturbation technique is employed to obtain approximate solutions for the limiting cases of slowly varying pressure and rapidly varying pressure. This is done in the two cases of specified pressure or velocity. Numerical solutions are then found for an oscillatory velocity flow and a step change in velocity. 4.1 Mixture Theory for Unsteady Flow The derivation of the governing equations is similar to that used in chapter two except that now I will examine the effect of including the inertial terms in the governing equations. The momentum equation for each phase is rewritten here from equation (2.11) as ( 4.1) where T 13 is a stress tensor and 1rfJ is the drag force between the constituents. Substitution of equations (2.13) and (2.14) for the drag and the stress tensors into the momentum equation leads to 82 u ps __ + ps€.s 5 - (4.3) where €8 = ( au •v) au ( 4.4) at at and ( 4.5) represent nonlinear inertial terms. For high values of vf the term €1 is more appropriately interpreted as a nonlinear drag term (4.6) where c is a constant dependent on the class of material, "' = k/ µ f is the permeability and µf is the viscosity of the fluid. This term, when added to Darcy's law, forms Forcheimer's law, equation (1.3). The latter expression for €1 comes from volume averaging the inertial term, (vf · '\l)vf, and is discussed in more detail in Ward (1964), Ahmed and Sunada (1969) and in chapter seven. Elimination of the pressure from ( 4.2) and ( 4.3) gives 2 8 8 a u . . a (V- f = - ps k 8t - v + ps V . a ' where . s pf p1 and p----...I.. (4.8) - <1>1PS - Pr · The result 1/k = K/( Attention is now restricted to deformations in one space dimension, z, only. The solid stress, a, can be written as ( 4.9) where HM = aa(l)/ay is the finite deformation analogue of the aggregate elastic modulus (Holmes 1986) for axial stretch y = l + au/az. In infinitesimal theory HM is replaced by Ha=>.+ 2µ, the sum of the Lame stress constants and F(8u/8z) = 8u/az. Equation ( 4. 7) is now expressed using equation( 4.6) as a2u au av a3u a1 at2 + a28t + biv + b28t + € = a3 az3, (4.10) where 1 ps a4>f V = v(t), a1 = 1, a2 = pSk - ( P a<1>1 1 a3 = HM F' (au) ' (4.11) ps az bl = ( . au a2u cp ( s8u) 2 1 €=8tozat- ,.fa v(t)-8t ( and where dash denotes differentiation with respect to the argument. Non-dimensionalising equations (4.10) and (4.11) with respect to a length scale L, taken as the initial length of the medium, the maximum displacement uo, the permeability in the absence of strain k0 and the typical period of a disturbance to the variables are written as Uo - u = u0 u, t = t0 t, z = Li, k = kok and v = -v. (4.12) to The magnitude of each term in equation ( 4.10) is now investigated. The hats, denoting non-dimensional variables will now be dropped as all variables are assumed non-dimensional hereafter. Multiplying (4.10) throughout by Prkoto/uo gives fJ2u _ 0 ( s ko) OU = O (1 Uo Prko) ot 2 - PT to ' ot + L to ' 2 8 u = 0(HMtoko) V = 0(1 + Uo Prko) (4. i:J) 8z2 L2 ' L to ' ov = O (koPy) ot lo ' It has been assumed that 5 , 4>f, c and pare all 0(1). The velocities are now assumed small enough so that the term lf can be taken as zero. For low frequencies (to~ Prko) equation {4.10) becomes (4.14) where and fJ = uo_ ( 4.1.S) L The parameter R represents the scaled distance into the medium that a disturbance travels in time to ( Holmes et al. 1985 ). The deformation is essentially uniform for R 2 ~ 1 ( very slow pressure variation). For R2 ~ 1 the period of oscillation is smaller than the time for the disturbance to travel the length of the medium and the medium appears as an infinite body at small times. Thus if lo ~ Pyko the problem can be regarded as inertia-free and governed by the diffusion equation (4.14). If to~ H~ko the compression is essentially uniform. If p1-ko ~ 10 and to ~ H~ko, then equation (4.14) defines a fast deformation problem. If lo ~ p}ko wave-like behaviour will dominate the compression. This has been dealt with in Biot ( 19.56) for acoustic propagation in soils but is unrealistic for most biological applications. It may be relevant, however, to transmission of high energy ultrasound in porous biological materials. If the porous medium is such that pyko » H~ko then to » pyko corresponds to slow compression rates in the inertia-free regime. If t0 ~ p'yko then the inertial terms cannot be ignored. 4. Unsteady Flow 45 Impermeable boundary I ~ ·I ~: P=LiP(t) I ~ .: U'(z,t) = 0 ~. ·I --- :..J Z=O Z=h Z=L Figure 4.1: Schematic diagram of uni-axial compression of a deformable porous medium by an applied pressure difference .6.P(t). The medium is constrained by side walls and a rigid filter. The free boundary, originally at z = L, is at z = h . From now on it is assumed that to ~ pTko and pTko ~ H~ko. Attention is now restricted to studying the inertia-free regime. 4.2 Nonlinear Diffusion Equation The geometry considered here is shown schematically in figure 4.1. This is a typical geometry for permeation experiments on soft tissues. The porous medium is confined between rigid walls. A constraining mesh at z = 0 allows free passage of fluid yet restrains the porous material. The boundary at z = h(t) is free to move under the action of a pressure gradient .6.P(t) applied to the fiuid. The medium is initially of unit non-dimensional length. The governing equation ( 4.14), for large time scales t0 , is op oa I (au ) (4 .16) oz = oz = R2k (b~~) 8t - v(t) . 4. Unsteady Flow 46 Appropriate boundary conditions are: u(0,t) = 0, p(0, t) = 0, 8u Bz (h(t), t) = 0, p(h(t), t) = L':iP(t), (4.17) u(z,0) = 0, p(z, 0) = 0. The length h(t) of the medium can be obtained from the implicit equation 1 + u(h(t), t) = h(t). ( 4.18) The nonlinearity of (4.18) can be avoided in a Lagrangian coordinate frame (X,t) with reference time t = 0 and displacement denoted U(X, t). The problem ( 4.16) becomes 2 au 2 ( au) a u 8t = R ( fJ 8X 8X2 + v(t), (4.19) with U(O, t) = 0, p(O, t) = 0, 8U BX(l,t) = 0, p(l,t) = L':iP, ( 4.20) U(X,O) = 0, and h(t) = 1 + U(l, t). (4.21) Here the function ( is defined by 8U) _ k (6fk) F'(y) ( ( 4.22) ( 6ax - y ' where y = 1 + 8U/8X and F(y) = a/HM- The pressure can be found from P(X, t) = a(X, t) - a(0, t). ( 4.23) 4.3 Infinitesimal Deformations The analysis for infinitesimal deformations is simpler and provides some useful insight into the deformation process. The linearization of ( 4.16) is possible for infinitesimal deformations where a (t~) ,.._, Hat~, k (t~) ,.._, 1 and the boundary conditions may be approximately applied at z = 1. This gives 8u 2 82 u 8t = R 8 z2 + v( t) ( 4 .24) with 8u u(0, t) = 0, Bz(l,t)=0, u(z,0) = 0, h(t) rv 1 + u(l, t). ( 4.25) The approximation in h(t) has been used previously by Kenyon (1976), Jayaraman (1983) and many others without noting the approximate nature of applying the boundary conditions at z = 1 rather than z = h( t ). This is discussed in more detail in chapter six. 4. Unsteady Flow 47 4.3.1 Exact Solution for given v(t) The simple transformation w(z,t) = - lot v(r)dr + u(z,t) ( 4.26) converts the problem to ( 4.27) with ow w(0, t) = J(t), oz(l,t)=0, w(z, 0) = 0, ( 4.28) and J(t)=-fotv(r)dr. ( 4.29) This transformation will also be used in later sections. The solution can be found easily (Carslaw and Jaeger 1959, p.104) to be ( 4.30) This solution can give u(z,t) and p(z,t) in terms of a given applied fluid velocity, v(t). 4.3.2 Exact Solution for given fl.P(t) A common experimental procedure is to apply a pressure difference to the porous material by means of a height of water or a pressure pump. Thus a more useful solution is to obtain u(z, t) in terms of l::i.P(t) directly rather than v(t). Taking the derivative of (4.24) with respect to z and setting (z,t) = 8u/8z(z,t) yields & = R2 a2 (4.31) 8t 8z 2 ' with (z,0) = 0, (0,t) = -l::i.P(t), and (1,t) = 0. The solution can then be derived from the solution for (Carslaw and Jaeger 1959, p.104) to be CX) t 2 2 2 2 2 2 2 u(z,t) = 2R I>xp (-R n 1r t) {cos(n1rz)- l} 1exp (R n 1r >.) l::i.P(>.)d>., ( 4.32) n=l 0 with 4. Unsteady Flow 48 4.3.3 Small Time Solution for given flP(t) A simpler solution suitable for small t may be obtained by finding the Laplace transform, .C{ 4>(z, t)} = ~(z, s), of ( 4.31) and finding the large s behaviour of ~(z, s) (Bender and Orszag 1972, p.263). The solution in the Laplace transform domain is _ _ sinh ( 1j- (z - 1)) 4>(z,s) = 6.P(s) . (-Li.) ( 4.34) smh R Therefore u(z,s) 1oz ~( r, s )dr ( 4.35) Rl:l.P(s) ( -z,./s/R ) Js e - 1 ass----+ oo. ( 4.36) Although this can be inverted for general l:l.P( s ), in terms of convolution integrals, it is sufficient for illustrative purposes to consider 6.P(t) = atm/2 , ma positive integer, for which, .!!!.±.l ( z ) t(m+l)/2 } u(z, t) ,.._, Raf(l + m/2) { (4t) m2 im+1erfc ~ - ( ) ( 4.37) 2Rvt 1 + m..±..1 r 2 as t ____, 0. Here i denotes error function integration, following Abramowitz and Stegun ( 1972, p 297) (see appendix A.4). 4.4 Perturbation Analysis for Slowly Varying Pressure For slow deformation rates, ~ ~ 1 can be used as a perturbation parameter. Holmes et al. ( 1985) used this approach to calculate the stress relaxation response of articular cartilage. Here a similar technique is used on equation (4.19) with the complication of the velocity term. 4.4.1 Solution for given v(t) Using the transformation ( 4.26) on equation ( 4.19) gives ( 4.38) with boundary conditions 8w w(0,t) = f(t), BX(l,t) = 0, w(X, 0) = 0. ( 4.39) Setting ( 4.40) 4- Unsteady Flow 49 and equating orders of 1/ R2 gives wo(X, t) = f( t), (4.41) w1(X,t) -v(t) (X _ l)2 + v(t). (4.42) 2 2 As 8w0 /8X = 0 the series 8w/8X is of order l/R2 , so ( can be expanded as a Taylor's series giving ( ow ) , 8w1 b ( 1 ) ( 4.43) ( b ax = 1 + ( (o) ax R2 + 0 R 4 · It follows readily that , { (X - 1) 2 ( X - 1 )4 - 2_ } - v2 ( t )(' ( 0 )b { ( X - l )3 + } . W2 ( X, t) = v ( t) ~ ( 4.44) 4 24 24 6 6 Here w0 (X, t) is the zero flow solution and w 1(X, t) represents the solution as a sequence of steady states. By allowing v(t) =tit can be seen that w2(X,t) combines a time lag, v'(t) term, with a nonlinear stress and permeability, v2 (t)('(0) term. For example if a "' gx, k rv emau/ax then ('(0) "' m - 1. We note that had the Eulerian formulation been used with approximate boundary conditions then ('(0) "' m. Thus the effect of the nonlinearity is overestimated if the Eulerian formulation had been used. This is discussed in more detail in section 4.4.3. 4.4.2 Small Time Solution for given v(t) As R2 -+ oo the time derivative in ( 4.38) becomes negligible and the initial condition, at t = 0, is unable to be satisfied as is apparent in equation ( 4.44) since v'( t) ::/ 0 in general. Hence these solutions are not valid fort~ 7b. By redefining T = R2 t the governing equations become 2 aw = ( (baw) 8 w ( 4.45) oT ax ax2 ' with w(X,0) = 0, w(0,T) = f(T/R2 ), and g;(l,T) = 0. Setting Vi ( X, T) V2 ( X, T) w(X,T)=vo(X,T)+ R2 + R 4 + ... ( 4.46) and assuming (4.47) gives = ( 4.48) with boundary conditions OVn Vn(X, 0) = 0, ax(l,T) = o. (4.49) The functions Vi(X,T) = 0, 0 ~ i < n since the boundary condition w(0,T) = anTn/R2n only becomes nonzero at order O(R2n). Equation (4.48) is simply the linear problem solved earlier in (4.30). 4. Unsteady Flow 50 4.4.3 Solution for given 6.P(t) Differentiating equation ( 4.19) with respect to X eliminates the velocity term giving 2 ~~ = R2 { ((04>) :~ + 0('(04>) (!t) }, (4.50) with OU 4>(X, t) ax(X,t), 4>( 1, t) 0, 4>(X, 0) 0, 4>(0, t) a-1 (-l:!.P(t)). ( 4.51) For convenience the notation Pa(t) = -a-1(-l:!.P(t)) is used for inverse function a-1 . This uses the result l:!.P(t) = -a([)u/[)X(0,t)). Hence if a rv [)u/oX then Pa(t) "'l:!.P(t). Setting (4.52) leads to 4>1(X, t) Pa(t)(X - 1), (4.53) 4>2(X, t) p;:t) {(x - 1)3 - (X - 1)}- o('(O~P;(t) {(x -1)2 + (X -1)} (4.54) which can be integrated to obtain u ( 1 t ) --_ 1 { ---Pa(t)} + -1 { --P;(t) + --u(P;(t) 1: '( 0 )} . ( 4.55) ' R2 2 R4 24 12 A singular perturbation analysis is required for small times. This can be seen in the schematic figure 4.2 which illustrates the displacement of the medium for a linear rise in pressure Pa(t) = t. The linear solution for steady pressures shown in figure 4.2 arises as the first order solution of the displacement expansion. The time lag and nonlinear deviations are also shown. If it is assumed that a ,...., %x and k '"" exp ( m %} ) then ('(0) '"" m - 1. The result in figure 4.2 illustrates the deformation when m > 1 which is normally the case. If, however, m < 1 then the displacement has the opposite curvature and increases relative to the 'steady state' solution. As the medium is compressed the length of material over which the pressure difference is applied decreases. Thus for a small increase in pressure difference a dispro portionately large increase in pressure gradient occurs. Thus if the permeability is almost constant or changes very slowly ( m < 1) then this leads to a relatively increased displacement. Normally, when m > 1, the permeability decreases enough so that the extra pressure gradient required to overcome the compaction of the tissue compensates for the decreased length of 4- Unsteady Flow 51 0.085 Steady solution .. · Nonlinear deviation ~ w ~ w Time lag ~--· (.) :5 a.. Cl) 0 . · _./ "Boundary layer" 0.02 0.06 0.1 TIME Figure 4.2: Schematic diagram of displacement versus time for an applied pressure tl.P(t) = t. A small boundary layer is followed by the solution for a succession of steady states with time lag and nonlinear effects indicated. material. If an Eulerian formulation was used, with the boundary conditions applied at the original position of the medium rather than the final position of the medium, then ('(O) = m and this effect would not be apparent. The pressure is now varied from a steady state value, A to 6.P(t) = A+ P(t), with the 1 time average of a periodically variable pressure, 6.P( t) = J1~"+t 6. P( t )dt = A, for t 1 ~ ~. If v* is the velocity corresponding to a steady state pressure A then by using the result 2 ( ) = au _ R2( (b au ) a u (4.56) v t at ax ax 2 ' at X = 0 gives v(t) - v* ,...., __l (6.P'(t) _ b('(O)~) (4.57) R2 3 2 Hence if the periodic pressure is such that the average of the gradient satisfies --- --2 .!!:_6.P 3b('(0) 6.P (4.58) dt > 2 2 then there is an increased permeability to flow. 4- Unsteady Flow 52 4.5 Singular Perturbation Analysis for Rapidly Varying Pres sue When the deformation rates are large so that R2 ~ 1, a singular perturbation problem arises. The distance is rescaled as x = X/ R and b = y1?_ ~ 1 is set as a small parameter. This assumes that the deformation is very small. 4.5.1 Solution for given v(t) Equation ( 4.38) now becomes ow = ( (tow) {J2w. ( 4.59) at ox 8x2 Expanding w in the form • ·2 w(x,t) = wo(x,t) + bw 1(x,t) + b w2(x,t) + ... ( 4.60) gives the following problem: (4.61) with 8wo wo(0, t) = f(t), ox (x, t)---. 0 as x---. oo, w0 (x, 0) = 0. (4.62) Since x E [O, oo) the similarity variable ( = -Ii can be used. If f ( t) ,...., atm/2 as t ...... 0 then set ting Wo = atmf2g(0, ( 4.63) gives mg(O - (g'(O = 2g"(O, g(O---. 0 as (---. oo, g(O) = 1. ( 4.64) Solving for g( 0 results in ( 4.65) This solution is equivalent to the infinitesimal solution for small times indicating the result that for small times the deformation is small enough to be considered infinitesimal and the medium infinite in extent. The first order solution is found from 2 2 OW1 = 8 w1 ('(O)OWo 8 wo ( 4.66) at 8x2 + &x 8x 2 ' which by setting w 1 = atm-½ h(O gives ( m - ½) h(O - ~h'(O = h"(O + ('(0)g'(Og''(O- ( 4.67) The above equation requires a numerical solution. 4- Unsteady Flow 53 4.5.2 Solution for given b..P(t) The same technique is applied to equation ( 4.50), where 84>--+ 8J, by setting J(x,t) = Jo(x,t) + 8J 1(x,t) + ... ( 4.68) giving (4.69) 8t 8x2 • If Pa-(t) = atm/2 , ma positive integer, this leads to 2 Jo(x, t) ,.._, -atmf r ( 1 + ; ) 2mimerfc ( 2/t) as t--+ 0. (4.70) The next order solution is found from 2 2 2 BJ1 = 8 1 + ('(0)c/>o o Jo + ('(0) (8Jo) ' ( 4. 71) Bt 8x2 8x2 Bx which with Ji(x,t) = tmh(fl gives mh(fl-!h'(() = h"(O + ('(0)g(fll'(O + ('(0) (g"(fl)2. (4.72) Although the displacement is difficult to find, the velocity can be related to the pressure by v(t) = -R2 ([84>(0,t)]:; lx=o (4.73) and therefore as t --+ 0 the velocity is v(t),...., -Ratm:;1 f(l + m/2) + 8 { f(l + ~) a2('(0) - h'(o)} tm-1/2_ (4.74) f (1 + m;-1) r(l + m;-1) 4.6 Numerical Solution for Finite Deformations In chapter three it was shown that a suitable stress relation is (Mow et al. 1985) a= !(l + d1(y- l)) (y2 _ _!_) = F(y), (4.75) 4 y y2 where y = l + i~, d1 = 0.9, m = 6.3 and the permeability (4.76) The numerical solution to ( 4.16) using the above stress-strain and permeability function was obtained using a method of lines routine, D03PBF, from the NAG library. This routine discretizes the spatial domain into discrete values using central finite difference routines thus obtaining a set of coupled ordinary differential equations which can be solved using standard routines such as Gear's method. 4. Unsteady Flow 54 2.5 2.0 Q) E 1.5 Q) u ' C ' ' ' 8, 1.0 ' ' ~ " Q) ' " > " .. C ' ' 0 ---- .. , .... _ O 0.5 .. ------... -- ...... 0 0.5 1 1.5 2 Velocity Figure 4.3: The time to reach equilibrium as a function of the magnitude Vm of a step change in velocity for the solid at X = 0.5 (dashed line) and X = I (solid line). The velocity v(t) was specified and the displacements and pressure subsequently found. For ease of calculation the time scale was set at to = h 2 / H M ko. Two velocity forms were used; a step increase v(O) = 0, v(t)=vm fort>0, ( 4. 77) and a sinusoidally varying form (4.78) where x is a constant. 4. 7 Results and Discussion For a step change in v the time for the solid matrix to reach equilibrium was calculated at X = 0.5 and X = 1.0. This time was based on reaching 99% of the corresponding steady state values. Figure 4.3 shows the convergence time as a function of Vm. These results indicate that for small velocities the medium deforms uniformly with all sections of the medium moving 'in phase'. For higher velocities the material at X = 1 takes longer to reach equilibrium than material further downstream. For Vm > 0.4 the speed of the fluid aiding deformation helps decrease convergence time. 4- Unsteady Flow The step increase contains high frequency components which the reduced equations used Ill this chapter were not designed to model. However since step function-like phenomena are often used experimentally, the numerical results for the step increase are included. For t ~ pj-k0 , wavelike compressions propagate down the material, hence our numerical results may be unreliable in this regime. The large speed of propagation of these waves often means these waves dissipate in less than the time scale of the compression. A pictorial description of the internal deformation pattern is shown in figure 4.4. The direction of flow is from top to bottom. Part A of this figure illustrates the deformation at various times after a step change in velocity. A consolidation region forms at the lower boundary even at small times. Part B indicates the deformation profile as a sequence of steady states. The velocity for each of these steady states is chosen to give the same height as the corresponding profile in part A. This would then be equivalent to a very slowly increasing velocity (or pressure) where the time lag is negligible. Part B illustrates that initially the medium deforms relatively uniformly (and linearly). The downstream material then reaches a limit of consolidation where any further deformation would unrealistically close off the pores. The deformation then continues upstream with more of the medium approaching maximum deformation. In figure 4.5 the displacement at X = 0.5 and X = 1.0 are shown for Vm = 0.3 in the sinusoidally varying velocity case. The displacements are scaled as u(X, t) = u(X, t)/um(X), where um(X) is the steady state displacement corresponding to flow velocity Vm- The velocity u(t) = v(t)/vm is also shown for reference. Note the linear nature of the deformation with both points of the medium deforming in phase with each other and with a constant time lag with respect to the velocity. This is expected from the previous analysis on slow compression rates. Figure 4.6 shows the displacements for Vm = 2.0. Comparing the results with figure 4.5 indicates that nonlinear effects are important at high velocities. The time lag at peak deformation is significantly less than that at the trough. The upstream medium shows a greater variation than the more aggregated downstream region. The high velocities in figure 4.6 also aid the solid to deform to 98% of its steady state value. This contrasts with the value of 60% for Vm = 0.3. Note also that there is a much greater variation for the upstream medium than for the medium at X = 0.5. The downstream medium is closer to the maximum consolidation region and so remains compressed while the upstream region can still fluctuate. 4.8 Conclusions For one-dimensional unsteady compression of porous sponges by fluid pressure the governing diffusion equation has been derived and a lower limit on the typical time scale found below 4. Unsteady Flow 56 8 A 0.0 0.24 0.48 0.72 0.96 1.2 Time Figure 4.4: An illustration of the porous medium consolidation. In part A a step change in velocity is applied and the deformation is shown for various times. The velocity was chosen to give a final height of 0.5. Part B shows a sequence of steady state values with velocities chosen so that the medium has the same heights as in part A. Note the increased consolidation at small times in part A. 4- Unsteady Flow 57 1.0 0.8 ,,. .. /' I '.a 0.6 I I '.a I I '& I / ':a ! / ':a I I / ':a I ! I ".l I I ',1 I , ':a I I , ':a I 0.4 I I ! I ".l I I I ',1 I I ".l I ! I , I I , I , .,. r I , ! ·. ':'•r 0.2 ! .. " ! ! I I I .,., 0 1 2 3 4 5 Time Figure 4.5: Displacement due to a sinusoidally varying velocity v = ½'-( l-cos(21rt/x)). Vm = 0.3. --, v/vm; ...... u(0.5, t)/um(0.5); ------u( l, t)/um(I). Um(X) denotes displacement for v = Vm. 1.0 ...... I.··,· .. \ ·. \ ·. \ ·. \ ·. \ ·. \ ·. \ : ' ·• ' ·. ' ·. ' ·. ' ·. '' :·. 0.8 ' : ' : ; I ' : ' •. ; I ' : : : 'I : ; I ' I : ' : ! I ; I I : ; I ' : ; I ; I 'I :-_ : I ' ·. ; I ; I ' : : I ' .. : ' ' : 0.6 ! I ' : ' ·. ; I ' ·. . ' ·. . ; I ' ·.... I • •• • ; I ; I '\ ' ; I ' ; I '\ .. , I 0.4 ; I ·.. , :,! I ; I ; I ; I :,; I :, 0.2 :, :, :, :, :, :, ~ r 0 1 2 3 4 5 Time Figure 4.6: Displacement due to a sinusoidally varying velocity v = ¾"-( 1 -cos(21rt/x)), Vm = 2.0. --, v/vm; ...... u(0.5, t)/um(0.5); ------u(I,-t)/um(I). um(X) denotes displacement for v = Vm. 4. Unsteady Flow 58 which inertial effects are important. The linear small time behaviour and the nonlinear behaviour for slow compression have also been investigated. This analysis led to the result that a periodic pressure difference can, under some circumstances, lead to an increase in the mean permeability. The nonlinear finite deformation problem was then solved numerically for a given pressure. This showed the different behaviour of the sponge for low and high fluid velocities, specifically that the solid deforms in phase for low velocities but that the upstream material takes longer to deform than the downstream material for higher velocities. Chapter 5 Fluid Flow Over a Thin Deformable Porous Layer This chapter deals with the unsteady flow of a viscous fluid in a channel or tube which is lined by a deformable porous material. The primary motivation for studying this problem is to understand the influence of the glycocalyx on flow within blood vessels. The glycocalyx is the thin layer of glycoproteins which lines the surface of endothelial cells in all blood vessels. This layer, which may be up to 1 µm thick, is obviously important in capillaries and other small blood vessels. It may also influence flow in the large arteries where local flow conditions, particularly the wall shear stress, have been linked to the development of atherosclerosis (Schettler et al. 1983). This study is also relevant to filtration technology, soil mechanics and to other biological problems such as the mechanics of articular cartilage. Krindel and Silberberg (1979) have considered experimentally the flow through a gel-walled tube in which the layer of gel is of the same order of magnitude as the width of the tube. The coupled equations for solid displacement and fluid velocity within the porous layer will be derived using biphasic mixture theory, specialised to the case of shear flow. For a rigid porous layer of nondimensional height E, exact solutions will be given for the fluid velocity inside and outside the porous layer. Boundary conditions for the velocity in the purely fluid region which can be applied at the fluid-porous layer interface will be found by solving the full coupled equations and expanding the solutions as asymptotic expansions in powers of E. The fluid velocity above the layer can then be found without the need to solve the full coupled equations. Comparisons are then made between the velocity profiles in the fluid for cases of rigid nonporous walls and walls lined with both rigid and deformable porous layers. The deformations are assumed infinitesimal and displacements are assumed predominantly in the x-direction (along the axis of the channel). The porosity and permeability will be assumed constant since they will not change significantly for the infinitesimal deformations considered here. The boundary conditions are applied at the original position of the boundary 59 5. Flow Over a Porous Layer 60 for ease of analysis. This is standard in infinitesimal theory but the ramifications of this approximation are discussed in chapters four and six. 5.1 Mathematical Model In chapter two the governing equations for the biphasic mixture were developed with an assumption that the fluid shear stress al in the porous medium was negligible. In this section we reintroduce al to develop new equations that are specialised to the case of purely tangential (shear) flow. The momentum equation can be written for both phases as ( 5.1) where Tt3 is the stress tensor and 7rt3 is an internal interaction force between the constituents. The stress tensors are expressed as (5.2) where p is the volume averaged pressure, as is the solid 'contact' stress, al is the viscous stress in the fluid and I is the identity tensor. The interaction force is modeled by (5.3) where I( is a drag coefficient. Making the assumption that deformations of the solid matrix are small, the solid stress can be written as as = Atrace( e )I + 2µe, (5.4) with e = ½(\7u + (\7uf), (5.5) solid displacement vector u and Lame constants A and µ. For the fluid (5.6) where µa is the apparent viscosity of the fluid in the porous material (see chapter two for a discussion of µa). In the fluid phase outside the porous medium, (j)l = 1 and the motion is described by the Navier-Stokes equations for an incompressible fluid, \7-q 0, ( 5. 7) pf ( ~~ + q. \7 q) ( 5.8) 5. Flow Over a Porous Layer 61 Porous Layer l .... h Fluid ... _i_ y=O ... Flow ... Impermeable Boundary Figure 5.1: Schematic diagram for flow past a porous layer lining the surface of a channel. The porous layer of width L = lh is small compared to the width of the channel. where q = vf, the fluid velocity in the fluid-only region and µf is the fluid viscosity. At the fluid-porous medium interface the boundary conditions derived by Hou et al. ( 1989) are [ef/(v1 - v5 )] • n 0, (5.9) [T5 + Ti - /vf(vf - v 5 )] • n 0, (5.10) where [·] is the jump going from the purely fluid phase to the porous medium and n is the normal unit vector at the interface pointing into the purely fluid phase. These two matching conditions represent the conservation of mass and momentum. 5.2 Flow Geometry and Governing Equations The geometry consists of fully developed flow through a symmetrical channel with solid walls at y =±hand a porous layer of thickness E attached to both walls as shown in figure 5.1. By symmetry only half of the channel y E [0,h] is considered. A pressure gradient fJp/fJx = G(t ) is applied, producing an axially directed flow. Due to the assumption of an infinite channel there is assumed to be no x dependence in any of the terms except the pressure. Substituting equations (5.2) and (5.3) into (5.1) and using equations (5.4) - (5.6) gives 5. Flow Over a Porous Layer 62 pi ( V1 + V2 ~~ ) pf ( v2 + v2 ~~ ) (5.11) where u = (u1,u2) is the displacement, vf = (v1,v2) is the fluid velocity and dot denotes differentiation with respect to time. Non dimensional variables are defined as 2 2 h Go A h Go A y = hy, t = t0 i, ql = - --qi, V1 = - --V1 (5.12) µf µf dh A dh A L U2 = dhuz, qz = -qz, Vz = -Vz, € = h, (5.13) to to for a typical pressure gradient Go, period to and infinitesimal deformation d ~ €. Assuming a small typical deformation d the boundary conditions (5.9) - (5.10) reduce to uncoupled conditions for the vertical and horizontal components up to the second order, O(d2 ). Equations (5.11) also uncouple to produce independent equations for the x and y components. The vertical components, u2, v2, q2, are then zero to this order of approximation. In later analysis boundary conditions will be found as power series expansions in E up to 0( c3 ). Truncation at order c3 is only valid provided the deformations are such that d::; O(c2 ). This is a reasonable assumption for sufficiently small velocities. The notation u = u1 , q = q1 and v = v1 is now used. With these assumptions the equations are linear and amenable to Fourier analysis. The hats ( A ) denoting nondimensional variables are now dropped as all variables are nondimen- sional hereafter. The Fourier components are defined by writing 00 u1(Y, t) = I: un(y)eiwnt' (5.14) n=-oo 00 q1(y,t) I: qn(y)eiwnt, (5.15) n=-oo (X) v1(Y, t) I: vn(y)eiwnt' (5.16) n=-= (X) G(t) cneiwnt (5.17) = I: ' n=-= for the displacement and velocity in the porous layer, the velocity in the purely fluid region and the applied pressure gradient respectively. Here Wn is the frequency of the nth component. The following nondimensional parameters arise from the nondimensionalisation of the Fourier expansion of equation ( 5.11 ): (5.18) 5. Flow Over a Porous Layer 63 J(h2 {j (5.19) µJ KµJ V (5.20) µpf' µJ TJ (5.21) µa psw?ih2 ,n (5.22) µ For ease of use the constants an and ,n are represented by a, 1 . The frequency parameter, a 2 , is generally known in haemodynamics as the Womersley parameter. A measure of the viscous drag of the outside fluid relative to drag in the porous medium is b. The ratio of viscous forces in the porous medium to the shear stress of the solid component is described by v. The ratio of the bulk fluid viscosity to the apparent fluid viscosity in the porous layer is TJ· The equations (5.11), for each Fourier component, can then be written in the form = AZ - B, yE(l-E,l), (5.23) y E (0, 1 - E), (5.24) where - 1 + iva2 A= [ (5.25) -iVTJ0.2 For the case of pure shear, the nondimensional boundary conditions are at y = 1: Un = Vn = 0, (5.26) dqn at y = 0: -=0, (5.27) dy at y = 1 - E: qn = q)vn + is/3un, (5.28) dqn 1 dvn - (5.29) dy TJ 5.3 Steady State Deformation For steady state deformation, wn = 0 for all n and the pressure gradient G 0 = 1, since the nondimensionalising pressure gradient Go is the applied steady pressure gradient. The 5. Flow Over a Porous Layer 64 equations of motion (5.23) - (5.24) for this simple case reduce to d2u 8 -bv - , (5.31) dy2 d2v bryv - 4>1 ry, (5.32) dy2 d2q -1. (5.33) dy2 By solving equations (5.32) and (5.33) first and then equation (5.31) the solution for the displacement and velocities are 2 u -ci sinh[((y - l)] - 4>f (1 - e((y-l)) - (y - l) + c2(Y - 1), (5.34) 'r/ bry 2 V c1 sinh[((y - 1)] + : ( 1 - e((y-l)), (5.35) -y2 q -+c32 , (5.36) where ( 2 = bry and the constants c1 ,c2 and c3 , found from the boundary conditions, are ( 1 - € )ryl (5.37) cosh(E()( ' C2 -1, (5.38) 2 c3 = (l - E) + } (1 - e-(() - J tanh(£() (e-(( - (l - E)7]). (5.39) 2 b b ( 5.4 Rigid Porous Layer The time dependent behaviour is now included in terms of the Fourier components. The general solutions to equations (5.23) - (5.24) with the solid deformation excluded are - -Vj~e-a< + tanh( 0o(l - E)) ( 1 - ~(l - e-a()) (5.42) :flg_ i~2a2 t cosh(aE) + ~ tanh(vfi°a(l - E))sinh(aE) i~2 a2 1 + ~(-c1 sinh(aE) - 1 + e-a() C5 = (5.43) cash( vfi°o( 1 - E)) Alternatively differentiating equation (5.40) and setting y = 1 - E gives - Rearranging the second of these equations to eliminate c4 from the first equation gives an expression for vn at y = 1 - fin terms of dvn / dy at y = 1- f. Using the boundary conditions (5.28, 5.29) it is then easy to obtain (5.46) This can be expanded in powers of f to give (5.47) as f -+ 0. The first two terms in the expansion are in phase with the pressure however the higher order terms show phase lag effects. This simple boundary condition can then be applied at the interface of the fluid with the porous layer without solving the full coupled equations. It can also be used to give approximate results for flow geometries that are too unwieldy to solve for both the porous layer and the fluid. As f-+ 0 or 5.5 Deformable Porous Layer The matrix A is diagonalisable provided (trA)2 -=p 4detA which guarantees distinct eigenval ues. This is assumed to be the case. The matrix A in equation (5.24) is expressed in terms of its complex eigenvalues >. 1,>.2 and eigenvectors e1 = (e11,e12), e2 = (e21,e22), A - CAc-1 - ' ( 5.48) where ( 5.49) The solution can be written as (5.50) where the notation exp (A1/2(y - 1)) = [ exp(v%(y- 1)) 0 ] , (5 ..51) 0 exp( A(y - 1)) has been used. The arbitrary constants c6 and c7 can be obtained from the boundary condi tions. This solution was obtained by decoupling equation (5.23) into two differential equations using standard spectral decomposition methods. As this full solution requires long calcula tions it would be useful to find a simple boundary condition like equation (5.47) which can be easily evaluated. 5. Flow Over a Porous Layer 66 Differentiating equation (5.50) and evaluating at y = 1 - f gives the constant vector Substituting this into equation (5.50) gives 2 B 3 Z) ,...., -E -dZ] + E - + E A -dZ] as f--+ 0. (5.53) 1-c dy 1-c 2 dy 1-c Applying the boundary conditions (5.28) - (5.30) now gives a new boundary condition in terms of the velocity in the fluid applied at the fluid-porous medium interface, (5.54) It should be noted that this condition reduces to the nonslip condition in the appropriate limits. Here the form of the boundary condition for a rigid layer is recovered when /3 = 0 and a phase lag, 0 = tan- 1(/3;/TJcp}), is noted. The complex coefficient indicates that the drag associated with the porous layer causes the layer to be out of phase with the pressure oscillations and hence causes a time lag in the boundary condition. 5.6 Results and Discussion The results for steady state flow will be described first. As can be seen from equations (5.36) and (5.39) the velocity profile in the fluid is parabolic plus a uniform flow given by the constant c3. The solutions for the displacement and the velocity, equations ( 5.34) - (5.35 ), are shown in figure 5.2. The parameters for figure 5.2A are fJ = 2.0, cpl = 0.5, T/ = 0.5 and E = 0.2. Figure 5.2B has the porosity changed to 4>1 = 0.8. These plots illustrate the almost linear nature of the velocity and the displacement in the porous medium. By increasing the porosity, the velocity in the porous medium is expected to increase as there is less solid to impede the flow. The displacement correspondingly decreases since there will be less drag on the solid component. The shape of the displacement profile is almost linear although there is a slight positive curvature not readily apparent at the scale of the plots. The variation of velocity profile with the parameters has been treated in some depth by Hou et al. ( 1989 ). In figure 5.3 the variation of the maximum velocity in the channel is found for steady flow over either a rigid or deformable gel layer. This is given by the constant c3 in equation (5.36). The variation with f and cpl are given for three different values for T/· Also included is the maximum velocity when the gel is completely solid, that is if the boundary condition q = 0 is applied at y = 1 - E. The variation with {J was found to be mimimal. This can easily be shown by expanding the equation (5.39) as a Taylor's series about small E. The dependence in fJ then disappears for small orders of E. The results indicate that as the width of the gel layer increases the velocity in the fluid also decreases. If the porosity increases then 5. Flow Over a Porous Layer 67 Fluid Flux 0.1 0.1 0.0 ~ _ _._ _ _._ _ __._--,-..... y -0.8 -r------; i i i i _; -1.0 0.1 0 0.1 Displacement A B Figure 5.2: Fluid velocity profile, solid line, and displacement of the solid, dot dash line, for steady flow over a deformable porous layer. The parameters are 8 = 2.0, TJ = 0.5, f = 0.2 with q;f = 0.5 in 5.2A and q;f = 0.8 in 5.2B. S. Flow Over a Porous Layer 68 £ 0 0.3 0.5 1.0 11 =0.66 0.33 0.1 0 1 f Figure 5.3: The maximum velocity in the fluid channel, c3 , is shown. The solid lines show the variation of c3 with £ E [0, 0.3] (top scale) when cpl = 0.5. The dashed lines show the variation of c3 with cpl E [0, 1] when £ = 0.2. Three plots with differing TJ are shown. The dotted line shows c3 when the porous gel is solid. the velocity, c3 , increases. Note also that when 7J = 1 and cpl = 1 the maximum velocity is 0.5. This is to be expected since then the 'gel' is effectively all fluid with the same viscosity as the fluid. The parameter 7J is a measure of the increased viscosity of the fluid in the porous layer. Thus by decreasing 7J, an increase in the viscosity in the porous layer, the velocity in the fluid only region decreases. In figures 5.4 - 5.6 the comparison is made between unsteady velocity profiles in the fluid only region for the three cases; ( 1) flow between rigid impermeable walls (solid lines), (2) with a rigid porous medium coating the walls (dashed lines) and (3) with a deformable porous material lining the walls (dotted lines). The pressure gradient applied to the system is a simple sinusoid. Since the porous layer is assumed thin here, I am not interested in the details of the flow and deformation in the porous layer and only the flow in the channel is shown. The exact solution, from equation ( 5.41 ), was used to calculate the flow over a rigid layer. The approximate boundary condition (5.54) was used to calculate the flow over a deformable porous layer. These graphs are plotted on different scales, indicated on the x axis. The velocity profiles are shown at four different times in the cycle, t = 0, 1r /4, 1r /2, 3r. /4. The parameters are set to /j = 2.0, 0.0 I j j i ·' y f -0.95; -1.0 0.5 1.8 Velocity Profile Figure 5.4: Velocity profile at four times, t = 0,1r/4,1r/2,31r/4, during an oscillatory cycle for flow in a channel with (1) No gel layer, solid line, (2) a rigid porous layer, dashed line and (2) a deformable porous layer, dotted line. The parameters are 6 = 2.0, In figure 5.4 the width of the gel layer is E = 0.05 and the frequency parameter is a = 2.0. This shows that for thin layers the results for deformable and rigid porous layers are virtually indistinguishable from each other although they are clearly different from the curves for flow without a porous layer. Note that at some time in the cycle the variation is restricted to a small region near the boundary while at other times the difference is almost constant over the entire width of the channel. The reverse part of the cycle is not shown since they are qualitatively similar to the results fort E (0,31r/4) except negative in direction and in the same order as that shown. In figure 5.5 the width of the gel is increased to E = 0.1 to illustrate that increasing f magnifies the difference between the gel layer solutions and the no gel layer velocity profile without changing the qualitative shape of the profile. This validates the perturbation series solution in terms of E. Note also that the differences between the rigid and deformable gel solutions are a little more apparent. The frequency parameter is increased from a = 2.0 to a = 6.0 in figure 5.6 to illustrate that the deformable porous layer is affected by the phase lag associated with the movement of the porous layer. When the frequency is increased the porous layer time lag becomes a significant fraction of the period and hence it produces a large relative effect on the velocity profile. The velocity for the deformable layer now differs significantly from that of the rigid layer solution. The time lag associated with the deformable layer is now clearly apparent. 5. Flow Over a Porous Layer 70 0.0 • ' •' -· i• 'j _i j j j y j , -0.9 ------ -1.0 0.5 1.8 Velocity Profile Figure 5.5: Velocity profile at four times, t = 0, 1r /4, 1r /2, 31r /4, during an oscillatory cycle for flow in a channel with (1) No gel layer, solid line, (2) a rigid porous layer, dashed line and (2) a deformable porous layer, dotted line. The width of the gel layer is increased to f = 0.1. 0.0 I I I y I I ' _,• ., -,:, ·I -i -0.9 -i . , ------.; ------.__ ------ -1.0 0.07 0.28 Velocity Profile Figure 5.6: Velocity profile at four times, t = 0,1r/4,1r/2,31r/4, during an oscillatory cycle for flow in a channel with (1) No gel layer, solid line, (2) a rigid porous layer, dashed line and (2) a deformable porous layer, dotted line. The frequency parameter is increased to ex = 6.0. 5. Flow Over a Porous Layer 71 5. 7 Conclusions The purpose of this chapter has been to develop the governing equations for flow over a porous layer and to give general solutions for the displacement and velocity in and above this layer. The theory of binary mixtures has been used to develop these equations assuming that the deformation is infinitesimal and predominantly in the flow direction. Exact solutions were obtained for the flow over both rigid and deformable layers. Boundary conditions were then developed for the velocity of the free fluid at the interface with the porous medium, which can be applied without solving the full coupled equations for flow in the porous layer. This greatly simplifies the analysis and is applicable to a number of situations beyond those considered here. These boundary conditions were then used to consider the difference between velocity profiles with and without a porous layer lining the channel. Chapter 6 Radial Flow Through Shells This chapter considers the flow of fluid radially out through both cylindrical and spherical shells. The aim is to develop a general theory covering planar, cylindrical and spherical geometries. These general schemes will then be illustrated by considering flow through a cylinder constrained by a rigid mesh at the outer boundary. The geometries under consideration will be outlined in the first section. The general governing equations for unsteady flow will then be developed in section 6.2 using infinitesimal deformation theory. Various forms for these equations will be given for a specified velocity or pressure difference. The boundary conditions will then be developed for the two cases of the outer boundary being free to expand and the case in which a rigid mesh covers the outer boundary. General solutions will be found for the steady state displacement and pressure in terms of a given velocity for spherical, cylindrical and planar geometries. These are in the form of exact integrals which can be evaluated analytically for various permeability functions. A perturbation technique is then used to find solutions for more difficult permeability relations. The time dependent displacement of the solid component is found for small times when the permeability is considered constant. Starting from the general formulation it is then possible to solve for planar, cylindrical and spherical geometries with constrained and unconstrained outer boundaries and different permeability functions. Detailed solutions will be given for radial flow through a cylindrical tube with a constrained outer boundary. This particular geometry is chosen to tie in with the previous chapters on steady and unsteady one dimensional flow. The unconstrained cylindri cal geometry has been considered as a model of flow through an artery wall by J ayaraman (1983), Jain and Jayaraman (1987), Klanchar and Tarbell (1987) and Kenyon (1976a, 1979). The primary motivation for radial flow is as a model of flow through the artery wall. The flux of proteins through this porous elastic tissue has been linked to the process of atherogenesis (Schettler et al. 1983), a build up of proteins in the intima (inner layer) of 72 6. Radial Flow 73 Constrained Unconstrained Figure 6.1: Schematic diagram of radial flow through cylindrical or spherical shells. In constrained flow the outer boundary is constrained by a rigid mesh. In unconstrained flow the outer boundary is free to expand. The radii of the inner and outer boundaries are shown. the artery wall. A related problem is the injection of fluid into tissue, often leading to a spherical cavity (Nicholson 1985) being formed. This may affect the rate of uptake into the tissue hence solutions for spherical geometries are needed. A typical experiment is the planar flow through slabs of tissue constrained by impermeable walls. Difficulty is often found with the porous material separating from the wall producing a fluid flux around the edges of the material. Alternatively excess drag may occur between the porous material and the side walls. By adopting a cylindrical geometry these effects can be minimised. 6.1 Geometry The two related problems under consideration here are shown in figure 6.1. The first shows a section of a cylinder or sphere with fluid moving radially out from the centre. The outer boundary is constrained by a rigid mesh that offers no resistance to the passage of the fluid . The inner radius is a 1 and the outer radius b1 . Unconstrained deformation is similar to constrained flow except that the outer boundary is free to expand. The radii are a2 and b2. The unconstrained planar flow problem is not physically feasible and only unconstrained cylindrical and spherical cases are considered. 6. Radial Flow 74 6.2 Governing Equations In developing the governing equations it is assumed that deformations are infinitesimal, that only the radial components of displacement and velocity are nonzero and that the permeability function is dependent on the local porosity only. From chapter two the governing equation is of the form ( 6.1) The components of the stress tensor for the solid are au u (,\ + 2µ)- + n,\-, (6.2) 8 r r u au u (,\ + 2µ)-:;: + ,\ Br + (n - l)-:;:,\, (6.3) where n = 0,1,2 correspond to planar (x,y,z), cylindrical (r,0,z) and spherical (r,0,cp) geometries respectively and u = Ur, the radial displacement. The Lame stress constants are ,\ and µ. In the case of planar flow it should be remembered that The divergence of the stress in the radial direction can be written as n ) _ 8crrr Clrr - cr00 ( v·G'r--a +n . ( 6.4) r r It can also be shown, with a little algebra, that substitution of equations (6.2) and (6.3) into ( 6.4) gives (6.5) where Integration of the continuity equation (2.10) leads to the velocity component in the radial direction v(t) Vr (6.7) = --.rn The governing equation (6.1) then becomes (6.8) The transformation, used previously in chapter four with n = 0, w(r,t) = -u(r,t) + -1 it v(r)dr, (6.9) rn 0 6. Radial Flow 75 allows the governing equation to be written as 8 1 8w or (L[w]) = k(L[w]) at' (6.10) where the operator L is given by (6.11) The velocity term will occur in the boundary conditions ( discussed in the next section) but has been removed from the governing equation. Alternatively equations in terms of can be derived. If equation ( 6.8) is rewritten as Hak()° (6.13) or upon expans10n H k(,1.,)_!_~ ( n8 6.3 Boundary Conditions In this section the boundary conditions will be derived for both problems outlined earlier. These boundary conditions will be expressed in terms of u, w or. 6.3.1 Constrained Flow The boundary condition for the displacement at the rigid porous mesh, r = b1 , is simply u = 0. Making use of the relation for At the free boundary, r = a 1 , the contact stress is zero since the flow is normal to the surface and so (6.16) 6. Radial Flow 76 This can be written as [-au + n>.--u] = 0, (6.17) or r r=a1 where "X = >./(>. + 2µ). In terms of this is u( a 1 , t) ( -) ( a1, t) = n___;,_~ 1 - >. . (6.18) a1 By integrating equation (6.6) the displacement can be written as (6.19) By substitution of u into equation (6.18) a boundary condition in terms of {l - "X) 1b1 n Making use of the relation between the pressure and the stress equation (6.8), the bound ary condition at the constrained boundary, r = b1 , is f).P(t) (a1,t) - ~- (6.21) Equation (6.20) and (6.21) then form the boundary conditions for equation (6.14). Boundary conditions in the variable w can be written simply as f( t) w(b1, t) bn , 1 8w1 w,] _ f(t) ("X _ 1) [-+n-A n+l ' (6.22) or r r=a1 al where J(t) = - Ji v(r) dr. 6.3.2 Unconstrained Flow By integrating equation (6.6) the displacement can be written in terms of (6.23) Su bsti tu ting this into the boundary condition at the inner boundary (6.24) gives 1 - "X 1b2 !1P bn+1 (6.26) 6. Radial Flow 77 The problem of the position of the boundaries is a potentially important one, usually avoided. In appendix C.5 the relationships between the original and final boundary position are derived as a a0 + u(a), b b0 + u(b), (6.27) in which the original position of the boundaries are a0 ,b0 • These are implicit equations for the new boundary positions, a, b. They can only be solved after the displacement function has been evaluated and will normally require a numerical solution. An approximate form would be to assume the explicit relations a a0 + u(a0 ), (6.28) b b0 + u(b0 ). (6.29) These are only valid for infinitesimal deformations and are considered in more detail in the results and discussion section where application of both exact and approximate boundary conditions are compared. 6.4 Steady Radial Flow The governing equations are nondimensionalised using (6.30) for typical pressure Pm, outer radius l, permeability ko and displacement uo. The governing equation (6.1) can then be written as (6.31) where the dimensionless parameters are b = Uo (6.32) l ' and also . _ 1 ·n . ) 6.4.1 General Solutions The parameter b is now taken to be b = 1, equivalent to nondimensionalising u with respect to I. In the next section this parameter will be reintroduced in a perturbation analysis. By using the function g( d V dr [g( (l-n)vr1-n+c1 for n = 0 or 2, g( 1 d n { g-1 (( 1 - n )vr1-n + c1) for n = 0 or 2, -- (r u) = (6.37) rn dr g-1(vln r + ci) for n = l. This can be integrated to obtain 1 1 - r: lb sng- ((1 - n)vs -n + c1) ds + ;! for n = 0 or 2, u(r) = { (6.38) --11b sg- 1(vlns + ci)ds + -C2 for n = 1, r r r where c2 is a constant of integration found from application of the boundary conditions. 6.4.2 Solution for Constrained Flow Using the boundary condition of u(b1 ) = 0 it is easy to see that c2 = 0. At r = a 1 , the inner boundary, g[ u(r) = - r: 1b1 sng-1 (u -n)v [sl-n - al-n] + g [ nu(a1!:l - "X) l) ds, (6.41) for n = 0 or 2, and u(r) = -~ ibi sg-1( vln(s/ai) + g [u~~1\1 - "X)]) ds, (6.42) for n = 1. 6. Radial Flow 79 This solution for u( r) is dependent on a 1 and u( a 1 ) which are not known prior to evaluation of the integral. By using the result a1 = a0 + u(ai), ( 6.43) where a0 is the initial inner radius, equation (6.42) gives an implicit equation for the radius a1, and hence u( a1). In general this implicit equation must be solved numerically. The integral, however, can be evaluated analytically for a number of permeability relations. These are included in appendix D.2.6. The pressure can be found from equation (6.26) which gives ( 6.44) The derivative ~~(bi) can be expressed in terms of the inner radius a 1 using the relationships u(ai). -) g ( du] ) _ v ln bi = g ( ~(l - >.) - vlna1, n = l, dr b1 du] ) 1 (nu(ai) - ) 1-n 9 ( dr bi -(l-n)v(b1)-n =g ai (1-,\) -(1-n)v(ai) , n = 0,2, ( 6.45) which are found by expressing the constant c1 in equation ( 6.37) in terms of both the inner and outer boundary conditions. Thus if the permeability function is known and equation ( 6.42) is analytically integrable then an implicit equation for a1 is found, which can be solved numerically, and the pressure can then be found directly. 6.4.3 Solution for Unconstrained Flow Following the same procedure the solution is found to be 2 2 u(r) = -- 1 1b S 2 9- 1 [ V ( ---1 1 ) + g ( 2--u( a ) ( 1 - -A) ) ] ds r 2 r a2 s a2 1 [ ( a2) - ) ] +-1 b~ - g- V - 1 - - 1 ) + g ( 2--(1u( - ,\) , (6.46) r 2 2(1-,\) a2 b2 a2 for n = 2, and 2 1 [ ( a 2 ) _ ) ] u( r) - -;: 1br s g - 1 v In ~s) + g (~ u( ( 1 - ,\) ds +~ b~_ g-1 [vln (!!3_) + g (u(a2 \1 - °X))], (6.47) r ( 1 - ,\) a2 a2 for n = l. 6. Radial Flow 80 In this geometry neither the inner or the outer boundary positions are known. Using the relationships between the displacement and the radii, equation (6.27), two coupled implicit equations occur of the form (6.48) for known functions F 1 , F2. These must then be solved numerically. The pressure is found from equation (6.21) using (6.18) to be b,.P = n(l - °X) (u(a2) _ u(b2)) . (6.49) Ha a2 b2 The value of u(b2 ) is given by the result u(b2) - ) (u(a2) - ) g ( ~(l-,\) -vlnb2=g ~(1-,\) -vlna2, (6.50) for n = 1 and g (nub~b 2)(1- 'X))- (l - n)v(b2)1-n = g (nua~2)(l - 1)) - (1- n)v(a2)1-n, (6.51) for n = 0, 2. 6.5 Perturbation Solution for Infinitesimal Displacements The parameter 8 is now reintroduced as a perturbation parameter assuming that the defor mation is small compared to the outer radius of the shell. The permeability can then be written in the form 2 k(!q,) = ((84>) = 1 + 8('(0)4> + 82 ("(0) ~ + 0(83 ). (6 ..52) The function ( is similar to that used in chapter four. In the limit of no deformation, k( ) = 1 as---> 0 and therefore ((0) = 1. By defining the displacement as (6.53) the governing equation (6.31) can be expanded in powers of 8. Taking coefficients of 8, the zeroth order equation is n (cfluo n duo n ) r --+----u0 = v, (6.54) dr2 r dr r 2 which has solutions C4 V c3r + r2 - 2' n = 2, C4 vr uo(r) = c3r + - + - ln r, n = 1, (6.55) r 2 n = 0, 6. Radial Flow 81 where c3 , c4 are constants of integration found from application of the boundary conditions. The constants c3, c4 for cylindrical constrained flow are given in appendix D.3. The first order equation then becomes 2 rn(d --u1 + --ndu1 - -uin) = v(, (0) (duo- + -uon) , (6.56) dr 2 r dr r2 dr r which can be easily integrated to yield solutions. 6.6 Small Time Solutions in Unsteady Flow U pan application of a pressure gradient the initial behaviour of the porous medium is governed by the linearised form of equation (6.13), since the deformation is not sufficiently large as to change the permeability significantly. The small time behaviour of the tissue upon application of a pressure gradient for the constrained problem is now considered. The governing equations can be written as :n:r (rn~:) = ~~, (6.57) with boundary and initial conditions cp(a,t) -(1 - "X) an:1 lb ancp(a, t)da, cp(b,t) Solutions to this linear system can be found using a number of approaches. For oscillatory pressure gradients the Fourier components of the system can be found. As the governing equations here are only valid for the infinitesimal displacements present in the initial stages of the deformation a Laplace transform technique is preferred. Taking the Laplace transform of the equation (6.57) gives r2ef>11 + nref>' - sef>r 2 = 0 (6.59) where £ { (6.60) where now 4>1 = 84;( T/, s )/ OTJ. 6.6.1 Cylindrical Geometry The solution of equation (6.60) for the case of cylindrical geometry is 4>(TJ,s) = A1(s)lo(TJ) + A2(s)Ko(TJ), (6.61) 6. Radial Flow 82 where [0 , 1(0 are modified Bessel functions of order zero and A 1 , A2 are functions found from application of the boundary conditions. By allowing the variable s -> oo the functional dependence of tp(r, t) can be found in the limit of small time. The analysis for this is shown in the appendix D.4. For an applied pressure gradient of the form b.P = tm/2 , m = 0, l, 2 ... the solution becomes 1 1 tp(r,t) = r (1 + ;) (1- °X)a~(4t(i im+ erfc (r2;,) 1 1 -r (1 + ;) f (4t)f (imerfc (b2Jt) + 2Jt ~r\m+ erfc (b2Jt)] ,(6.62) where a b - c1(r)=- ( -+-(1-,\)--.a) (6.63) b a 8r The notation i erfc denotes integration of the complementary error function, ( appendix A.4 ). The displacement is found from numerical integration of tp. 6. 7 Perturbation Solutions for Slow Compression Rates In this section a perturbation solution for the case in which the compression rate is slow is considered. The method is similar to that used in chapter four. The governing equation for this case is l 8tp _.!._~ (k(otp)rn° Since the compression is slow the parameter # ~ 1 and so the problem is expressed as a power series 1 1 tp(r,t) = R2 2 k(o The coefficient of the first order of 1/ R 2 is the steady state equation _.!._~ (rn Otp1) = O, (6.67) rn or or giving Cg ( t) ( ) -- + Cg t, for n = 2, r cs(t)r + c9 (t), for n = 0, 6. Radial Flow 83 0.0 u(r) 0.0 1.0 0.5 Radial distance Figure 6.2: Equilibrium displacement, u(r), for a cylindrical shell with 0.5 :Sr :S l and a constrained outer boundary. Solid line: Exact solution with k = l; Dotted line: Exact Solution with k = 1/(1 - 4 where the constants of integration are found from application of the boundary conditions. The constants for the cylindrical situation are given in appendix D.S. The equation for the second order (6.69) which can be solved easily by successive integrations. 6.8 Results and Discussion This section includes the results for the radial flow through a cylindrical shell of porous material constrained at the outer boundary by a rigid porous mesh. The results for the other geometries have not been included but can similarly be calculated using the techniques illustrated. The equations and solution methods outlined in the previous sections can easily be em ployed to find the relations between velocity and pressure similar to the results in chapter three. Without adequate experimental data, however, this will not add to our understanding of the dynamics of fluid-solid interactions. The results shown here are chosen to illustrate a few important points in the analysis. 6. Radial Flow 84 0.3 r .•.···•··· ... '· .... t=0.005 ...... ··...... ··· ..... ·· ... ·· .... ··-...... t=0.02 -1.0 Figure 6.3: Change in porosity, (r), for a constrained cylindrical shell 0.5 :S r :S l. The solid lines are at four small times, t = 0.005,0.01,0.015,0.02. The dotted line is the final equilibrium state of the shell. Note the local expansion of the medium at the inner boundary. When solving the governing equations many authors apply the boundary conditions at the original position of the boundaries a0 , b0 even though they use an Eulerian coordinate frame. The assumption, often not stated, is that the error in applying this boundary condition incorrectly will be negligible since the theory is for infinitesimal deformations. In figure 6.2 the displacement, u(r), for a nondimensional velocity of v = -0.l is shown. Note that a different scale is used on both axes. The solid line indicates the true solution obtained using the integral (6.42) with the position of the boundary found from the iterative technique outlined in section 6.4.2. The dotted line shows the displacement when the approximate permeability relation is used, k = 1/(1 - m) for m = 4. The dashed line shows the displacement for constant permeability but with the boundary conditions applied at the original positions. The point to note is that the effect on the displacement of using the incorrect boundary conditions, is of the same order of magnitude as that for a complicated permeability relation. This is when the maximum displacement is only 2% of the outer radius. This serves to illustrate that correct application of the boundary conditions should be used before resorting to complicated permeability relations. Although the full time dependent behaviour of radial flow may be obtained using a nu merical scheme similar to that employed in chapter four much of the useful information of the deformation can be found analytically from the small time behaviour of the displacement. 6. Radial Flow 85 The NAG library special function routine S15ADF was used to evaluate the error function in equation (6.62). Figure 6.3 shows the porosity change at four different times after applica tion of a unit step change in pressure. The inner radius was initially chosen to be 0.5. The final equilibrium result is shown as the dotted curve and is found from solving the steady state equations. The most important point to note is that there is an expansion of the medium at the inner boundary. The medium is displaced in the radial direction but there is little radial compaction, 8u/ 8r ~ 0, at the inner boundary. The increase in radial distance then causes a corresponding increase in volume due to stretching in the angular direction. The results also show that for small times a highly compressed region forms at the constrained boundary and then expands inwards. This is similar to the result in chapter four. The expansion of the inner section of the medium for small times can play an important role in the dynamics of the solid-fluid interaction. Firstly care must be used in choosing a permeability relation that gives feasible values for an expanded medium. Secondly it is as yet unknown whether the permeability should be a function of the total porosity or a function of the radial strain du/ dr. At the inner boundary the medium is compressed radially and expanded in the angular direction producing an overall net increase in porosity. For a random porous material the dependence of the permeability relation is not clear and would require some further experimentation. The effect on the apparent permeability due to this combined expansion and contraction has not been calculated here. Figure 6.4 which shows the displacement as a function of radial distance for various times, t = 0.005, 0.01, 0.015, 0.02, confirms the existence of the consolidation region growing in time. A maximum displacement gradient is reached at r = 1.0 representing the limit of consolidation. The dotted line represents the displacement at equilibrium. This displacement was obtained by numerical integration of using the NAG cubic spline integration routine D03GAF. In figures 6.5 and 6.6 the porosity and the displacement are shown for the case of a thin shell where 0.8 :S r :S 1. Since the thin walled shell reaches equilibrium faster than the thick walled shell the maximum time plotted is t = 0.005. Note that there is little expansion of the inner region of the shell here and that the displacement is remarkably uniform for all four times. That is the initial displacement profiles do not show the slight inflection apparent in figure 6.4. Uniform results should be expected since the shell is essentially thin enough to react like a uniform medium for most values of time. 6. Radial Flow 86 0.0 u(r) . . 0.0 ------,.----T""---,------r------1 1.0 0.5 radial distance Figure 6.4: Displacement, u(r), for a constrained cylindrical shell with 0.5 ~ r ~ 1. The solid lines are u at four times, t = 0.005, 0.01, 0.015, 0.02. The dotted line is the final equilibrium state of the shell. Note the formation of a maximum consolidation region near r = 1 that expands inwards with time. 0.3 -1.0 Figure 6.5: Change in porosity, at four small times, t = 0.00125,0.0025,0.0375,0.005. The dotted line is the final equilibrium state of the shell. Note that very little local expansion occurs with this thin shell. 6. Radial Flow 87 0.0 u(r) ····-··- t=0.00125 0.0 0.8 radial distance 1.0 Figure 6.6: Displacement, u( r), for a constrained cylindrical shell occupying 0.8 :=; r :=; l. The solid lines are u at £our times t = 0.00125, 0.0025, 0.0375, 0.005. The dotted line is the final equilibrium state of the shell. 6. 9 Conclusions In this chapter the general equations that govern radial flow through cylindrical and spherical shells have been given. Various forms of the equations have been developed for either a spec ified velocity or a specified pressure. Methods of solution were illustrated for the governing nonlinear equations. For steady flow an integral solution is obtained and a solution in terms of the small parameter 8 was shown. For unsteady flow a solution at small times using a Laplace transform technique was given. A perturbation solution for slow compression rates was also outlined. Results were obtained for the case of flow through a cylindrical shell of porous material that is constrained at its outer boundary by a rigid porous mesh. It was shown that it is necessary to apply the boundary conditions at the correct Eulerian position of the boundary rather than the initial position even when deformations are very small. It was also shown that for a thick cylindrical shell there is local expansion at the inner boundary due to the increase in radial distance with relatively little radial compression. As with the one dimensional situation a maximum consolidation region which grows with time forms at the constrained boundary. Chapter 7 High Speed Flow in a Rigid Porous Medium This chapter deals with the high speed flow of a fluid through a rigid porous material. The aim is to evaluate whether inclusion of nonlinear drag terms has a large effect on the pattern of streamlines and on the pressure gradient. Since in some biological flows the relationship between pressure and velocity becomes nonlinear at high velocity, it is useful to evaluate the effect of the nonlinearity before including deformation effects and nonlinear drag in the same model. In the introduction I will review some of the literature in this field and explain the governing equation that will be used. The papers referenced come mainly from the soil mechanics literature and hence the notation and method of analysis differ slightly from the previous chapters. In particular the permeability used in the soil literature K is equal to k /µf. I will use this notation to conform to that standard and hope that this does not cause confusion. After the derivation and discussion of the governing equation I will consider the flow of fluid through a box filled with a rigid porous material. The motivation for this is as a crude model of a placental subunit, which is discussed in more detail in the next section. A comparison will then be made between high and low speed flows. 7.1 Introduction The low speed steady flow of fluid through a rigid fluid saturated porous medium is accurately described by Darcy's law, Vp=-µ1v, (7 .1) K, 88 7. High Speed Flow 89 where p is the pressure, vis the macroscopic velocity vector, µJ is the viscosity of the fluid and K. is the permeability of the medium. At higher speeds the flow of fluid deviates from Darcy's law due to inertial drag effects. Using the dimensionless Reynolds' number (Re), equal to a characteristic velocity by a characteristic length divided by the kinematic viscosity of the fluid, UL Re= (7.2) v as a parameter of the flow, the flow of fluid can be divided into two main regions (Bear 1972, p176-184, p125). For Re< 1 Darcy's law is valid and when Re> 1 the inertial terms come into effect. This chapter will consider the difference between the flow in the inertial region, termed 'high speed flow', and Darcy flow. 7 .1.1 Inertial Effects It has been noted for some time that at high Reynolds' numbers the fluid flow deviates from Darcy's law but the reason for this was often misunderstood. Since porous media may be considered a collection of capillary tubes, it was thought that since the flow in tubes became turbulent at a distinct Reynolds' number, then the deviation from Darcy's law was due to a transition to turbulence. Hence numerous experiments were carried out to determine this critical Reynolds' number (see Scheidegger 1963, p159 for a comprehensive review). The values found for this critical value ranged from 0.1 to 75, well below a Reynolds' number of 2000 observed for the transition to turbulence in tubes (Scheidegger 1963, p159). However nonlinear effects occur in curved tubes at much lower Reynolds' numbers due to the influence of inertial effects, specifically the ( v · 'v )v term in the N avier-Stokes equation. The departure from Darcy's law can then be thought to be due to the increase in inertial effects rather than turbulence (Scheidegger 1963, pl60 and 173, Beavers and Sparrow 1969, Hubbert 1956). Ahmed and Sunada (1969) refer to the experimental work of Schneebeli (1955) who 'injected dye into the flow at various velocities ( steady-state conditions) and found that, even though measurements ... indicated nonlinear flow, the dye assumed laminar characteristics ... .' reaffirming this conclusion. 7 .1.2 Forcheimer's Law The first modification to Darcy's law to account for high Reynolds' number flow was by Forcheimer in 1901. He gave a nonlinear equation for steady one-dimensional flow of the form 'vp = av + bv2 , (7.3) where p is the pressure, vis the macroscopic velocity and a, bare independent of the velocity. This equation is known as Forcheimer's law. Forcheimer gave two modifications to his law by adding a cubic term to the right hand side and later by modifying the exponent 2 on the 7. High Speed Flow 90 right hand side to be a fitted constant between 1.6 and 2. Neither of these extensions are accepted today as they are too empirical and without theoretical basis. Both Bear ( 1972) and Scheidegger ( 1963) give useful summaries of the forms of nonlinear equations used to account for inertial effects. Many of these are of the Forcheimer type ( equation 7.3) with different forms for the constants a, bin terms of the properties of the fluid and solid medium (porosity, viscosity, density and others). Various methods have been used to obtain Forcheimer type relationships. Blick ( 1966), Coulaud ( 1986) and others have used idealised models of porous media, such as an array of spheres and capillary-orifice models, to solve the Navier-Stokes equation exactly. Dullien and Azzam (1973) formed an idealised model of a porous medium by constructing a capillary tube with non uniform diameter using a series of different diameter disks. From experimental data for flow through this system they were able to verify the Forcheimer relationship. Various experiments have been performed by Schwartz and Probstein (1969), Beavers and Sparrow (1969), Ergun (1952), Engelund (1953) and Ward (1964) on random porous media which leave no doubt about the accuracy of a Forcheimer type relation. Ahmed and Sunada ( 1979) and Macdonald et al. ( 1979) have also examined a range of experimental data with the same conclusion. It seems then that the Forcheimer's law is the correct equation to account for the inertial drag effects that occur in porous media at moderately large Reynolds' numbers. It remains then to put this law on a firmer theoretical footing and find the appropriate forms for the constants a, b. 7.1.3 Theoretical Backing for Forcheimer's Law Derivations of Forcheimer's law for random porous media have been attempted by Dullien and Azzam (1973) and Ahmed and Sunada (1969). These involve volume averaging of the Navier-Stokes' equations where the volume average of a quantity '1/J is given by (1/;) = _!_ / 1/; dV, (7.4) V Jv1 where V is a given volume of medium and V1 is the volume of the fluid rn that medium. An important theorem regarding the averaging of a gradient, known as Slattery's Averaging Theorem, is ('v'I/J) = 'v('I/J) + t JA,f .1/;ndA, (7.5) where A is the area bounding V, Ai the area of interface between the solid and the fluid and n is the normal vector to the surface A. Slattery (1972, pl94-196) and Whitaker (1969) give clear proofs for this theorem. Using this Dullien and Azzam (1973) found that 'v(P) µv { D;V it 'v*2u* dV} -pv2 {-1 _!_ / u*. 'v*u* dV + _!__!_ f P*ndA}, (7.6) D 2V Jv1 D V JA, 7. High Speed Flow 91 where u• = v /v, p• = P / pv2 and V* = DV represent nondimensionalisations with respect to reference velocity v and length D. This is then in the form of Forcheimer's law. At first glance this seems to be little more than the recognition that v · Vv has dimensional units v2 / D since the term involving the volume averaging of u• · V*u* yields not much more information than the original volume averaging of (v · Vv). Ahmed and Sunada (1969) give more detail in their analysis by splitting the nondimensional velocity u* into average and fluctuating components u• = (u) + u'. Their analysis also differs in that the term JA; P*ndA does not appear. These derivations, averaging the N avier-Stokes equations, appear then to give some theo retical backing to Forcheimer's law but due to the limitations of averaging over an unknown porous medium do not give much insight into the correct macroscopic forms for the constants a,b in equation (7.3). 7.1.4 Gradient Form for Forcheimer's Law A particularly clear and useful form for Forcheimer's law has been developed by Ward ( 1964) and subsequently used by Ahmed and Sunada ( 1969), Schwartz and Probstein (1969), Beavers and Sparrow (1969), Joseph et al. (1982), Nield and Joseph (1985) and Coulaud et al. (1986). Ward showed that if the gradient of pressure Vp is assumed a function of v, K-,PJ,µf then the only combination to give the correct physical dimensions is ( 7. 7) for some constant c. Although Ward calculated the constant c to be 0.55 to have 'the same value for all porous media' it has been found by other authors (Beavers and Sparrow 1969, Schwartz and Probstein 1969) to be universal only for a class of materials but may vary between materials. For instance c ~ 0.1 for foamed metal fibres and 0.26 for compacted polyethylene particles of random shape. This is to be expected since c measures the relative importance of the inertial losses which will depend on the structure of the medium. A medium made up of straight capillaries will have a. lower value for c than a similar medium with curved capillaries. For a useful discussion on the constant c as predicted by the models of Brinkman (1947) and Irmay (1958) see Joseph et al. (1982). 7.1.5 Darcy Flow versus Nonlinear Flow Although sufficient evidence exists to suggest that the nonlinear equation (7 .7) is the correct form for high Reynolds' number flow, there has been little work on the differences between the nonlinear equation and Darcy's law in practical flow geometries. Rao and Das (1978) have considered the problem of 'seepage to a fully penetrating well in an unconfined aquifer'. 7. High Speed Flow 92 By using a finite element package they were able to show that the rate of discharge into a well is lower for nonlinear flow than for Darcy flow. It was also only necessary to consider a small nonlinear region close to the well with Darcy flow being an accurate representation for the rest of the region. Schwartz and Probstein ( 1969) applied Forcheimer 's law to the flow into a vertical column of a slurry separator used in desalination. In both of these cases the pressure and flow rates were found but the streamlines for the flow were not given. To study the differences between Forcheimer's and Darcy's laws the flow into a confined box is now considered. 7 .1.6 Flow into Boxes The flow into a confined box or column ( figure 7 .1) has also been considered by B uevich and Minaev (1974) as a general problem relating to heat exchangers and catalytic chemical reac tors. In this problem they solved only the linear Darcy's law for an axi-symmetric geometry to find both the streamline and the velocity profiles. Chen and Lam ( 1985) also considered a similar problem relating to a spouting bed. They considered a rigid medium and calculated the minimum pressure required to induce spouting (lifting of the porous grains). Rather than solving Darcy's law they considered a classical streamline approach by combining a uniform stream flow and a source as an approximation to flow into a column. This approximation will be worst at the base of the box. The flow into a confined box has also been studied for the case of blood flow into a human placental sub-unit. The placenta consists of small circulatory units where exchange of nutrients, oxygen and waste occurs between the foetal blood and the maternal blood. From the 'roof' of one of these units grows a thick bush-like structure whose branches (villae) contain the foetal capillaries. Maternal blood then spurts into the unit from an artery at the base and then flows out into veins also at the base. The maternal and foetal blood do not come into contact so nutrient exchange relies on a good flow of maternal blood between the villae. This process has been modeled as flow into a box filled with a porous medium with injection flow from the base of the box. For good discussions of the modeling of the placenta in this way see Aifantis (1978) and Erian et al. (1977). Aifantis (1978) not only gives an excellent background to the placenta but attempts to set the groundwork for a model of the placenta that includes deformation effects by virtue of a mixture theory. Erian et al. ( 1977) has attempted to model the placenta as an extension of Darcy's law in which the permeability is a function of the velocity. He then proposed an ad hoe permeability that attempted to take into account the deformation of the tissue. To analyse the relationship of (7.7) with Darcy's law jetting-like flows of fluid into a box filled with a saturated porous medium are considered. Two flow geometries are chosen: the injection of fluid from a small opening into a two-dimensional semi-infinite box and an 7. High Speed Flow 93 Figure , . l : Schematic diagrams of geometries for flow into a rigid porous medium. Fluid flows in through the bottom of the box and then out through either the top or bottom. injection of fluid into a finite box where the outflow occurs on the base o f the box. 7.2 Method of Solution T he nondimensionalisation of equation (7. 7) using a cha racteristic velocity v· and length /, g ives B'vp = V + Alvlv, ( ,.S ) where v = vv·, x = i L and pressure scales a.s p = ppv· 2 . The nondimensional constant [J is equal to µJp/kl a nd t he important nondimensional constant A is given by ( v" Lp) A = (Lc/k ) -----;:; = Pm.R e. ( , .9) T he constant Pm is fixed for a given porous medium a nd t he Reyno ld s' 11u m ber, !le, is \,1 ri(·d to re present diffe rent speeds of flow. Taki ng the c url o f (7.8) gives Vx(v +Alvlv) = O, (,. lO ) (all variables a re now nondimensional a nd t he hats will be dropped). A two-dime nsio na l stream fun ctio n, ·l/J , is defin ed in the usua l way by (,. l l ) 7. High Speed Flow 94 where v = (v1 ,v2 ) and the subscripts on '1/J represent partial differentiation with respect to that variable. Equation (7.10) then yields a nonlinear partial differential equation in '1/J: A 2 2 } (1 + Alvl)( 1Pxx + 1Pyy) + ~ {1Px 1Pxx + 21/ix1Py1Pxy + 1Py 1Pyy = 0, (7.12) where (7.13) It is easy to show that the pressure contours are always perpendicular to the streamlines with the pressure spacing dependent on the magnitude of A. For A = 0, equation (7.12) reduces to "v 2 1/, = 0 which is equivalent to decreasing the Reynolds' number to the Darcy's law limit. In the limit as A--+ oo, equation (7.12) gives 1 { 2 2 } (1Pxx + 1Pyy) + lvl 2 1Px1Pxx + 21/ix1Py1Pxy + 1/Jy1Pyy = 0, (7.14) the same as solving "v P = (A/ B)lvlv. A simple solution to equation (7.12) is the purely radial flow field from a point source. In radial coordinates, 'lj; is then dependent linearly only on the angle. This flow is the same in both high and low speed flows although the pressure gradient is larger in high speed flow. A physical situation of interest is that of an injection of fluid from a small hole in the base of a two dimensional semi-infinite box with impermeable side walls (figure 7.la). The inflow of fluid has velocity v* in the y direction. The jet is centered along the axis of the box so the problem is solved on the right hand side only by symmetry. The slip of fluid is allowed along the boundaries. The top boundary is assumed to be a large distance from the inlet jet so that a uniform outflow across the width of the box can be assumed. The boundary conditions are then 1/,( X, 0) = -x, x E (0,1); 1/,(x, 0) = -1, x E (1, N); 1/,(N, y) = -1, y E (0,L); 1/1(0, y) = 0, y E (0, L ); (7.15) 1/,(x, L) = -x/ N, x E (0, N), where x = ±N are the side walls of the box and y = L is the height of the box. These are illustrated in figure 7.2 Darcy's law (A = 0) is equivalent to 'v21/, = 0, which can be solved analytically for the above geometry. The solution for the situation of a jet into a porous medium half space is simply obtained from Poisson 's integral formulae. An infinite sum of such jets at intervals of x = ±2N, ±4N, ... give the required flow by method of images. In practice only 1001 jets were evaulated in the summation. 7. High Speed Flow 95 >, l L T'"" 0 I II ~ 'lf=-X 'If= -1 Figure 7.2: Boundary conditions for flow into a rigid porous medium. For a single jet centred at the origin 'l/;(x,y) = -; [(x + l)arctan (x; 1 )- (x - l)arctan (x; 1) - J!.10 ( ( x + 1 )2 + y2 ) l (7.16) 2 g (x - 1)2 + y2) · The other 1000 jets are simply found by translating this solution. Another form for the solution is 1 ·'·('// x,y )= 1 arctan [ ------cos(x1r/a)sinh(y1r/a) ] db . (7.17) b=-1 sin(x1r/a)cosh(y1r/a)- sin(b1r/a) This was obtained using the conformal transformation (=sin (:z) (7.18) to map the region x E (-a/2, a/2), y ~ 0 to the upper half plane. A uniform distribution of sources were placed along the y = 0 line going from x = -1 to x = l. To solve the full equation (7.12) an iterative nine point finite difference scheme was em ployed with successive over relaxation and a Shank's transformation (Bender and Orszag, 1978, p. 370) used to speed convergence. An initial guess for the solution solution was spec ified at every point in the domain. By application of fini te differences to all the derivatives and rearrangement to obtain the central coordinate in terms of the surrounding eight co ordinates a new value can be established for the central point. By repeating this for every 7. High Speed Flow 96 100,r------ 75 'iJ2. -050 0 ::J -+ (") ::r ~ 25 0 L---.I.---...I..---...L..___;=------' -5 -3 -1 1 3 5 Log(A) Figure 7.3: Percentage change of the stream fun ction at four central points with fl ow speed, A. point in the domain, and iterating, the solution can be found. The convergence was checked numerically and it was found that the initial guess for the starting solution did not affect the final result. The solution was also tested against the analytical solution for Darcy's law and also by specifying different simpler geometries with known analytical solutions (such as a uniform stream). 7 .3 Results and Discussion The value of '!jJ was obtained for different A and scaled so that at A = 0, '!jJ = 100% and the limiting solution for '!jJ as A --+ oo was '!jJ = 0% for each point in the domain. The variation of four typical points with A is shown in figure 7.3. This shows that for A < 0.1 the solution is equivalent to Darcy's law, if A > 30 then the solution is that of the high A limit, equation (7.14), but in the region 0.1 Figure 7.4 shows the streamlines for both of these flows. It is of interest to note that although a high speed flow is being considered there is no extra 'jetting' of the fluid into the box except for a marginal amount in a very small region close to the fluid inlet. The high speed flow did, however, produce a definite increase in the flow into the corner of the box. With this typical geometry the use of the complicated nonlinear equation, as opposed to the 7. High Speed Flow 97 8 y I I I I I I 4 I I I I I I X Figure 7.4: Streamlines for flow into a porous medium. --, high speed flow A = 45. - - - , Darcy flow. simple Darcy's law, had little effect on the streamlines. Figure 7 .5 shows the streamlines for the geometry in which fluid is entering the base of the box from the left and leaving on the right. This geometry does show a more pronounced jetting of the fluid with a corresponding increase in the amount of fluid flowing through the upper reaches of the box. The main difference between use of Darcy's law and the nonlinear formulation is not so much in the change to the streamline but in the required pressure to reach a given fluid velocity. Darcy's law would underestimate the necessary pressure. High speed flow also creates an increase of flow into the 'corners' of the region. The original model of blood flow in the placental subunit predicted that over 50 percent of the blood moves directly from the maternal artery to the vein without passing through the villae significantly. The high speed model does not change this significantly although it does predict an increased flow of blood through the upper regions of the placental unit. 7. High Speed Flow 98 4 y 2 0 1 2 3 4 X Figure 7.5: Streamlines for flow into a porous medium. high speed flow A 45. - - - , Darcy flow. Chapter 8 Summary and Discussion A number of related problems dealing with the flow-induced deformation of porous materials have been considered. These are now summarised and the future directions of this work are discussed. 8.1 Summary The flow of a fluid through a deformable porous medium has been studied in depth over the past ten years, mostly in the field of biological modeling. In chapter two a theory for the coupled motion of the fluid and solid was given. This was based upon the theory of mixtures and formed an overview of the theoretical development of the field since 1979. The applications of poro-elastic modeling to various biological systems was also considered, specifically to some viscoelastic problems. Some of these solutions, specifically the unsteady analysis of stress relaxation, were given in some detail since the methods related to later work in the thesis. The fundamental equation governing the fluid-solid interaction was shown in chapter two to be 1 (8u ) (8.1) V p = V . (T = k( ) 8t - V ' for solid contact stress u, pressure p, displacement u, velocity v and k the permeability dependent upon the strain or local dilatation . In chapter three this equation was investigated for the case of steady flow through a one dimensional medium constrained downstream by a rigid mesh. Solutions were found for differing functional forms of k and u and compared with the data of Parker et al. (1987). It was found that a finite deformation model incorporating nonlinear stress-strain relationships 99 8. Summary 100 was necessary and that a suitable form of the permeability was (8.2) The nonlinear behaviour of the deformation was also illustrated in that a highly consolidated region forms at the downstream boundary with relatively little compaction upstream. In chapter four the unsteady analysis of this geometry was considered. In this chapter the governing equation was rederived. The effect of inertia was included in the equations and a lower bound on the typical time scale of deformation was found above which (to ~ pyko) these inertial effects could be ignored. An alternative form for the governing equation was given that was independent of the velocity hence allowing solution for a specified applied pressure gradient which is more commonly given in permeation experiments. This equation in terms of the dilatation is the nonlinear diffusion equation 2 ~~ = R2 { ((b) :; + b('(b) (%i) }, ( 8.~.3) for diffusion parameter R and nonlinear function (, a combination of the permeability and the stress relation. Solutions were obtained for the linearised version of the governing equations and for small times. A perturbation technique was then used to find the solution for a slowly varying pres sure gradient. This was used to show that if a periodic pressure difference is applied, with nonzero mean, the permeability may, under some circumstances, be increased from the per meability for the corresponding steady pressure. Numerical solutions were found for different specified velocity functions to illustrate the effect of the nonlinearities on the dynamics of the system. It was shown that for small velocities the medium deforms in unison, that is all sections of the medium move in phase. As the velocity is increased the downstream solid component reaches a state of near maximum deformation with the deformation continuing upstream. This leads to a nonuniform deformation profile with definite phase lag effects. In chapter five the flow over a thin deformable layer lining a channel was considered. The coupled equations for the two components of the solid and fluid were developed and simplified by assuming small deformations. Solutions were then found for the velocity of the fluid in the channel and the solid and fluid velocities in the porous layer. Both cases of rigid and deformable porous layers were considered. In many practical cases the solution in the porous layer is not required, but only the effect that the layer has on the flow in the channel. By reworking the exact solutions a boundary condition was developed for the fluid velocity in the channel that can be applied at the boundary with the porous medium. This boundary condition takes into account the dynamics of the porous layer without the necessity of solving the full equations. In terms of the Fourier components of the fluid velocity q = "E,qn exp( iwnt) the boundary condition is (8.4) 8. Summary 101 where £ is the nondimensional width of the gel layer and 17, (3 are parameters of the flow. This expression, the first term in a perturbation series, indicates a phase lag effect due to the deformation of the layer. This boundary condition can then be applied to a variety of situations where full solutions cannot be found. In chapter six the flow of fluid radially out through cylindrical and spherical shells is considered. The theory was developed generally and applied to the cases of the outer bound ary free to expand or constrained by a rigid mesh. Solutions for steady flow with a general permeability were found as exact integrals and also as perturbation solutions. Time depen dent solutions were also found for small times. The techniques used were essentially those of chapter three and four generalised to other geometries. The solutions were then examined for the case of flow through a cylinder constrained at the outer boundary. It was shown that due to the increase in the inner radius there is a local expansion of the medium at the inner boundary. The position of the inner boundary was also considered since numerous authors apply the boundary conditions at the original position of the boundary rather than the true position. It was shown that even for small deformations (2% of the radius) the effect can be as significant as employing a nonlinear permeability function. The effect of inertia in steady flow is considered in chapter seven for the flow of fluid through a box filled with a rigid porous medium. A rigid porous medium is examined so that the effect of the noninertial terms can be isolated from the effect of the deformation. The difference between the linear low speed flow equations and the high speed nonlinear equations was examined. By solving the resulting nonlinear equations in terms of the stream function it was found that the high speed flow produced no significant 'jetting' of fluid into the medium but that the fluid was pushed further into the corners of the box by the pressure. In summary this thesis has investigated the deformation of a porous material by applied pressure differences in a number of situations. The aim has been to illustrate the fundamental interactions of the fluid and solid. I see my original contributions to this field of study as being: • Establishing that the nonlinear permeability and stress relations used in the cartilage literature are versatile enough for use in polyurethane sponges and thus are indicative of the fundamental solid-fluid interaction. • Illustrating that the governing equations for permeation problems can be written and solved without reference to the velocity, hence allowing solutions in terms of the pres sure. • Showing that the equations for unsteady flow predict that a pressure gradient periodic around a mean value may lead to an increased mean permeability. • Finding a lower limit on the typical time scale of the permeation problem above which 8. Summary 102 inertial forces can be ignored. • Developing and solving the poroelastic equations for flow over a deformable layer where shear stress now dominates. • Establishing a boundary condition for fluid flow over a solid covered with a thin layer of porous material. This boundary condition incorporates information about the in teraction with the porous medium, without requiring a solution of the full coupled equations. • Generalising to flow in cylindrical and spherical geometries and showing a general so lution for steady flow. • Showing that expansion occurs m a region near the mner boundary in radial flow through cylindrical shells. • Showing that use of the nonlinear Forcheimer's law does not necessarily produce extra jetting into a porous medium but pushes more fluid into the 'dead water' regions of the flow region. Some of these contributions have already been published. Chapter three forms the basis of Barry and Aldis (1990a) while chapter four is the basis of a paper accepted for publication by International Journal of Non-linear Mechanics (Barry and Aldis 1990b ). Chapter five is in the final stages of preparation as a paper soon to be submitted. Chapter seven has been presented at the Australian Applied Mathematics Conference in Leura, 1988. Chapter three was presented at the Australian Applied Mathematics Conference in Ballarat, 1989. Chapter four formed the basis of a poster presentation at the First World Congress of Biomechanics, La Jolla, California, 1990. 8. 2 Future Research Poroelastic modeling is already being used very successfully for a number of biological situa tions. The developments of the theory of poroelasticity can be thought to progress on three possible fronts. (I) The present theory can be extended to consider physiologically important factors such as intrinsic viscoelasticity of the solid component, anisotropy, osmotic forces or three component swelling systems. Work has already begun on each of these problems and progress looks promising. (11) Extensions of the theory to more specific systems is now required. Biological systems contain numerous components and the effect of each of these components on the whole tissue should be considered. Elastin is normally considered to be simply part of the elastic solid 8. Summary 103 matrix yet recent work has shown that elastin itself can act as a poroelastic medium on a microscale. Thus the theory should first be developed to model elastin poroelasticity and then to include this into a macroscopic poroelastic model, a poroelastic model on multiple scales. The glycosaminoglycans (GAG) form an integral part of many tissues and form the gel layer that lines the blood vessels. Hyaluronic acid (HA), a type of GAG, plays a major part in the permeability of tissues and hence the permeability and compression properties of HA need to be studied in more detail. Once the properties of this are understood it can be incorporated into a more detailed model of tissue deformation. (III) Application of the present theories to a variety of new tissues and geometries is an obvious future direction of research. Already the theory has been applied to lung tissue, carti lage, intervertebral disks and chick hearts. The application of improved finite element models will make the unusual geometries of biological systems accessible to poroelastic modeling. This thesis has also shown some direct problems that need further research. It was shown in chapter four that in unsteady flow it was possible to increase the flux of fluid by periodically varying the pressure difference. This is an important result which needs some experimental analysis to find the extent of this increase or verify the conditions under which an increase occurs. This verification would have implications to not only filtration technology but also to artery wall dynamics where such an increase in permeability may explain some unusual aspects of atherogenesis. The boundary condition found in chapter five for flow over a layer of thin porous material also requires experimental verification and could lead to some useful results for flow in arteries. This analysis could also be extended to include the vertical components of the displacement and the velocity and so lead to the possibility of wave like solutions propagating down the porous layer. The expansion of the medium in the inner region of a cylindrical shell of porous material also needs to be considered in more detail. As the flow is radially out, should the permeability be dependent on the local porosity or on the strain component du/dr"? One can imagine a situation in which the medium is compressed in the r direction sufficiently to close off the pore space yet have no change in local porosity due to angular expansion. The permeability may then be a function of du/ dr not The effect of inertia should also be considered in more detail with the nonlinear equation of chapter seven included in deformable porous medium modeling. For a one dimensional porous medium a nonlinear drop in pressure will occur due to Forcheimer's law and the deformation of the material. Which of these effects will dominate and the nonlinear interaction of both effects will be useful to consider. 8. Summary 104 I hope to be able to answer some of these questions in the future. Appendix A Mathematical Relations A.I Modified Bessel Functions I 11 , !(11 Solutions to 2d2w dw 2 2 z dz 2 + zdz - (z + v )w = 0, (A.l) are J,,,(z), J(,,,(z), the modified Bessel functions. The asymptotic expansions for z -+ oo are: ez { µ - 1 (µ - 1) (µ - 9) } J,,,(z),..., ~ 1 - --+ ( )2 + · · · , (A.2) y21rz 8z 2 8z 0 -z { µ - 1 (µ - 1)(µ - 9) } K,,,(z),..., V2ze 1 + s:;- + 2(8z)2 + ... ' (A.3) whereµ= 4v 2 • Some integrals used in chapter six are: 1b zlo(z)dz = afi(a)- bfi(b), (A.4) 1b zKo(z)dz = -aK1(a) + bK1(b). (A.5) A.2 Laplace Transforms The Laplace transform is defined as: f(s) = £ {J(t)} = fo 00 e-st f(t)dt. (A.6) 105 Appendix 106 Some useful pairs of functions and their transfoms are: .c{l-1}, 3 .C exp (- ) }, {-k2~ k4t 1 -k 1s -e V" .C { erfc ~ }, (A.7) s 2 -e1 -k v~ 1s 2 .C exp (- k )} Js i_l.Jii, 4t ' 1 -k 1s --e v~ .C ( 4t r1 2 in erfc_!:_}. 8 1+n/2 2.Jt A.3 Error Function The complementary error function is defined as 00 erfcz= ..fir2 1z e-t2 dt. (A.8) For large arguments 2 c e-z ( ~( )m1·3 ... (2m-1)) er1cz rv r.;; 1 + L., -1 ( 2 ) (A.9) y7rZ m=l 2z m as z ---> oo. Integrals of the error function are given by (A.10) where i0 erfcz = erfcz, (A.11) and also (A.12) A.4 Exponential Integral For positive z: oo -t Ei(z) = - J _e - dt. (A.13) -z t In chapter three evaluations of Ei(z) are always of the form Ei(a)-Ei(b), hence the integration through the origin can be disregarded. Appendix B Notation This appendix gives a list of the notation used within the thesis. Only those notations common throughout the thesis will be shown. These are also defined within the text. /3: superscript denoting solid or fluid phases /3 = s, f. 8: typical displacement as a fraction of the typical length, u0 / h. e: strain tensor: e = (Vu+ (Vuf )/2. e : trace of e. F(y ): nonlinear stress relation: u = F(y) J(t): integral of the velocity: J(t) = -l/rn fv(r)dr. h: final length of a one dimensional medium. Ha: aggregate elastic modulus, Ha = (>, + 2µ). HM: finite deformation form for aggregate elastic modulus, H M = t(y = l). I: identity tensor. k: intrinsic permeabilty of the medium. ka: apparent permeability. ko : permeability of the medium in the absence of strain. ~= permeability k/µJ· I( : drag coefficient I( = ( .X, µ: Lame stress constants. 107 Appendix 108 L: initial length of a one dimensional medium. m: parameter in the permeability function. k = exp(mdu/dz). µ1: viscosity of the fluid. µa: apparent viscosity of the fluid in the porous medium. n: normal vector to a surface. 1rs : interaction force of the solid on fluid. : change in porosity, dilatation: r~ %r(rnu). p(z, t): pressure of fluid in porous medium in Eulerian coordinates. P(X, t): pressure of fluid in porous medium in Lagrangian coordinates. bi.P(t): applied pressure difference. r/3: apparent density of the /3 phase. p~: intrinsic density of f3 phase. p: total density of mixture, p = p 8 + pf. R2 : scaled length in the medium traveled in time t0 : R 2 = Hak0 t0 /h2 . s: Laplace transform variable. af3 : stress in the f3 phase, minus hydrostatic pressure terms. a : solid contact stress, a = as. a: solid contact stress in one dimension. t: time. T{J: total stress: T{J = - uo: typical displacement. u(z, t): Eulerian displacement of the solid medium. U(X, t ): Lagrangian displacement of the solid medium. vf3: velocity of (3 = solid, fluid phase. v: macroscopic velocity of fluid: v = X: Lagrangian Position. X is the vector form. Appendix 109 y: stretch ratio y = 1 + dU / dX, in one dimension. z: Eulerian position. z is the vector form. (: combination of permeability and stress relations, y/[k(U'(X))F(U'(X))]. Appendix C Theoretical Relations C.1 Proof of k = c/>}/ J{ In the derivation of the governing equations by the theory of mixtures the following proposition was made in equation (2.23). Proposition: 1 J( --2· (C.l) k The following is similar to that given in Lai and Mow (1980). Proof: Consider the steady-flow geometry outlined in figure 3.1 where the one-dimensional medium is compressed by the fluid flow against a porous rigid mesh. From the derivation of the flow equations we can write dp J( dz = - 4>JVf, (C.2) where v J is the velocity of the fluid in the pore space. This can be integrated to yield I( p = 4>fv1(h - z). (C.3) The apparent permeability across the medium is given by Darcy's relation v = -kab..P. Therefore k -- 4>f v1h a - b..P ' (C.4) and from equation (C.3) with z = 0 (C.5) Therefore 1 J( --2· (C.6) k 110 Appendix 111 C.2 Relation between apparent permeability and intrinsic permeability Equation (2.25) was given as a relationship between the apparent and intrinsic permeabilities. The proof of this relation is due to Holmes ( 1985 ). It is reproduced since the general technique is used in this thesis. For a one dimensional steady flow, with small deformations, the governing equation for the medium is dp _ H d2 u _ _ v (C.7) dz - a dz2 - k(du/dz) · Defining a function g as d drg(r) = k(r), (C.8) the governing equation can be written as (C.9) By integrating equation (C.9) du) -v g ( - = -z+co, (C.10) dz Ha where co is a constant. This is then written as du -1 ( V ) dz = g co - Ha z . (C.11) Hence u (C.12) (C.13) where Ep = 6.P / Ha and the apparent permeability is ka = vh/ 6.P. The boundary condition u( h) = 0 has been used. Integrating the governing equation (C.7) gives ( du dul) (C.14) p(z) = Ha dz - dz z=h ' which uses the boundary condition p = 0 at z = h. From equation (C.14) at z = 0 and with p(O) = 6.P it can be seen that (C.15) so that by subsitution for du/dz from equation (C.11) (C.16) Appendix 112 However in the case in which the pressure is sufficient for the material at the upstream end to break free from the surface the upstream boundary condition is du/dz = 0. Then from equation (C.11) (C.17) implying c0 = g(O). Substitution of the above into equation (C.16) gives (C.18) or upon rearrangement 1 ka = -[g(O) - g(-Ep)]. (C.19) €p Since d~g(r) = k(r) an equation that relates the apparent permeability as an integral over the intrinsic permeability is found, 1 ka=- l(p k(-r)dr. (C.20) Ep 0 Therefore ka can be found from an integral of the intrinsic permeability over the range of the strains in the medium. C.3 Relation between the porosity and strain in one dimen- . SIOn The strain du/dx is related to the porosity by the following proposition given in equation (2.24). An alternative proof is given in Holmes et al. (1985). Proposition: The porosity in the final state 4>f is related to the initial porosity 6 by the relation (C.21) for a one dimensional deformation. Proof: Consider a small strip of the medium 6z0 deformed to length ~z as shown in figure C.l. As the strip has moved with the solid then the increase (decrease) in the 'volume' of the medium is due to an influx of fluid only. Hence 4>f 6z = Dividing by ~z --> 0 gives ,iJ _ ,1J 6zo du '+-' - '+-'o 6z + dz · (C.23) Also note that the length of the element is given by 6z = ~zo + [u(6z) - u(O)] (C.24) Appendix 113 I u(O) - LlZO I l LlZ u(Z) l Figure C.l: A small strip of material deformed from length .6.z0 to length .6. z. so that by dividing by /j.z in the limit as /j.z --. 0 .6. zo du -=1--. (C.25) .6.z dz Substituting this into equation (C.22) for f f s du = where 6 = 1 - g has been used. C.4 Relation between the porosity and strain in two dimen- . SIOnS Proposition: For two dimensions the porosity is related to the strai n ( to fi rst order) by (C.27) where the displacements are u1(x1,x2) in the x 1 direction and u2(x1, x2) in the x2 direction. Proof: The Eulerian position (xi) is related to the Lagrangian position X; by the relation Xj=Xj-Uj. (C.28) Hence the small length au dX dx - - 1 dx · (C.29) I = l OX . ) l J by the chain rule. Appendix 114 The two dimensional 'volume' of a small initial element is then dVo = dX1 dX2 and the final volume is dVi = dx 1 dx2. The two volumes are related by (C.30) (C.31) As the increase in the volume can only be due to an increase in the fluid content then (C.33) This shows that the final porosity is related to the initial porosity by a compression factor plus a shear factor plus second order terms. In most applications the shear terms are small compared to the compressive terms and hence in many instances the the permeability is taken to be a function k( e;; ). C.5 Relation between initial and final radii in cylindrical deformations This section describes the relationship between the initial radii a0 , b0 and the final radii a, bin the radial (cylindrical or spherical) geometry shown in figure 6.1. These were given in equation (6.27). Consider a small volume dV0 deformed to volume dVi where dVo = dVi ( 1 - :: ) ( 1 - ; ) n , (C.34) where u is the radial displacement and n = 1, 2 corresponds to either cylindrical or spherical geometry. Since compression is entirely due to the loss of fluid in the control volume then Thus substitution of equation (C.34) gives to a first order approximation. Taking n = I and conserving the total volume of solid in the material before and after deformation indicates that (C.37) and so b0 b ( au u) 21r Therefore this is satisfied approximately by the expressions a a0 +u(a), (C.40) b b0 +u(b), (C.41) which are more intuitively obvious. Appendix D Algebraic Results D .1 Integrals from chapter 3 In chapter 3, equations (3.22) and (3.24) require evaluation of j k(y)t(y) dy and j k(y)F'(y)dy (D.l) for various F(y) and k(y). These integrals were done using the REDUCE and MATHEMAT ICA (Wolfram 1988) algebraic manipulation packages. For 1 F(y) = {[1 + di(Y - l)] (y - y3 ) (D.2) and (D.3) these integrals become emy ( 1 y) 4 j k(y)F'(y)dy = -(1 - di)+ 2diemy --+ - m m 2 m my( 1 m )] m2Ei(my)l +2 d 1 [ - ( e - + - + --'-~ 2y 2 2y 2 +3(1 - di) (-emy [-1 + .!!.!:__ + m2] + m3Ei(my)) (D.4) 3y3 6y2 6y 6 and 2 3 +2d1 (-emy [-1- + _m_ + _m_ ] + _m_E_i_(1_ny_)) 3y3 6y2 6y 6 +3(1 - di) (-emy [-1- + _m_ + _m_2 + _m_3 l 4y4 12y3 24y2 24y m4 Ei(my)) + 24 . (D.5) 116 Appendix 117 If the permeability function is 1 k(y)=--- (D.6) 1 + m- my then d1 - 1 2di + 3m - dim 2dim + 3m2 - dim2 2d1y 4 j k(y)F'(y)dy = y3 (1 + m) 2y2(1 + m)2 y(l + m)3 m 2dim2 + 3m3 - dim3 + log(y) (l + m)3 - (5 + 9di + 2difm + 4m + 16d1m 2 2 3 3 4)log(-l-m+my) + 6m + 14dim + 4m + 8d1m + 4m m(l + m)4 (D.7) and 3(d1 - 1) 2di + 3m - dim) 4 Jk(y)t(y) dy = m---- 4(l+m)y4 3(1 + m)2y3 2dim + 3m2 - dim2 2dim2 + 3m3 - dim3 2(l+m)3 y2 (l+m)4y + (1 - d1 + 4m(l - d1) + 6m2 (1- di)+ 2ni3(2 - di)+ 2m4(2 - di)) log(y) (-2di - m - 9d1 m - 4m2 - 16dim2 - 6m3 - 14d1m3 (1 + m) 5 4 d 4 5 )log(-l-m+my) -4m -8 im -4m . m(l+m)5 • (D.8) D.2 Integrals from chapter 6 on steady radial flow This section evaluates the integrals in chapter six concerning the steady state displacement for a given velocity. D.2.1 Constrained flow, k = 1, n = 1 For the permeability relation k() = 1, (D.9) then g()= jk()d=. (D.10) The integral to be evaluated for this cylindrical geometry is u(r) = -f jb 1 sg-1(vln(s/a1) + g [u~~i)(l -1)]) ds. (D.11) For this function of k, 2 u(r)=--v [-;--lns--8 2 8 lbi -:--1 (u(ai)-(l->.)-vlnai - ) [s 2 ]b1 . (D.12) r 2 4 2r a 1 r r The implicit equation for a1 is found from (D.13) Appendix 118 D.2.2 Constrained flow, k = exp(m), n = 1 For (D.14) the integral of k is 1 g() = -em, (D.15) m with inverse 1 g-1 (() = -ln(m(). (D.16) m Equation (D.11) becomes u(r) = - -1 1b1 sln(mvln(s) + c2) ds, (D.17) ffi T where mu(ai)(l - -:X)) c2 = -mvlna1 + exp ( ai . (D.18) Defining a new variable y = m v ln s + c2 this becomes exp(-2c2) 1y(b1) ( 2y) u(r) = --2=m~v- exp - ln(y)dy, (D.19) m v y(r) mv which can be integrated by parts to yield exp(-2CZ) [ 21L ] y(bi) exp(-2c2) J,y(b1 l exp( -11L) u(r) = - mv e=v ln Y + mv mv dy. (D.20) 2m y(r) 2m y(r) y In terms of the exponential integral Ei this is exp(-2c2) [ 21L ]y(bi) [ . ( mvy)]y(bi) u(r) = - mv emv lny + E1 --- . (D.21) 2m y(r) 2 y(r) This is an abuse of the notation for the Exponential integral, Ei, since the arguments y(r) cannot be predetermined as positive or negative hence a choice of either £ 1 or Ei must be used when this sign has been evaluated. D.2.3 Constrained flow, k = 1/(1 - m), n = 1 For 1 k( ) = 1 - m' (D.22) then 1 g() = -- ln(l - m), (D.23) m and (D.24) Appendix 119 The displacement is then (D.25) which simply becomes u(r)=--1 [s2 _ l-mu(a1)(l-°X) (!.-)l-vmlb1 (D.26) mr 2 1 - vm a1 r D.2.4 Constrained flow, k = l, n = 2 The integral required is (D.27) which for this k is (D.28) This simply becomes 1 [ s2 sn+l ( nu( a ) _ ) bi u(r)=-- (1-n)v-+-- -(l-n)va1-n+ 1 (1->.) l (D.29) rn 2 n + 1 1 a1 r D.2.5 Constrained flow, k = exp(m The integral to evaluate in this case is u(r) = --1 1b 1 s 2 ln [ -mv ( ---1 1 ) +exp (2m----- u(a)(l - X))] ds, (D.30) mr2 r s a1 a which becomes u(r) = __1_ [- ((mv)2s) _ mvs 2 + s3 log(c - 7:v) _ (mv)3 log(-mv + cs)l bi mr2 3c2 6c 3 3c3 (D.3l) T where mv u(a)(l - c= -+exp (2m----- X)) . (D.32) a 1 a D.2.6 Constrained flow, k = 1/(1 - m The integral to solve here is Appendix 120 which implies 2 1 [ 3 ] b1 m e-m/ai ( 2mu( ai) ( ')) u(r) = - -- S + ---- l ------'---'- 1 - A 3mr2 r r 2 a1 X [- {eY (-1 + _1 + ~)} + Ei(y)] m/r (D.34) 3 3y 6y2 6y 6 y=m/b1 ' where the substitution of y = m/ s has been used. D.3 Constants from equation (6.55) The constants c3 , c4 are from section 6.5 for radial flow through a constrained cylindrical shell. These are found from making C4 VT u(r) = c3r + - + - ln r, (D.35) r 2 satisfy the boundary conditions u(b) = 0, [-+A-du -U] = 0. (D.36) dr T r=a The constants are found to be 2 (a 2 v+ a 2 vlog(a) + a 2'Xvlog(a)) b2 (-2 + 2-X) vlog(b) C - ----'------=------c=-~ - --~---=---= (D.37) 3 - -4a2 - 4b2 - 4a2 A + 4b2 A -4a2 - 4b2 - 4a2 ..\ + 4b2 ..\' and -2b2 ( a2v + a2v log(a) + a2'Xv log(a)) b2 (-2a2 + 2a2X) vlog(b) C4 = (D.38) -4a2 - 4b2 - 4a2 ..\ + 4b2 A -4a2 - 4b2 - 4a 2 A + 4b2 ..\ · D.4 Solution for unsteady radial flow, equation (6.57) This section finds the solution to equation (6.57) (D.39) with boundary conditions 4>(a,t) + 2.x 1b a(a,t)da = 0, (D.40) a a (b,t) = (a,t)- b.P, (D.41) and initial condition 4>(r, 0) = 0. (D.42) The constant ~ = 1 - X. Appendix 121 Taking Laplace transforms with respect to t and using the variables 77 = r,/s, 4> = 4>( 77, s) equation (D.39) becomes (D.43) This has solution (D.44) where 10 , 1(0 are modified Bessel functions. Application of the boundary conditions gives (D.45) (D.46) Ko('TJa) - Ko('T/b) - ((s)(Io(17a) - Io(17b))' ry~Ko(TJa)- 5'(-17aK1(1Ja) + 1]bK1(1Jb)) ((s) = (D.47) 1]~lo(17a) - 5' (17afi(17a) - 17bfi(17b)) where the integral given in appendix A.4 has been used to evaluate the integral boundary condition and 'T/a = a,/s, 'T/b = b,/s. Since this is difficult to invert only the small time solution is found. By taking the limit s -+ oo and finding the asymptotic form for the solution above invertible approximate solutions can be found for Using the asymptotic results for I,,, K,, shown in appendix A.4 it can be shown that as s - CX) 1]~Ko(17a) + 5'T/aK1(1Ja) ((s) ~ (D.48) 5'T/bli ( T/b) rr ( C1 ~ e-11a-1lb l -::-1]a 1 + - + 2C2) . (D.49) A 17a 17a Since a < b and s -+ oo then 17a -+ oo and 1Jb -+ oo so e 110 « e7lb. It is easy to see that 'T/b = 17a!· The constants c1,c2 were determined using MATHEMATICA (Wolfram 1988) and are (D.50) (D.51) Also (D.52) where ( D .53) Appendix 122 The constants A1 , A2 can then be determined giving the final solution as (D.54) The 'constant' c1(r) = - -a + -b~ - -a) , (D.55) (8b a br was determined using MATHEMATICA (Wolfram 1988) and is necessary to allow both parts of the solution to be given to same order of accuracy. Equation (D.54) can then be inverted as a convolution integral although for small times it can be assumed that t::.P = tm/2 for some integer m = 0, 1, 2 .... Making use of the inverse Laplace transforms in appendix A.4 equation (D.54) becomes 1 -r ( 1 + ; ) /f.(4t)1r [imerfc ( b2Jt) + 2vt: im+lerfc ( b2Jt)] ,(D.56) where r is the Gamma function. D.5 Constants from equation (6.68) From section 6. 7 the solution for 1(r,t) = cs(t)ln r + c9 (t), (D.57) where cs, c9 are determined from the boundary conditions These constants are found to be t::.P (-2a2 - 2b2 - 2a2"X + 2b2"X) Cs=-~------~ (D.59) c* ' l:iP (a 2 - b2 - a2"X + b2"X + 2a2 log(a) + 2a2"Xlog(a) + 2b 2 log(b) - 2b 2°Xlog(b)) Cg = ----'------_;_ c* ' (D.60) where -2a2 log(a) - 2b2 log(a) - 2a2"Xlog(a) + 2b2°Xlog(a) + 2a2 log(b) +2b2 log(b) + 2a2"Xlog(b) - 2b2"Xlog(b). (D.61) Appendix E Computational Scheme E.1 Euler-Cauchy Numerical Scheme The third order Euler-Cauchy numerical scheme used in chapter 3 solves the differential equation dw dX = ((w), (E.l) with boundary condition w(l) = 0. (E.2) The domain x E [O, l] is discretised into N divisions of length h1. Taylor's theorem is used to define dw h 12 d2 w w(X) + hi dX + 2 dX2 ••• (E.3) h 2 ~ w(X) + h1((w) + +('(w)((w) + ht [c"(w)((w) + (('(w))2 ] ((w). 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The symbol K- will be used for the perme ability in chapter seven and the symbol k = K-µ 1 will be used in the other chapters on deformable porous materials but will still be referred to as the 'permeability'. This is to confirm to the notation which varies between the soil and the biological literature. p6 : Equation (1.3) should read: Vp =_µIv - cpl vlvl (1) K, ,.fa and following this the expression K- = kµ 1 should be used. p9 : Included after the first paragraph. ... of the mixture. Assumption 5 follows directly from assumption 1 but is explicitly stated since some biological materials, particularly cartilage, possess strong osmotic potentials. pll : In the first line the following phrase is included . ... resultant external body force (neglected herin by assumption 3) and ... pll : In the third line of paragraph two the phrase: Excluding the effect of body forces, is deleted. pll : Equation (2.15) should read: u 1 = µa (vv1 + (Vv1f) (2) p13 : After equation (2.26) insert: ••• fp = 6.P/ Ha, where Ha is the aggregate elastic modulus of the Lame stress constants. 1 p19 : After equation (2.61) the following paragraph should replace the single sentence These represent . . . . . medium. The first equation equates the fluid velocity at the interface with the volume averaged velocity of the porous layer. The final two equations arise from the conservation of axial momentum across the fluid-porous layer interface and the assumption that the pro portion of the total stress in the porous layer borne by each com ponent is proportional to its volume fraction. p26 After equation (2.93) insert. This result assumes that k is a well behaved smooth monotonic function. p28 After equation (2.97) the sentence should read: ... radial displacement, v = v( t) / r in the radial direction using the conservation of fluid mass and .. p36 After the first sentence in section 3.4 the following line is inserted: .... shown in figure 3.2. This is discussed in more detail in section 3.5. p43 After equation (4.6) the expression K- = kµJ should be used. p44 : In the last two paragraphs and in the first line in the following page replace h by Lin the terms H~ko. p60 Equation (5.6) should read (3) p62 A factor of two should appear on the right hand side of equation ( 5.11) before all appearances of µa. p62 Replace in the third paragraph the word zero by 'negligible'. p63 After equation (5.30) delete from the last line These and insert: The first two of these conditions represent the no slip .... solid boundary and the symmetry ... channel. Delete the rest of the line and replace by: 2 The other conservation conditions are discussed on page 19. p65 In section 5.5 the paramater A would be better written as A. p68 In the first paragraph delete the last two lines and write: Since 'f/ is the ratio of the intrinsic viscosity of the fluid to the apparent viscosity of the fluid in the porous layer, increasing the apparent viscosity decreases 'fJ resulting in a decrease in the velocity in the purely fluid region. p91 In section 7.1.4 before equation (7. 7) it should read: ... the only combination to give the correct physical dimensions in the one dimensional equation are if a = µ J / l'i, and b = cp f / ,JK, for some dimensionless constant c. The appropriate extension to three dimensions is then "vp= --v--vvµf CPJ I I (4) l'i, VK, which then preserves the quadratic nature of (7 .1) as well as keep ing the correct vector form. Although ... p93 : In section 7.2 from equation (7.8) to (7.9) the constant L should be written as h0 • p95 : In the last paragraph, second line delete the second occurence of the word solution. 3