Oldroyd Viscoelastic Model Lecture Notes
|
|
- Beatrice Francis
- 6 years ago
- Views:
Transcription
1 Oldroyd Viscoelastic Model Lecture Notes Drew Wollman Portland State University Maseeh College of Engineering and Computer Science Department of Mechanical and Materials Engineering ME 510: Non-Newtonian Fluids Dr. Derek Tretheway February 16, 2011
2 Putting the Oldroyd Model Into Context Linear Models Maxwell Model τ - stress tensor λ 1 - relaxation time η 0 - zero-strain rate viscosity γ - shear-rate tensor τ + λ 1 = η 0 γ 1 We can see a clear linear relationship between τ and γ in the equation 1. See Zach Wilson s notes for details. Jeffrey Model If we add additional linear relationships, i.e. the time derivative of γ, then we arrive at the Jeffrey Model. τ + λ 1 = η γ 0 γ + λ 2 2 λ 2 - retardation time Limitations For flows where λ max 1 the linear models can not describe shear-rate dependence of viscosity. non-linear effects can not be described by linear model normal stresses if small-strain phenomena require large displacement gradients to manifest them the linear models will not work in steady shear free flows the model produces an infinite elongational viscosities This motivates non-linear models. Quasi-linear Models Before we can introduce Non-linear models, there are a few building blocks we need to acquire. Convected Time Derivative For the convected time derivative of the stress tensor, τ, we will use the symbol τ 1. If τ is symmetric, then equation 3 becomes: τ 1 = Dτ { Dt v T τ + τ v 3 τ 1 = Dτ Dt {τ vt {τ v 4 2 of 7
3 Retarded-Motion Expansion Asymptotic expansion of τ used to describe small departures from Newtonian model. We could use the series expansion [ γ { 1 τ = b 1 γ 1 + b 2 + b 1 1 γ 1 γ 1 +b 3 2 γ b 1 2 { γ 1 γ However, this expansion does not lend itself to modeling certain flow very well. Also, we know from continuum mechanics, that the partial time derivative of γ 1 should not be in constitutive equations. Only Convected time derivative of γ 1 should appear in constitutive equations. We, therefore, need a more suitable expansion. Assuming: And: incompressible fluid stress tensor is symmetric stress tensor can be expresses as polynomial in γ n arranging terms of polynomial in increasing order collecting terms of equal order We end up with the Retarded-Motion Expansion. τ = [b 1 γ 1 + b 2 γ 2 + b 1 1 {γ 1 γ 1 +b 3 γ 3 + b 1 2 { γ 1 γ 2 + γ 2 γ 1 + b 1 : 1 1 γ 1 : γ 1 γ b n - retarded-motion constant material properties these are dependent on the fluid, flow, and model What is important here is understanding vocabulary. Ordered fluid model names come from truncating the Retarded-Motion Expansion. If we keep only the: first order terms, Oγ 1, then we have the Newtonian model again with b 1 = viscosity. second order terms, Oγ 2, then we have an incompressible second-order fluid third order terms, Oγ 3, then we have an incompressible third-order fluid You will not often see a higher order then third-order fluids Limitations of Retarded-Motion Expansion: Can predict a shear stress vs shear rate maximum Can predict a negative viscosity or elongational viscosity Can predict an unstable behavior for second-order fluids at rest 3 of 7 ] ]
4 May require additional boundary conditions - if perturbation expansion is used, this is not the cases Can not qualitatively describe entire flows i.e. stress relaxation experiment Plus more... The point is to reinforce the small De restriction. This comes out in the scaling Not done in this lecture.. Convected Jeffrey Model Oldroyd-B If we plug equation 4 into the Jeffrey model equation 2 we get τ 1 n n γ γ n+1 Jeffrey Model Convected Jeffrey Model Making the substitutions yields the Convected Jeffrey Model: This is also known as Oldroyd-B Model τ + λ 1 τ 1 = η 0 γ 1 + λ 2 γ 2 5 η 0 - zero shear-rate viscosity λ 1 - relaxation time λ 2 - retardation time kinematic tensors defined previously in retarded-motion expansion. One cool thing about the Convected Jeffrey Model is that it contains other models as special cases. So if 1. λ 2 = 0 then equation 5 transforms into the Maxwell Model equation 1 2. λ 1 = 0 then equation 5 transforms into a second order fluid with a normal stress coefficient 0 3. λ 1 = λ 2 then equation 5 transforms into the Newtonian Model with viscosity, η 0 Non-Linear Differential Models Recall For a Newtonian fluid: τ = η γ 6 η is a function of scalar γ The exact function is unknown but there are models. One famous model is the power law, η = m γ n 1. While there are other models, the point is that for a Newtonian fluid, η = f γ. If we apply the same principle for non-newtonian fluid, we can say: 1 γ = γ 2 1 : γ of 7
5 If we plug equation 7 into 5 to get a new model called the White-Metznel Model. Benefits G - constant modulus Simple τ + Produces reasonable solutions for shear-rate dependent viscosity Can handle the first-order normal stress coefficient Can be used to model fast time dependent motions η γ G τ 1 = η γγ 1 8 It is good at ballparking interaction of shear thinning and memory on flow fields Shortcomings Poor model of fast time dependent motions in steady shear free flows the model produces an infinite elongational viscosities same as in Maxwell Model It is also important to note that instead of useing equation 7 we could have also used other models ie Phan-Thien-Taner or FENE-P but those are not within the scope of this lecture. * * * Oldroyd Model Fits Right Here.* * * Non-Linear Integral Models While there are many Integral Models in the literature, the details of each fall outside the scope of this lecture. That said, to name a few: Lodge K-BKZ Rivlin-Sawyers Wagner Doi-Edwards Oldroyd Models τ + λ 1 τ {{ 2 λ 3 term I {γ 1 τ + τ γ λ 5 tr τ γ λ 6 τ : γ 1 δ { = η 0 [γ 1 + λ 2 γ 2 +λ 4 γ 1 γ ] 9 {{ 2 λ 7 γ 1 : γ 1 δ term II Taking a closer look we may recognize that terms I and II are the terms of the convected Jeffreys model. 5 of 7
6 Restrictions on Oldroyd Model 1. λ 1 > λ 2 > 0 in order to ensure η decreases with increasing ω. 2. σ 1 > σ 2 >... σ i > 0 to ensure viscosity is generally a monotone decreasing function of γ σ i = λ 1 λ 3 + λ 5 + λ 1+2 λ 1 λ 3 λ 5 + λ i+5 λ1 λ λ 5 3. for steady shear flow, σ σ 1 to ensure τ xy is a monotone increasing function of γ 4. σ 1 σ 2 < λ 1 λ 1 λ 2 to ensure that η curve is above η curve where η γ and η ω are plotted together λ 5 + λ [4λ2 6 11λ 5 λ 6 + 4λ 2 3] 1/2 < λ 1 λ 3 < 2 3 λ 5 + λ [4λ2 6 11λ 5 λ 6 + 4λ 2 3] 1/2 to ensure that elongational viscosity is bounded Benifits of Oldroyd Model More variety in rheological response can be described for Oldroyd then Jeffreys. Wider range of of properties can be correctly described for Oldroyd then White-Wetzner. Easier to solve analytically then White-Wetzner because of it s form. Like with the generalized Maxwell equation, more terms of λ i yield a more accurate result. Limitations of Oldroyd Model Strong presence of singularities in η 1 and η 2 for ɛ < 0 and 0 < b < 1. No reason why a constitutive equation must be linear in stress. This modivates the Giesekus Model which is outside the scope of this lecture. 6 of 7
7 Summary Name Newtonian Maxwell Jeffrey Convected Jeffrey Oldroyd B White-Metznel Oldroyd Model τ = η γ τ + λ 1 = η 0 γ τ + λ 1 = η 0 γ + λ 2 γ τ + λ 1 τ 1 = η 0 γ 1 + λ 2 γ 2 η γ τ + G τ 1 = η γγ 1 τ + λ 1 τ λ 3 {γ 1 τ + τ γ λ 5 tr τ γ λ 6 τ : γ 1 δ = [ { η 0 γ 1 + λ 2 γ 2 + λ 4 γ 1 γ ] 2 λ 7 γ 1 : γ 1 δ Type Linear Linear Linear Quisi-linear Non-Linear Differential Non-Linear Differential Nomenclature Latin Letters b n - retarded-motion constant G - constant modulus Greek Letters γ - shear-rate tensor ɛ - extensional rate η 1 - dynamic viscosity η 2 - η 0 - zero-strain rate viscosity λ 1 - relaxation time λ 2 - retardation time τ - stress tensor 7 of 7
Lecture 2: Constitutive Relations
Lecture 2: Constitutive Relations E. J. Hinch 1 Introduction This lecture discusses equations of motion for non-newtonian fluids. Any fluid must satisfy conservation of momentum ρ Du = p + σ + ρg (1) Dt
More information5 The Oldroyd-B fluid
5 The Oldroyd-B fluid Last time we started from a microscopic dumbbell with a linear entropic spring, and derived the Oldroyd-B equations: A u = u ρ + u u = σ 2 pi + η u + u 3 + u A A u u A = τ Note that
More informationEVALUATION OF NONLINEAR DIFFERENTIAL MODELS FOR THE SIMULATION OF POLYMER MELTS
1 th Fall Rubber Colloquium EVALUATION OF NONLINEAR DIFFERENTIAL MODELS FOR THE SIMULATION OF POLYMER MELTS Jochen Kroll, Stefan Turek, Patrick Westervoß Institute of Applied Mathematics (LS III), TU Dortmund
More informationHEAT TRANSFER OF SIMPLIFIED PHAN-THIEN TANNER FLUIDS IN PIPES AND CHANNELS
HEAT TRANSFER OF SIMPLIFIED PHAN-THIEN TANNER FLUIDS IN PIPES AND CHANNELS Paulo J. Oliveira Departamento de Engenharia Electromecânica, Universidade da Beira Interior Rua Marquês D'Ávila e Bolama, 600
More informationvs. Chapter 4: Standard Flows Chapter 4: Standard Flows for Rheology shear elongation 2/1/2016 CM4650 Lectures 1-3: Intro, Mathematical Review
CM465 Lectures -3: Intro, Mathematical //6 Chapter 4: Standard Flows CM465 Polymer Rheology Michigan Tech Newtonian fluids: vs. non-newtonian fluids: How can we investigate non-newtonian behavior? CONSTANT
More informationOn the congruence of some network and pom-pom models
Korea-Australia Rheology Journal Vol 8, No, March 2006 pp 9-4 On the congruence of some network and pom-pom models Roger I Tanner* School of Aerospace, Mechanical and Mechatronic Engineering, University
More informationMadrid, 8-9 julio 2013
VI CURSO DE INTRODUCCION A LA REOLOGÍA Madrid, 8-9 julio 2013 NON-LINEAR VISCOELASTICITY Prof. Dr. Críspulo Gallegos Dpto. Ingeniería Química. Universidad de Huelva & Institute of Non-Newtonian Fluid Mechanics
More informationNon-linear Viscoelasticity FINITE STRAIN EFFECTS IN SOLIDS
FINITE STRAIN EFFECTS IN SOLIDS Consider an elastic solid in shear: Shear Stress σ(γ) = Gγ If we apply a shear in the opposite direction: Shear Stress σ( γ) = Gγ = σ(γ) This means that the shear stress
More informationFEM techniques for nonlinear fluids
FEM techniques for nonlinear fluids From non-isothermal, pressure and shear dependent viscosity models to viscoelastic flow A. Ouazzi, H. Damanik, S. Turek Institute of Applied Mathematics, LS III, TU
More informationRheology and Constitutive Equations. Rheology = Greek verb to flow. Rheology is the study of the flow and deformation of materials.
Rheology and Constitutive Equations Rheology = Greek verb to flow Rheology is the study of the flow and deformation of materials. The focus of rheology is primarily on the study of fundamental, or constitutive,
More informationOutline. Motivation Governing equations and numerical methods Results: Discussion:
Bifurcation phenomena in strong extensional flows (in a cross-slot geometry) F. A. Cruz 1,*, R. J. Poole 2, F. T. Pinho 3, P.J. Oliveira 4, M. A. Alves 1 1 Departamento de Engenharia Química, CEFT, Faculdade
More informationExcerpt from the Proceedings of the COMSOL Users Conference 2006 Boston
Using Comsol Multiphysics to Model Viscoelastic Fluid Flow Bruce A. Finlayson, Professor Emeritus Department of Chemical Engineering University of Washington, Seattle, WA 98195-1750 finlayson@cheme.washington.edu
More informationModelling the Rheology of Semi-Concentrated Polymeric Composites
THALES Project No 1188 Modelling the Rheology of Semi-Concentrated Polymeric Composites Research Team Evan Mitsoulis (PI), Professor, NTUA, Greece Costas Papoulias (Research Student), NTUA, Greece Souzanna
More informationChapter 3: Newtonian Fluid Mechanics. Molecular Forces (contact) this is the tough one. choose a surface through P
// Molecular Constitutive Modeling Begin with a picture (model) of the kind of material that interests you Derive how stress is produced by deformation of that picture Write the stress as a function of
More informationChapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature
Chapter 1 Continuum mechanics review We will assume some familiarity with continuum mechanics as discussed in the context of an introductory geodynamics course; a good reference for such problems is Turcotte
More informationChristel Hohenegger A simple model for ketchup-like liquid, its numerical challenges and limitations April 7, 2011
Notes by: Andy Thaler Christel Hohenegger A simple model for ketchup-like liquid, its numerical challenges and limitations April 7, 2011 Many complex fluids are shear-thinning. Such a fluid has a shear
More informationQUIZ 2 OPEN QUIZ WHEN TOLD THERE ARE TWO PROBLEMS OF EQUAL WEIGHT. Please answer each question in a SEPARATE book
2.341J MACROMOLECULAR HYDRODYNAMICS Spring 2012 QUIZ 2 OPEN QUIZ WHEN TOLD THERE ARE TWO PROBLEMS OF EQUAL WEIGHT Please answer each question in a SEPARATE book You may use the course textbook (DPL) and
More informationLinear viscoelastic behavior
Harvard-MIT Division of Health Sciences and Technology HST.523J: Cell-Matrix Mechanics Prof. Ioannis Yannas Linear viscoelastic behavior 1. The constitutive equation depends on load history. 2. Diagnostic
More informationStability analysis of constitutive equations for polymer melts in viscometric flows
J. Non-Newtonian Fluid Mech. 103 (2002) 221 250 Stability analysis of constitutive equations for polymer melts in viscometric flows Anne M. Grillet 1, Arjen C.B. Bogaerds, Gerrit W.M. Peters, Frank P.T.
More informationMechanical properties of polymers: an overview. Suryasarathi Bose Dept. of Materials Engineering, IISc, Bangalore
Mechanical properties of polymers: an overview Suryasarathi Bose Dept. of Materials Engineering, IISc, Bangalore UGC-NRCM Summer School on Mechanical Property Characterization- June 2012 Overview of polymer
More informationStability of two-layer viscoelastic plane Couette flow past a deformable solid layer
J. Non-Newtonian Fluid Mech. 117 (2004) 163 182 Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer V. Shankar Department of Chemical Engineering, Indian Institute of Technology,
More informationNumerical study of flow of Oldroyd-3-Constant fluids in a straight duct with square cross-section
Korea-Australia Rheology Journal Vol. 19, No. 2, August 2007 pp. 67-73 Numerical study of flow of Oldroyd-3-Constant fluids in a straight duct with square cross-section Mingkan Zhang, Xinrong Shen, Jianfeng
More informationViscoelastic Flows in Abrupt Contraction-Expansions
Viscoelastic Flows in Abrupt Contraction-Expansions I. Fluid Rheology extension. In this note (I of IV) we summarize the rheological properties of the test fluid in shear and The viscoelastic fluid consists
More informationLos Alamos. Nonlocal Models in Continuum Mechanics. Norman L. Johnson and Nhan Phan-Thien
LA-UR-93-3124 Approved for public release; distribution is unlimited. Title: Nonlocal Models in Continuum Mechanics Author(s): Norman L. Johnson and Nhan Phan-Thien Published in: Advanced Computational
More informationSimple constitutive models for linear and branched polymers
J. Non-Newtonian Fluid Mech. 116 (2003) 1 17 Simple constitutive models for linear and branched polymers Roger I. Tanner, Simin Nasseri School of Aerospace, Mechanical and Mechatronic Engineering, University
More informationChapter 1 Introduction
Chapter 1 Introduction This thesis is concerned with the behaviour of polymers in flow. Both polymers in solutions and polymer melts will be discussed. The field of research that studies the flow behaviour
More informationContinuum Mechanics and Theory of Materials
Peter Haupt Continuum Mechanics and Theory of Materials Translated from German by Joan A. Kurth Second Edition With 91 Figures, Springer Contents Introduction 1 1 Kinematics 7 1. 1 Material Bodies / 7
More informationRheology of Soft Materials. Rheology
Τ Thomas G. Mason Department of Chemistry and Biochemistry Department of Physics and Astronomy California NanoSystems Institute Τ γ 26 by Thomas G. Mason All rights reserved. γ (t) τ (t) γ τ Δt 2π t γ
More informationSUMMARY A STUDY OF VISCO-ELASTIC NON-NEWTONIAN FLUID FLOWS. where most of body fluids like blood and mucus are non-newtonian ones.
SUMMARY A STUDY OF VISCO-ELASTIC NON-NEWTONIAN FLUID FLOWS Non-Newtonian fluids abound in many aspects of life. They appear in nature, where most of body fluids like blood and mucus are non-newtonian ones.
More informationThis article was originally published in a journal published by Elsevier, and the attached copy is provided by Elsevier for the author s benefit and for the benefit of the author s institution, for non-commercial
More informationViscoelasticity. Basic Notions & Examples. Formalism for Linear Viscoelasticity. Simple Models & Mechanical Analogies. Non-linear behavior
Viscoelasticity Basic Notions & Examples Formalism for Linear Viscoelasticity Simple Models & Mechanical Analogies Non-linear behavior Viscoelastic Behavior Generic Viscoelasticity: exhibition of both
More informationFlow of Non-Newtonian Fluids in Porous Media
Flow of Non-Newtonian Fluids in Porous Media Taha Sochi 2010 University College London, Department of Physics & Astronomy, Gower Street, London, WC1E 6BT. Email: t.sochi@ucl.ac.uk. i Contents Contents
More informationRock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth
Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth References: Turcotte and Schubert, Geodynamics, Sections 2.1,-2.4, 2.7, 3.1-3.8, 6.1, 6.2, 6.8, 7.1-7.4. Jaeger and Cook, Fundamentals of
More informationIn the name of Allah the most beneficent the most merciful
In the name of Allah the most beneficent the most merciful Transient flows of Maxwell fluid with slip conditions Authors Dr. T. Hayat & Sahrish Zaib Introduction Non-Newtonian Newtonian fluid Newtonian
More informationMonolithic FEM multigrid techniques for the simulation of viscoelastic flow
Monolithic FEM multigrid techniques for the simulation of viscoelastic flow A. Ouazzi, H. Damanik, S. Turek, J. Hron Institute of Applied Mathematics, LS III, TU Dortmund http://www.featflow.de European
More informationPhan-Thien-Tanner Modeling of a Viscoelastic Fluid in the Stick-Slip Scenario. Andrew Griffith
Phan-Thien-Tanner Modeling of a Viscoelastic Fluid in the Stick-Slip Scenario Andrew Griffith Supervisor: Bruce A. Finlayson Department of Chemical Engineering University of Washington March 1, 7 Introduction
More informationSimulation Tests of the Constitutive Equation of a Nonlinear Viscoelastic Fluid
Simulation Tests of the Constitutive Equation of a Nonlinear Viscoelastic Fluid A. Czirák, Z. Kőkuti, G. Tóth-Molnár, G. Szabó University of Szeged, Hungary P. Ailer, L. Palkovics Kecskemét College, Hungary
More informationFlow of Non-Newtonian Fluids in Porous Media
Flow of Non-Newtonian Fluids in Porous Media TAHA SOCHI Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT Received 9 June 2010; revised 17 July 2010; accepted
More informationChapter 2 Rheological Models: Integral and Differential Representations
Chapter 2 Rheological Models: Integral and Differential Representations Viscoelastic relations may be expressed in both integral and differential forms. Integral forms are very general and appropriate
More informationExplaining and modelling the rheology of polymeric fluids with the kinetic theory
Explaining and modelling the rheology of polymeric fluids with the kinetic theory Dmitry Shogin University of Stavanger The National IOR Centre of Norway IOR Norway 2016 Workshop April 25, 2016 Overview
More informationConstitutive equation and damping function for entangled polymers
Korea-Australia Rheology Journal Vol. 11, No. 4, December 1999 pp.287-291 Constitutive equation and damping function for entangled polymers Kunihiro Osaki Institute for Chemical Research, Kyoto University
More informationContinuum Mechanics. Continuum Mechanics and Constitutive Equations
Continuum Mechanics Continuum Mechanics and Constitutive Equations Continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform
More informationNon-Newtonian Fluids and Finite Elements
Non-Newtonian Fluids and Finite Elements Janice Giudice Oxford University Computing Laboratory Keble College Talk Outline Motivating Industrial Process Multiple Extrusion of Pastes Governing Equations
More informationUniversity Graz / Austria Institut für Chemie Volker Ribitsch
University Graz / Austria Institut für Chemie Volker Ribitsch 1 Rheology Oscillatory experiments Dynamic experiments Deformation of materials under non-steady conditions in the linear viscoelastic range
More informationThe Polymers Tug Back
Tugging at Polymers in Turbulent Flow The Polymers Tug Back Jean-Luc Thiffeault http://plasma.ap.columbia.edu/ jeanluc Department of Applied Physics and Applied Mathematics Columbia University Tugging
More informationChapter 9: Differential Analysis
9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control
More informationDirect Simulation of the Motion of Solid Particles in Couette and Poiseuille Flows of Viscoelastic Fluids
Direct Simulation of the Motion of Solid Particles in Couette and Poiseuille Flows of Viscoelastic Fluids by P. Y. Huang 1, J. Feng 2, H. H. Hu 3 and D. D. Joseph 1 1 Department of Aerospace Engineering
More informationDept. Engineering, Mechanical Engineering, University of Liverpool Liverpool L69 3GH, UK,
R. J. Poole pt. Engineering, Mechanical Engineering, University of Liverpool Liverpool L69 3GH, UK, robpoole@liv.ac.uk M. A. Alves partamento de Engenharia uímica, CEFT, Faculdade de Engenharia da Universidade
More informationSplash singularity for a free-boundary incompressible viscoelastic fluid model
Splash singularity for a free-boundary incompressible viscoelastic fluid model Pierangelo Marcati (joint work with E.Di Iorio, S.Spirito at GSSI) Workshop 2016 Modeling Computation of Shocks Interfaces
More informationLes Houches School of Foam: Rheology of Complex Fluids
Les Houches School of Foam: Rheology of Complex Fluids Andrew Belmonte The W. G. Pritchard Laboratories Department of Mathematics, Penn State University 1 Fluid Dynamics (tossing a coin) Les Houches Winter
More informationThe Non-Linear Field Theories of Mechanics
С. Truesdell-W.Noll The Non-Linear Field Theories of Mechanics Second Edition with 28 Figures Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest Contents. The Non-Linear
More informationChapter 6 Molten State
Chapter 6 Molten State Rheology ( 流變學 ) study of flow and deformation of (liquid) fluids constitutive (stress-strain) relation of fluids shear flow shear rate ~ dγ/dt ~ velocity gradient dv 1 = dx 1 /dt
More informationMicrostructural studies for rheology. Chapter 7: Microstructural studies for rheology. Separation of length scales. Micro & macro views
Chapter 7: Microstructural studies for rheology Microstructural studies for rheology To calculate the flow of complex fluids, need governing equations, in particular, the constitutive equation relating
More informationChapter 9: Differential Analysis of Fluid Flow
of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known
More informationChapter 2: Fluid Dynamics Review
7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading
More informationDYNAMIC STABILITY OF NON-DILUTE FIBER SHEAR SUSPENSIONS
THERMAL SCIENCE, Year 2012, Vol. 16, No. 5, pp. 1551-1555 1551 DYNAMIC STABILITY OF NON-DILUTE FIBER SHEAR SUSPENSIONS by Zhan-Hong WAN a*, Zhen-Jiang YOU b, and Chang-Bin WANG c a Department of Ocean
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Thursday 1 June 2006 1.30 to 4.30 PAPER 76 NONLINEAR CONTINUUM MECHANICS Attempt FOUR questions. There are SIX questions in total. The questions carry equal weight. STATIONERY
More informationMathematical Models and Numerical Simulations for the Blood Flow in Large Vessels
Mathematical Models and Numerical Simulations for the Blood Flow in Large Vessels Balazs ALBERT 1 Titus PETRILA 2a Corresponding author 1 Babes-Bolyai University M. Kogalniceanu nr. 1 400084 Cluj-Napoca
More informationPore-Scale Modeling of Non-Newtonian Flow in Porous Media
Pore-Scale Modeling of Non-Newtonian Flow in Porous Media Taha Sochi A dissertation submitted to the Department of Earth Science and Engineering Imperial College London in fulfillment of the requirements
More informationModeling the Rheology and Orientation Distribution of Short Glass Fibers Suspended in Polymeric Fluids: Simple Shear Flow
Modeling the Rheology and Orientation Distribution of Short Glass Fibers Suspended in Polymeric Fluids: Simple Shear Flow Aaron P.R. berle, Donald G. Baird, and Peter Wapperom* Departments of Chemical
More informationMHA042 - Material mechanics: Duggafrågor
MHA042 - Material mechanics: Duggafrågor 1) For a static uniaxial bar problem at isothermal (Θ const.) conditions, state principle of energy conservation (first law of thermodynamics). On the basis of
More informationCM4650 Polymer Rheology
CM4650 Polymer Rheology CM4650 Polymer Rheology Michigan Tech Professor Faith A. Morrison Department of Chemical Engineering Michigan Technological University Text: Faith A. Morrison, Understanding Rheology
More informationPolymer Dynamics and Rheology
Polymer Dynamics and Rheology 1 Polymer Dynamics and Rheology Brownian motion Harmonic Oscillator Damped harmonic oscillator Elastic dumbbell model Boltzmann superposition principle Rubber elasticity and
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 2012-13 MODULE TITLE: Introduction to Fluids DURATION OF
More informationElements of Continuum Elasticity. David M. Parks Mechanics and Materials II February 25, 2004
Elements of Continuum Elasticity David M. Parks Mechanics and Materials II 2.002 February 25, 2004 Solid Mechanics in 3 Dimensions: stress/equilibrium, strain/displacement, and intro to linear elastic
More informationNumerical simulation of generalized second-grade fluids using a 1D hierarchical model
Proceedings of the 10th WEA International Confenrence on APPLIED MATHEMATIC, Dallas, Texas, UA, November 1-3, 2006 337 Numerical simulation of generalized second-grade fluids using a 1D hierarchical model
More informationReview of fluid dynamics
Chapter 2 Review of fluid dynamics 2.1 Preliminaries ome basic concepts: A fluid is a substance that deforms continuously under stress. A Material olume is a tagged region that moves with the fluid. Hence
More informationKINEMATICS OF CONTINUA
KINEMATICS OF CONTINUA Introduction Deformation of a continuum Configurations of a continuum Deformation mapping Descriptions of motion Material time derivative Velocity and acceleration Transformation
More informationInterfacial hoop stress and viscoelastic free surface flow instability. Michael D. Graham University of Wisconsin-Madison
Interfacial hoop stress and viscoelastic free surface flow instability Michael D. Graham University of Wisconsin-Madison Free surface instabilities of viscoelastic flows Eccentric cylinders (Varela-Lopez
More informationLecture 8: Tissue Mechanics
Computational Biology Group (CoBi), D-BSSE, ETHZ Lecture 8: Tissue Mechanics Prof Dagmar Iber, PhD DPhil MSc Computational Biology 2015/16 7. Mai 2016 2 / 57 Contents 1 Introduction to Elastic Materials
More informationA viscoelastic constitutive model for thixotropic yield stress fluids: asymptotic and numerical studies of extension
A viscoelastic constitutive model for thixotropic yield stress fluids: asymptotic and numerical studies of extension Holly Grant Dissertation submitted to the Faculty of the Virginia Polytechnic Institute
More informationNumerical simulation of large amplitude oscillatory shear of a high-density polyethylene melt using the MSF model
J. Non-Newtonian Fluid Mech. 130 (2005) 63 76 Numerical simulation of large amplitude oscillatory shear of a high-density polyethylene melt using the MSF model P. Wapperom a,, A. Leygue b, R. Keunings
More informationSOME THOUGHTS ON DIFFERENTIAL VISCOELASTIC MODELS AND THEIR NUMERICAL SOLUTION
CONFERENCE ON COMPLEX FLOWS OF COMPLEX FLUIDS University of Liverpool, UK, March 17-19, 2008 SOME THOUGHTS ON DIFFERENTIAL VISCOELASTIC MODELS AND THEIR NUMERICAL SOLUTION Paulo J. Oliveira Universidade
More informationStress Overshoot of Polymer Solutions at High Rates of Shear
Stress Overshoot of Polymer Solutions at High Rates of Shear K. OSAKI, T. INOUE, T. ISOMURA Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan Received 3 April 2000; revised
More informationA numerical approximation with IP/SUPG algorithm for P-T-T viscoelastic flows
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 6 (2016, 152 161 Research Article A numerical approximation with IP/SUPG algorithm for P-T-T viscoelastic flows Lei Hou a, Yunqing Feng a,, Lin
More informationA Multigrid LCR-FEM solver for viscoelastic fluids with application to problems with free surface
A Multigrid LCR-FEM solver for viscoelastic fluids with application to problems with free surface Damanik, H., Mierka, O., Ouazzi, A., Turek, S. (Lausanne, August 2013) Page 1 Motivation Polymer melts:
More informationPolymerization Technology Laboratory Course
Polymerization Technology Laboratory Course Viscometry/Rheometry Tasks 1. Comparison of the flow behavior of polystyrene- solution and dispersion systems 2. Determination of the flow behaviour of polyvinylalcohol
More informationControllability of Non-Newtonian Fluids under Homogeneous Extensional Flow
Applied Mathematical Sciences, Vol. 2, 2008, no. 43, 2145-2156 Controllability of Non-Newtonian Fluids under Homogeneous Extensional Flow Lynda Wilson, Hong Zhou 1, Wei Kang Department of Applied Mathematics
More informationChapter 2. General concepts. 2.1 The Navier-Stokes equations
Chapter 2 General concepts 2.1 The Navier-Stokes equations The Navier-Stokes equations model the fluid mechanics. This set of differential equations describes the motion of a fluid. In the present work
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationChapter 2 Constitutive Formulations
Chapter 2 Constitutive Formulations Abstract Controversy about the frame indifference principle, the concept of non-local continuum field theories, local constitutive formulations, differential constitutive
More informationarxiv: v1 [cs.na] 21 Jul 2010
DEVELOPMENT OF THREE DIMENSIONAL CONSTITUTIVE THEORIES BASED ON LOWER DIMENSIONAL EXPERIMENTAL DATA Satish Karra, K. R. Rajagopal, College Station arxiv:1007.3760v1 [cs.na] 1 Jul 010 Abstract. Most three
More informationBoundary Conditions in Fluid Mechanics
Boundary Conditions in Fluid Mechanics R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkson University The governing equations for the velocity and pressure fields are partial
More informationStudy of steady pipe and channel flows of a single-mode Phan-Thien Tanner fluid
J. Non-Newtonian Fluid Mech. 101 (2001) 55 76 Study of steady pipe and channel flows of a single-mode Phan-Thien Tanner fluid Manuel A. Alves a, Fernando T. Pinho b,, Paulo J. Oliveira c a Departamento
More informationPart III. Polymer Dynamics molecular models
Part III. Polymer Dynamics molecular models I. Unentangled polymer dynamics I.1 Diffusion of a small colloidal particle I.2 Diffusion of an unentangled polymer chain II. Entangled polymer dynamics II.1.
More informationViscoelastic Structures Mechanics of Growth and Aging
Viscoelastic Structures Mechanics of Growth and Aging Aleksey D. Drozdov Institute for Industrial Mathematics Ben-Gurion University of the Negev Be'ersheba, Israel ACADEMIC PRESS San Diego London Boston
More informationME FINITE ELEMENT ANALYSIS FORMULAS
ME 2353 - FINITE ELEMENT ANALYSIS FORMULAS UNIT I FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS 01. Global Equation for Force Vector, {F} = [K] {u} {F} = Global Force Vector [K] = Global Stiffness
More informationUnsteady Flow of a Newtonian Fluid in a Contracting and Expanding Pipe
Unsteady Flow of a Newtonian Fluid in a Contracting and Expanding Pipe T S L Radhika**, M B Srinivas, T Raja Rani*, A. Karthik BITS Pilani- Hyderabad campus, Hyderabad, Telangana, India. *MTC, Muscat,
More informationKinematics of fluid motion
Chapter 4 Kinematics of fluid motion 4.1 Elementary flow patterns Recall the discussion of flow patterns in Chapter 1. The equations for particle paths in a three-dimensional, steady fluid flow are dx
More informationCHAPTER 8 ENTROPY GENERATION AND TRANSPORT
CHAPTER 8 ENTROPY GENERATION AND TRANSPORT 8.1 CONVECTIVE FORM OF THE GIBBS EQUATION In this chapter we will address two questions. 1) How is Gibbs equation related to the energy conservation equation?
More informationENGR 292 Fluids and Thermodynamics
ENGR 292 Fluids and Thermodynamics Scott Li, Ph.D., P.Eng. Mechanical Engineering Technology Camosun College Jan.13, 2017 Review of Last Class Course Outline Class Information Contact Information, Website
More informationFAILURE AND RECOVERY OF ENTANGLED POLYMER MELTS IN ELONGATIONAL FLOW
FAILURE AND RECOVERY OF ENTANGLED POLYMER MELTS IN ELONGATIONAL FLOW Yogesh M. Joshi and Morton M. Denn Benjamin Levich Institute for Physico-Chemical Hydrodynamics and Department of Chemical Engineering
More informationA CONTINUUM MODEL OF PLATELET AGGREGATION: CLOSURE, COMPUTATIONAL METHODS AND SIMULATION
A CONTINUUM MODEL OF PLATELET AGGREGATION: CLOSURE, COMPUTATIONAL METHODS AND SIMULATION by Robert David Guy A dissertation submitted to the faculty of The University of Utah in partial fulfillment of
More informationFlow and Transport. c(s, t)s ds,
Flow and Transport 1. The Transport Equation We shall describe the transport of a dissolved chemical by water that is traveling with uniform velocity ν through a long thin tube G with uniform cross section
More informationViscoelasticity in mantle convection
Mgr. et Mgr. Vojtěch Patočka Supervised by: RNDr. Ondřej Čadek, CSc. [patocka@karel.troja.mff.cuni.cz] 5th June 2015 Content Method of implementation Model Method Testing the method Thermal convection
More informationSOME DYNAMICAL FEATURES OF THE TURBULENT FLOW OF A VISCOELASTIC FLUID FOR REDUCED DRAG
SOME DYNAMICAL FEATURES OF THE TURBULENT FLOW OF A VISCOELASTIC FLUID FOR REDUCED DRAG L. Thais Université de Lille Nord de France, USTL F9 Lille, France Laboratoire de Mécanique de Lille CNRS, UMR 817
More informationLecture 5. Rheology. Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm
Lecture 5 Rheology Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm WW Norton; unless noted otherwise Rheology is... the study of deformation and flow of
More informationDifferential relations for fluid flow
Differential relations for fluid flow In this approach, we apply basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of a flow
More informationV. Electrostatics Lecture 24: Diffuse Charge in Electrolytes
V. Electrostatics Lecture 24: Diffuse Charge in Electrolytes MIT Student 1. Poisson-Nernst-Planck Equations The Nernst-Planck Equation is a conservation of mass equation that describes the influence of
More informationPEAT SEISMOLOGY Lecture 2: Continuum mechanics
PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a
More information