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Table of contents
1 Introduction
2 Taylor-vortex flow in shear-thinning fluids
2.1 Introduction
2.1.1 Brief Review on Taylor-Couette flow of shear-thinning fluids
2.1.2 Objectives, methodology and outline of the paper
2.2 Physical and mathematical model
2.2.1 Basic formulation
2.2.2 Carreau model
2.2.3 Base flow
2.2.4 Perturbation equations
2.3 Linear stability analysis
2.3.1 Direct mode
2.3.2 Characteristic time
2.3.3 Adjoint mode
2.4 Weakly nonlinear stability analysis
2.4.1 Principle and formulation
2.4.2 Solution procedure
2.5 Results and discussion
2.5.1 Modification of the base flow
2.5.2 Second harmonic mode
2.5.3 Modification of the fundamental mode at cubic order : Cubic Landau constant
2.5.4 Features of the perturbation near the threshold
2.5.5 Validation by computing higher-order Landau constants
2.5.6 Description of the flow field
2.5.7 Harmonics
2.6 Conclusion
2.A Validation
2.B Contribution of nonlinear inertial and nonlinear viscous terms
2.C Landau constants
2.D Flow structure and viscosity field for η = 0.9
3 Secondary instabilities in Taylor Couette flow of shear-thinning fluids
3.1 Introduction
3.1.1 Taylor vortex flow in Newtonian fluids
3.1.2 Stability of Taylor vortices in Newtonian fluids
3.1.3 Brief Review on Taylor-Couette flow of shear-thinning fluids
3.1.4 Stability of Taylor vortices in shear-thinning fluids
3.1.5 Objectives, methodology and outline of the paper
3.2 Mathematical formulation
3.3 Numerical method
3.3.1 Weak formulation
3.3.2 Time and space discretization
3.4 Validation of the numerical method
3.4.1 Comparison with linear theory
3.4.2 Comparison with weakly nonlinear theory
3.4.3 Comparison with literature in strongly nonlinear regime
3.5 Experimental setup, Fluids used and Protocol
3.5.1 Experimental cell
3.5.2 Flow visualization
3.5.3 Velocity measurements
3.5.4 Fluids used : preparation and rheology
3.5.5 Experimental protocol
3.6 Primary bifurcation : onset of Taylor vortex flow
3.6.1 Influence of the endwalls : Numerical results
3.6.2 PIV measurements
3.6.3 Visualization
3.6.4 Discussion
3.7 Secondary bifurcations
3.7.1 Newtonian fluid : Instability to travelling azimuthal wave mode
3.7.2 Shear-thinning fluids : creation and merging of vortices
3.7.3 Possible mechanisms of instability of the TVF regime
3.8 Conclusion
4 Secondary instabilities in Taylor Couette flow of shear-thinning fluids
4.1 Introduction
4.2 Problem formulation
4.2.1 Basic flow
4.2.2 Perturbation equations
4.2.3 Yield-stress terms
4.3 Linear stability analysis
4.3.1 Boundary conditions
4.3.2 Normal mode approach
4.3.3 Results
4.4 Weakly nonlinear analysis
4.4.1 Multiple scales method
4.4.2 Derivation of Ginzburg Landau equation
4.4.3 Results and discussion
4.5 Conclusion
4.A Operators and matrix coefficients
4.A.1 The operator C
4.A.2 The operator LI
4.A.3 The operator LV
4.B Boundary conditions
4.B.1 Case (III) of the base solution
4.B.2 Case (II) of the base solution
4.C Contribution of nonlinear inertial and nonlinear viscous terms
Bibliographie
