Chapter 7: Generalized Newtonian Fluids
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Analysis of Deformation
Chapter 7: Constitutive Equations Definition: In the previous chapters we’ve learned about the definition and meaning of the concepts of stress and strain. One is an objective measure of load and the other is an objective measure of deformation. In fluids, one talks about the rate-of-deformation as opposed to simply strain (i.e. deformation alone or by itself). We all know though that deformation is caused by loads (i.e. there must be a relationship between stress and strain). A relationship between stress and strain (or rate-of-deformation tensor) is simply called a “constitutive equation”. Below we will describe how such equations are formulated. Constitutive equations between stress and strain are normally written based on phenomenological (i.e. experimental) observations and some assumption(s) on the physical behavior or response of a material to loading. Such equations can and should always be tested against experimental observations. Although there is almost an infinite amount of different materials, leading one to conclude that there is an equivalently infinite amount of constitutive equations or relations that describe such materials behavior, it turns out that there are really three major equations that cover the behavior of a wide range of materials of applied interest. One equation describes stress and small strain in solids and called “Hooke’s law”. The other two equations describe the behavior of fluidic materials. Hookean Elastic Solid: We will start explaining these equations by considering Hooke’s law first. Hooke’s law simply states that the stress tensor is assumed to be linearly related to the strain tensor. -
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). -
Overview of Turbulent Shear Flows
Overview of turbulent shear flows Fabian Waleffe notes by Giulio Mariotti and John Platt revised by FW WHOI GFD Lecture 2 June 21, 2011 A brief review of basic concepts and folklore in shear turbulence. 1 Reynolds decomposition Experimental measurements show that turbulent flows can be decomposed into well-defined averages plus fluctuations, v = v¯ + v0. This is nicely illustrated in the Turbulence film available online at http://web.mit.edu/hml/ncfmf.html that was already mentioned in Lecture 1. In experiments, the average v¯ is often taken to be a time average since it is easier to measure the velocity at the same point for long times, while in theory it is usually an ensemble average | an average over many realizations of the same flow. For numerical simulations and mathematical analysis in simple geometries such as channels and pipes, the average is most easily defined as an average over the homogeneous or periodic directions. Given a velocity field v(x; y; z; t) we define a mean velocity for a shear flow with laminar flow U(y)^x by averaging over the directions (x; z) perpendicular to the shear direction y, 1 Z Lz=2 Z Lx=2 v¯ = lim v(x; y; z; t)dxdz = U(y; t)^x: (1) L ;L !1 x z LxLz −Lz=2 −Lx=2 The mean velocity is in the x direction because of incompressibility and the boundary conditions. In a pipe, the average would be over the streamwise and azimuthal (spanwise) direction and the mean would depend only on the radial distance to the pipe axis, and (possibly) time. -
Guide to Rheological Nomenclature: Measurements in Ceramic Particulate Systems
NfST Nisr National institute of Standards and Technology Technology Administration, U.S. Department of Commerce NIST Special Publication 946 Guide to Rheological Nomenclature: Measurements in Ceramic Particulate Systems Vincent A. Hackley and Chiara F. Ferraris rhe National Institute of Standards and Technology was established in 1988 by Congress to "assist industry in the development of technology . needed to improve product quality, to modernize manufacturing processes, to ensure product reliability . and to facilitate rapid commercialization ... of products based on new scientific discoveries." NIST, originally founded as the National Bureau of Standards in 1901, works to strengthen U.S. industry's competitiveness; advance science and engineering; and improve public health, safety, and the environment. One of the agency's basic functions is to develop, maintain, and retain custody of the national standards of measurement, and provide the means and methods for comparing standards used in science, engineering, manufacturing, commerce, industry, and education with the standards adopted or recognized by the Federal Government. As an agency of the U.S. Commerce Department's Technology Administration, NIST conducts basic and applied research in the physical sciences and engineering, and develops measurement techniques, test methods, standards, and related services. The Institute does generic and precompetitive work on new and advanced technologies. NIST's research facilities are located at Gaithersburg, MD 20899, and at Boulder, CO 80303. -
Circular Birefringence in Crystal Optics
Circular birefringence in crystal optics a) R J Potton Joule Physics Laboratory, School of Computing, Science and Engineering, Materials and Physics Research Centre, University of Salford, Greater Manchester M5 4WT, UK. Abstract In crystal optics the special status of the rest frame of the crystal means that space- time symmetry is less restrictive of electrodynamic phenomena than it is of static electromagnetic effects. A relativistic justification for this claim is provided and its consequences for the analysis of optical activity are explored. The discrete space-time symmetries P and T that lead to classification of static property tensors of crystals as polar or axial, time-invariant (-i) or time-change (-c) are shown to be connected by orientation considerations. The connection finds expression in the dynamic phenomenon of gyrotropy in certain, symmetry determined, crystal classes. In particular, the degeneracies of forward and backward waves in optically active crystals arise from the covariance of the wave equation under space-time reversal. a) Electronic mail: [email protected] 1 1. Introduction To account for optical activity in terms of the dielectric response in crystal optics is more difficult than might reasonably be expected [1]. Consequently, recourse is typically had to a phenomenological account. In the simplest cases the normal modes are assumed to be circularly polarized so that forward and backward waves of the same handedness are degenerate. If this is so, then the circular birefringence can be expanded in even powers of the direction cosines of the wave normal [2]. The leading terms in the expansion suggest that optical activity is an allowed effect in the crystal classes having second rank property tensors with non-vanishing symmetrical, axial parts. -
Soft Matter Theory
Soft Matter Theory K. Kroy Leipzig, 2016∗ Contents I Interacting Many-Body Systems 3 1 Pair interactions and pair correlations 4 2 Packing structure and material behavior 9 3 Ornstein{Zernike integral equation 14 4 Density functional theory 17 5 Applications: mesophase transitions, freezing, screening 23 II Soft-Matter Paradigms 31 6 Principles of hydrodynamics 32 7 Rheology of simple and complex fluids 41 8 Flexible polymers and renormalization 51 9 Semiflexible polymers and elastic singularities 63 ∗The script is not meant to be a substitute for reading proper textbooks nor for dissemina- tion. (See the notes for the introductory course for background information.) Comments and suggestions are highly welcome. 1 \Soft Matter" is one of the fastest growing fields in physics, as illustrated by the APS Council's official endorsement of the new Soft Matter Topical Group (GSOFT) in 2014 with more than four times the quorum, and by the fact that Isaac Newton's chair is now held by a soft matter theorist. It crosses traditional departmental walls and now provides a common focus and unifying perspective for many activities that formerly would have been separated into a variety of disciplines, such as mathematics, physics, biophysics, chemistry, chemical en- gineering, materials science. It brings together scientists, mathematicians and engineers to study materials such as colloids, micelles, biological, and granular matter, but is much less tied to certain materials, technologies, or applications than to the generic and unifying organizing principles governing them. In the widest sense, the field of soft matter comprises all applications of the principles of statistical mechanics to condensed matter that is not dominated by quantum effects. -
Introduction to FINITE STRAIN THEORY for CONTINUUM ELASTO
RED BOX RULES ARE FOR PROOF STAGE ONLY. DELETE BEFORE FINAL PRINTING. WILEY SERIES IN COMPUTATIONAL MECHANICS HASHIGUCHI WILEY SERIES IN COMPUTATIONAL MECHANICS YAMAKAWA Introduction to for to Introduction FINITE STRAIN THEORY for CONTINUUM ELASTO-PLASTICITY CONTINUUM ELASTO-PLASTICITY KOICHI HASHIGUCHI, Kyushu University, Japan Introduction to YUKI YAMAKAWA, Tohoku University, Japan Elasto-plastic deformation is frequently observed in machines and structures, hence its prediction is an important consideration at the design stage. Elasto-plasticity theories will FINITE STRAIN THEORY be increasingly required in the future in response to the development of new and improved industrial technologies. Although various books for elasto-plasticity have been published to date, they focus on infi nitesimal elasto-plastic deformation theory. However, modern computational THEORY STRAIN FINITE for CONTINUUM techniques employ an advanced approach to solve problems in this fi eld and much research has taken place in recent years into fi nite strain elasto-plasticity. This book describes this approach and aims to improve mechanical design techniques in mechanical, civil, structural and aeronautical engineering through the accurate analysis of fi nite elasto-plastic deformation. ELASTO-PLASTICITY Introduction to Finite Strain Theory for Continuum Elasto-Plasticity presents introductory explanations that can be easily understood by readers with only a basic knowledge of elasto-plasticity, showing physical backgrounds of concepts in detail and derivation processes -
2 Review of Stress, Linear Strain and Elastic Stress- Strain Relations
2 Review of Stress, Linear Strain and Elastic Stress- Strain Relations 2.1 Introduction In metal forming and machining processes, the work piece is subjected to external forces in order to achieve a certain desired shape. Under the action of these forces, the work piece undergoes displacements and deformation and develops internal forces. A measure of deformation is defined as strain. The intensity of internal forces is called as stress. The displacements, strains and stresses in a deformable body are interlinked. Additionally, they all depend on the geometry and material of the work piece, external forces and supports. Therefore, to estimate the external forces required for achieving the desired shape, one needs to determine the displacements, strains and stresses in the work piece. This involves solving the following set of governing equations : (i) strain-displacement relations, (ii) stress- strain relations and (iii) equations of motion. In this chapter, we develop the governing equations for the case of small deformation of linearly elastic materials. While developing these equations, we disregard the molecular structure of the material and assume the body to be a continuum. This enables us to define the displacements, strains and stresses at every point of the body. We begin our discussion on governing equations with the concept of stress at a point. Then, we carry out the analysis of stress at a point to develop the ideas of stress invariants, principal stresses, maximum shear stress, octahedral stresses and the hydrostatic and deviatoric parts of stress. These ideas will be used in the next chapter to develop the theory of plasticity. -
Fluid Mechanics Linear Viscoelasticity
1 Fluid Mechanics Stress Strain Strain rate Shear vs. Extension Apparent Viscosity Oversimplified Models: Maxwell Model Voigt Model Continuity Equation Navier-Stokes Equations Boundary Conditions Volumetric Flow Rate Linear Viscoelasticity Boltzmann Superposition Step Strain: Relaxation Modulus Generalized Maxwell Model Viscosity Creep/Recovery: Creep Compliance Recoverable Compliance Steady State Compliance Terminal Relaxation Time Oscillatory Shear: Storage Modulus Loss Modulus Phase Angle Loss Tangent Time-Temperature Superposition 1 2 Molecular Structure Effects Molecular Models: Rouse Model (Unentangled) Reptation Model (Entangled) Viscosity Recoverable Compliance Diffusion Coefficient Terminal Relaxation Time Terminal Modulus Plateau Modulus Entanglement Molecular Weight Glassy Modulus Transition Zone Apparent Viscosity Polydispersity Effects Branching Effects Die Swell 2 3 Nonlinear Viscoelasticity Stress is an Odd Function of Strain and Strain Rate Viscosity and Normal Stress are Even Functions of Strain and Strain Rate Lodge-Meissner Relation Nonlinear Step Strain Extra Relaxation at Rouse Time Damping Function Steady Shear Apparent Viscosity Power Law Model Cross Model Carreau Model Cox-Merz Empiricism First Normal Stress Coefficient Start-Up and Cessation of Steady Shear Nonlinear Creep and Recovery 3 4 Stress and Strain SHEAR F Shear Stress σ ≡ A l Shear Strain γ ≡ h dγ Shear Rate γ˙ ≡ dt Hooke’s Law σ = Gγ Newton’s Law σ = ηγ˙ EXTENSION F Tensile Stress σ ≡ A ∆l Extensional Strain ε ≡ l dε Extension Rate ε˙ ≡ dt Hooke’s Law σ =3Gε Newton’s Law σ =3ηε˙ 1 5 Viscoelasticity APPARENT VISCOSITY σ η ≡ γ˙ 1.Apparent Viscosity of a Monodisperse Polystyrene. 2 6 Oversimplified Models MAXWELL MODEL Stress Relaxation σ(t)=σ0 exp( t/λ) − G(t)=G0 exp( t/λ) − Creep γ(t)=γ0(1 + t/λ) 0 0 J(t)=Js (1 + t/λ)=Js + t/η 2 G0(ωλ) Oscillatory Shear G (ω)=ωλG”(ω)= 0 1+(ωλ)2 The Maxwell Model is the simplest model of a VISCOELASTIC LIQUID. -
Multiband Homogenization of Metamaterials in Real-Space
Submitted preprint. 1 Multiband Homogenization of Metamaterials in Real-Space: 2 Higher-Order Nonlocal Models and Scattering at External Surfaces 1, ∗ 2, y 1, 3, 4, z 3 Kshiteej Deshmukh, Timothy Breitzman, and Kaushik Dayal 1 4 Department of Civil and Environmental Engineering, Carnegie Mellon University 2 5 Air Force Research Laboratory 3 6 Center for Nonlinear Analysis, Department of Mathematical Sciences, Carnegie Mellon University 4 7 Department of Materials Science and Engineering, Carnegie Mellon University 8 (Dated: March 3, 2021) Dynamic homogenization of periodic metamaterials typically provides the dispersion relations as the end-point. This work goes further to invert the dispersion relation and develop the approximate macroscopic homogenized equation with constant coefficients posed in space and time. The homoge- nized equation can be used to solve initial-boundary-value problems posed on arbitrary non-periodic macroscale geometries with macroscopic heterogeneity, such as bodies composed of several different metamaterials or with external boundaries. First, considering a single band, the dispersion relation is approximated in terms of rational functions, enabling the inversion to real space. The homogenized equation contains strain gradients as well as spatial derivatives of the inertial term. Considering a boundary between a metamaterial and a homogeneous material, the higher-order space derivatives lead to additional continuity conditions. The higher-order homogenized equation and the continuity conditions provide predictions of wave scattering in 1-d and 2-d that match well with the exact fine-scale solution; compared to alternative approaches, they provide a single equation that is valid over a broad range of frequencies, are easy to apply, and are much faster to compute. -
Stress, Cauchy's Equation and the Navier-Stokes Equations
Chapter 3 Stress, Cauchy’s equation and the Navier-Stokes equations 3.1 The concept of traction/stress • Consider the volume of fluid shown in the left half of Fig. 3.1. The volume of fluid is subjected to distributed external forces (e.g. shear stresses, pressures etc.). Let ∆F be the resultant force acting on a small surface element ∆S with outer unit normal n, then the traction vector t is defined as: ∆F t = lim (3.1) ∆S→0 ∆S ∆F n ∆F ∆ S ∆ S n Figure 3.1: Sketch illustrating traction and stress. • The right half of Fig. 3.1 illustrates the concept of an (internal) stress t which represents the traction exerted by one half of the fluid volume onto the other half across a ficticious cut (along a plane with outer unit normal n) through the volume. 3.2 The stress tensor • The stress vector t depends on the spatial position in the body and on the orientation of the plane (characterised by its outer unit normal n) along which the volume of fluid is cut: ti = τij nj , (3.2) where τij = τji is the symmetric stress tensor. • On an infinitesimal block of fluid whose faces are parallel to the axes, the component τij of the stress tensor represents the traction component in the positive i-direction on the face xj = const. whose outer normal points in the positive j-direction (see Fig. 3.2). 6 MATH35001 Viscous Fluid Flow: Stress, Cauchy’s equation and the Navier-Stokes equations 7 x3 x3 τ33 τ22 τ τ11 12 τ21 τ τ 13 23 τ τ 32τ 31 τ 31 32 τ τ τ 23 13 τ21 τ τ τ 11 12 22 33 x1 x2 x1 x2 Figure 3.2: Sketch illustrating the components of the stress tensor. -
The Dynamics of Parallel-Plate and Cone–Plate Flows
The dynamics of parallel-plate and cone– plate flows Cite as: Phys. Fluids 33, 023102 (2021); https://doi.org/10.1063/5.0036980 Submitted: 10 November 2020 . Accepted: 09 December 2020 . Published Online: 18 February 2021 Anand U. Oza, and David C. Venerus COLLECTIONS This paper was selected as Featured Phys. Fluids 33, 023102 (2021); https://doi.org/10.1063/5.0036980 33, 023102 © 2021 Author(s). Physics of Fluids ARTICLE scitation.org/journal/phf The dynamics of parallel-plate and cone–plate flows Cite as: Phys. Fluids 33, 023102 (2021); doi: 10.1063/5.0036980 Submitted: 10 November 2020 . Accepted: 9 December 2020 . Published Online: 18 February 2021 Anand U. Oza1 and David C. Venerus2,a) AFFILIATIONS 1Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA 2Department of Chemical and Materials Engineering, New Jersey Institute of Technology, Newark, New Jersey 07102, USA a)Author to whom correspondence should be addressed: [email protected] ABSTRACT Rotational rheometers are the most commonly used devices to investigate the rheological behavior of liquids in shear flows. These devices are used to measure rheological properties of both Newtonian and non-Newtonian, or complex, fluids. Two of the most widely used geome- tries are flow between parallel plates and flow between a cone and a plate. A time-dependent rotation of the plate or cone is often used to study the time-dependent response of the fluid. In practice, the time dependence of the flow field is ignored, that is, a steady-state velocity field is assumed to exist throughout the measurement.