Brief Review on Taylor-Couette flow of shear-thinning fluids
Circular Couette flow of a shear-thinning fluid is mainly characterized by a viscosity stratification in the annular space, which is more significant as the shear-thinning eﬀects are stronger and the annular space is wider. With increasing shear-thinning eﬀects, the shear rate increases at the inner wall and decreases at the outer one. Furthermore, the nonlinear variation of the viscosity with the shear rate introduces at the linear level an anisotropy in the deviatoric tensor associated to the perturbation. This latter point is discussed in section 2.2.4.
The mechanism of instability of CCF of shear-thinning fluids is the same as for a Newtonian fluid and results in axisymmetric counter rotating vortices separated by radial inflow and outflow jets of angular momentum emanating from the fluid layers adjacent to the cylinders’ wall. However, the critical conditions are diﬀerent because of the radial viscosity stratification and the modification of the azimuthal velocity profile. In the case where the inner cylinder is rotating and the outer one is at rest, the critical Reynolds and axial wave numbers are given in the literature for power-law and Carreau fluids, for wide and narrow annular spaces, see for instance Agbessi et al  and Alibenyahia et al  and the references therein. When both the inner and the outer cylinders are rotating, the critical conditions were determined by Agbessi et al  for a narrow and a wide annular space.
It is shown that when the Reynolds number is defined using the inner wall-shear viscosity, the shear-thinning delays the appearance of Taylor vortices. It is explained that this delay is due to the reduction of the energy exchange between the base flow and the perturbation. A radically diﬀerent conclusion may be reached if one uses the zero-shear viscosity of the fluid as viscosity scale. In the narrow gap-limit and weakly shear-thinning behavior of the fluid, Li & Khayat  found that the critical Reynolds number defined with the zero-shear viscosity becomes lower as shear-thinning eﬀects increase. Similar tendency is observed when free (slip) boundary conditions are used [18, 19, 20]. Recently Masuda et al.  suggested to use an average viscosity weighted by the strain-rate squared. They found that the critical Reynolds number defined with this average viscosity is the same as for a Newtonian fluid. However, this result is limited only to a narrow annular space with a radius ratio η > 0.7.
From experimental point of view, Escudier et al.  suggested to determine the critical conditions by focusing on the development of the axial velocity component, near the inner wall at a radial position r such (R2 − r)/(R2 − R1) = 0.8.
Sinevic et al.  measured the torque acting on the inner cylinder for three shear-thinning fluids described by a power-law model (np = 0.4, 0.45 and 0.57). They found that in the Taylor-vortex flow region, the power number P o behaves as P o ∝ Re−w0.7, where, Rew is the Reynolds number defined with the inner wall shear-viscosity. Concerning the flow structure, for a wide gap, it is shown theoretically [16, 15] and experimentally  that with increasing shear-thinning eﬀects, the vortex eye is shifted toward the inner cylinder, because of the viscosity stratification : the viscosity increases from the inner cylinder to the outer one. Escudier et al.  investigated the flow structure in a Taylor-Couette geometry with a radius ratio of 0.5. Axial and tangential velocity measurements were made using Laser Doppler Anemometry for a 0.15% aquesous solution of xanthan gum, whose rheological behavior is described by a power-law model with a shear-thinning index np ≈ 0.45. The results show an axial shift of the vortices towards the radial outflow boundary slightly more pronounced for a shear-thinning fluid than for a Newtonian fluid. Except for this issue dealing with the position of the vortex, results are very sparse. For instance, there is no indication on the influence of shear-thinning eﬀects on the strength of the radial outflow and radial inflow, nor on the azimuthal streaks in outflow and inflow regions, nor on the modification of the viscosity field by Taylor vortices particularly in a wide annular space and strong shear-thinning eﬀects. It is clear that a more clear understanding and characterization of supercritical Taylor vortex flow of a shear-thinning fluid is needed.
To our best knowledge, there are no theoretical nor numerical studies on shear-thinning eﬀects in Taylor vortex flow structure.
Modification of the fundamental mode at cubic order : Cubic Lan-dau constant
The nonlinear interactions between the fundamental, the second harmonic and the modification of the base flow lead to a cubic correction O A3 to the fundamental mode. The first Landau coeﬃcient g1 accounts for these nonlinear interactions on the fundamental mode. The modification of the fundamental mode at order |A|2A is governed by (2.70) and (2.71) with m = n = 1, i.e.
L113F13 + L213V13 = [NI1]|A|2AE1 + [NV 1]|A|2AE1 − g1S1 F11 , (2.80).
L313F13 + L413V13 = [NI2]|A|2AE1 + [NV 2]|A|2AE1 − g1V11 . (2.81).
The boundary conditions are F13 = DF13 = V13 = 0 at r = R1, R2 . (2.82).
The system (2.80) and (2.81) can be written L · X13 = −g1M · X11 + N I + N V with X13 = (F13, V13) . (2.83).
At the critical conditions, (2.83) has a non-trivial solution if the Fredholm solvability condition is satisfied, i.e. orthogonality of the inhomogeneous part of (2.83) to the null-space of the adjoint operator of L. The cubic Landau constant is then readily obtained, g1 = gI + gV = gI + gI + gV + gV + gV , (2.84).
Validation by computing higher-order Landau constants
Figures 2.15 and 2.16 indicate that the kinetic energy of the perturbation and the pseudo-Nusselt number N u∗ decrease with increasing shear-thinning eﬀects. This result was obtained by truncating the series (2.68) to the first Landau constant, at cubic order in A. For a significant deviation from the critical condition, terms of higher order become large and should be taken into account. A weakly nonlinear expansion was then carried out up to seventh-order in amplitude. As in the Newtonian case, for shear-thinning fluids, the Landau constants are of the same sign and increase very fast, i.e. A has to be very small to satisfy the assumption of the weakly nonlinearity. This increase is stronger with increasing shear-thinning eﬀects as it is shown by the data in table 2.5 (Appendix 2.C). The representation of the equilibrium amplitude versus ǫ at figure 2.27 (Appendix 2.C), at the third-, fifth- and seventh-order shows that the correction brought by adding a new term decreases.
Taylor vortex flow in Newtonian fluids
As demonstrated by Taylor , as inertial eﬀects start to dominate over viscous ones, CCF becomes unstable giving rise to the Taylor vortex flow (TVF) characterized by sta-tionary counter-rotating vortices stacked along the axial direction. The onset of instability can be parameterized by the Reynolds number (3.5). Equivalently, the Reynolds number can be used. In most common configurations the outer cylinder is at rest and the inner one rotates with angular velocity Ω1. This defines the velocity scale Uref = R1Ω1. (3.4).
In an incompressible Newtonian flow with uniform viscosity µ and density ρ where kine-matic viscosity ν = µ/ρ is well defined, the standard definition of the Reynolds number using the gap d as length scale results in : Re = R1Ω1d/ν. (3.5).
The critical Reynolds number of the onset of TVF will be denoted Rec. The canonical configuration, most convenient for theoretical study, consists in considering a small gap η → 1 and an infinite aspect ratio L → ∞ (Taylor ). However, in practical cases, in particular, experimental implementations, large gaps and finite aspect ratios occur. The values of the critical Reynolds for several values of the radius ratio are available in the literature (see for instance Table 1 in DiPrima et al.). Approximate expressions of Rec(η) can be found in Esser & Grossmann  and Dutcher & Muller . Concerning the influence of the aspect ratio, Cole  has shown experimentally that there is practically no eﬀect of the annulus length on the critical Reynolds number for an aspect ratio L as low as 8. The interaction between the endwall boundary layer and the centrifugal Taylor instability has been studied numerically by Czarny et al. (2003) for a particularly low value of aspect ratio L = 6.
Stability of Taylor vortices in Newtonian fluids
For small gap width, the range of Reynolds number Re, in which the axisymmetric vortices remain stable is small. For instance, in the experiments of Cole  where η = 0.95 and L = 60, the Taylor vortex flow becomes unstable with respect to azimuthal disturbances at Res = 1.05Rec. A bifurcation from TVF to the wavy vortex flow (WVF) is observed. The structure of the WVF and the doubly periodic (axially and in time) motion was first studied experimentally by Coles (1965). Unlike the transition to TVF, the Reynolds number Res of the onset of the secondary instability yielding WVF depends significantly on the aspect ratio L. It increases strongly when the aspect ratio L is reduced below 40, as it has been shown experimentally by Cole . For L ≥ 40, with η > 0.89, Res changes by only few percent. The azimuthal wavenumber varies on a much wider range. It ranges from 2 to 8 depending on the conditions by which the second transition is approached (Coles , Cole , Mullin , Dutcher & Muller . The non-uniqueness of this flow has been also observed through the existence of hysteresis phenomena Coles . In other words, multiple stable flow states could be reached for a given Reynolds number. Concerning the physical mechanisms that drive the transition from TVF to WVF, they were discussed by Martinand et al.  and Dessup et al.. For η < 0.75, Res increases rapidly as η decreases ). For instance, for η = 0.67, Res ≈ 5Rec . This tendency is in agreement with the experimental results of Snyder & Lambert, Meincker & Egbers and King et al.. A recent direct numerical simulation by Razzak et al. in a wide gap setup η = 0.5 yielded Res ≈ 8.45Rec. In their study, a four wavelengths fluid column is considered (L = 4 Λ = 7.944) with periodic boundary conditions in the axial direction. For this relatively small aspect ratio, Razzak et al.  evidenced an intermediate step between TVF and WVF in the interval ≤ Re ≤ 8.45 Rec. They found that the flow becomes non-axisymmetric with a strong azimuthal wave in the inflow region (inward oriented flow of the TVF vortex array) as compared to the outflow region. Three decades earlier, the linear stability of the Taylor vortices was investigated by Jones  for the same radius ratio η = 0.5. The axisymmetric solution was determined using a Fourier expansion in the axial direction with a period of one or two axial wavelengths, and a Chebychev polynomials in the radial direction. He found that the results depend on the axial wavelength Λ selected. For Λ < 2 (1.6 and 1.7 in the table 1 of his paper ), Jones  detected a wavy outflow boundary (WOB) mode at Re ≈ 5 Rec. In this mode, the oscillation amplitude is localized in the outflow boundary jet and adjacent outflow boundaries jets oscillate in antiphase, i.e. the flow is axial subharmonic with respect to the period of Taylor vortices. If Λ > 2 a direct transition to wavy vortex flow is observed. Still earlier, Lorenzen et al.  observed experimentally for η around 0.5 a transition from TVF to WOB mode when the axial wavelength is less than 2 (the size of one vortex is less that the gap width). In their experiments, the number of vortices was kept constant as L is varied so that the size of individual vortices varied.
Hence for a wide gap, η around 0.5, wavy modes diﬀerent from the conventional wavy vortex flow are obtained numerically and experimentally. The type of wavy mode observed is probably very sensitive to the aspect ratio, the size of vortices and may be also to the type of boundary conditions. Furthermore, the wavy mode obtained by Razzak et al.  was not predicted by the linear stability analysis done by Jones . Therefore, we believe that additional experimental or numerical data are needed for a wide gap geometry. Concerning, the eﬀect of endwalls on the wavy vortex flow, it has been shown numerically by Czarny et al. , that this eﬀect does not penetrate far from the endwall. The waviness is already present one or two vortices far from the endwall.
Table of contents :
1.1 Instabilit´e du r´egime TVF
1.1.1 Instabilit´e vis-`a-vis de perturbations azimuthales : bifurcation vers le r´egime WVF
1.1.2 Instabilit´e vis-`a-vis de perturbations axisym´etriques
1.2 Ecoulement de Taylor-Couette pour des fluides non-Newtoniens
1.2.1 Influence du comportement visco´elastique
1.2.2 Influence du comportement rh´eofluidifiant sur la stabilit´e d’un ´ecoulement de Taylor-Couette
1.3 Objectifs et m´ethodologie
1.4 Organisation du manuscrit
2 Taylor-vortex flow in shear-thinning fluids
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.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
3.1.1 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
3.7 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
4 Secondary instabilities in Taylor Couette flow of shear-thinning fluids
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
4.4 Weakly nonlinear analysis
4.4.1 Multiple scales method
4.4.2 Derivation of Ginzburg Landau equation
4.4.3 Results and discussion
4.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