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Shear Flow Instabilities in Viscoelastic Fluids

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Shear Flow Instabilities in Viscoelastic Fluids Shear Flow Instabilities in Visco elastic Fluids Helen Jane Wilson Clare College September A dissertation submitted to the University of Cambridge for the degree of Do ctor of Philosophy Preface The work describ ed in this dissertation was carried out b etween Octob er and September while the author was a research student in the Department of Applied Mathematics and Theoretical Physics University of Cambridge The dissertation is the result of my own work and includes nothing which is the outcome of work in collab oration Except where explicit reference is made to the work of others the contents of this dissertation are b elieved to b e original No part of this dissertation has b een or is b eing submitted for a degree diploma or other qualication at any other University It is a pleasure to record my thanks to my sup ervisor Dr John Rallison for all his help Noone could wish for a b etter sup ervisor I would also like to thank Professor John Hinch for useful discussions and Jeremy Bradley for pro ofreading This work was funded by a research studentship from the Engineering and Physical Sciences Research Council To my parents Contents Introduction Outline Shear rheology of nonNewtonian uids Dilute p olymer solution Boger uid Entangled p olymer melt Constitutive equations theoretical mo dels Governing equations Newtonian uid OldroydB mo del Upp erConvected Maxwell mo del Nonlinear dumbbell mo dels Other rate equations Retarded motion expansions nth order uid DoiEdwards mo del Mo dels with viscometric functions of Summary Observations of instabilities CONTENTS Interfacial instabilities in channel and pip e ows Viscometric ows having curved streamlines Instabilities in extensional ows More complex ows Extrudate distortions and fracture Constitutive instability Theoretical analyses of instability Instabilities in inertialess parallel shear ows Interfacial instabilities Flows with curved streamlines Stagnation p oint ow Scop e of this dissertation Stability of Channel Flows Introduction Geometry Constitutive equations Base solution Linear stability Temporal stability Weakly nonlinear stability Perturbation equations for channel ow Alternative form for the equations Existence of ro ots Numerical metho d CONTENTS Co extrusion Instability Introduction Statement of the problem Base state Perturbation equations The longwave limit k The shortwave limit k The eect of layer thickness on short waves The eect of p olymer concentration on short waves The eect of ow strength on short waves Numerical results Conclusions Continuously Stratied Fluid Introduction Geometry and constitutive mo del Perturbation ow Longwave asymptotics Outer ow Inner solution Numerical results App endix Longwave asymptotics Outer region Inner region Matching CONTENTS Concentration Stratication Introduction Base state Stability problem for two uids Governing equations Longwave limit Numerical results Single uid with varying concentration Sp ecic C prole Conclusions App endix Details of the expansion for long waves Channel Flows of a WhiteMetzner Fluid Introduction Governing equations for a channel ow Statement of the equations A mathematical diculty Long waves Order calculation Order k calculation Long waves and low n Short waves Analytic scalings for short waves Numerical results for short waves Intermediate wavelengths numerical results Scalings for highly shearthinning limit n CONTENTS Governing equation Physical meaning of the reduced equation Numerical results for small n Conclusions App endix Preliminary asymptotics for n Outer Wall layer Matching Criterion for Co extrusion Instability Introduction A mo del problem WhiteMetzner uid OldroydB uid Base state WhiteMetzner uid OldroydB uid Choice of parameters WhiteMetzner uid OldroydB uid Perturbation equations WhiteMetzner uid OldroydB uid Numerical results Conclusions CONTENTS Conclusions Chapter Introduction CHAPTER INTRODUCTION Outline Visco elastic liquids for example p olymeric melts and solutions have ow prop erties that are in part viscous and in part elastic The presence of elastic stresses can generate instabilities even in inertialess ow that do not arise in Newtonian liquids These are often describ ed as purely elastic instabilities The principal aim of this dissertation is to examine the class of such instabilities that arise in shear ows and the mechanism resp onsible for them We start with a review of the principal nonNewtonian features of uids of interest section then consider how these may b e captured in a con stitutive mo del section We then turn to a review of the exp erimental literature on purely elastic instabilities section and the current theoret ical understanding of these phenomena section An outline of the new instabilities discussed in this thesis is given in section Fuller descriptions of nonNewtonian phenomena are given by Bird et al and of constitutive mo dels by Larson and Bird Wiest We select here the items of greatest concern for the understanding of shear instabilities Shear rheology of nonNewtonian uids In steady simple shear having a velocity vector u y it may b e shown that any incompressible simple uid will have a stress tensor with in general four nonzero comp onents SHEAR RHEOLOGY OF NONNEWTONIAN FLUIDS N C B C B C B A N As a result of the incompressibility the stress is undetermined to within addition of an isotropic pressure term For a Newtonian uid N N and In a visco elastic uid however these three quantities may vary with In considering the dierent categories of exp erimentally relevant uid b elow we dene N and N We note in passing that b ecause of the nonzero N the stress tensor for a planar ow is not entirely twodimensional This may aect the stability of the ow to p erturbations in the third dimension For Newtonian uids Squires theorem states that the most unstable disturbance is always two dimensional and hence stability calculations may b e restricted to the two dimensional domain For a general nonNewtonian uid this may not b e true Dilute p olymer solution Boger uid For a dilute p olymer solution the rst normal stress dierence N is found to b e p ositive and may b e large compared with Both and show very little shearthinning at mo derate shearrates ie they are more or less constant However at higher shearrates there is some thinning often with The second normal stress dierence N is almost zero Figure sketches the shear b ehaviour of typical dilute p olymeric uids A basic way to start characterising visco elastic uids is by their zero shearrate shear viscosity and their longest relaxation time which we CHAPTER INTRODUCTION Shearrate Figure Generic b ehaviour of b oth the viscometric functions and against shear 1 rate for dilute solutions Both axes use a logarithmic scale There is a lowshearrate plateau followed by a region of weak shearthinning For a Boger uid the plateau would extend across a very large range of shearrates shall call This time can b e made large by using high molecular weight p olymers or a very viscous solvent Dilute solutions in which b oth of these metho ds are used to create a very long relaxation time are called Boger uids They use as solvent a very low molecular weight melt of the same p olymer as the solute These uids have a very long plateau of constant .
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