DOCSLIB.ORG

Fluid Inertia and End Effects in Rheometer Flows

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

FLUID INERTIA AND END EFFECTS IN RHEOMETER FLOWS by JASON PETER HUGHES B.Sc. (Hons) A thesis submitted to the University of Plymouth in partial fulfilment for the degree of DOCTOR OF PHILOSOPHY School of Mathematics and Statistics Faculty of Technology University of Plymouth April 1998 REFERENCE ONLY ItorriNe. 9oo365d39i Data 2 h SEP 1998 Class No.- Corrtl.No. 90 0365439 1 ACKNOWLEDGEMENTS I would like to thank my supervisors Dr. J.M. Davies, Prof. T.E.R. Jones and Dr. K. Golden for their continued support and guidance throughout the course of my studies. I also gratefully acknowledge the receipt of a H.E.F.C.E research studentship during the period of my research. AUTHORS DECLARATION At no time during the registration for the degree of Doctor of Philosophy has the author been registered for any other University award. This study was financed with the aid of a H.E.F.C.E studentship and carried out in collaboration with T.A. Instruments Ltd. Publications: 1. J.P. Hughes, T.E.R Jones, J.M. Davies, *End effects in concentric cylinder rheometry', Proc. 12"^ Int. Congress on Rheology, (1996) 391. 2. J.P. Hughes, J.M. Davies, T.E.R. Jones, ^Concentric cylinder end effects and fluid inertia effects in controlled stress rheometry, Part I: Numerical simulation', accepted for publication in J.N.N.F.M. Signed ...^.^Ms>3.\^^. Date Ik.lp.^.m FLUH) INERTIA AND END EFFECTS IN RHEOMETER FLOWS Jason Peter Hughes Abstract This thesis is concerned with the characterisation of the flow behaviour of inelastic and viscoelastic fluids in steady shear and oscillatory shear flows on commercially available rheometers. The first part of this thesis is concerned with a linear viscoelastic theory to describe the oscillatory shear flow behaviour of fluids on a Weissenberg rheogoniometer. A fluid inertia perturbation analysis is used to produce analytical formulae for correcting complex viscosity data for first and second order fluid inertia effects. In order to validate the perturbation theory we perform a simulation of the oscillatory shear flow behaviour of Newtonian and single element Maxwell fluids on a Weissenberg rheogoniometer. A theoretical prediction of end effects and fluid inertia effects on steady shear viscosity measurements of Newtonian fluids in a recessed concentric cylinder geometry is developed for a GSR controlled stress rheometer and a Weissenberg rheogoniometer. The relevant equations are solved using a perturbation analysis which is valid for low Reynolds number flows. From this theory correction formulae are produced to compensate for end effects and second order fluid inertia effects in steady shear flows on these instruments. End effects and fluid inertia effects are also investigated for power law shear thinning fluids. The final part of the thesis is concerned with a theoretical prediction of the end effect of a recessed concentric cylinder geometry on complex viscosity measurements of a generalised linear viscoelastic fluid. The linear viscoelastic theory is carried out for oscillatory shear flows on a CSR controlled stress rheometer and a Weissenberg rheogoniometer. A fluid inertia perturbation analysis is used to produce analytical formulae to correct complex viscosity data for end effects and second order fluid inertia effects. Numerically simulated oscillatory shear data is used to establish the limitations of the second order fluid inertia correction formulae which include end effects. CONTENTS Page Chapter 1 INTRODUCTION 1 Chapter 2 RHEOLOGICAL EQUATIONS OF STATE AND EQUATIONS OF MOTION 6 2.1 The characterisation of fluids 6 2.2 Equations of motion 7 2.2.1 Equation of conservation of mass 7 2.2.2 Equation of conservation of momentum 8 2.3 . Equations of state 9 2.3.1 The generalised Newtonian fluid 9 2.3.2 The Maxwell model fluid 10 2.3.3 The generalised linear viscoelastic model fluid 12 2.4 Definition of complex viscosity 13 Chapter 3 NUMERICAL METHODS FOR THE SOLUTION OF THE EQUATIONS OF MOTION 15 3.1 Introduction to numerical methods 15 3.2 The finite difference method 16 3.2.1 Finite difference representation of partial derivatives 17 3.2.2 Finite difference boundary conditions 18 3.2.3 The finite difference method applied to Laplace's equation 18 3.2.4 Iterative methods for solving the finite difference equations 20 3.2.5 Irregular finite difference meshes and transformations 21 3.3 The Polyflow package 22 3.3.1 Polyflow boundary conditions 23 3.4 Non-dimensional form of the equations of motion 24 Chapter 4 THE EFFECT OF FLUID INERTIA ON COMPLEX VISCOSITY PREDICTIONS OBTAINED FROM A WEISSENBERG RHEOGONIOMETER 26 4. i Introduction 26 4.2 Parallel plate geometry 30 4.2.1 Governing equations 30 4.2.2 Perturbation method of solution to predict complex viscosity 33 4.2.3 Complex viscosity prediction for a strain gauge torsion head system on the Weissenberg rheogoniometer 35 4.2.4 Complex viscosity prediction for a CSR controlled stress rheometer 36 4.3 Concentric cylinder geometry 37 4.3.1 Governing equations 37 4.3.2 Perturbation method of solution to predict complex viscosity 41 4.3.3 Complex viscosity prediction for a strain gauge torsion head system on the Weissenberg rheogoniometer 44 4.3.4 Complex viscosity prediction for a CSR controlled stress rheometer 45 4.4 Cone and plate geometry 46 4.4.1 Governing equations 46 4.4.2 Perturbation method of solution to predict complex viscosity 50 4.4.3 Complex viscosity prediction for a strain gauge torsion head system on the Weissenberg rheogoniometer 54 4.4.4 Complex viscosity prediction for a CSR controlled stress rheometer 55 4.5 Inaccuracies in complex viscosity predictions near the natural frequency of the constrained member 56 4.6 Comments 57 Chapter 5 SIMULATION OF OSCILLATORY SHEAR FLOW ON A WEISSENBERG RHEOGONIOMETER AND A CSR CONTROLLED STRESS RHEOMETER 59 5.1 Introduction 59 5.2 Parallel plate geometry constrained by a torsion bar 60 5.2.1 Newtonian fluids 61 5.2.2 Single element Maxwell fluids 62 5.3 Parallel plate geometry connected to a strain gauge torsion head 64 5.3.1 Newtonian fluids 65 5.3.2 Single element Maxwell fluids 66 5.4 Concentric cylinder geometry constrained by a torsion bar 67 5.4.1 Newtonian fluids 68 5.4.2 Single element Maxwell fluids 70 5.5 Concentric cylinder geometry connected to a strain gauge torsion head 71 5.5.1 Newtonian fluids 72 5.5.2 Single element Maxwell fluids 73 5.6 Cone and plate geometry constrained by a torsion bar 74 5.6.1 Newtonian fluids 76 5.6.2 Single element Maxwell fluids 76 5.7 Cone and plate geometry connected to a strain gauge torsion head 77 5.7.1 Newtonian fluids 78 5.7.2 Single element Maxwell fluids 79 5.8 Comments 79 5.9 Conclusions 82 Chapter 6 CONCENTRIC CYLINDER END EFFECTS AND FLUID INERTIA EFFECTS ON STEADY SHEAR VISCOSITY PREDICTIONS IN CONTROLLED STRESS RHEOMETRY 85 6.1 Introduction 85 6.2 Steady shear theory 87 6.2.1 Shear viscosity prediction using infinite cylinder theory 87 6.2.2 Governing equations and boundary conditions 88 6.3 Fluid inertia perturbation theory for a Newtonian fluid 91 6.3.1 Zero order equations and boundary conditions 95 6.3.2 First order equations and boundary conditions 95 6.3.3 Second order equations and boundary conditions 98 6.3.4 Modified formulae for predicting the shear viscosity of a Newtonian fluid 99 6.4 Finite difference method of solution for the equations obtained from the fluid inertia perturbation theory 100 6.4.1 Transformation from physical domain to a computational space 101 6.4.2 Transformations of the derivative terms 104 6.4.3 Finite difference equations for the zero order component of the primary velocity v^Q(r,z) 105 6.4.4 Finite difference equations for the streamfunction y^{r,z) 110 6.4.5 Determination of the secondary flow velocities v^i(^.r) and v^,(/-,z) from the finite difference solution for the streamfunction y/{r,z) 118 6.4.6 Finite difference equations for the second order component of the primary velocity Vg^ (r, z) 121 6.4.7 Calculation of the geometry dependent constants from the finite difference solutions for ^'eoi^'y^) v^2(r,r) 125 6.4.8 Selection of the mesh constants in the finite difference equations 126 6.5 Steady shear results 128 6.6 Flow simulation of a Newtonian fluid in a 1:2 ratio gap geometry 133 6.7 Conclusions 134 Chapter 7 CONCENTRIC CYLINDER END EFFECTS AND FLUID INERTIA EFFECTS ON STEADY SHEAR VISCOSITY PREDICTIONS OBTAINED FROM A WEISSENBERG RHEOGONIOMETER 136 7.1 Introduction 136 7.2 Steady shear theory 137 7.2.1 Governing equations and boundary conditions 138 7.2.2 Stability of concentric cylinder flow 138 7.3 Fluid inertia perturbation theory for a Newtonian fluid 139 7.4 Relationship between concentric cylinder end effects on a Weissenberg rheogoniometer and a CSR controlled stress rheometer 140 7.5 Steady shear results 141 7.6 Flow simulation of a Newtonian fluid in the three CSR geometries 145 7.7 Conclusions 147 Chapter 8 CONCENTRIC CYLINDER END EFFECTS AND FLUID INERTIA EFFECTS ON COMPLEX VISCOSITY PREDICTIONS IN CONTROLLED STRESS RHEOMETRY 148 8.1 Introduction 149 8.2 Oscillatory shear theory 150 8.3 Fluid inertia perturbation theory for a generalised linear viscoelastic fluid 152 8.4 Finite difference method of solution for the perturbation theory equations 155 8.4.1 Finite difference equations for the velocity function V(,(r,z) 156 8.4.2 Finite difference equations for the velocity function v,(r,r) 156 8.4.3 Finite difference equations for the velocity function V2(r,z) 158 8 .4 .4 Calculation of the geometry dependent constants from the finite difference solutions for v^{f^z)^ v,(r,r)and y^iir^z) 160 8.5
Recommended publications
  • Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow Through Porous Media

    Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow Through Porous Media

    Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory Title Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow through Porous Media Permalink https://escholarship.org/uc/item/6zv599hc Author Wu, Y.S. Publication Date 1990-02-01 eScholarship.org Powered by the California Digital Library University of California Lawrence Berkeley Laboratory e UNIVERSITY OF CALIFORNIA EARTH SCIENCES DlVlSlON Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow through Porous Media Y.-S. Wu (Ph.D. Thesis) February 1990 TWO-WEEK LOAN COPY This is a Library Circulating Copy which may be borrowed for two weeks. r- +. .zn Prepared for the U.S. Department of Energy under Contract Number DE-AC03-76SF00098. :0 DISCLAIMER I I This document was prepared as an account of work sponsored ' : by the United States Government. Neither the United States : ,Government nor any agency thereof, nor The Regents of the , I Univers~tyof California, nor any of their employees, makes any I warranty, express or implied, or assumes any legal liability or ~ : responsibility for the accuracy, completeness, or usefulness of t any ~nformation, apparatus, product, or process disclosed, or I represents that its use would not infringe privately owned rights. : Reference herein to any specific commercial products process, or I service by its trade name, trademark, manufacturer, or other- I wise, does not necessarily constitute or imply its endorsement, ' recommendation, or favoring by the United States Government , or any agency thereof, or The Regents of the University of Cali- , forma. The views and opinions of authors expressed herein do ' not necessarily state or reflect those of the United States : Government or any agency thereof or The Regents of the , Univers~tyof California and shall not be used for advertismg or I product endorsement purposes.
  • On Exact Solution of Unsteady MHD Flow of a Viscous Fluid in An

    On Exact Solution of Unsteady MHD Flow of a Viscous Fluid in An

    Rana et al. Boundary Value Problems 2014, 2014:146 http://www.boundaryvalueproblems.com/content/2014/1/146 R E S E A R C H Open Access On exact solution of unsteady MHD flow of a viscous fluid in an orthogonal rheometer Muhammad Afzal Rana1, Sadia Siddiqa2* and Saima Noor3 *Correspondence: [email protected] Abstract 2Department of Mathematics, COMSATS Institute of Information This paper studies the unsteady MHD flow of a viscous fluid in which each point of Technology, Attock, Pakistan the parallel planes are subject to the non-torsional oscillations in their own planes. Full list of author information is The streamlines at any given time are concentric circles. Exact solutions are obtained available at the end of the article and the loci of the centres of these concentric circles are discussed. It is shown that the motion so obtained gives three infinite sets of exact solutions in the geometry of an orthogonal rheometer in which the above non-torsional oscillations are superposed on the disks. These solutions reduce to a single unique solution when symmetric solutions are looked for. Some interesting special cases are also obtained from these solutions. Keywords: viscous fluid; MHD flow; orthogonal rheometer; eccentric rotation; exact solutions 1 Introduction Berker [] has defined the ‘pseudo plane motions’ of the first kind that: if the streamlines in a plane flow are contained in parallel planes but the velocity components are dependent on the coordinate normal to the planes. Berker [] has obtained a class of exact solutions to the Navier-Stokes equations belonging to the above type of flows.
  • Lecture 1: Introduction

    Lecture 1: Introduction

    Lecture 1: Introduction E. J. Hinch Non-Newtonian fluids occur commonly in our world. These fluids, such as toothpaste, saliva, oils, mud and lava, exhibit a number of behaviors that are different from Newtonian fluids and have a number of additional material properties. In general, these differences arise because the fluid has a microstructure that influences the flow. In section 2, we will present a collection of some of the interesting phenomena arising from flow nonlinearities, the inhibition of stretching, elastic effects and normal stresses. In section 3 we will discuss a variety of devices for measuring material properties, a process known as rheometry. 1 Fluid Mechanical Preliminaries The equations of motion for an incompressible fluid of unit density are (for details and derivation see any text on fluid mechanics, e.g. [1]) @u + (u · r) u = r · S + F (1) @t r · u = 0 (2) where u is the velocity, S is the total stress tensor and F are the body forces. It is customary to divide the total stress into an isotropic part and a deviatoric part as in S = −pI + σ (3) where tr σ = 0. These equations are closed only if we can relate the deviatoric stress to the velocity field (the pressure field satisfies the incompressibility condition). It is common to look for local models where the stress depends only on the local gradients of the flow: σ = σ (E) where E is the rate of strain tensor 1 E = ru + ruT ; (4) 2 the symmetric part of the the velocity gradient tensor. The trace-free requirement on σ and the physical requirement of symmetry σ = σT means that there are only 5 independent components of the deviatoric stress: 3 shear stresses (the off-diagonal elements) and 2 normal stress differences (the diagonal elements constrained to sum to 0).
  • Rheometry SLIT RHEOMETER

    Rheometry SLIT RHEOMETER

    Rheometry SLIT RHEOMETER Figure 1: The Slit Rheometer. L > W h. ∆P Shear Stress σ(y) = y (8-30) L −∆P h Wall Shear Stress σ = −σ(y = h/2) = (8-31) w L 2 NEWTONIAN CASE 6Q Wall Shear Rateγ ˙ = −γ˙ (y = h/2) = (8-32) w h2w σ −∆P h3w Viscosity η = w = (8-33) γ˙ w L 12Q 1 Rheometry SLIT RHEOMETER NON-NEWTONIAN CASE Correction for the real wall shear rate is analogous to the Rabinowitch correction. 6Q 2 + β Wall Shear Rateγ ˙ = (8-34a) w h2w 3 d [log(6Q/h2w)] β = (8-34b) d [log(σw)] σ −∆P h3w Apparent Viscosity η = w = γ˙ w L 4Q(2 + β) NORMAL STRESS DIFFERENCE The normal stress difference N1 can be determined from the exit pressure Pe. dPe N1(γ ˙ w) = Pe + σw (8-45) dσw d(log Pe) N1(γ ˙ w) = Pe 1 + (8-46) d(log σw) These relations were calculated assuming straight parallel streamlines right up to the exit of the die. This assumption is not found to be valid in either experiment or computer simulation. 2 Rheometry SLIT RHEOMETER NORMAL STRESS DIFFERENCE dPe N1(γ ˙ w) = Pe + σw (8-45) dσw d(log Pe) N1(γ ˙ w) = Pe 1 + (8-46) d(log σw) Figure 2: Determination of the Exit Pressure. 3 Rheometry SLIT RHEOMETER NORMAL STRESS DIFFERENCE Figure 3: Comparison of First Normal Stress Difference Values for LDPE from Slit Rheometer Exit Pressure (filled symbols) and Cone&Plate (open symbols). The poor agreement indicates that the more work is needed in order to use exit pressures to measure normal stress differences.
  • A Fluid Is Defined As a Substance That Deforms Continuously Under Application of a Shearing Stress, Regardless of How Small the Stress Is

    A Fluid Is Defined As a Substance That Deforms Continuously Under Application of a Shearing Stress, Regardless of How Small the Stress Is

    FLUID MECHANICS & BIOTRIBOLOGY CHAPTER ONE FLUID STATICS & PROPERTIES Dr. ALI NASER Fluids Definition of fluid: A fluid is defined as a substance that deforms continuously under application of a shearing stress, regardless of how small the stress is. To study the behavior of materials that act as fluids, it is useful to define a number of important fluid properties, which include density, specific weight, specific gravity, and viscosity. Density is defined as the mass per unit volume of a substance and is denoted by the Greek character ρ (rho). The SI units for ρ are kg/m3. Specific weight is defined as the weight per unit volume of a substance. The SI units for specific weight are N/m3. Specific gravity S is the ratio of the weight of a liquid at a standard reference temperature to the o weight of water. For example, the specific gravity of mercury SHg = 13.6 at 20 C. Specific gravity is a unit-less parameter. Density and specific weight are measures of the “heaviness” of a fluid. Example: What is the specific gravity of human blood, if the density of blood is 1060 kg/m3? Solution: ⁄ ⁄ Viscosity, shearing stress and shearing strain Viscosity is a measure of a fluid's resistance to flow. It describes the internal friction of a moving fluid. A fluid with large viscosity resists motion because its molecular makeup gives it a lot of internal friction. A fluid with low viscosity flows easily because its molecular makeup results in very little friction when it is in motion. Gases also have viscosity, although it is a little harder to notice it in ordinary circumstances.
  • Rheology of PIM Feedstocks SPECIAL FEATURE

    Rheology of PIM Feedstocks SPECIAL FEATURE

    Metal Powder Report Volume 72, Number 1 January/February 2017 metal-powder.net Rheology of PIM feedstocks SPECIAL FEATURE Christian Kukla, Ivica Duretek, Joamin Gonzalez-Gutierrez and Clemens Holzer Introduction constant viscosity is called zero-viscosity h0 (Fig. 1). After a certain > g Powder injection molding (PIM) is a cost effective technique for shear rate ( ˙1), viscosity starts to decrease rapidly as a function of producing complex and precise metal or ceramic components in shear rate, this is known as shear thinning or pseudoplastic behav- mass production [1]. The used raw material, referred as feedstock, ior. For highly filled compounds like PIM feedstocks a yield stress consists of metal or ceramic powder and a polymeric binder can be observed. Thus the viscosity increases dramatically when mainly composed of thermoplastics. The thermoplastic binder decreasing the shear stress and the zero shear viscosity is hard to composition gives plasticity to the feedstock during the molding measure and thus shear thinning is observed even at very low shear g process and holds together the powder grains before sintering. rates. Around a certain higher shear rate ˙2 a second Newtonian > g Most binder systems are made of multi-component systems plateau can be observed and at very high shear rates ( ˙3) the with a range of modifiers which fulfill the above mentioned plateau can change to an increasing viscosity curve due to formation requirements. The flow behavior of the feedstock is the result of of particle agglomerates that can restrict the flow of the binder complex interactions between its constituents. The viscosity of the system.
  • Rheology of Petroleum Fluids

    Rheology of Petroleum Fluids

    ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 20, 2012 Rheology of Petroleum Fluids Hans Petter Rønningsen, Statoil, Norway ABSTRACT NEWTONIAN FLUIDS Among the areas where rheology plays In gas reservoirs, the flow properties of an important role in the oil and gas industry, the simplest petroleum fluids, i.e. the focus of this paper is on crude oil hydrocarbons with less than five carbon rheology related to production. The paper atoms, play an essential role in production. gives an overview of the broad variety of It directly impacts the productivity. The rheological behaviour, and corresponding viscosity of single compounds are well techniques for investigation, encountered defined and mixture viscosity can relatively among petroleum fluids. easily be calculated. Most often reservoir gas viscosity is though measured at reservoir INTRODUCTION conditions as part of reservoir fluid studies. Rheology plays a very important role in The behaviour is always Newtonian. The the petroleum industry, in drilling as well as main challenge in terms of measurement and production. The focus of this paper is on modelling, is related to very high pressures crude oil rheology related to production. (>1000 bar) and/or high temperatures (170- Drilling and completion fluids are not 200°C) which is encountered both in the covered. North Sea and Gulf of Mexico. Petroleum fluids are immensely complex Hydrocarbon gases also exist dissolved mixtures of hydrocarbon compounds, in liquid reservoir oils and thereby impact ranging from the simplest gases, like the fluid viscosity and productivity of these methane, to large asphaltenic molecules reservoirs. Reservoir oils are also normally with molecular weights of thousands.
  • Lecture 2: (Complex) Fluid Mechanics for Physicists

    Lecture 2: (Complex) Fluid Mechanics for Physicists

    Application of granular jamming: robots! Cornell (Amend and Lipson groups) in collaboration with Univ of1 Chicago (Jaeger group) Lecture 2: (complex) fluid mechanics for physicists S-RSI Physics Lectures: Soft Condensed Matter Physics Jacinta C. Conrad University of Houston 2012 Note: I have added links addressing questions and topics from lectures at: http://conradlab.chee.uh.edu/srsi_links.html Email me questions/comments/suggestions! 2 Soft condensed matter physics • Lecture 1: statistical mechanics and phase transitions via colloids • Lecture 2: (complex) fluid mechanics for physicists • Lecture 3: physics of bacteria motility • Lecture 4: viscoelasticity and cell mechanics • Lecture 5: Dr. Conrad!s work 3 Big question for today!s lecture How does the fluid mechanics of complex fluids differ from that of simple fluids? Petroleum Food products Examples of complex fluids: Personal care products Ceramic precursors Paints and coatings 4 Topic 1: shear thickening 5 Forces and pressures A force causes an object to change velocity (either in magnitude or direction) or to deform (i.e. bend, stretch). � dp� Newton!s second law: � F = m�a = dt p� = m�v net force change in linear mass momentum over time d�v �a = acceleration dt A pressure is a force/unit area applied perpendicular to an object. Example: wind blowing on your hand. direction of pressure force �n : unit vector normal to the surface 6 Stress A stress is a force per unit area that is measured on an infinitely small area. Because forces have three directions and surfaces have three orientations, there are nine components of stress. z z Example of a normal stress: δA δAx x δFx τxx = lim τxy δFy δAx→0 δAx τxx δFx y y Example of a shear stress: δFy τxy = lim δAx→0 δAx x x Convention: first subscript indicates plane on which stress acts; second subscript indicates direction in which the stress acts.
  • Chapter 3 Newtonian Fluids

    Chapter 3 Newtonian Fluids

    CM4650 Chapter 3 Newtonian Fluid 2/5/2018 Mechanics Chapter 3: Newtonian Fluids CM4650 Polymer Rheology Michigan Tech Navier-Stokes Equation v vv p 2 v g t 1 © Faith A. Morrison, Michigan Tech U. Chapter 3: Newtonian Fluid Mechanics TWO GOALS •Derive governing equations (mass and momentum balances •Solve governing equations for velocity and stress fields QUICK START V W x First, before we get deep into 2 v (x ) H derivation, let’s do a Navier-Stokes 1 2 x1 problem to get you started in the x3 mechanics of this type of problem solving. 2 © Faith A. Morrison, Michigan Tech U. 1 CM4650 Chapter 3 Newtonian Fluid 2/5/2018 Mechanics EXAMPLE: Drag flow between infinite parallel plates •Newtonian •steady state •incompressible fluid •very wide, long V •uniform pressure W x2 v1(x2) H x1 x3 3 EXAMPLE: Poiseuille flow between infinite parallel plates •Newtonian •steady state •Incompressible fluid •infinitely wide, long W x2 2H x1 x3 v (x ) x1=0 1 2 x1=L p=Po p=PL 4 2 CM4650 Chapter 3 Newtonian Fluid 2/5/2018 Mechanics Engineering Quantities of In more complex flows, we can use Interest general expressions that work in all cases. (any flow) volumetric ⋅ flow rate ∬ ⋅ | average 〈 〉 velocity ∬ Using the general formulas will Here, is the outwardly pointing unit normal help prevent errors. of ; it points in the direction “through” 5 © Faith A. Morrison, Michigan Tech U. The stress tensor was Total stress tensor, Π: invented to make the calculation of fluid stress easier. Π ≡ b (any flow, small surface) dS nˆ Force on the S ⋅ Π surface V (using the stress convention of Understanding Rheology) Here, is the outwardly pointing unit normal of ; it points in the direction “through” 6 © Faith A.
  • 7. Fluid Mechanics

    7. Fluid Mechanics

    7. Fluid mechanics Introduction In this chapter we will study the mechanics of fluids. A non-viscous fluid, i.e. fluid with no inner friction is called an ideal fluid. The stress tensor for an ideal fluid is given by T1= −p , which is the most simplest form of the stress tensor used in fluid mechanics. A viscous material is a material for which the stress tensor depends on the rate of deformation D. If this relation is linear then we have linear viscous fluid or Newtonian fluid. The stress tensor for a linear viscous fluid is given by T = - p 1 + λ tr D 1 + 2μ D , where λ and μ are material constants. If we assume the fluid to be incompressible, i.e. tr D 1 = 0 and the stress tensor can be simplified to contain only one material coefficient μ . Most of the fluids can not be modelled by these two simple types of stress state assumption and they are called non-Newtonian fluids. There are different types of non-Newtonian fluids. We have constitutive models for fluids of differential, rate and integral types. Examples non-Newtonian fluids are Reiner-Rivlin fluids, Rivlin-Ericksen fluids and Maxwell fluids. In this chapter we will concentrate on ideal and Newtonian fluids and we will study some classical examples of hydrodynamics. 7.1 The general equations of motion for an ideal fluid For an arbitrary fluid (liquid or gas) the stress state can be describe by the following constitutive relation, T1= −p (7.1) where p= p (r ,t) is the fluid pressure and T is the stress.
  • Equation of Motion for Viscous Fluids

    Equation of Motion for Viscous Fluids

    1 2.25 Equation of Motion for Viscous Fluids Ain A. Sonin Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 2001 (8th edition) Contents 1. Surface Stress …………………………………………………………. 2 2. The Stress Tensor ……………………………………………………… 3 3. Symmetry of the Stress Tensor …………………………………………8 4. Equation of Motion in terms of the Stress Tensor ………………………11 5. Stress Tensor for Newtonian Fluids …………………………………… 13 The shear stresses and ordinary viscosity …………………………. 14 The normal stresses ……………………………………………….. 15 General form of the stress tensor; the second viscosity …………… 20 6. The Navier-Stokes Equation …………………………………………… 25 7. Boundary Conditions ………………………………………………….. 26 Appendix A: Viscous Flow Equations in Cylindrical Coordinates ………… 28 ã Ain A. Sonin 2001 2 1 Surface Stress So far we have been dealing with quantities like density and velocity, which at a given instant have specific values at every point in the fluid or other continuously distributed material. The density (rv ,t) is a scalar field in the sense that it has a scalar value at every point, while the velocity v (rv ,t) is a vector field, since it has a direction as well as a magnitude at every point. Fig. 1: A surface element at a point in a continuum. The surface stress is a more complicated type of quantity. The reason for this is that one cannot talk of the stress at a point without first defining the particular surface through v that point on which the stress acts. A small fluid surface element centered at the point r is defined by its area A (the prefix indicates an infinitesimal quantity) and by its outward v v unit normal vector n .
  • The Design and Analysis of a Micro Squeeze Flow Rheometer

    The Design and Analysis of a Micro Squeeze Flow Rheometer

    The Design and Analysis of a Micro Squeeze Flow Rheometer By David Cheneler A thesis submitted to The University of Birmingham for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering College of Engineering and Physical Sciences The University of Birmingham September 2009 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. ABSTRACT This thesis describes the analysis and design of a micro squeeze flow rheometer. The need to analyse the rheology of complex liquids occurs regularly in industry and during research. However, frequently the amount of fluid available is too small, precluding the use of conventional rheometers. Conventional rheometers also tend to have the disadvantage of being too massive, preventing them from operating effectively at high frequencies. The investigation carried out in this thesis has revealed that current microrheometry techniques also have their own disadvantages. The proposed design is a stand-alone device capable of measuring the dynamic properties of nanolitre volumes of viscoelastic fluid at frequencies up to the kHz range, an order of magnitude greater than conventional rheometers. The device uses a single piezoelectric component to both actuate and sense its own position.