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Fluid Inertia and End Effects in Rheometer Flows

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Fluid Inertia and End Effects in Rheometer Flows 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
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