S Topayev

C Nouar

D Bernardin

A Neveu

S A Bahrani

Taylor-vortex flow in shear-thinning fluids

The present paper deals with the Taylor-Couette flow of shear-thinning fluids. It focuses on the first principles understanding the influence of the viscosity stratification and the nonlinear variation of the effective viscosity µ with the shear rate γ on the flow structure in the Taylor vortex flow regime. A wide gap configuration (η = 0.4) is mainly considered. A weakly nonlinear analysis, using the amplitude expansion method at high order is adopted as a first approach to study nonlinear effects. For the numerical computation, the shear-thinning behavior is described by the Carreau model. The rheological parameters are varied in a wide range. The results indicate that the flow field undergoes a significant change as shear-thinning effects increase. First, vortices are squeezed against the inner wall and the center of the patterns are shifted axially towards the radial outflow boundaries (z = 0, z/λ z = 1). This axial shift leads to increasing concentration of vorticity at these positions. The outflow becomes more stronger than the inflow and the inflow zone, where the vorticity is low, increases accordingly. Nevertheless, the strength of the vortices relative to the velocity of the inner cylinder is weaker. Second, the pseudo-Nusselt number, ratio of the torque to that obtained in the laminar flow, decreases. Third, higher harmonics become more relevant and grow faster with Reynolds number. Finally, the modification of the viscosity field is described.

I. INTRODUCTION

Instabilities of Couette flow between two coaxial cylinders are considered as prototypes for general studies in hydrodynamic instability and transition to turbulence. A survey of the literature on the Taylor-Couette problem can be found in Koschmieder [START_REF] Koschmieder | Bénard cells and Taylor vortices[END_REF] and Tagg [START_REF] Tagg | The Couette-Taylor problem[END_REF].

In the classical configuration, the inner cylinder is rotating and the outer one is fixed. The basic state, a circular Couette-flow (CCF), is purely azimuthal with a balance between the centrifugal force and the pressure gradient pointing inward. In this situation, the angular momentum decreases with the radius. According to Rayleigh criterion [START_REF] Drazin | Hydrodynamic stability[END_REF], CCF is unstable in the inviscid limit with respect to axisymmetric three-dimensional perturbation. Taylor [START_REF] Taylor | Stability of a viscous liquid contained between two rotating cylinders[END_REF] determined theoretically and experimentally the influence of the fluid viscosity on the stability boundary. The primary instability results in axisymmetric meridional rolls (Taylor vortices) periodically spaced in the axial direction and separated by radial jets. When the inner cylinder rotation rate is increased, a Hopf bifurcation to a rotating wave occurs, i.e. the Taylor vortex flow (TVF) acquires a periodicity along the azimuthal direction θ.

The resulting state is the wavy vortex flow (WVF). The Reynolds number R for onset of time-dependent wavy vortices was computed numerically by Jones [START_REF] Jones | The transition to wavy Taylor vortices[END_REF]. It is shown that for a radius ratio, η = R 1 /R 2 , close to 1, say η ≥ 0.85, the supercritical Reynolds number regime in which the axisymmetric vortices are stationary is very small. As the radius ratio is decreased, the wavy vortices set in at higher and higher Reynolds numbers. This result was confirmed experimentally. For instance, in experiments performed by Snyder & Lambert [START_REF] Snyder | Harmonic generation in Taylor vortices between rotating cylinders[END_REF] with a gap of radius ratio R 1 /R 2 = 0.5, the first appearance of significant azimuthal waviness was reported at R ≈ 10 R c . Furthermore, they suggested that this was probably due to end effects of the cylinders. For the same radius ratio, i.e. η = 0.5, Meincke & Egbers [START_REF] Meincke | Routes into chaos in small and wide gap Taylor-Couette flow[END_REF] observed experimentally a transition to WVF at R ≈ 16 R c . Thus, from experimental point of view, it is not clear at which Reynolds number the flow in a wide gap undergoes a transition to wavy-mode. The disagreement between experimental results could be related to the influence of the aspect ratio (AR = column length/gap width). Indeed in [START_REF] Snyder | Harmonic generation in Taylor vortices between rotating cylinders[END_REF] AR ≈ 30, whereas in [START_REF] Meincke | Routes into chaos in small and wide gap Taylor-Couette flow[END_REF], AR ≈ 4.

For moderate or small gap width (say η > 0.75), increasing further R, the system undergoes a bifurcation from wavy-vortex-flow (WVF) to modulated wavy-vortex-flow (MWVF), characterized by the presence of a second frequency modulating the rotating waves as described by Andereck et al. [START_REF] Andereck | Flow regimes in a circular Couette system with independently rotating cylinders[END_REF]. Subsequent bifurcations lead generally to turbulence after few steps. This orderly progression of nonlinear flow states makes Taylor-Couette flow as an attractive model for studying transition to turbulence.

In the case of non-Newtonian fluids (polymer solutions, suspensions, emulsions, ...), mechanisms of the instability and transition to turbulence may be modified by the nonlinearity of the rheological behavior. Most non-Newtonian fluids have two common properties, viscoelasticity and shear-thinning. Polymer solutions, colloïdal suspensions as well as particulate dispersions display this behavior above a certain concentration threshold. There was a significant interest in inertialess viscoelastic Taylor-Couette instability since the pioneering work of Giesekus [START_REF] Giesekus | Zur stabilität von strömungen viskoelastischer flüssigkeiten[END_REF]. This instability was then analyzed by Muller et al. [START_REF] Muller | A purely elastic transition in Taylor-Couette flow[END_REF], Larson et al. [START_REF] Larson | A purely elastic instability in Taylor-Couette flow[END_REF] and Shaqfeh et al. [START_REF] Shaqfeh | The effects of gap width and dilute solution properties on the viscoelastic Taylor-Couette instability[END_REF]. In the laminar state, the rotation produces a shear which stretches the polymer molecule along the curved stream lines. This leads to a first normal stress difference which acts against the centrifugal force. Groisman and Steinberg (1998) [START_REF] Groisman | Mechanism of elastic instability in Couette flow of polymer solutions[END_REF] showed experimentally that elastic instability leads to a strong nonlinear flow transition at vanishing inertia.

Hereafter, we focus on shear-thinning fluids, for which the elastic response can be neglected.

Typically, stiff polymer solutions, show significant nonlinear decrease of the viscosity with the shear-rate, with almost negligible elastic effects [START_REF] Lindner | Viscous fingering in a shear-thinning fluid[END_REF]. The shear-thinning behavior, i.e. a nonlinear decrease of the viscosity with the shear rate, arises from the reorganization of the internal fluid structure to reduce the viscous dissipation. The characteristic time of the reorganization of the flow structure is supposed much smaller than all characteristic times of the problem. In other words, the fluid is assumed purely viscous and its response to an applied shear-stress is instantaneous.

I.1. 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 II. [START_REF] Taylor | Stability of a viscous liquid contained between two rotating cylinders[END_REF].

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 [START_REF] Agbessi | Linear stability of Taylor-Couette flow of shear-thinning fluids: modal and non-modal approaches[END_REF] and Alibenyahia et al [START_REF] Alibenyahia | Revisiting the stability of circular Couette flow of shear-thinning fluids[END_REF] and the references therein. When both the inner and the outer cylinders are rotating, the critical conditions were determined by Agbessi et al [START_REF] Agbessi | Linear stability of Taylor-Couette flow of shear-thinning fluids: modal and non-modal approaches[END_REF] 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 [START_REF] Li | A non-linear dynamical system approach to finite amplitude Taylor-Vortex flow of shear-thinning fluids[END_REF] 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 [START_REF] Ashrafi | Shear-thinning-induced chaos in Taylor-Couette flow[END_REF][START_REF] Ashrafi | Stability analysis of shear-thinning flow between rotating cylinders[END_REF][START_REF] Ashrafi | Effect of nonlinearity on the Taylor-Couette flow in the narrow-gap[END_REF]. Recently Masuda et al. [START_REF] Masuda | Prediction of onset of Taylor-Couette instability for shear-thinning fluids[END_REF] 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. [START_REF] Escudier | Taylor vortices in Newtonian and shearthinning liquids[END_REF] 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 (R 2 -r)/(R 2 -R 1 ) = 0.8. Sinevic et al. [START_REF] Sinevic | Power numbers, Taylor numbers and Taylor vortices in viscous Newtonian and non-Newtonian fluids[END_REF] measured the torque acting on the inner cylinder for three shear-thinning fluids described by a power-law model (n p = 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 -0.7 w , where, Re w is the Reynolds number defined with the inner wall shear-viscosity. Concerning the flow structure, for a wide gap, it is shown theoretically [START_REF] Agbessi | Linear stability of Taylor-Couette flow of shear-thinning fluids: modal and non-modal approaches[END_REF][START_REF] Alibenyahia | Revisiting the stability of circular Couette flow of shear-thinning fluids[END_REF] and experimentally [START_REF] Escudier | Taylor vortices in Newtonian and shearthinning liquids[END_REF] 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. [START_REF] Escudier | Taylor vortices in Newtonian and shearthinning liquids[END_REF] 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 described by a power-law model with a shear-thinning index n p ≈ 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.

I.2. Objectives, methodology and outline of the paper

The objective of the present work is to examine the effects of the nonlinear relation between the viscosity and the shear rate on the flow structure, the viscosity field and the torque applied on the inner cylinder. Actually, there is an interplay between nonlinear inertia terms and nonlinear viscous terms. Based on the experimental results such the streamline patterns in Taylor vortices represented by Escudier et al. [START_REF] Escudier | Taylor vortices in Newtonian and shearthinning liquids[END_REF] and the spatiotemporal plots reported by Cagney and Balabani [START_REF] Cagney | Taylor-Couette flow of shear-thinning fluids[END_REF] for different shear-thinning fluids, the Taylor vortex flow observed can be considered as periodic with a wavenumber k close to the critical value k c . Here we take k = k c and we concentrate on solutions that consist of this mode plus modes that can be generated from it through the nonlinearity. A weakly nonlinear analysis based on the amplitude expansion method is used as a first approach to study nonlinear effects. This method was pioneered by Watson [START_REF] Watson | On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. part 1. the development of a solution for plane Poiseuille flow and for plane Couette flow[END_REF] and Stuart [START_REF] Stuart | On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. part 1. the basic behaviour in plane Poiseuille flow[END_REF] who used such expansion to describe the subcritical modes in plane Poiseuille flow below the linear instability threshold. Herbert [START_REF] Herbert | On perturbation methods in nonlinear stability theory[END_REF] and Sen & Venkateswarlu [START_REF] Sen | On the stability of plane Poiseuille flow to finite-amplitude disturbances, considering the higher-order Landau coefficients[END_REF] improved Watson's expansion method by making a definition of the disturbance amplitude clear. An other alternative is the multiple scales method.

Fujimura [START_REF] Fujimura | The equivalence between two perturbation methods in weakly nonlinear stability theory for parallel shear flows[END_REF] demonstrated the complete equivalence between these two methods. Here, we focus on the case of a wide gap with η = 0.4, where shear-thinning are more pronounced.

The case of a narrow gap is briefly discussed in the last section.

The article is organized as follows. In § 2, we formulate the physical problem, state the governing equations and define the dimensionless parameters. The velocity and viscosity profiles of the base state are discussed and the perturbation equations are derived. Subsequently, the linear stability analysis is presented in section 3. In § 4, the main steps of the weakly nonlinear analysis are outlined. The results are presented and discussed in § 5. The contribution of the nonlinear viscous terms on the saturated coefficient at cubic order in the amplitude equation is analysed. Departing from the onset, the weakly nonlinear analysis is carried out up to seventh-order in the amplitude equation. The influence of shear-thinning effects on the flow structure, the viscosity field and the torque applied on the inner cylinder is highlighted. Finally, § 6 is devoted to a concluding discussion.

II. PHYSICAL AND MATHEMATICAL MODEL

II.1. Basic formulation

We consider the flow of an incompressible shear-thinning fluid between two coaxial cylinders of inner and outer radii R1 and R2 respectively. The radius ratio is η = R1 / R2 . The outer cylinder is at rest and the inner cylinder rotates with an angular velocity Ω1 . The cylinders are assumed to be infinite. The incompressibility condition and the momentum equations read div U = 0

(1)

∂ t U + R (∇U ) • U = -∇P + div (τ ) , (2)
where U = Ue r + V e θ + W e z is the velocity vector in cylindrical coordinates (r, θ, z), P a generalized pressure which includes the effect of gravity, τ the deviatoric extra-stress tensor, and R is the Reynolds number:

R = ρ Ω1 R1 d/μ 0 . (3)
The quantities defined with a hat (.) are dimensional, while quantities without (.) are dimensionless. To scale these equations, we have followed Chossat & Iooss [START_REF] Chossat | The Couette-Taylor Problem[END_REF], with the aim of maintaining some compatibility between the notations used here and those found in the literature for Newtonian fluids. Lengths are scaled with the annular gap d = R2 -R1 .

Velocities are scaled with R1 Ω1 . Time is scaled with a viscous diffusion time ρ d2 /μ 0 , where ρ and μ0 are the density a characteristic viscosity of the fluid. The pressure and the deviatoric stress are scaled with μ0 R1 Ω1 / d. To the previous equations, we add the no-slip and impermeability conditions at the walls,

U = W = 0, V = 1 at r = R 1 , (4)
U = V = W = 0 at r = R 2 , (5)
where

R 1 = η/(1 -η) and R 2 = 1/(1 -η).
The fluid is assumed to be purely viscous. The constitutive equation reads:

τ = µ(R, Γ) γ with γ = ∇U + (∇U) T , (6)
where the second invariant of the strain-rate tensor is expressed as

Γ = 1 2 γ : γ = 1 2 γij γij . (7)
In Equation 7, the Einstein's summation convention for repeated indices is used.

II.2. Carreau model

Although the equations derived here are valid for any nonlinear purely viscous fluid, for numerical applications, a rheological model has to be chosen to describe the shear-thinning behavior. We have chosen the Carreau model [START_REF] Carreau | Rheological equations from molecular network theories[END_REF] because it has a sound theoretical basis and is C ∞ with respect to Γ, unlike the power-law model or Carreau-Yasuda model which are singular at Γ = 0. The Carreau's law reads [START_REF] Bird | Dynamics of polymeric liquids[END_REF] μ -μ∞ μ0 -μ∞ = 1 + λ2 Γ

(nc-1)/2 , (8)
where μ0 and μ∞ are the dynamic viscosity at low and high shear rate, n c < 1 the shearthinning index and λ the characteristic time of the fluid. The location of the transition from the Newtonian plateau to the shear-thinning regime is determined by λ since 1/ λ defines the characteristic shear rate for the onset of shear-thinning. Increasing λ reduces the Newtonian plateau to lower shear rates. The infinite shear viscosity μ∞ is generally associated with the breakdown of the fluid and is frequently significantly smaller, (10 3 -10 4 ) times smaller than μ0 see Bird et al [START_REF] Bird | Dynamics of polymeric liquids[END_REF] and Tanner [START_REF] Tanner | Engineering rheology[END_REF]. The ratio μ∞ /μ 0 will be thus neglected in the following. This leaves three rheological parameters: μ0 , λ and n c . The Newtonian behavior, μ = μ0 is recovered by setting n c = 1 or λ = 0. It is interesting to note that when the characteristic time λ of the fluid is larger than the advection time d/ R1 Ω-1 1 , the Carreau model tends towards the power-law model μ = K Γ(nc-1)/2 with a consistency K = μ0 λ(nc-1)/2 . Examples of the values of the Carreau parameters can be found in the literature, particularly for solutions of xanthan gum, stiff-rode-like polymer [START_REF] Masuda | Prediction of onset of Taylor-Couette instability for shear-thinning fluids[END_REF][START_REF] Darbouli | Natural convection in shear-thinning fluids: Experimental investigations by MRI[END_REF][START_REF] Haase | Inelastic non-Newtonian flow over heterogeneously slippery surfaces[END_REF].

Typically, the characteristic time of the fluid λ is of order of seconds or tens of seconds and the zero shear viscosity is of order 0.1 Pa.s or 10 Pa.s. The shear thinning index n c ∈ ]0, 1[.

In dimensionless form the Carreau's law reads

µ (R, Γ) = μ μ0 = 1 + (λR) 2 Γ nc-1 2 with λ = λ ρ d2 /μ 0 . (9)

II.3. Base flow

For sufficiently small Reynolds number, the circular Couette flow is a solution of the system (1), (2), satisfying the boundary conditions ( 4) and [START_REF] Jones | The transition to wavy Taylor vortices[END_REF]. It is given by :

U b = (0, V b (r), 0) , d dr r 2 τ rθ b = 0 with V b (R 1 ) = 1 and V b (R 2 ) = 0 , (10)
where,

τ rθb = µ b dV b dr - V b r with µ b = µ (R, Γ b ) = 1 + (λR) 2 Γ b (nc-1)/2 . (11)
The system [START_REF] Muller | A purely elastic transition in Taylor-Couette flow[END_REF], [START_REF] Larson | A purely elastic instability in Taylor-Couette flow[END_REF] is solved numerically using an iterative process combined with a polynomial Chebyshev approximation of the azimuthal velocity profile [START_REF] Agbessi | Linear stability of Taylor-Couette flow of shear-thinning fluids: modal and non-modal approaches[END_REF][START_REF] Alibenyahia | Revisiting the stability of circular Couette flow of shear-thinning fluids[END_REF]. One must notice that contrarily to the Newtonian case, for a non-Newtonian fluid, the dimensionless base flow U b also depends on R. However, for the sake of simplicity, this dependence is left implicit. Figure 1 shows the effect of the shear-thinning index on the velocity and viscosity profiles at η = 0.4 and λR = 100. With increasing shear-thinning effects, the curvature of the velocity profile becomes more pronounced. The shear rate increases at the inner wall and decreases at the outer one. This results in a viscosity stratification within the gap. The viscosity as well as the degree of viscosity stratification defined by dµ b /dr increases from the inner to the outer wall. For sufficiently strong shear-thinning effects, a thick layer of the fluid with low shear-rate and large viscosity forms at the outer wall as shown by the curve (4) in Fig. 1(b). In the figure 1(a), the velocity profiles are almost the same as those obtained for a power-law fluid,

V b (r) = r R 1   1 - 1 R 2/np 2 - 1 R 2/np 1 -1 1 r 2/np - 1 R 2/np 1 .   (12)
It is the same for the viscosity profiles except for n c = 0.2 and η = 0.4, where a difference is observed near the outer cylinder. Actually at η = 0.4, the numerical results show that there is no significant difference between Carreau and power-law velocity profiles when λR ≥ 10.

However, for the viscosity profiles a larger value of λR is needed to reduce the difference observed near the outer cylinder, particularly for low values of n c .

II.4. Perturbation equations

The velocity U and the pressure P are splitted into the basic field (with the subscript b) and the disturbance:

U = U b + u and P = P b + p . (13)
Substituting U and P by their expressions in (1) and (2) leads to

∇ • u = 0 , (14)
∂ t u = -R [u • ∇u + U b • ∇u + u • ∇U b ] - (15)
∇p

+ ∇ • (τ (U b + u) -τ (U b )) ,
where,

τ ij (U b + u) = µ (U b + u) γij (U b + u).
The boundary conditions are:

u = 0 and ∂ r (u • e r ) = 0 at r = R 1 , R 2 . ( 16
)
The second condition arises from the continuity equation. For a small amplitude disturbance, the viscosity of the perturbed flow can be expanded around the base flow as:

µ (U b + u) = µ b + µ 1 (u) + µ 2 (u, u) + µ 3 (u, u, u) + ... , (17)
where,

µ 1 (u) = ∂µ ∂ γij b γij (u) , (18)
µ 2 (u, u) = 1 2 ∂ 2 µ ∂ γij ∂ γkℓ b γij (u) γkℓ (u) , (19)
µ 3 (u, u, u) = 1 6 ∂ 3 µ ∂ γij ∂ γkℓ ∂ γpq b γij (u) γkℓ (u) γpq (u) , (20)
The deviatoric stresses in the disturbed flow can also be written as

τ ij (U b + u) = τ ij (U b ) + τ 1,ij (u) + τ 2,ij (u, u) + τ 3,ij (u, u, u) + ... , (21)
with,

τ 1,ij (u) = µ b γij (u) + µ 1 (u) γij (U b ) , (22)
τ 2,ij (u, u) = µ 2 (u , u) γij (U b ) + µ 1 (u) γij (u) , (23)
τ 3,ij (u, u) = µ 3 (u , u , u) γij (U b ) + µ 2 (u, u) γij (u) . (24)
In

µ 1 = 2 dµ dΓ b γb rθ γrθ (u) , (25)
µ 2 = dµ dΓ b Γ 2 + 2 d 2 µ dΓ 2 b Γ b γ2 rθ (u) , (26)
µ 3 = 2 d 2 µ dΓ 2 b γb rθ γrθ (u)Γ 2 + 4 3 d 3 µ dΓ 3 b γb rθ 3 γ3 rθ (u) . (27)
Replacing µ 1 , µ 2 , µ 3 by their expressions (25)-27) into equations ( 22)-( 24), we obtain

τ 1,ij = µ b γij (u) if ij = rθ, θr , (28)
τ 1,rθ = µ b + 2 dµ dΓ b γb rθ 2 = µ t γrθ (u) , (29)
τ 2,ij = µ 1 γrθ (u) if ij = rθ, θr , (30)
τ 2,rθ = µ 2 γb rθ + µ 1 γrθ (u) , (31)
τ 3,ij = µ 2 γrθ (u) if ij = rθ, θr , (32)
τ 3,rθ = µ 3 γb rθ + µ 2 γrθ (u) . (33)
In equation ( 29), µ t is the tangent viscosity. For one dimensional shear flow with the velocity V b (r) in the azimuthal direction, the tangent viscosity is given by µ t = (∂µ/∂ γrθ ) b . Note that for simplicity of writing, we preferred to write the viscosity as a function of U b and u rather than R, Γ b and Γ 2 .

Here, we consider axisymmetric disturbances. For generalized Newtonian fluids and in the case where the outer cylinder is fixed and the inner one is rotating, the numerical tests performed by Alibenyahia et al [START_REF] Alibenyahia | Revisiting the stability of circular Couette flow of shear-thinning fluids[END_REF] and Agbessi et al [START_REF] Agbessi | Linear stability of Taylor-Couette flow of shear-thinning fluids: modal and non-modal approaches[END_REF], for a large range of rheological parameters, showed that the least stable mode is axisymmetric. In this case, the continuity simplifies and is satisfied via introduction of a streamfunction

ψ u = - 1 r ∂ψ ∂z and w = 1 r ∂ ∂r (ψ) . (34)
However, it is found that it is more convenient to work with φ = ψ r in terms which

u = - ∂φ ∂z and w = 1 r ∂ ∂r (rφ) (35)
Cross-differentiating r-and z-momentum equations and eliminating the pressure, we obtain

∂Ω ∂t = R ∂φ ∂z DΩ -D * φ ∂Ω ∂z - 1 r ∂φ ∂z Ω - 2 r v ∂v ∂z -2 V b r ∂v ∂z + ( 36
)
DD * - ∂ 2 ∂z 2 τ rz + ∂ ∂z Dτ zz -D * τ rr + τ θθ r ∂v ∂t = R ∂φ ∂z D * v -D * φ ∂v ∂z + ∂φ ∂z D * V b + 1 r 2 D r 2 τ rθ + ∂ ∂z τ θz , (37)
where,

D ≡ ∂ ∂r , D * ≡ D + 1 r
and Ω = DD * + ∂ 2 ∂z 2 φ the azimuthal vorticity.

III. LINEAR STABILITY ANALYSIS

The linearized version of the disturbance equations ( 36) and ( 37) is

∂Ω ∂t = -2R V b r ∂v ∂z + DD * - ∂ 2 ∂z 2 τ 1,rz + ∂ ∂z Dτ 1,zz -D * τ 1,rr + τ 1,θθ r , (38)
∂v ∂t = R(D * V b ) ∂φ ∂z + 1 r 2 D r 2 τ 1,rθ + ∂ ∂z τ 1,θz . (39)

III.1. Direct mode

Using the invariance of equations ( 38), [START_REF] Crumeyrolle | Experimental study of inertioelastic Couette-Taylor instability modes in dilute and semidilute polymer solutions[END_REF] (by translation along the axial direction, reflection z → -z ), we seek the eigenmodes in the following form:

(φ, v) = (F 11 (r), V 11 (r)) exp (st + ikz) , (40)
where k ∈ R, is the axial wavenumber and s = s r + is i a complex eigenvalue. Its real part s r is the growth-rate of the disturbance and the imaginary part allows to define the axial phase velocity. Substituting ( 40) into ( 38) and ( 39):

s DD * -k 2 F 11 = -2ikR V b r V 11 + DD * -k 2 µ b DD * -k 2 F 11 + (41) 2k 2 D 2 µ b F 11 , sV 11 = ikR (D * V b ) F 11 + 1 r 2 D r 2 µ t DV 11 - V 11 r -k 2 µ b V 11 , (42)
with

F 11 = DF 11 = V 11 = 0 at r = R 1 , R 2 . ( 43
)
The set of differential equations ( 41) and ( 42) is an eigenvalue problem where s is an eigenvalue and X 11 = (F 11 , V 11 ) the eigenvector. It can be written formally as

sM • X 11 = L • X 11 . (44)
Since any multiple of the eigenvector X 11 is also solution of ( 43), X 11 can be normalized

such that max(V 11 ) = 1. (45)
The eigenvalue problem [START_REF] Esmael | Transitional flow of a non-Newtonian fluid in a pipe: Experimental evidence of weak turbulence induced by shear-thinning behavior[END_REF] with the boundary conditions ( 43) is solved using a spectral collocation method based on Chebyshev polynomials evaluated at N +1 collocation points of the Gauss-Lobatto method. The matrix eigenvalue problem that results is solved using QZ algorithm with Matlab. To test the convergence of the numerical method, the computations

ξ 0 = 1 2R c ∂ 2 R ∂k 2
Rc,kc .

To evaluate the second derivative, we have followed the methodology described in [START_REF] Dominguez-Lerma | Marginal stability curve and linear growth rate for rotating Couette-Taylor flow and Rayleigh-Bénard convection[END_REF]. The with n c = 0.2 and λ = 44. The eigenvalues are real or complex conjugate [START_REF] Diprima | Complex eigenvalues for the stability of Couette flow[END_REF]. However, the first eigenvalue that crosses the real axis is always real in agreement with the principle of stability exchange [START_REF] Diprima | Complex eigenvalues for the stability of Couette flow[END_REF]. One can note that with increasing shear-thinning effects, the relaxation times of the eigenmodes become closer one to another. For instance, for a Newtonian fluid at the critical conditions, the first eigenvalue is s 1 = -8.0 × 10 -6 and the second one 4)

n c = 0.2, R c = 2.282.
circular Couette flow is inertial. Therefore, the oscillating mode observed for instance by Crumeyrolle et al. [START_REF] Crumeyrolle | Experimental study of inertioelastic Couette-Taylor instability modes in dilute and semidilute polymer solutions[END_REF] at the onset for some polymer solutions which are viscoelastic and shear-thinning is a signature of the emergence of elastic effects.

III.1.1. Linear energy equation

The linear energy equation is derived by multiplying ( 41), (42) by r times the complex conjugate (F * 11 , V * 11 ) T then integrating between the two cylinders. One obtains

Re(s) k 2 |F 11 | 2 + |V 11 | 2 + |D * F 11 | 2 = RJ I -J µ b + J µ b -µt , (46) where
(.) = R 2 R 1
(.)r dr, J I , J µ b and J µ b -µt denote inertial and viscous contributions to the kinetic energy growth of the perturbation. They are defined by:

J I = -γb rθ (-ik c F 11 ) V * 11 , (47)
J µ b = µ b 1 2 | γ′ θθ | 2 + | γ′ zz | 2 + | γ′ rr | 2 + | γ′ rz | 2 + | γ′ rθ | 2 + | γ′ θz | 2 , (48)
J µ b -µt = (µ b -µ t ) | γ′ rθ | 2 , (49)
with γ′ ij = γij (u). At criticality, the transfer of energy from the base flow to the perturbation R J I is exactly balanced by the viscous dissipation (J µ b -J µ b -µt ). The decrease of the critical Reynolds number with increasing shear-thinning effects means that 3), the viscosity perturbation is taken into account. In the curves (2') and (3'), the viscosity perturbation is not taken into account. Curve ( 1) is represented as a reference and corresponds to the Newtonian case.

R c = J µ b -J µ b -µt J I (50
decreases with increasing shear-thinning effects. The term J µ b -µt originates in the viscosity perturbation. It is positive definite and produces a reduction of the viscous dissipation and thus a decrease of the critical Reynolds number. This decrease is quite modest as it is shown by figure 7, where R c is compared with that obtained when the viscosity perturbation is not taken into account.

The numerical results show that J I and J µ b decrease with increasing shear-thinning effects. However, J µ b decrease faster than J I . Thus R c decreases with increasing shear-thinning effects. The integrand of J I contains γrθb and the quadratic product of the eigenfunctions.

The integrand of J µ b contains µ b , the quadratic product of the eigenfunctions and their first derivatives. The eigenfunctions are normalized such that max (V 11 ) = 1.

The variations of J I and J µ b with shear-thinning effects, has to be due to change with n c (for instance), in either γrθb , µ b or the eigenfunctions.

III.2. Characteristic time

In the neighborhood of the critical conditions such that (R -R c ) /R c = ǫ << 1, the growth rate can be approximated using Taylor expansion

s = ǫ τ 0 + O ǫ 2 , ( 51
)
where τ 0 is the characteristic time for the instability to grow. The determination of τ 0 can be obtained by evaluating ds dǫ ǫ=0 (s is calculated for different values of ǫ, around ǫ = 0). Figure 10 shows for a Carreau fluid, the variation of τ 0 as a function of the shearthinning index n c for different values of λ. The characteristic time for the instability to grow increases significantly with λ, reaches a maximum at n c ≈ 0.3 then decreases slightly as n c

decreases. These results reveal that the onset of Taylor vortex flow is much more gradual for a shear-thinning fluid than for a Newtonian fluid.

III.3. Adjoint mode

To the direct problem ( 44) is associated the adjoint problem

sM + • X ad = L + • X ad with X ad = (F ad , V ad ) . (52)
The adjoint operators M + and L + are defined by

X ad , M • X = M + • X ad , X and X ad , L • X = L + • X ad , X , (53)
with the inner product between two vector functions f (r) and g(r),

f , g = R 2 R 1 f * • g r dr , (54)
where f * is the complex conjugate of f . The adjoint problem associated to [START_REF] Generalis | Range of validity of weakly nonlinear theory in the Rayleigh-Bénard problem[END_REF], ( 42) is

s DD * -k 2 F ad = ikR V b r V ad + DD * -k 2 µ b DD * -k 2 F ad + (55) 2k 2 D 2 µ b F ad , sV ad = -2ikR (D * V b ) F ad + 1 r 2 D r 2 µ t DV ad - V ad r -k 2 µ b V ad , (56)
with

F ad = DF ad = V ad = 0 at r = R 1 , R 2 . ( 57
)
IV. WEAKLY NONLINEAR STABILITY ANALYSIS

IV.1. Principle and formulation

A weakly nonlinear analysis using the amplitude expansion method is used as a first approach to investigate nonlinear effects. Near the bifurcation point, the dynamics are assumed to be determined by the fundamental mode with wavenumber k = k c , its higher harmonics generated by the the nonlinear self-interactions and the modification of the base state due to the interactions with the complex conjugate. It is natural to write the nonlinear perturbation as the Fourier series

[φ (r, z, t) , v (r, z, t)] = +∞ n=-∞ [φ n (r, t) , v n (r, t)] E n with E n = e inkcz . ( 58
)
The transient evolution of the perturbation is taken into account by the temporal evolution of the Fourier coefficients φ n and v n . Because φ and v are real, we have φ -n = φ * n and v -n = v * n , where the star denotes the complex conjugate. Substituting (58) into ( 36) and (37) and separating out the coefficients of similar exponentials, we obtain a set of nonlinear partial differential equations for the Fourier components φ n and v n :

∂ ∂t S n φ n = S n [S n (µ b φ n )] -2inkRe V b r v n + [NI1] E n + [NV 1] E n , (59)
∂ ∂t v n = ink c Re (D * V b ) φ n + 1 r 2 D r 2 µ t (Dv n -v n /r) + [NI2] E n + [NV 2] E n , (60)
with

S n = DD * -n 2 k 2 c , (61)
[NI] E n and [NV ] E n the coefficients of E n in the nonlinear inertial and nonlinear viscous terms respectively. The nonlinear viscous terms arise from the shear-thinning behavior.

As in references [25-28, 40, 41], the amplitude A(t) of the fundamental mode (φ 1 , v 1 ) is considered as small, therefore the Fourier components φ n and v n can be sought using a perturbation method expanding around the solution of the linear problem:

[φ 1 (r, t), v 1 (r, t)] = A(t) [F 1 (r, t), V 1 (r, t)] . (62)
The amplitude of the perturbation is defined by setting

A(t) = max(v 1 (r, t)) . (63)
It is clear that if the linear solution is O(A), the leading term of (

φ 2 , v 2 ) is O(A 2 ) because
of the interaction of the fundamental with itself. The same reasoning applied to higher harmonics shows that (φ n , v n ) can be written as

[φ n (r, t), v n (r, t)] = A n (t) [F n (r, t), V n (r, t)] if n > 0, (64)
and

[φ 0 (r, t), v 0 (r, t)] = A 2 (t) [F 0 (r, t), V 0 (r, t)] . (65)
Substituting ( 64) and ( 65) into (59) and (60) and equating similar powers of A(t), leads to the following set of equations for F n and

V n ∂ ∂t + ng S n F n = S n [S n (µ b F n )] -2inkRe V b r V n + [NI1] A n E n + [NV 1] A n E n , (66)
∂ ∂t + ng V n = ink c Re (D * V b ) F n + 1 r 2 D r 2 µ t (DV n -V n /r) + [NI2] A n E n + [NV 2] A n E n . ( 67
)
The time evolution of the amplitude A(t) is given by the Stuart-Landau equation

g = 1 A dA dt = m=0 g m |A| 2m , (68)
where g 0 = s is the linear eigenvalue, and g m with m > 1, the mth Landau coefficient. Since

F n (V n ) is O(1)
or O(A 2 ) as A → 0, the nonlinearities generate terms in ascending power of A 2 . Hence, F n and V n are expanded as follows

F n (r, t) = +∞ m=0 F n,2m+n (r)A 2m , V n (r, t) = +∞ m=0 V n,2m+n (r)A 2m . (69)
F n,2m+n and V n,2m+n ,

L1 nm F n,2m+n + L2 nm V n,2m+n = [NI1] E n A 2m+n + [NV 1] E n A 2m+n - m j=1 [2(m -j) + n] g j S n F n,2(m-j)+n (70) L3 nm F n,2m+N + L4 nm V n,2m+n = [NI2] E n A 2m+n + [NV 2] E n A 2m+n - m j=1 [2(m -j) + n] g j V n,2(m-j)+n , (71)
with

L1 nm = (2m + n) sS n -S n (µ b S n ) , L2 nm = -2ink c RV b /r , (72)
L3 nm = -ink c R (D * V b ) , L4 nm = (2m + n) s - 1 r 2 D r 2 µ t (D -1/r) (73)
IV.2. Solution procedure

The system of differential equations ( 70) and ( 71) is solved sequentially beginning from n = 1 and m = 0. The problem n = 1, m = 0 is the linear problem ( 41) and [START_REF] Dubrulle | Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection[END_REF]. The problem n = 0, m = 1 yields the first correction to the base flow, the problem n = 2, m = 0 yields the second harmonic mode; n = 1, m = 1 yield the coefficient g 1 of feedback on the fundamental mode. More precisely, g 1 is determined using the solvability condition to the equation which gives the modification of the fundamental mode. For each (n, m), the system of differential equations (70), (71) with the associated boundary conditions is solved numerically using a spectral collocation method based on Chebyshev polynomials as in the linear problem ( 41)- [START_REF] Eckhardt | Torque scaling in turbulent Taylor-Couette flow between independently rotating cylinders[END_REF].

V. RESULTS AND DISCUSSION

V.1. Modification of the base flow

The interaction of the fundamental AF 11 , AV 11 with its complex conjugate through the nonlinear quadratic terms produces a correction of the basis state: A 2 F 02 and A 2 V 02 . Equations for F 02 and V 02 are obtained by setting n = 0 and m = 1 in (70)-(73). The factor of A 2 E 0 arising from the nonlinear inertial and nonlinear viscous terms in (70) vanish, therefore

F 02 = 0 . ( 74
)
The modification of the base flow satisfies the following equation

1 r 2 D r 3 µ t D V 02 r -2sV 02 =-R ∂φ ∂z D * v -D * φ ∂v ∂z A 2 E 0 (75) - 1 r 2 D r 2 τ 2,rθ A 2 E 0 ,
with the boundary conditions

V 02 = 0 at r = R 1 , R 2 . ( 76
)
In equation ( 75), (•) A 2 E 0 means the coefficient of A 2 E 0 in the nonlinear inertial or nonlinear viscous terms. Figure 11 shows the modification of the base flow at order A 2 for λ = 1 and different values of n c . One notes that with increasing shear-thinning effects (decreasing n c ),

V 02 (r) becomes more concentrated in the neighborhood of the inner cylinder because of the viscosity stratification. Furthermore, the numerical results indicate that from λ = 1, there is practically no influence of this parameter.

The profiles of V 02 are related to the radial inflow and outflow jets. The radial inflow jet carries fluid particles with low azimuthal momentum from the outer cylinder inward, decreasing the azimuthal velocity near the inner cylinder. The radial outflow jet carries fluid particles with high azimuthal momentum from the inner cylinder outward, increasing the azimuthal velocity near the outer cylinder. This reorganization of the azimuthal flow tends to increase

|dV /dr| at the walls and to reduce |dV /dr| in the fluid interior. Note also that the deficit of the azimuthal velocity near the inner cylinder is higher than the surplus near the outer cylinder. Canceling artificially, the nonlinear viscous terms in (75) allows to highlight the contribution of the nonlinear inertial terms and vice-versa, to highlight the contribution of the nonlinear viscous terms on the modification of the base flow.

The contribution of the nonlinear viscous terms, which arise from the viscosity perturbation is shown in Fig. 12(a). The azimuthal velocity is increased in practically all the annular space, and mainly near the inner cylinder for strong shear-thinning effects. This is a consequence of the reduction of the viscous dissipation described by the term J µ b -µt in equation [START_REF] Coles | Transition in circular Couette flow[END_REF]. The modification of the base flow by the nonlinear viscous terms is weaker, but of the same magnitude order, than that induced by the nonlinear inertial terms shown in figure 12(b). The difference between these two contributions decreases with increasing shear-thinning effects.

r -R 1 V v 02 (4) (3)
(2) 0 0.2 0.4 0.6 0.8 1 (70) and (71) reduce to

-12 -8 -4 0 4 r -R 1 V i 02 (1) (2) (3) (4)
L1 22 F 22 + L2 22 V 22 = [NI1] A 2 E2 + [NV 1] A 2 E 2 , (77)
L3 22 F 22 + L4 22 V 22 = [NI2] A 2 E2 + [NV 2] A 2 E 2 , . (78)
The boundary conditions read

F 22 = DF 22 = V 22 = 0 at r = R 1 , R 2 . ( 79
)
Radial profiles of Im(F 22 ) and V 22 are shown in figure 13 for λ = 10 and different values of n c . As for the fundamental mode, with increasing shear-thinning effects, the variations of F 22 and V 22 are confined in a thin layer adjacent to the inner wall, i.e the second harmonic is squeezed against the inner cylinder. Furthermore, with increasing shear-thinning effects the maximum of the azimuthal velocity V 22 is amplified, whereas the radial velocity is reduced.

V.3. Modification of the fundamental mode at cubic order: Cubic Landau constant

The nonlinear interactions between the fundamental, the second harmonic and the modification of the base flow lead to a cubic correction O A 3 to the fundamental mode. The first Landau coefficient g 1 accounts for these nonlinear interactions on the fundamental mode.

The modification of the fundamental mode at order A 3 is governed by ( 70) and ( 71) with m = n = 1, i.e.

L1 13 F 13 + L2 13 V 13 = [NI1] A 3 E 1 + [NV 1] A 3 E 1 -g 1 S 1 F 11 , (80)
L3 13 F 13 + L4 13 V 13 = [NI2] A 3 E 1 + [NV 2] A 3 E 1 -g 1 V 11 . (81)
The boundary conditions are

F 13 = DF 13 = V 13 = 0 at r = R 1 , R 2 . ( 82
)
The system ( 80) and ( 81) can be written

L • X 13 = -g 1 M • X 11 + N I + N V with X 13 = (F 13 , V 13 ) .
At critical conditions, (83) has a non-trivial solution if the Fredholm solvability condition is satisfied, i.e. orthogonality the inhomogeneous part of (83) to the null-space of the adjoint operator of L. The cubic Landau constant is then readily obtained,

g 1 = g I 1 + g V 1 = g I 10 + g I 12 + g V 10 + g V 12 + g V 1-11 , (84)
with

g I 10 = N I (X 02 |X 11 ) , X ad M • X 11 , X ad , g V 10 = N V (X 02 |X 11 ) , X ad M • X 11 , X ad (85)
g I 12 = N I (X 22 |X -11 ) , X ad M • X 11 , X ad , g V 12 = N V (X 12 |X -11 ) , X ad M • X 11 , X ad (86) g V 1-11 = N V (X 11 , X 11 |X -11 ) , X ad M • X 11 , X ad , (87)
where X -11 = X * 11 , g I 10 and g V 10 are the feedback of the mean flow correction onto the fundamental through the nonlinear inertial and nonlinear viscous terms respectively, g I 12 is the feedback of the second harmonic onto the fundamental, etc. The vertical bar notation in ( 85)-( 87) may be better defined through an example of a nonlinear term. For instance

D * φ ∂v ∂z (X 22 |X -11 ) → D * F 22 (-ikV * 11 ) + D * F * 11 (2ikV 22
). The integrals in (85)-( 87) are evaluated by means of the Clenshaw and Curtis method. In figure 14(a), g 1 is plotted as a function of n c for different values of λ. The sign of g 1 is negative indicating a supercritical nature of the bifurcation. As expected for weakly shear-thinning effects, the contribution of the nonlinear viscous terms g V 1 , on the first Landau coefficient, play a minor role and

g I 1 >> g V 1 .
With increasing the shear-thinning effects (decreasing n c ), the contribution of the nonlinear inertial terms g I 1 and nonlinear viscous terms g V 1 become of the same order, however |g I 1 | remains larger than g V 1 , as shown in figure 14(b). Note that g V 1 is positive, i.e. nonlinear viscous terms favor a subcritical bifurcation. Contributions of the different terms g I 10 , g I 12 , g V 12 ... that control the value of g 1 are given in Appendix B. The data show that for a Newtonian fluid, g 1 is dominated by the feedback of the base flow correction onto the fundamental. With increasing shear-thinning effects, the feedback of the second harmonic becomes more significant.

V.4. Features of the perturbation near the threshold

Besides the Landau coefficient, the amplitude of the perturbation A is an important quantity in the nonlinear stability analysis. It is obtained by setting dA/dt = 0 in (68). In the neighbourhood of the critical conditions, to lowest order in ǫ, the amplitude is

A = -ǫ τ 0 g 1 . ( 88
)
The numerical values of the Landau coefficient and hence the values the perturbation amplitude A depend upon the normalization condition used for the eigenfunction in the linear theory. However, the physical velocity components, i.e. the product of the amplitude with the eigenfunctions of the linear theory are independent of the normalization. For instance, we can consider the mean kinetic energy of the perturbation or the torque applied on the inner cylinder.

V.4.1. Mean kinetic energy

At order A 2 , the mean kinetic energy is defined by

ξ = A 2 R 2 R 1 k 2 c |F 11 | 2 + |V 11 | 2 + |D * F 11 | 2 rdr . ( 89
)
Figure 15(a) plots ξ as a function of ǫ for different values of the shear-thinning index n c . The dimensionless constant time is fixed λ = 10. The curves are practically identically to those obtained for a power-law fluid. As it can be observed the kinetic energy of the perturbation decreases with increasing shear-thinning effects. Figure 15(b) shows that the azimuthal kinetic energy is much larger than the radial and axial ones. For instance at n c = 0.2, more than 95% of the kinetic energy is concentrated on the azimuthal velocity.

The decrease of the kinetic energy with increasing shear-thinning effects is due to the viscosity stratification. Indeed, with increasing shear-thinning effects, the Taylor vortices are confined in the thin region near the inner wall where the viscosity is low (the perturbations are suppressed in the highly viscous fluid at a short distance from the inner cylinder) and their strength becomes weaker as shown in figure 6a.

V.4.2. Torque

The determination of the torque T applied on the inner cylinder is of great interest. It provides information about the energy dissipation since the energy injected in the fluid per unit of time is obtained by multiplying the torque by the velocity of the inner cylinder.

Dubrulle [START_REF] Dubrulle | Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection[END_REF] and Eckhardt [START_REF] Eckhardt | Torque scaling in turbulent Taylor-Couette flow between independently rotating cylinders[END_REF] demonstrated that the dimensionless torque G = T /2π ρν 2 ĥ where ν is the kinematic viscosity and ĥ the height of the inner cylinder, is related to the angular momentum flux. By analogy with the heat transfer in Rayleigh-Bénard convection, the previous authors defined a pseudo-Nusselt number, Nu * for a Taylor Couette flow as the ratio of G to the dimensionless torque of the laminar flow G b . Hence, for a purely viscous fluid, Using ( 17) and ( 28)-( 33), Nu * can be written as

Nu * = G G b = 2π/kc 0 (µ γrθ ) r=R 1 dz (2π/k c ) (µ b γrθb ) r=R 1 . (90)
Nu * = 1 + µ t µ b R 1 2π/kc 0 ( γrθ ) r=R 1 dz ( γrθb ) r=R 1 + 2π/kc 0 (µ 1 γrθ + µ 2 γrθb ) r=R 1 dz (µ b γrθb ) r=R 1 + ... (91)
In the laminar regime, the angular momentum flux is diffusive and Nu * = 1. In the TVF regime, the pseudo-Nusselt number at order A 2 is given by

Nu * = 1 + A 2 µ t µ b R 1 DV 02 DV b R 1 + 2 A 2 (µ b ) r=R 1 dµ b dΓ b R 1 3 |DV 11 | 2 + D 2 F 11 2 R 1 + 4 A 2 (µ b ) r=R 1 d 2 µ b dΓ 2 b R 1 |DV 11 | 2 R 1 + O A 4 . (92)
Using (88), the pseudo-Nusselt number can be written formally as

Nu * = 1 + Kǫ . (93)
For a Newtonian fluid, the expression of the slope K reduces to

K = - 1 τ 0 g 1 (DV 02 ) R 1 (DV b ) R 1 with DV b = - 2 η (1 + η) (94)
In this case, the increase of the torque applied on the inner cylinder is due to the increase of the inner wall shear rate of the mean flow, via the term (DV 02 ) R 1 . For non-Newtonian shear-thinning fluids, there is an additional term arising from the viscosity perturbation Reynolds-stress" in [START_REF] Esmael | Transitional flow of a non-Newtonian fluid in a pipe: Experimental evidence of weak turbulence induced by shear-thinning behavior[END_REF] contributes to the reduction of the viscous dissipation.

V.5. Validation by computing higher-order Landau constants Nusselt number Nu * decrease with increasing shear-thinning effects. This result was obtained by truncating the series (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 shearthinning effects as it is shown by the data in table V (Appendix C). The representation Figure 17 shows the variation of the pseudo-Nusselt number as a function of ǫ for different values of n c . As it can be observed, the decrease of Nu * with increasing shear-thinning effects is confirmed at least for a reasonable distance from the critical conditions. Curves for Carreau and power-law models are not distinguishable except for n c = 0.2. In this last case, a higher value of λ is needed.

Similarly, the numerical results show that for a reasonable departure from the critical conditions, the kinetic energy of the perturbation decreases with increasing shear-thinning effects and it is mainly concentrated on the azimuthal velocity component.

Remarks

-Our results for Nu * are compared with those obtained experimentally by Donnelly & Simon [START_REF] Donnelly | An empirical torque relation for supercritical flow between rotating cylinders[END_REF] for a Newtonian fluid with η = 0.5. A good agreement is found at least up to ǫ = 0.5 as it is illustrated by Tab. III (Appendix A). With the exception of one point, the relative difference is less than 1 %.

-For n c = 0.2 and 0.4, the curves Nu * versus ǫ are dashed when ǫ exceeds a particular value say ǫ 0 . Actually, harmonics analysis done in §V.7 shows that for sufficiently strong shear-thinning effects, the fourth harmonic overtakes the third one when ǫ > ǫ 0 , which may delimit the range of validity of the results.

-Departing sufficiently from the critical conditions, we can no longer assume s = 0 in the equations that describe the modification of the fundamental mode. Hence, these equations become unconditionally solvable. To calculate the Landau coefficients, an iterative process was used as suggested by Sen & Venkateswarlu [START_REF] Sen | On the stability of plane Poiseuille flow to finite-amplitude disturbances, considering the higher-order Landau coefficients[END_REF] (See also Bouteraa et al. [START_REF] Bouteraa | Weakly nonlinear analysis of Rayleigh-Bénard convection in shear-thinning fluids: nature of the bifurcation and pattern selection[END_REF] and Bouteraa & Nouar [START_REF] Bouteraa | Weakly nonlinear analysis of Rayleigh-Bénard convection in a non-Newtonian fluid between plates of finite conductivity: Influence of shear-thinning effects[END_REF]).

V.6. Description of the flow field

The change in the flow structure with increasing shear-thinning effects is described by contours of stream function ψ, azimuthal vorticity and velocity components u, v and w. to the viscosity stratification discussed in § II.3. The displacement of the vortex center in the direction of higher shear-stress and lower viscosity leads to an increasing concentration of vorticity near the inner wall at z/λ z = 0 and 1 as it is shown in figure 19 where isolines of contours of Ω are displayed for different rheological parameters. In contrast with the build-up of vorticity near the inner cylinder and at the corners z = 0 and z = λ z , the region between the two cells, around z/λ z = 0.5, which is practically void of vorticity increases in area with increasing shear-thinning effects. This build-up of the vorticity with increasing shear-thinning effects and the change in the vortices can be related to the evolution of the radial flow from the inner to the outer cylinder (outflow) at z/λ z = 0 and z/λ z = 1 and from the outer to the inner one (inflow), around z/λ z = 0.5. Figure 20 shows the radial velocity component at the vortex eye center, versus z for Newtonian and Carreau fluids. It can be observed that the radial outflow is stronger than the radial inflow.

The ratio r u between the intensity (max|u|) of the radial outflow and that the radial inflow, increases with increasing shear-thinning effects. At ǫ = 0.1, r u increases from 1.6 for a New- (u, v, w) can be written as sum of Fourier components. For instance for v:

v(r, z) = v 0 (r) + ∞ n=1 v n cos nk c z (95) with v 0 = ∞ m=0 A 2m V 0,2m and v n = 2 ∞ m=0 A 2m+n V n,2m+n . (96)
The amplitude A is the stationary solution of the amplitude equation (68). The form of the u-component is similar to v except that v 0 is omitted. The w-component is like u except that the z-dependency is in sin nk c z. In order to describe the growth of harmonics with Reynolds number, the Fourier amplitude v n in (95) are averaged over the gap width, as in [START_REF] Fasel | Numerical investigation of supercritical Taylor-vortex flow in a wide gap[END_REF] and [START_REF] Snyder | Harmonic generation in Taylor vortices between rotating cylinders[END_REF],

v n = 1 R 2 -R 1 R 2 R 1 v n (r)rdr. ( 97
= 1.1 R c , (2)R = 1.2 R c , (3) R = 1.3 R c
We recall that v 0 corresponds to the modification of the base flow, v 1 to the distorted fundamental mode, v 2 to the second harmonic, etc ... In Figure 26 All in all, nonlinearities of the viscous terms combined with that of the inertial terms lead to a stronger amplification of higher harmonics. this, analysis of the Fourier coefficients shows that even at moderate distance from the onset, higher harmonics become more relevant comparatively to the Newtonian case.

Additional computations were performed for a narrow gap configuration. In this case, the azimuthal velocity profiles of the base flow get close to the linear profile, the viscosity stratification is weak and the range of R where the TVF regime is stable is quite limited.

According to Coles [START_REF] Coles | Transition in circular Couette flow[END_REF] and Jones [START_REF] Jones | The transition to wavy Taylor vortices[END_REF], for η ≥ 0.9, transition to wavy vortex flow is observed at R ≈ 1.2 R c (ǫ = 0.2). Some results are reported in Appendix D for radius ratio η = 0.9.

For strong shear-thinning effects, the marginal stability curve flattens and the coherence length of the perturbation decreases significantly. In this case, modes of different wavenumbers (delimited by the Eckhaus boundary) exist close to the onset of the primary bifurcation.

These modes may interact with each other leading to a secondary bifurcation with a more complex dynamic. The interaction of modes of different wavenumber limits the range of validity of the weakly nonlinear analysis. Nevertheless, we think that the modifications of the flow structure described in this study plays a fundamental role in triggering an eventual secondary bifurcation. To understand the interaction between different modes, a fully nonlinear method based on the numerical continuation can be used [START_REF] Meyer-Spasche | Some bifurcation diagrams for Taylor vortex flows[END_REF]. This approach could be useful to clarify the influence of shear-thinning on the coupling between the fundamental mode and the second harmonic observed experimentally by Crumeyrolle et al. [START_REF] Crumeyrolle | Instabilities with shear-thinning polymer solutions in the Couette-Taylor system[END_REF] for semi-dilute aqueous solutions of high molecular weight polyethylenoxide.

FIG. 1 : 1 (

 11 FIG. 1: (Color online) Base flow for Carreau fluids at η = 0.4 and λR = 100. Influence of shear-thinning index n c on the velocity (a) and viscosity (b) profiles. (1) n c = 1 (Newtonian fluid); (2) n c = 0.7; (3) n c = 0.5 and (4) n c = 0.2. Dashed line is the viscosity profile for a power-law fluid with n p = 0.2.

  the case of a circular Couette flow of a viscous fluid, we have γb ij = 0, if ij = rθ, θr and γb rθ = DV b -V b /r, where D ≡ d/dr. Setting Γ b = γb rθ 2 and Γ 2 = (1/2) γij (u) γij (u), the expressions of µ 1 , µ 2 and µ 3 can be simplified,

FIG. 2 :

 2 FIG. 2: (Color online) Normalized marginal stability curves at η = 0.4 for a Newtonian fluid (1) where R c = 68.3, and k c = 3.183 and Carreau fluids: (2) n c = 0.5, λ = 1, R c = 16.57, k c = 3.487 ; (3) n c = 0.2, λ = 44, R c = 2.282, k c = 8.355.

variation of ξ 0

 0 as a function of the shear-thinning index n c for different values of λ is shown in figure 3. It can be observed that the coherence length of perturbations decreases with decreasing n c . This effect is particularly significant for very low values of n c . A second feature of shear-thinning effects is highlighted by eigenvalues spectra. They are shown in Fig. 4 at the critical conditions with η = 0.4, for a Newtonian and a Carreau-fluid

is s 2 =FIG. 3 :

 23 FIG. 3: (Color online) Variation of characteristic length ξ 0 as a function of the shear-thinning index n c for different values of λ: λ = 1(⋆), λ = 10(△), λ = 100(•).

FIG. 4 :FIG. 5 :FIG. 6 :

 456 FIG. 4: (Color online) Eigenspectra for axisymmetric perturbations at critical conditions with η = 0.4. (a) Newtonian fluid at R c = 68.296, k c = 3.183. (b) Carreau fluid with n c = 0.2, λ = 44 at R c = 2.282, k c = 8.355 .

) 2 FIG. 7 :

 27 FIG. 7: (Color online) Critical Reynolds number as a function of the shear-thinning index for Carreau fluids at λ = 1, curves (2) and (2') and λ = 100, curve (3) and (3'). In the curves (2) and (3), the viscosity perturbation is taken into account. In the curves (2') and

Figure 10 (

 10 Figure 10(a) shows that the integrand of J µ b decreases with increasing shear-thinning effects.

Figure 10 (FIG. 8 :FIG. 9 :

 1089 Figure 10(b) shows that the eigenfunctions contribute to increase the viscous dissipation. Therefore the decrease of J µ b is due to µ b . Analysis of figures 9(a) and 10(a), show that the eigenfunctions contribute to a decrease of J I . The shear rate γrθb attenuates this decrease and amplifies the transfer of energy from the

6 FIG. 10

 610 FIG. 10: (Color online) Variation of the characteristic time of circular Couette flow instability as function of the shear-thinning index for different values of the dimensionless constant-time of the fluid. The radius ratio is η = 0.4.

4 FIG. 11 :

 411 FIG. 11: (Color online) Modification of the base flow at η = 0.4, for Carreau fluids with λ = 1 and different values of the shear-thinning index: (1) n c = 1 Newtonian fluid; (2) n c = 0.7; (3) n c = 0.5 and (4) n c = 0.2.

FIG. 12 :V. 2 .FIG. 13 :

 12213 FIG. 12: (Color online) Modification of the base flow at η = 0.4, for Carreau fluids with λ = 100 and different values of the shear-thinning index: (1) n c = 1; (2) n c = 0.7; (3) n c = 0.5 and (4) n c = 0.2. (a) Contribution of nonlinear viscous terms; (b) contribution of nonlinear inertial terms.

FIG. 14 :

 14 FIG. 14: (Color online) (a) Cubic Landau constant, g 1 , as a function of n c for different values of the dimensionless constant time λ with η = 0.4. (b) Contribution of the nonlinear inertial terms (-g I 1 ) and nonlinear viscous terms g V 1 at λ = 100.

FIG. 15 :

 15 FIG. 15: (Color online) (a) Kinetic energy at order A 2 for different values of the shear-thinning index n c with λ = 10 and η = 0.4. (b) Ratio of azimuthal ξ θ (curve 1), radial ξ r (curve 2) and axial ξ z (curve 3) kinetic energy to the total kinetic energy ξ.

FIG. 16 :

 16 FIG. 16: (Color online) Torque applied on the inner cylinder, at order A 2 , versus ǫ for different values of the shear-thinning index n c with λ = 10.

Figures 15 and 16

 16 Figures 15 and 16 indicate that the kinetic energy of the perturbation and the pseudo-

3 FIG. 17

 317 FIG. 17: (Color online) Influence of shear thinning effects on the pseudo-Nusselt number at η = 0.4 and λ = 10. The symbols correspond to a power-law model for the same shear-thinning index.

Figure 18

 18 Figure 18 shows isolines of constant ψ on one wavelength λ z = 2π/k c (two cells), for Newtonian and Carreau fluids at a reduced Reynolds number ǫ = 0.1. Positive and negative values of ψ correspond to anticlockwise and clockwise rotation. The vortices lose the symmetry along the horizontal axis through the cell center. The eye of vortices is shifted towards upper and lower cell boundaries, i.e. z/λ z = 0 and 1. This effect is weak for a Newtonian fluid (figure 18(a) and becomes more pronounced with decreasing the shear-thinning index n c . This axial shift of the vortex eye is coupled with the radial shift towards the inner cylinder due

FIG. 18 :FIG. 19 :FIG. 20 :FIG. 21 :FIG. 22 :

 1819202122 FIG. 18: (Color online) Contours of streamfunction at ǫ = 0.1 for (a) Newtonian fluid, with (ψ max = 2.2 10 -2 ); (b) Carreau fluid with n c = 0.5, λ = 1 (ψ max = 1.05510 -2 ); (c) Carreau fluid with n c = 0.2, λ = 1, (ψ max = 2.082 10 -3 ) ; (d) Carreau fluid with n c = 0.2, λ = 10, (ψ max = 2.1 10 -3 ). The increment between lines is ∆ ψ = ψ max /10.

)FIG. 23 :

 23 FIG. 23: (Color online) Azimuthal component of velocity versus z at fixed radial position (center of pattern) and different Reynolds number. (a) n = 1 Newtonian fluid, (b) Carreau fluid with n c = 0.5, λ = 1, (c) Carreau fluid with n c = 0.2, λ = 1, (d) Carreau fluid with n c = 0.2, λ = 10. (1) R = 1.1 R c , (2)R = 1.2 R c , (3) R = 1.3 R c

FIG. 24 :

 24 FIG. 24: (Color online) Contour of axial velocity component w at ǫ = 0.1 for (a) Newtonian fluid, w max = 0.082 ; (b) Carreau fluid with n c = 0.5, λ = 1, w max = 0.063 ; (c) Carreau fluid with n c = 0.2, λ = 1, w max = 0.05 , (d) Carreau fluid with n c = 0.2, λ = 10, w max = 0.048. The increment between lines is ∆ w = w max /10 for positives values and ∆w = -w max /10 for negative values.

FIG. 25 :FIG. 26 :

 2526 FIG. 25: (Color online) (a)Viscosity contours at ǫ = 0.1 for: Carreau fluid with n c = 0.5, λ = 1, µ max = 0.63, µ min = 0.094, the increment between lines is ∆ µ = 0.015. (b) Carreau fluid with n c = 0.2, λ = 10, µ min = 0.0046, µ max = 0.86, the increment between lines is ∆ µ = 0.02.

FIG. 27 :

 27 FIG. 27: (Color online) Evolution of the amplitude A versus ǫ = (R -R c ) /R c for a shear-thinning fluid with n c = 0.5, λ = 10, at cubic, fifth and seventh-order in the amplitude expansion.

FIG. 28

 28 FIG. 28: (Color online) Narrow gap, η = 0.9. Flow field structure for a Newtonian fluid and a Carreau fluid with n c = 0.2, λ = 10. (a) Contours of the stream function for a Newtonian fluid, ψ max = 0.12, ∆ψ = ψ max /10. (b) Contours of the stream function for the Carreau fluid,ψ max = 0.072, ∆ψ = ψ max /10 . (c) Contours of vorticity for a Carreau fluid, Ω max = 0.35, ∆Ω = Ω max /10. (d), Contours of the viscosity for the Carreau fluid, µ max = 0.0843, µ min = 0.02, ∆µ = 0.004.

  R N u * Our results N u * Donnelly & Simon[START_REF] Donnelly | An empirical torque relation for supercritical flow between rotating cylinders[END_REF]

	69.1	1.012422	1.016190
	70.2	1.027175	1.024306
	73.0	1.063204	1.053082
	75.3	1.090811	1.076942
	78.8	1.129259	1.111199
	85.1	1.188594	1.167744
	94.8	1.260240	1.246044
	107.0	1.326927	1.279206
	121.0	1.383261	1.394628

TABLE III :

 III Newtonian fluid. Variation of the Pseudo-Nusselt number Nu * versus Reynolds number at η = 0.5. Comparison between our results and those obtained experimentally by Donnelly & Simon[START_REF] Donnelly | An empirical torque relation for supercritical flow between rotating cylinders[END_REF]

	n c	g 1	g I 10	g V 10	g I 12	g V 12	g V 1-11
	0.2 -38.235 -29.561 6.721 -28.098 25.534 -12.831
	0.3 -53.698 -33.737 5.724 -42.354 26.268 -9.598
	0.4 -76.801 -45.024 3.527 -52.555 24.690 -7.438
	0.5 -108.301 -65.101 9.766 -60.968 22.549 -5.758
	0.6 -151.688 -95.215 -1.276 -71.688 20.446 -3.956
	0.7 -214.583 -140.846 -3.006 -86.899 18.018 -1.849
	0.8 -315.127 -215.481 -4.105 -110.776 14.775 4.592
	0.9 -496.243 -351.458 -3.970 -152.723 9.693 2.216
	1.0 -867.033 -632.705 0.000 -234.329 0.000 0.000

TABLE IV :

 IV Cubic Landau constant and contributions of nonlinear inertial and nonlinear viscous terms at the critical conditions

	n c	g 1	g 2	g 3
	0.2 -38.217 -11496.709 -2411946.180
	0.3 -53.698 -16939.108 -5604408.660
	0.4 -76.647 -21517.202 -8480701.020
	0.5 -108.271 -25742.819 -10761822.517
	0.6 -151.684 -31643.773 -13182598.598
	0.7 -214.583 -33736.107 -12116764.641
	0.8 -315.127 -38383.606 -11625135.458
	0.9 -496.243 -48609.312 -12000787.340
	1.0 -867.033 -62111.466 -12310212.318

TABLE V :

 V Landau constants at the critical conditions for different values of the

			shear-thinning index
	0.015
	0.01
	0.005
	0	0	0.1	0.2	0.3	0.4	0.5	0.6

Appendix A: Validation

Radius ratio η Chandrasekhar [START_REF] Chandrasekhar | Hydrodynamic and hydromagnetic stability[END_REF] .

Appendix B: Contribution of nonlinear inertial and nonlinear viscous terms

The feedback of the mean flow correction and that of the first harmonic on the fundamental mode through the nonlinear inertial and nonlinear viscous terms are given in table

IV for λ = 10 and different values of the shear-thinning index n c .

Appendix C: Landau constants

The Landau coefficients g 1 , g 2 , g 3 are given in table V for λ = 10 and different values of the shear-thinning index.

In the figure 28 we have represented contours of the stream function, the vorticity and the viscosity distribution at ǫ = 0.1, for a Carreau fluid with n c = 0.2 and λ = 10.


Azeem Haider
email: aahaider@jazanu.edu.sa

Usman Ali
email: uali@bzu.edu.pk

Moin A Ansari
email: maansari@jazanu.edu.sa

Properties of Tiny Braids and the Associated Commuting Graph

Keywords: Fibonacci numbers, centralizer, commuting graph, positive braid, tiny braid. 2000 Mathematics Subject Classification: 11B39, 05A15, 05A05

In this text, we focus on a subset (called the set of tiny braids) of factors of the Garside braid and generalize some known results related to tiny braids. These generalized results, along with some combinatorics, strengthen the existing relationship between this subset and Fibonacci numbers. We also associate a commuting graph with the subset and explore its fundamental identities, including its order, diameter, girth and degree-related properties.

Introduction

The monoid of positive n-braids B + n is defined in [START_REF] Artin | Theory of braids[END_REF] as

B + n = σ 1 , σ 2 , • • • , σ n-1 : σ i+1 σ i σ i+1 = σ i σ i+1 σ i σ i σ j = σ j σ i for |i -j| ≥ 2 . (1)
In fact, a positive braid α is a class of words in the set of generators {σ 1 , σ 2 , . . . , σ n-1 } :

1 σ i 2 i i + 1 n -1 n . . . . . . r r r ¨1
The monoid B + n canonically embedds in the braid group B n , which is defined by the same presentation as B + n , but considered as a presentation of a group. The monoid B + n is embedded in B + n+1 , and two braids α, β ∈ B + n commute in B + n+1 if and only if α and β commute in B + n . Three kinds of divisors of α ∈ B + n are defined as factors of α: (γ|α), Div(α) = {γ ∈ B + n : there exist δ, ε ∈ B + n , α = δγε}; left divisors of α (γ| L α), Div L (α) = {γ ∈ B + n : there exists ε ∈ B + n , α = γε}; and right divisors of α (γ| R α), Div R (α) = {γ ∈ B + n : there exists δ ∈ B + n , α = δγ}. Clearly, Div L (α) ∪ Div R (α) ⊆ Div(α). Let T B n be the set of all positive braids in which a letter σ i occurs at most once (see [START_REF] Ashraf | Simple braids[END_REF]). Clearly, any word representing a braid in the set T B n must contain σ i at most once. The set T B n is a proper subset of Div(∆ n ), where

∆ n = σ 1 (σ 2 σ 1 ) • • • (σ n-1 σ n-2 • • • σ 2 σ 1 )
is the Garside braid (see [START_REF] Garside | The braid groups and other groups[END_REF]). The set Div(∆ n ) plays a vital role in the solutions to the word and conjugacy problems given in [START_REF] Garside | The braid groups and other groups[END_REF]. In the solution to the latter problem, the factors of the braid ∆ n are used to generate an invariant (called the summit set) of a conjugacy class. Note that in the classical literature on braid theory, every factor of the Garside braid is known as a simple braid, but in [START_REF] Ashraf | Simple braids[END_REF][START_REF] Ashraf | Fibonacci numbers and positive braids[END_REF], the elements from the set T B n are called simple braids. Previous articles [START_REF] Ali | Braids with trivial simple centralizer[END_REF][START_REF] Ali | Centralizer of braids and Fibonacci numbers[END_REF] follow the same naming convention for the braids in T B n . To avoid confusion, we call an element from the set T B n a tiny braid instead of a simple braid. [START_REF] Ashraf | Fibonacci numbers and positive braids[END_REF] shows that the number of elements in the set T B n is a Fibonacci number F 2n-1 , where (F 0 , F 1 , F 2 , F 3 , F 4 , F 5 . . .) = (0, 1, 1, 2, 3, 5, . . .). For further details regarding and the motivation behind defining the set T B n , see [START_REF] Ashraf | Simple braids[END_REF] or [START_REF] Ashraf | Fibonacci numbers and positive braids[END_REF]. A subset of the centralizer of an element α ∈ T B n is defined in [START_REF] Ali | Centralizer of braids and Fibonacci numbers[END_REF] as follows:

C n (α) = {γ ∈ T B n : αγ = γα}.

An element α ∈ T B n is said to have a trivial tiny centralizer (in [START_REF] Ali | Centralizer of braids and Fibonacci numbers[END_REF], the set C n (α) is called a simple centralizer) if

C n (α) = {e, α}.
Let I n be the set of all α ∈ T B n that have a trivial tiny centralizer. In the first part of this article, we discuss the different properties of tiny centralizers and generalize a few of the results. Note that throughout this article, we use letters h, i, j, k, l, m to denote positive integers (for further details on braids, see [START_REF] Kassel | Braid Groups. Grad. Texts in Math[END_REF]).

For graph G, we denote the set of vertices by V (G), the set of edges by E(G), the degree of a vertex v ∈ V (G) by deg(v) and the number of vertices (order) of a graph G by |V (G)|. Graph K n denotes the complete graph on n vertices. The maximum value of the shortest distance between any two vertices of G is called its diameter. Graph G is said to be connected if a path exists between any two vertices of G. The girth of G, denoted by girth(G), is the length of its shortest nontrivial cycle, if it exists. The minimum and maximum degrees of a vertex in G are denoted by δ(G) and ∆(G), respectively. For further details on graph theory, see, e.g., [START_REF] West | Introduction to Graph Theory[END_REF].

The commuting graph Γ(H) associated with a finite subset H of group G is a simple graph whose vertices are the elements of H \ {e}. An edge exists between g and h if and only if gh = hg in G (see [START_REF] Bates | Commuting involution graphs for symmetric groups[END_REF][START_REF] Bondy | The connectivity of commuting graphs[END_REF][START_REF] Iranmanesh | On the commuting graph associated with the sym-metric and alternating groups[END_REF]).

For n ≥ 4, we define a commuting graph Γ(T B n ) associated with the set T B n , the vertex set of which is the set

T B n \ (I n ∪ {e}); that is, V (Γ(T B n )) = T B n \ (I n ∪

{e}). An edge exists between two distinct vertices if and only if they commute in B +

n . In fact, the graph Γ(T B n ) defined here is the major component of the commuting graph defined in [START_REF] Ali | Centralizer of braids and Fibonacci numbers[END_REF], where the vertex set is the entire set T B n . Graphs Γ(T B 4 ) = K 3 and Γ(T B 5 ) are shown in Figure 1. Fifty-two vertices exist in Γ(T B 6 ), and the graph

x 3 x 1 x 2 x 4 x 4 x 3 x 2 x 1 x 1 x 2 x 4 x 4 x 1 x 1 x 4 x 2 x 1 x 4 x 2 x 1 x 2 x 3 x 3 x 4 x 1 x 3 x 4 x 1 x 4x3 Figure 1: Γ(T B 5 )
structure becomes more complex for n ≥ 7.

In the second section of this article, we focus on the graphical aspect of this association and investigate some properties of the commuting graph Γ(T B n ), such as its order, connectedness, diameter, girth and degree-related results.

To develop our text, we need the following known results:

Theorem 1.1. [1] If α ∈ T B n with σ i |α for all 1 < i < n -1, then the tiny centralizer C n (α) = {e, α}. Lemma 1.2. [2] If α ∈ T B n and σ n-1 |α, then either σ n-1 | R α or σ n-1 | L α. Lemma 1.3. [2] If β(γ 1 γ 2 ) = (γ 1 γ 2 )β and βγ 1 = γ 1 β or βγ 2 = γ 2 β, then βγ 2 = γ 2 β or βγ 1 = γ 1 β, respectively. Lemma 1.4. [2] If β ∈ T B n-1 and α ∈ T B n such that σ n-2 |β and σ n-1 |α, then βα = αβ.

Centralizer of Tiny Braids

We define a set T B (0,n) = T B n . Generally,

T B (i,j) = {α ∈ T B n : σ k α for all k ≤ i and k ≥ j}, i ≤ j < n.
Correspondingly, we define, for any α ∈ T B n , the set

C (i,j) (α) = {γ ∈ T B (i,j) : αγ = γα}.
We rewrite the following lemma from [START_REF] Ali | Centralizer of braids and Fibonacci numbers[END_REF] using the above construction.

Lemma 2.1. [2] If β ∈ T B i , i > 1 and σ i-1 |β, then γ ∈ C n (β) if and only if γ = γ 1 γ 2 , where γ 1 ∈ C i (β) and γ 2 ∈ C (i,n) (β) = T B (i,n) .
Now, consider the set C 0 (α) = {e} for any α ∈ T B n , T B 0 = {e} and T B (n,n) = {e}. Due to the symmetric nature of σ i and σ n-i in set T B n , and using Lemma 2.1, we obtain the following result.

Lemma 2.2. If β ∈ T B (i,n) and σ i+1 |β, then γ ∈ C n (β) if and only if γ = γ 1 γ 2 , where γ 1 ∈ C i (β) = T B i and γ 2 ∈ C (i,n) (β).
Lemma 2.1 and Lemma 2.2 lead to the following result.

Theorem 2.3. For any β ∈ T B n , if i + 1 = min{k : σ k |β} and j -1 = max{k : σ k |β} with i < j < n, then γ ∈ C n (β) if and only if γ = γ 1 γ 2 γ 3 , where γ 1 ∈ T B i , γ 2 ∈ C (i,j) (β) and γ 3 ∈ T B (j,n) .
This can be written symbolically as

C n (β) = T B i × C (i,j) (β) × T B (j,n) = {γ 1 γ 2 γ 3 : γ 1 ∈ T B i , γ 2 ∈ C (i,j) (β), γ 3 ∈ T B (j,n) }.
Proof. Because β ∈ T B j when σ j-1 |β, by Lemma 2.1, γ ∈ C n (β) if and only if γ = γ γ 3 , where γ ∈ C j (β) and γ 3 ∈ C (j,n) (β) = T B (j,n) . Additionally, the given conditions indicate that β ∈ T B (i,j) , with σ i+1 |β. Therefore, by Lemma 2.2, γ ∈ C j (β) if and only if γ = γ 1 γ 2 , where

γ 1 ∈ C i (β) = T B i and γ 2 ∈ C (i,j) (β).
For any α ∈ T B n , |C n (α)| denotes the number of elements in the tiny centralizer C n (α). [START_REF] Ashraf | Fibonacci numbers and positive braids[END_REF] shows that

|T B n | = F 2n-1 . Furthermore, |T B (j,n) | = |T B n-j | = F 2(n-j)-1 , as per our consideration, we take |T B 0 | = |{e}| = F -1 = 1.
As a consequence, we have the following corollary:

Corollary 2.4. For any β ∈ T B n , if i + 1 = min{k : σ k |β} and j -1 = max{k : σ k |β}, with i < j < n, then |C n (β)| = |C (i,j) (β)|F 2i-1 F 2(n-j)-1 .
Theorem 2.3 and Corollary 2.4 are generalizations of the previous results found in [START_REF] Ali | Centralizer of braids and Fibonacci numbers[END_REF], which were obtained only for the generators of B + n . We define a segment set in set T B n as

[σ i , σ i+j ] = {α ∈ T B n : σ k |α if and only if i ≤ k ≤ i + j} .
The construction of these segments is significant for set T B n . A complete characterization of C n (σ i ) is given in [START_REF] Ali | Centralizer of braids and Fibonacci numbers[END_REF], which shows that γ ∈ C n (σ i ) if and only if σ k γ for any k satisfying |k -i| = 1. As a generalization, we characterize the tiny centralizer

C n (α) for α ∈ [σ i , σ i+j ]. Theorem 2.5. If α ∈ [σ l , σ k ] ⊂ T B n , l ≤ k < n, then C n (α) = T B l-1 × {e, α} × T B (k+1,n) . Moreover, |C n (α)| = 2F 2l-3 F 2(n-k)-3 . Proof. Theorem 1.1 states that C n (α) = {e, α} for any α ∈ [σ 1 , σ n-1 ].
Therefore, for any α ∈ [σ l , σ k ], we have C (l-1,k+1) (α) = {e, α} in the set T B (l-1,k+1) . By substituting i -1 = l and j + 1 = k into Theorem 2.3, we obtain

C n (α) = T B l-1 × {e, α} × T B (k+1,n) . Consequently, |C n (α)| = |T B l-1 |×|{e, α}|×|T B (k+1,n) | = 2F 2l-3 F 2(n-k)-3 . Let I n = {α ∈ T B n : C n (α) = {e, α}}.
Clearly, all non-identity elements in the set T B 2 = {e, σ 1 } belong to I 2 , and those in the set T B 3 = {e, σ 1 , σ 2 , σ 1 σ 2 , σ 2 σ 1 } belong to I 3 . In the following result, we completely characterize the set I n for n ≥ 4.

Theorem 2.6. For n ≥ 4, we have

I n = [σ 1 , σ n-1 ] [σ 1 , σ n-2 ] [σ 2 , σ n-1 ] [σ 2 , σ n-2 ].
(2)

Proof. Theorem 1.1 states that an element α ∈ T B n \ {e} with σ i |α for all 1 < i < n -1 has a tiny centralizer C n (α) = {e, α}. Therefore, α ∈ I n .

If σ i α for some 1 < i < n -1, then we can write α = α 1 α 2 such that α 1 ∈ T B i and α 2 ∈ T B (i,n) . Without loss of generality, if α 1 = e, then α = α 2 , which implies that σ 1 α = ασ 1 . Consequently, if neither α 1 nor α 2 is an identity, then α 1 α = αα 1 and α 2 α = αα 2 . Hence, α ∈ I n . Lemma 1.2 gives the following:

I n = {α ∈ T B n : σ i |α for all 1 < i < n -1} = {α ∈ T B n : β|α for any one β ∈ [σ 2 , σ n-2 ]} = [σ 2 , σ n-2 ] [σ 1 , σ n-2 ] [σ 2 , σ n-1 ] [σ 1 , σ n-1 ].
Lemma 2.7. For n ≥ 2, in the set of tiny braids T B n , we have

|[σ 1 , σ 1+j ]| = |[σ i , σ i+j ]| = 2 j . Moreover, |I n | = 9 × 2 n-4 for any n ≥ 4. Proof. Clearly, |[σ 1 , σ 1+j ]| = |[σ i , σ i+j ]|. Furthermore, we use induction on j to show that |[σ 1 , σ 1+j ]| = 2 j . For j = 1, [σ 1 , σ 2 ] = {σ 1 σ 2 , σ 2 σ 1 }; hence, |[σ 1 , σ 2 ]| = 2. Let |[σ 1 , σ j ]| = 2 j-1 . By Lemma 1.2, [σ 1 , σ 1+j ] = σ 1+j [σ 1 , σ j ] [σ 1 , σ j ]σ 1+j . Therefore, |[σ 1 , σ 1+j ]| = |σ 1+j [σ 1 , σ j ]|+|[σ 1 , σ j ]σ 1+j | = |[σ 1 , σ j ]|+|[σ 1 , σ j ]| = 2×2 j-1 = 2 j .
By Equation ( 2), we obtain

|I n | = |[σ 1 , σ n-1 ] [σ 1 , σ n-2 ] [σ 2 , σ n-1 ] [σ 2 , σ n-2 ]| = |[σ 1 , σ n-1 ]| + |[σ 1 , σ n-2 ]| + |[σ 2 , σ n-1 ]| + |[σ 2 , σ n-2 ]| = 2 n-2 + 2 n-3 + 2 n-3 + 2 n-4 = 4 × 2 n-4 + 2 × 2 n-4 + 2 × 2 n-4 + 2 n-4 = 9 × 2 n-4 .
For any α ∈ I n ⊂ T B n , |C n (α)| = 2, and for every α ∈ I n , we have the following result. Proof. Clearly, {e, α} ⊆ C n (α) for every α ∈ T B n . If α ∈ I n , then by Lemma 2.6, σ i α for some 1 < i < n -1; we can write α = α 1 α 2 such that α 1 ∈ T B i and α 2 ∈ T B (i,n) . Following the proof of Lemma 2.6, we can see that either {e, α, σ 1 , σ

1 α} ⊆ C n (α), {e, α, σ n-1 , σ n-1 α} ⊆ C n (α) or {e, α, α 1 , α 2 } ⊆ C n (α).
C n (β) = T B 0 × {e, β} × T B (n-2,n) = {e, β, σ n-1 , βσ n-1 }. Therefore, |C n (β)| = |{e, σ n-1 , β, βσ n-1 }| = 4.
The shifting property of Fibonacci numbers, i.e.,

F m+n-1 = F m F n + F m-1 F n-1 , yields F m F n ≤ F m+n-1 . (3)
This inequality enables us to find the maximum cardinality of

C n (α) for α ∈ [σ i , σ i+j ]. Proposition 2.9. For all n > 3, |C n (σ 1 )| = |C n (σ n-1 )| = 2F 2n-5 ,

and for any other

α ∈ [σ l , σ k ] ⊂ T B n , l ≤ k < n, max α∈T Bn |C n (α)| = 2F 2n-7 . Proof. For α = σ 1 or α = σ n-1 , Theorem 2.5 yields |C n (α)| = 2F 2n-5 . For any other α ∈ [σ l , σ k ] ⊂ T B n , l ≤ k < n, Theorem 2.5 implies |C n (α)| = 2F 2l-3 F 2(n-k)-3
. By the Fibonacci number shifting property given in (3) and because k -l ≥ 0, we obtain

|C n (α)| ≤ 2F 2l-3+2(n-k)-3-1 = 2F 2(n-(k-l))-7 ≤ 2F 2n-7 .
3 Properties of the Commuting Graph Γ(T B n )

The following characterization of the vertex set V (Γ(T B n )) follows from the definition of Γ(T B n ) still be deleted to avoid being too repetitive or "wordy". and the proof of Lemma 2.6.

Lemma 3.1. A vertex α ∈ V (Γ(T B n )) if and only if σ i α for some 1 < i < n -1 and α = e.

Lemma 2.7 enables us to find the number of vertices (order) of Γ(T B n ).

Theorem 3.2. The order of

Γ(T B n ) is F 2n-1 -9 × 2 n-4 -1. Proof. By definition, V (Γ(T B n )) = T B n \ (I n ∪ {e}) implies that |V (Γ(T B n ))| = |T B n | -|I n | -1 = F 2n-1 -|I n | -1. By Lemma 2.7, |V (Γ(T B n ))| = F 2n-1 -9 × 2 n-4 -1. Proposition 3.3. For n ≥ 4, Γ(T B n ) is connected.
Proof. Consider two vertices α, β ∈ V (Γ(T B n )). By Lemma 3.1, σ i α for some 1 < i < n -1. Hence, we can write α = α 1 α 2 such that α 1 ∈ T B i and α 2 ∈ T B (i,n) . Similarly, σ j β for some 1 < j < n -1. Hence, β = β 1 β 2 such that β 1 ∈ T B j and β 2 ∈ T B (j,n) . We must consider two cases. Case 1: If none of α 1 , α 2 , β 1 β 2 is the identity, then for i ≤ j, the path ((α, α 1 ), (α 1 , β 2 ), (β 2 , β)) exists, while for i > j, the path ((α, α 2 ), (α 2 , β 1 ), (β 1 , β)) exists.

Case 2: Without loss of generality, we suppose that α 1 = e, which implies that α = α 2 . Consider β 2 = e; then, we have the path ((α, σ 1 ), (σ 1 , β 2 ), (β 2 , β)) .

If β 2 = e, then β = β 1 , and we have the path ((α, σ 1 ), (σ 1 , σ n-1 ), (σ n-1 , β)) .

Consequently, Γ(T B n ) is connected.

Since Γ(T B 4 ) = K 3 ,, the diameter of Γ(T B 4 ) is 1. For n ≥ 5, we have the following result. Theorem 3.4. For any n ≥ 5, the diameter of Γ(T B n ) is 3.

Proof. The proof of Proposition 3.3 shows that Γ(T B n ) has a diameter ≤ 3. To show that it is exactly 3, we show the existence of a path of length 3 between two vertices of Γ(T B n ). Since n ≥ 5, we consider

α = σ 2 σ 3 . . . σ n-3 , β = σ 3 σ 4 . . . σ n-2 ∈ V (Γ(T B n )), where, in fact, α ∈ [σ 2 , σ n-3 ] and β ∈ [σ 3 , σ n-2 ]. Therefore, by Theorem 2.5, C n (α) = T B 1 × {e, α} × T B (n-2,n) = {e, α, σ n-1 , ασ n-1 } and C n (β) = T B 2 × {e, β} × T B (n-1,n) = {e, β, σ 1 , σ 1 β}.
Clearly, the most likely shortest path between α and β is ((α, σ n-1 ), (σ n-1 , σ 1 ), (σ 1 , γ)).

Theorem 3.5. For α ∈ V (Γ(T B n )), n ≥ 4, K 3 is a subgraph of Γ(T B n ) such that α ∈ V (K 3 ). Moreover, girth(Γ(T B n )) = 3.
Proof. By Lemma 3.1, σ i α for some 1 < i < n -1. We can write α = α 1 α 2 such that α 1 ∈ T B i and α 2 ∈ T B (i,n) . If α 1 = e, then α = α 2 ∈ V (K 3 ) = {α, σ 1 , σ 1 α}; a similar existence can be shown for α 2 = e. If neither α 1 nor α 2 is an identity, then α ∈ V (K 3 ) = {α, α 1 , α 2 }. Clearly, girth(Γ(T B n )) = 3.

Based on the definition of the vertices of Γ(T B n ), the identity element {e} and element α ∈ V (Γ(T B n )) themselves do not contribute to the degree of α, which gives deg(α) = |C n (α)| -2. This motivates us to discuss the degree-related properties of the commuting graph as a consequence of the results obtained in the previous section. Recall that (F k ) is the Fibonacci sequence and that F -1 = 1. Moreover, if deg(α) = 0, then α ∈ V (Γ(T B n )). By setting l = k in Proposition 3.6, we obtain the following result. The following Corollaries are obtained from Theorem 2.8 and Proposition 2.9, respectively.

Theorem 2 . 8 .

 28 For n ≥ 4, min α ∈In |C n (α)| = 4 in T B n .

  Therefore, |C n (α)| ≥ 4 for all α ∈ T B n \ I n . To obtain the required result, we need to prove the existence of an element β ∈ T B n \ I n with |C n (β)| = 4. As n ≥ 4, we can consider β = σ 1 σ 2 . . . σ n-3 ∈ T B n , where β ∈ [σ 1 , σ n-3 ]. Hence, by Theorem 2.5,

Proposition 3 . 6 .

 36 For α ∈ [σ l , σ k ] ⊂ T B n , l ≤ k < n, n ≥ 4, we have deg(α) = 2(F 2l-3 F 2(n-k)-3 ) -2.Proof. It follows from Theorem 2.5.

Corollary 3 . 7 .

 37 For σ k ∈ T B n , n ≥ 4, we have deg(σ k ) = 2(F 2k-3 F 2(n-k)-3 ) -2.

Corollary 3 . 8 .Corollary 3 . 9 .

 3839 For n ≥ 4, the minimum degree δ(Γ(T B n )) = 2. For any n ≥ 4, deg(σ 1 ) = deg(σ n-1 ) = 2F 2n-5 -2, and for any other α ∈ [σ l , σ k ] ⊂ T B n , l ≤ k < n, max(deg(α)) = 2F 2n-7 -2.

Open Problems

The following questions related to Γ(T B n ) remain to be explored.