Y A G B E S S I

B A L I B E N Y A H I A

C L E M A I T R E

L C H O P L I N

Y Agbessi

B Alibenyahia

C Nouar

C Lemaitre

L Choplin

Linear stability of Taylor-Couette flow of shear-thinning fluids: modal and non-modal approaches

In this paper, the receptivity of circular Couette flow of shear-thinning fluids between two infinitely long coaxial cylinders to weak disturbances is addressed. It is highlighted by pseudospectra and transient growth analysis. Power-law model and Carreau model are used to describe the rheological behavior of the fluid.

The first part of the paper deals with the asymptotic long-time behavior of threedimensional infinitesimal perturbations. Using the normal-mode approach, an eigenvalue problem is derived and solved with spectral collocation method. Extensive description and classification of eigenspectra is presented. The influence of shear-thinning effects on the critical Reynolds numbers as well as on the critical azimuthal and axial wavenumbers is analyzed.

The second part investigates the short-time behavior of the disturbance using the non-

Introduction

The Taylor-Couette flow of a viscous incompressible fluid between two infinitely long coaxial cylinders rotating at different angular velocities is considered as a paradigm for the study of hydrodynamic instabilities and transition in nonlinear dynamical systems.

A survey of the literature on the Taylor-Couette problem can be found in [START_REF] Koschmieder | Bénard Cells and Taylor Vortices[END_REF] and [START_REF] Tagg | The Couette-Taylor problem[END_REF]. The basic state, i.e. a Circular Couette Flow, CCF, is purely azimuthal and presents only a radial variation across the annular gap. The linear instability of the base flow with respect to axisymmetric perturbations is governed in the inviscid limit by the Rayleigh criterion: the CCF flow is unstable with respect to axisymmetric perturbations when the square of the angular momentum, L 2 , decreases with the radius, r, at some radial position [START_REF] Drazin | Hydrodynamic stability[END_REF]. In the plane (Re 1 , Re 2 ),

where Re 1 and Re 2 are respectively the inner and outer Reynolds numbers, the Rayleigh stable and unstable domains are separated by the Rayleigh line. For Newtonian fluids, the Rayleigh line is given by Re 2 /Re 1 = η, where η = R 1 /R 2 is the radius ratio. In this case, the angular momentum is constant in the annular gap. In the Rayleigh stable domain, dL 2 /dr > 0 in the whole annular space. On the contrary in the unstable domain and in the counter-rotating configuration, dL 2 /dr is negative near the inner cylinder and positive near the outer one. This appears to have a great effect on the wavelength of the critical mode [START_REF] Taylor | Stability of a viscous liquid contained between two rotating cylinders[END_REF]. Generalized Rayleigh criterion for non axisymmetric perturbations has been derived for a free axisymmetric vortex using a large axial wavenumber WKB approximation [START_REF] Billant | Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities[END_REF].

For a viscous fluid, the same instability mechanism applies, except that the instability occurs if the destabilizing force overpasses the viscous force on a time scale smaller than the diffusion time scale. When the outer cylinder is fixed, the angular momentum decreases from the inner to the outer cylinder. The instability of the CCF of a viscous incompressible Newtonian fluid appears via a stationary axisymmetric disturbance [START_REF] Taylor | Stability of a viscous liquid contained between two rotating cylinders[END_REF] in the form of toroidal counter-rotating vortices regularly spaced along the axis of the cylinders. These vortices are separated by radial inflow and outflow.

For counter-rotating configuration, L 2 decreases with the radius r near the inner cylinder and increases near the outer one. It is shown from linear stability analysis that for sufficient counter-rotation of the outer cylinder, the CCF becomes unstable first to a non-axisymmetric mode [START_REF] Krueger | On the relative importance of Taylorvortex and non-axisymmetric modes in flow between rotating cylinders[END_REF][START_REF] Snyder | Stability of a rotating Couette flow. i Asymmetric waveform[END_REF][START_REF] Langford | Primary instabilities and bicriticality in flow between counter rotating cylinders[END_REF]. The azimuthal wavenumber of the new flow increases as the outer Reynolds number increases [START_REF] Langford | Primary instabilities and bicriticality in flow between counter rotating cylinders[END_REF]. These spiral vortices are traveling waves in the axial direction and rotating waves in the azimuthal direction [START_REF] Altmeyer | Islands of instability for growth of spiral vortices in the Taylor-Couette system with and without axial through flow[END_REF][START_REF] Tagg | The Couette-Taylor problem[END_REF]). In the unstable Rayleigh regime and co-rotating cylinders, where L 2 decreases from the inner to the outer cylinder, the CCF becomes centrifugally unstable with respect to axisymmetric toroidal vortices.

Increasing the Reynolds number above the critical values leads to the occurrence of different patterns depending on whether both cylinders are rotating or only the inner one is rotating. [START_REF] Andereck | Flow regimes in a circular Couette system with independently rotating cylinders[END_REF] investigated by visualization the diversity of flow states depending on the inner and outer Reynolds numbers. They identified approximately 20 flow types displayed in the plane (Re 1 , Re 2 ). When the outer cylinder is fixed, or at low Y. Agbessi, B. Alibenyahia, C. Nouar, C. Lemaitre and L. Choplin Re 2 , the transition to turbulence occurs through successive linear instabilities. The flow becomes more and more complex [START_REF] Andereck | Flow regimes in a circular Couette system with independently rotating cylinders[END_REF]). The transition is supercritical.

For counter-rotating cylinders with sufficiently large Re 2 , the subcritical nature of this transition has been shown experimentally by [START_REF] Coles | Transition in circular Couette flow[END_REF] and [START_REF] Atta | Exploratory measurements in spiral turbulence[END_REF]. These authors observed intermittent spiral turbulence below the linear stability boundary of the basic CCF.

The transient energy growth of a perturbation is considered as a prerequisite for a subcritical bifurcation. In this mechanism, small perturbations interact with the shear profile to create much more energetic disturbance. It is argued that despite linear stability, large growth of perturbation may trigger nonlinear effects which can sustain its energy away from zero for all times. From mathematical point of view, the transient growth is associated to the non-normality of the linear operators involved in the linear stability problem [START_REF] Trefethen | Threshold amplitudes for transition to turbulence in a pipe[END_REF]. Such effect was pointed out first by [START_REF] Gebhart | The Taylor-Couette eigenvalue problem with independently rotating cylinders[END_REF] for the Taylor-Couette problem. The features of this transient growth have been investigated by [START_REF] Hristova | Transient growth in Taylor-Couette flow[END_REF] by the computation of the pseudospectra for axisymmetric perturbation and fixed axial periodicity, with counter-rotating cylinders such that the net effect of the Coriolis force vanishes. The authors show that the curvature increases the transient growth. [START_REF] Meseguer | Energy transient growth in the Taylor-Couette problem[END_REF] determined, for a fixed radius ratio (η = 0.881), the optimal axial and azimuthal wavenumbers which provide the maximum transient growth. The computations were performed in the linear stable regime with counter-rotating cylinders. Recently results of transient growth calculations covering all linearly stable regimes of Taylor-Couette flow were presented by [START_REF] Maretzke | Transient growth in linearly stable Taylor-Couette flows[END_REF].

To do this, they have used the shear Reynolds number and the rotation number introduced by [START_REF] Dubrulle | Stability and turbulent transport in Taylor-Couette flow from analysis of experimental data[END_REF]. The authors found that optimal transient growth, in linearly stable Taylor-Couette flow, varies as Re 2/3 for large Re, suggesting that those effects might be significant at large Re.

To our knowledge, the transient growth in Taylor-Couette flow of non-Newtonian fluids has not been considered before in the literature. Actually, comparatively to Newtonian fluids, few studies have been devoted to the Taylor-Couette problem for non-Newtonian fluids despite the fact that they are involved in a wide variety of industrial applications.

Their material properties have a strong impact on the flow behavior which may modify the conditions and the mechanisms of flow instabilities. Most non-Newtonian fluids have two common properties, viscoelasticity and shear-thinning. Polymer and colloid solutions as well as particulate dispersions display this behavior above a certain concentration threshold. There was a significant interest in inertialess viscoelastic Taylor-Couette instability, observed initially by [START_REF] Giesekus | Zur Stabilität von Strömungen viskoelastischer Flüssigkeiten. 1.Ebene und kreisförmige Couette-Strömung[END_REF] and then analyzed by [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], [START_REF] Shaqfeh | The effect of gap width and dilutesolution properties on the viscoelastic Taylor-Couette instability[END_REF]. In the laminar state, the rotation produces a shear which stretches the polymer molecules along the curved stream lines. This leads to a first normal stress-difference which acts against the centrifugal force. [START_REF] Groisman | Mechanism of elastic instability in Couette flow of polymer solutions: Experiments[END_REF] showed experimentally that the elastic instability leads to a strong nonlinear flow transition at vanishing inertia. Hereafter, we neglect the elastic response.

We focus only on the shear-thinning effects, i.e. the influence of nonlinear decrease of the effective viscosity with the shear rate. The shear-thinning behavior arises from the re-organization of the internal fluid structure in order to reduce the viscous dissipation. where µ b denote the viscosity of the base flow, is even more significant as the shearthinning is more pronounced and the radius ratio η is small. The azimuthal velocity profile is also modified: with increasing shear-thinning effects, the shear rate increases at Y. Agbessi, B. Alibenyahia, C. Nouar, C. Lemaitre and L. Choplin the inner wall and decreases at the outer one. The mechanism of instability described for a Newtonian fluid applies for shear-thinning fluids, however, the critical conditions are modified due to the viscosity stratification and the variation of the velocity profile.

Almost all the papers dealing with the CCF of shear-thinning fluids consider the case where the inner cylinder is rotating and the outer is fixed. The critical conditions were determined experimentally or numerically for different radius ratios and different shearthinning degrees.

From an experimental point of view, [START_REF] Sinevic | Power numbers, Taylor numbers and Taylor vortices in viscous Newtonian and non-Newtonian fluids[END_REF] determined the onset of Taylor vortices by measuring the torque exerted on the inner rotating cylinder. The results were obtained for two radius ratios η = 0.7 and 0.9. The rheological behavior of the fluids used, CMC and Carbopol solutions were described by the power-law model. [START_REF] Escudier | Taylor vortices in Newtonian and shear-thinning liquids[END_REF] investigated the flow structure in a Taylor-Couette geometry with radius ratio η = 0.5. The fluids used was an aqueous solution of Xanthan gum 0.15 %w, which is shear-thinning and slightly viscoelastic and an aqueous solution of Laponite which is also shear-thinning and thixotropic. Axial and tangential velocity measurements were made using a laser Doppler anemometer. The critical Reynolds number, Re c defined with the inner wall shear viscosity, was determined by monitoring the development of the axial velocity component at a fixed radius with increasing the rotation speed of the inner cylinder. The authors observed that: (i) the onset of Taylor vortices is much more gradual than that for a Newtonian fluid and (ii) the shear-thinning behavior induces a significant radial shift in the location of the vortex eye towards the centerbody.

From a numerical point of view, the critical conditions were either determined using 2D direct numerical simulation [START_REF] Lockett | The stability of inelastic non-Newtonian fluids in Couette flow between concentric cylinders[END_REF][START_REF] Coronado-Malutti | Instability of inelastic shear-thinning liquids in a Couette flow between concentric cylinders[END_REF] or by solving a generalized eigenvalue problem which results from a linear stability analysis [START_REF] Jastrzebski | Stability of Couette flow of liquids with power-law viscosity[END_REF][START_REF] Ashrafi | Shear-induced chaos in Taylor-vortex flow[END_REF][START_REF] Caton | Linear stability analysis of circular Couette flow of inelastic viscoplastic fluids[END_REF]. A critical review of these studies was done by [START_REF] Alibenyahia | Revisiting the stability of circular Couette flow of shear-thinning fluids[END_REF]. It was shown in this paper that when the Reynolds number is defined using the inner-wall shear viscosity, shear-thinning delays the appearance of Taylor vortices. This delay is due to a reduction in the energy exchange between the base flow and the perturbation and not to the modification of the viscous dissipation.

In all the previous studies, the outer cylinder is fixed. The case where the two cylinders are both rotating was considered , to our knowledge, only by [START_REF] Pascal | Stability of power law fluid flow between rotating cylinders[END_REF]. These authors determined the critical Reynolds number for power-law fluids at two radius ratios η = 0.7 and 0.9. The results are limited to two values of shear-thinning index (equation 2.8) n = 0.8 and 1.2. The perturbation is assumed to be axisymmetric and the viscosity perturbation was not taken into account.

Objectives, methodology and outline of the paper

The objectives of the present study are to investigate the influence of the shear-thinning behavior characterized by the stratification of the viscosity and the modification of the velocity profile on a three-dimensional linear stability analysis of CCF. In this study, both cylinders are rotating independently and thin and wide gaps are considered. Short-and long-time evolution of the perturbation will be considered. Many models are available to describe the dependence of the viscosity on the shear rate. All the governing equations of the present paper are given for a general nonlinear purely viscous fluid. For numerical computations we adopt two models: (i) the power-law and (ii) the [START_REF] Carreau | Rheological equations from molecular network theories[END_REF] model.

Power-law model is the most popular. It has the advantage to provide simple analytical expressions. However for very large annular spaces, shear rate at the vicinity of the outer cylinder could be very low and, in this situation, the power-law model is not very accurate. The Carreau model has a sound theoretical basis and is C ∞ (differentiable for all degrees of differentiation) with respect to the second invariant of the strain-rate tensor Γ. The main disadvantage is that the base flow has to be calculated numerically.

Note that the Carreau model approaches the power-law as the ratio of the infinite-shear viscosity µ ∞ to the zero-shear viscosity µ 0 can be neglected and the characteristic time of the fluid λ is very large.

The article is organized as follows. In section 2, the governing equations and dimensionless parameters are given. The effect of the rheology on the base flow and on the Rayleigh line is described in section 3 and 4. The linear perturbation equations are derived in section 5.

In section 6, the long-time behavior of the perturbation is considered. The influence of the shear-thinning effect on the eigenspectra and critical conditions are analyzed. The linear stability problem is presented. The resulting eigenvalue problem is solved numerically.

Section 7 deals with the pseudo-spectra, transient growth and optimal perturbation.

Conclusions of this study are presented in section 8.

Mathematical formulation

Governing equations

We consider the flow of a purely viscous non-Newtonian fluid between two infinitely long concentric cylinders. The inner cylinder of radius R1 is rotating with a constant angular velocity Ω1 > 0. The outer cylinder of radius R2 is rotating with a constant angular velocity Ω2 , with Ω2 > 0 for co-rotating cylinders and Ω2 < 0 for counterrotating cylinders. The radius ratio is η = R1 / R2 . The incompressibility condition and the momentum equations are

∇ • U = 0,
(2.1)

∂U ∂t + Re 1 (U • ∇) U = -∇P + ∇ • τ , (2.2)
where U = U e r + V e θ + W e z is the velocity vector in cylindrical coordinates (r, θ, z), P the pressure and τ the deviatoric extra-stress tensor. The quantities denoted with a hat (.) are dimensional while quantities without a hat are non-dimensional. We define the Reynolds number for the inner Re 1 and the outer Re 2 cylinders as

Re 1 = ρ R1 Ω1 d μref , Re 2 = ρ R2 Ω2 d μref .
(2.3)

The governing equations have been non-dimensionalized using the width of the annular space d = R2 -R1 as the reference length scale, the velocity of the inner cylinder Ω1 R1

as the velocity scale, the viscous diffusion time ρ d2 /μ ref for time scale and the quantity μref R1 Ω1 / d for stresses and pressure scale where μref is a reference viscosity which will be specified later for each rheological model. To the previous equations, we add the no-slip and impermeability conditions at the walls,

U (R 1 , θ, z) = (0, 1, 0) , U (R 2 , θ, z) = 0, Re 2 Re 1 , 0 , (2.4)
where the dimensionless radii are expressed as R 1 = η/(1 -η) and R 2 = 1/(1 -η).

Rheological models

We consider a purely viscous shear-thinning fluid. The viscous stress tensor reads

τ = µ (Γ) γ with γ = ∇u + (∇u) T (2.5)
where the second invariant of the strain-rate tensor is expressed as

Γ = 1 2 γij γij .
(2.6)

-The power-law model is given by

μ = K Γ n-1 2 , (2.7)
where K is the consistency, and n 1 the shear-thinning index. In dimensionless form, the viscosity becomes

µ = Γ n-1 2 , (2.8)
and the reference viscosity reads

μref = K R1 Ω1 / d n-1
.

(2.9)

-The Carreau model is given by

μ -μ∞ μ0 -μ∞ = 1 + λ2 Γ n-1 2 , (2.10)
with μ0 and μ∞ the viscosities at low and high shear rates 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 1/ λ. The infinite-shear viscosity μ∞ is generally associated with a breakdown of the fluid, and is frequently significantly smaller (10 -3 to 10 -4 times as small) than μ0 , see [START_REF] Bird | Dynamics of polymeric liquids[END_REF] and [START_REF] Tanner | Engineering rheology[END_REF]. The ratio μ∞ /μ 0 will thus be neglected in the following. Increasing λ reduces the Newtonian plateau to lower shear rates and the model thus tends to a power-law model. The dimensionless effective viscosity is then .11) and the viscosity scale reads in this case μref = μ0 .

µ = 1 + λ 2 Γ n-1 2 with λ = λ ρ μref d2 , ( 2
(2.12)

Note that the reference viscosities adopted here (2.9) and (2.12) are those widely used in the literature.

Governing parameters

The dimensionless governing parameters are the rheological parameters, n for the powerlaw model, n and λ for the Carreau model, the radius ratio η and the inner and outer

Re = ρ γb rθ r d2 μb ( r) , R Ω = 2 Ωf ( r) γb rθ ( r)
.

(2.13)

The superscript b refers to the base flow which is purely azimuthal, γb rθ ( r) is the base shear rate evaluated at a radial position r. Here, r is the radial position of a frame rotating with an angular velocity Ωf ( r) such that

[ V b ( R1 ) -Ωf R1 ] = -[ V b ( R2 ) -Ωf R2 ]. (2.14)
With respect to this rotating frame, the cylinders are seen to rotate at exact opposite velocities. For a Newtonian fluid, it can be shown straightforwardly [START_REF] Dubrulle | Stability and turbulent transport in Taylor-Couette flow from analysis of experimental data[END_REF])

that r = R 1 R 2 , Re = 2|ηRe 2 -Re 1 | 1 + η , R Ω = (1 -η) (Re 1 + Re 2 ) ηRe 2 -Re 1 . (2.15)
For power-law fluids, using equations (2.13) and (2.14), we obtain

r = (R 1 + R 2 ) R 2 n -1 1 R 2 n -1 2 R 2 n -1 1 + R 2 n -1 2 n 2 , Re = Re 1 γb rθ (r) (n-2) , R Ω = - 2 n B 0 r2/n ,(2.16)
where B 0 is given by equation (3.5).

= η -1 R Ω =(1 -η)/η R Ω = (1 -η)/η R Ω = -1 R Ω = ± ∞ R Ω = 0 .1 R Ω = 0 . 3 (
R Ω = ± ∞ R Ω = -1 R Ω = η -1 R Ω = 0. 1 R Ω = 0.13 R Ω = 0.13 R Ω = 0 . 3 (

Base flows

The base flow is azimuthal and is solution of

d dr r 2 τ b rθ = 0 , (3.1)
with the boundary conditions

V b (R 1 ) = 1 and V b (R 2 ) = Re 2 Re 1 . (3.2)
Hereafter, we describe the velocity and the viscosity profiles for power-law and Carreau fluids in the case of a thin and a wide gap, η = 0.881 and η = 0.4 respectively.

Power-law fluid

The velocity profile is [START_REF] Bird | Dynamics of polymeric liquids[END_REF])

V b (r) = A 0 r + B 0 r r 2/n (3.3) with A 0 = 1 R 1   1 + 1 R 2/n 1 1 - Re 2 Re 1 η 1 R 2/n 2 - 1 R 2/n 1 -1   (3.4) B 0 = - 1 R 1   1 - Re 2 Re 1 η 1 R 2/n 2 - 1 R 2/n 1 -1   (3.5)
Figure 2 shows the effect of the shear-thinning index on the velocity and viscosity profiles for η = 0.4. With increasing shear-thinning effects, the curvature of the velocity profile becomes more pronounced, figure 2(a). Shear rate increases at the inner wall and decreases at the outer one. The wider the gap the more this effect is significant. This results in a viscosity stratification within the gap, figure 2(b). The viscosity as well as the degree of the viscosity stratification increases from the inner to the outer wall. This is more pronounced with increasing shear thinning effects (decreasing n). Figure 3 shows an example of velocity profile for co-rotating cylinders. The same characteristics as for counter-rotating cylinders are observed, except that the velocity profile is convex instead of concave.

Carreau fluid

Since no analytical expression of the velocity profile exists in that case, equation (3.1) is solved numerically using an iterative process combined with a Chebyshev polynomial approximation of the velocity profile. Details can be found in [START_REF] Alibenyahia | Revisiting the stability of circular Couette flow of shear-thinning fluids[END_REF].

For a given value of λ, the effect of index n is similar to the effect of the power-law fluid index. When increasing λ, the base flow tends to that of a power-law fluid. This is observed for values of λ as low as λ = 10 (figure 2).

0.6 0.8 1 1.2 1.4 1.6 -10 -8 -6 -4 -2 0 1 r V b (1) (2) (3) (a) 0.6 0.8 1 1.2 1.4 1.6 -10 0 20 40 60 (1) (2) r µ b /µ b (r=R 1 ) (3)

Solid rotation and Rayleigh lines

The solid rotation line corresponds to the situation where Ω1 = Ω2 and Re 1 = ηRe 2 .

It is represented with dashed lines in figure 4. In the parametrization of [START_REF] Dubrulle | Stability and turbulent transport in Taylor-Couette flow from analysis of experimental data[END_REF], the solid line is given by R Ω = ±∞.

As indicated in the introduction, the instability of the Taylor-Couette flow is due to the stratification of the angular momentum. When the cylinders are co-rotating, the Rayleigh stable regime for an inviscid fluid is defined by

d dr (L 2 ) > 0, ∀ r ∈ [R 1 , R 2 ],
where L = rV b is the angular momentum [START_REF] Drazin | Hydrodynamic stability[END_REF]. This regime is bounded Following the same approach for the non-Newtonian models considered here, the Rayleigh criterion is derived straightforwardly.

For power-law model, we obtain

Re 1 = 1 n(η 2/n -1) + 1 ηRe 2 (4.2)
For the Carreau model, the Rayleigh line is computed numerically. As expected, for sufficiently large λ, say λ > 10, we recover the power-law results for the same n. Decreasing λ reduces shear-thinning effects and tends progressively to the Newtonian case.

As illustrated in figures 4(a)-(b), with increasing shear-thinning effects (decreasing n or increasing λ) the Rayleigh line approaches the solid rotation line, thus reducing the socalled quasi-Keplerian zone which is delimited by these two lines in plane (Re 1 , Re 2 ).

In the quasi-Keplerian regime, the angular velocity decreases radially while the angular momentum increases radially.

Note that for a Newtonian fluid, the Rayleigh line corresponds to R Ω = -1. This effect is more significant for a large gap and strong shear-thinning effects. For instance, for a power-law fluid with n = 0.3 and η = 0.4, the Rayleigh line corresponds to R Ω = -3.17.

Linear perturbation equations

In a classical way, we superimposed to the base flow an infinitesimal perturbation

{U , P, τ } = {U b , P b , τ b } + ǫ{u ′ , p ′ , τ ′ } .
(5.1)

Substituting (U , P, τ ) by their expression (5.1) in equations (2.1)-( 2.2) and linearizing around the base flow, we obtain the perturbation equations

∇ • u ′ = 0 , (5.2) ∂u ′ ∂t + Re 1 U b • ∇ u ′ + (u ′ • ∇) U b = -∇p ′ + ∇ • τ ′ , (5.3)
where

τ ′ = 1 ǫ τ U b + ǫu ′ -τ U b = µ b γ (u ′ ) + µ t -µ b A , (5.4)
is the perturbation of the deviatoric shear-stress tensor. The components of tensor A are all zero except for A 1,2 = A 2,1 = γrθ (u ′ ). We have introduced in equation (5.4) the tangential viscosity µ t

µ t = µ U b + ∂µ ∂ γrθ U b γrθ U b .
(5.5)

We obtain the system

0 = D * u ′ + 1 r ∂v ′ ∂θ + ∂w ′ ∂z , (5.6) ∂u ′ ∂t = -Re 1 V b r ∂u ′ ∂θ -2v ′ -Dp ′ + µ b ∆u ′ -2 1 r 2 ∂u ′ ∂θ - u ′ r 2 + 2Dµ b Du ′ + 1 r ∂(µ t -µ b ) ∂θ Dv ′ + 1 r ∂u ′ ∂θ , (5.7) ∂v ′ ∂t = -Re 1 u ′ D * V b + V b r ∂v ′ ∂θ - 1 r ∂p ′ ∂θ + µ b ∆v ′ + 2 r 2 ∂u ′ ∂θ - v ′ r 2 + Dµ b Dv ′ - v ′ r + 1 r ∂u ′ ∂θ + D * µ t -µ b Dv ′ + 1 r ∂u ′ ∂θ + µ t -µ b 1 r Dv ′ + 1 r ∂u ′ ∂θ , (5.8) ∂w ′ ∂t = -Re 1 V b r ∂w ′ ∂θ - ∂p ′ ∂z + µ b ∆w ′ + Dµ b Dw ′ + ∂u ′ ∂z , (5.9)
where

D = ∂ ∂r , D * = D + 1 r , D = D - 1 r and ∆ = D 2 + 1 r D + 1 r ∂ ∂θ 2 + ∂ ∂z 2 .
The boundary conditions are obtained by assuming no slip and impermeability at inner and outer walls,

u ′ = 0 at r = R 1 , R 2 .
(5.10)

The linear system (5.6) -(5.9) is homogeneous in the axial and azimuthal directions.

Perturbations can thus be decomposed into Fourier modes with an azimuthal wavenumber m ∈ Z and an axial wavenumber k ∈ R:

{u ′ , p ′ } = {ũ(r, t), p(r, t)} exp [i(mθ + kz)] + c.c., (5.11)
where, c.c. stands for the complex conjugate and ũ(r, t) = ũ(r, t) e r +ṽ(r, t) e θ + w(r, t) e z .

By eliminating the pressure and one velocity component, the resulting system can be written in terms of two components of the velocity (ũ, ṽ) for k = 0 or (ũ, w) for m = 0.

The different formulations lead to an initial value problem which can be written formally as Lq = M ∂q ∂t (5.12)

where q = (ũ, ṽ) T or (ũ, w) T depending on the formulation adopted. The linear operators L and M are given in appendix A.

Long-time behavior of the disturbance: eigenvalue problem

The initial value problem can be turned into an eigenvalue problem by the assumption of the exponential time dependence, i.e. q(r, t) = q(r) exp(σt). The resulting generalized eigenvalue problem reads

Lq = σ M q (6.1)
with the complex frequency σ as the eigenvalue. The real part of σ is the growth rate of the perturbation and its imaginary part is its oscillations frequency. The eigenvalue problem (6.1) is discretized using a spectral collocation method based on Chebyshev polynomials evaluated at N collocation points of Gauss-Lobatto. The matrix eigenvalue problem that results is solved using QZ algorithm with Matlab. To ensure the convergence of the numerical method, computations were performed for increasing N . It was found that for N 40, the five first eigenvalues remain unchanged with three digits accuracy. Each computation was then performed for N and N + 5 and only the overlapping eigenvalues were retained. This procedure allows one to remove spurious eigenvalues.

Eigenvalue spectra

In this section, eigenspectra are analyzed for axisymmetric and non axisymmetric perturbations. For both situations, the inner Reynolds number is fixed and the outer Reynolds number is varied. Then, Re 2 is fixed and Re 1 is increased until the marginal stability is reached.

6.1.1. Axisymmetric perturbation m = 0

The perturbation is two-dimensional and the (ũ, ṽ) formulation is used. As m = 0, the linear operators L and M, given by the equations (A 2-A 7) in the appendix A are real. Therefore, the eigenvalues are real or complex conjugate. Spectra for that case are presented in figure 5 for a Newtonian fluid.

On the Rayleigh line, the angular momentum is constant. The spectrum, figure 5(a), consists of a vertical branch with real eigenvalues. It is independent of Re 1 , as indicated by [START_REF] Gebhart | The Taylor-Couette eigenvalue problem with independently rotating cylinders[END_REF]. These properties are demonstrated in appendix B.

When the outer cylinder is fixed, Re 2 = 0, most of the eigenvalues are real, figure 5(b). [START_REF] Diprima | Complex eigenvalues for the stability of couette flow[END_REF] have shown that the first eigenvalue that crosses the real axis is always real, in agreement with the principle of stability exchange.

For counter-rotating cylinders, the real vertical branch, which is made of paired eigenvalues, splits into two lateral curved branches, i.e. one pair of eigenvalues separates into two complex conjugate eigenvalues, figure 6(a). The higher the absolute value of Re 2 , the lower the locus of the branch separation. The imaginary part of the spectra are bounded by the epicyclic frequency κ evaluated at the outer wall, r = R 2 [START_REF] Shu | The Physical Universe: An Introduction to Astronomy[END_REF]. The epicyclic frequency is defined by Additional computations not presented here were conducted for co-rotating cylinders and spectrum evolutions similar to the counter-rotating situation have been obtained.

κ 2 (r) = - 1 r 3 d dr (rV ) 2 . (6.2) For a Newtonian fluid at r = R 2 , κ 2 (R 2 ) = 4Re 2 (Re 2 -ηRe 1 ) 1 -η 1 + η . ( 6
For shear-thinning fluids, eigenspectra are quite similar to the Newtonian case. However,

Non-axisymmetric perturbation m = 0

Numerical computations show no fundamental differences in the eigenspectra shape for Newtonian and shear-thinning fluids. In order not to overload the paper, we have chosen to present only the power-law results. The classification of the eigenspectra is less evident than in the axisymmetric case. According to [START_REF] Gebhart | The Taylor-Couette eigenvalue problem with independently rotating cylinders[END_REF], the structure of the eigenspectra depends on the relative value of the axial wavenumber compared to the modified azimuthal wavenumber m/r m with r m = (R 1 + R 2 )/2. If k m/r m , the spectrum structure is close to that of an azimuthal invariant flow. On the contrary, if k m/r m , features of an axial invariant flow dominate. Figure 8 shows that with increasing m, the eigenvalue spectrum evolves from a "jet-like" structure (figure 8(a))

characteristic of axisymmetric perturbation flow to a "Y -shaped" spectrum characteristic of axial invariant perturbation [START_REF] Gebhart | The Taylor-Couette eigenvalue problem with independently rotating cylinders[END_REF]. Figure 8(d) shows that when Re 1 is increased until marginal stability (σ r = 0), the critical mode corresponds to time-dependent spiral vortices, with an angular phase velocity ω = -σ i /m.

Marginal stability curves

First of all, a validation test was performed by confronting our results to studies previously published in the literature. In figure 9 our results are compared to those obtained by [START_REF] Meseguer | Energy transient growth in the Taylor-Couette problem[END_REF] for 3D perturbations in a Newtonian fluid. A very good agreement is observed. In the case of counter-rotating cylinders, the critical Re 1c is lower for non axisymmetric modes (m = 0) than for axisymmetric ones (m = 0), in agreement with the literature [START_REF] Krueger | On the relative importance of Taylorvortex and non-axisymmetric modes in flow between rotating cylinders[END_REF][START_REF] Langford | Primary instabilities and bicriticality in flow between counter rotating cylinders[END_REF]. For power-law fluids, figure 10 shows the influence of the shear-thinning index on the critical conditions. Globally, for fixed Re 2 , shear-thinning reduces Re 1c except in the neighborhood of Re 2 = 0 with a large gap and for co-rotating cylinders with small gaps, where Re 1c increases slightly.

Similar evolutions are observed for Carreau fluids. For λ 10, we recover the power-law results. The decrease of Re 1c with increasing shear-thinning effects can be analyzed using the energy equation obtained by multiplying the linearized momentum equations with the complex conjugate u * and by integrating between the two cylinders

Re(σ)||u|| 2 = -Re 1 R2 R1 f (r)rdr - R2 R1 g 1 (r)rdr + R2 R1
g 2 (r)rdr with (6.4)

f (r) = γb rθ u * v + uv * 2 (6.5) g 1 (r) = µ b 1 2 | γrr | 2 + | γθθ | 2 + | γzz | 2 + | γrθ | 2 + | γrz | 2 + | γθz | 2 (6.6) g 2 (r) = µ b -µ t | γ′ rθ | 2 . (6.7) with ||u|| 2 = R2 R1 1 2 (uu * + vv * + ww * ) rdr.
In equation ( 6 At criticality, the production term is exactly balanced by the dissipation term, Γ + = Γ -, as shown in figure 11 in the case of a Newtonian fluid. The influence of shear-thinning effects on the energy budget can be appreciated by comparing the results obtained for a Newtonian fluid with those of a power-law at the same value of Re 1 , i.e. Re 1 = Re 1c (Newtonian). The critical azimuthal and axial wavenumbers are evaluated for the parameters Re 1 = Re 1c (Newtonian), Re 2 and n. With increasing shear-thinning effects, the average viscous dissipation remains close to that of a Newtonian fluid, while the production term increases significantly, rendering the flow less stable compared to the Newtonian case. The main factor determining stability or instability of the flow is thus the exchange of energy between the base flow and the perturbation. In figure 12, the location of the maxima f and (g 1 +g 2 ), functions defined in equations (6.5 -6.7) are plotted against Re 2 . It appears that the maxima of the energy exchange and of viscous dissipation occur near the inner cylinder. The increase of Γ + with increasing shear-thinning effects may be attributed to the increase of the inner velocity gradient. Concerning the critical azimuthal and axial wavenumbers, in the co-rotating regime, the critical mode is axisymmetric (figure 10(c)(d)). The primary bifurcation leads to time independent Taylor-vortex flow.

In the case of counter-rotating regime, the critical azimuthal wavenumber increases with |Re 2 |. There is a limit value of Re 2 , say Re 2s under which the primary bifurcation leads to axisymmetric toroidal structures (m c = 0) and over which the bifurcation leads to spiral vortices (m c 1). Spiral vortices are traveling waves in axial and azimuthal directions.

Figure 13 shows the variation of Re 2s with shear-thinning index for both a wide and a thin gap. For strong shear-thinning effects, the critical mode remains axisymmetric for large negative Re 2 . The axial wavelength decreases with increasing negative Re 2 and increasing shear-thinning effects. This may be attributed to the decrease of the "effective gap" where the Rayleigh instability crietrion holds, i.e. the gap between the inner cylinder and the "nodal surface" where V b = 0.

Short-time behavior: transient growth and optimal perturbation

The traditional approach of stability based on eigenvalues is not sufficient to describe the temporal behavior of the disturbance at all times because of the non-normality of the linear stability operator. The non-normality of the linear stability operator L = M -1 L is characterized by using the ε-pseudospectrum tools (see for instance [START_REF] Trefethen | Hydrodynamic stability without eigenvalues[END_REF]; [START_REF] Reddy | Pseudospectra of the Orr -Sommerfeld Operator[END_REF], and references therein). More details are given in appendix E. This mathematical property means that there is a potential for energy extraction from the base flow by a subspace of perturbations leading to transient growth, despite the absence of exponential instability.

Transient growth and characteristics of optimal perturbation

The transient evolution of a perturbation kinetic energy in the linear regime is determined by following the methodology described by [START_REF] Schmid | Stability and transition in shear flows[END_REF]. It is presented here only for the (ũ, ṽ) formulation, but similar development applies to the (ũ, w) formulation. For a given Fourier mode, the disturbance instantaneous kinetic energy is given by

||q(t)|| 2 E = R2 R1 |ũ| 2 + 1 k 2 1 r 2 d dr (rũ) 2 + m 2 r 2 k 2 + 1 |ṽ| 2 + im kr ṽ r d dr (rũ * ) - ṽ * r d dr (rũ) r dr (7.1)
which is a function of time and of the initial condition, q 0 = (ũ 0 , ṽ0 ) T = q(r, t = 0, m, k).

The maximum kinetic energy amplification at time t over all initial conditions is defined by

G(t, m, k) = sup q 0 =0 ||q(t)|| 2 E ||q(0)|| 2 E . ( 7
G max (m, k) = sup t 0 G(t, m, k) . (7.
3)

The maximum of G max over all pairs (m, k) is denoted G opt and is reached by the optimal perturbation at time t opt .

The quantity G(t) is obtained as follows. Consider the linear subspace S M spanned by the M eigenfunctions q j = (ũ j , ṽj ) corresponding to the first M eigenvalues σ j of the spectrum of L , sorted in order of decreasing growth rate. Any admissible perturbation can be written as a linear combination of q j , q = j=M j=1

K j q j = Q • K, (7.4)
where Q is the eigenvector matrix. One can approximate G(t) by computing the maximum possible energy amplification at time t over all possible initial combinations of the M eigenfunctions.

The initial value problem (5.12) can be expressed as

K = e Σ t K 0 , (7.5)
with Σ = diag (σ 1 , ..., σ M ) the diagonal matrix of eigenvalues and K 0 the vector of expansion coefficients that defines the initial perturbation. The energy norm of q(t) is

||q(t)|| 2 E = K * M K = K * F H F K = ||F K|| 2 2 , (7.6)
where the superscript H stands for the complex conjugate transpose, M is a positive definite Hermitian matrix, M = F H F , whose elements are obtained from the inner product of the M eigenfunctions, associated to the energy norm (7.1). Hence, As F K 0 is a vector, the energy growth can be expressed using the matrix 2-norm as

G(t) = sup K0 =0 ||F K(t)|| 2 2 ||F K 0 || 2 2 = sup K0 =0 F e Σt F -1 F K 0 2 2 ||F K 0 || 2 2 . ( 7
G(t) = e Σt 2 E = F e Σt F -1 2 2 = s 2 1 , (7.8)
where s 1 is the largest singular value of the matrix F e Σt F -1 . Since the eigenfunctions have been expressed in terms of Chebyshev polynomials, all the calculations are recasted in terms of Chebyshev expansion coefficients. The number of collocation points needed for convergence to be achieved depends on the shear-thinning degree and on the magnitude of the Reynolds numbers. Globally, a resolution of N = 40 is sufficient. However, for very strong shear-thinning effects say n 0.3 in the case of the power-law model and n 0.3 with λ 10 for the Carreau model, the number of collocation points necessary to achieve convergence is increased up to N = 150.

To begin with, we performed validation tests for a Newtonian fluid. In table 1, our results are compared to those of [START_REF] Meseguer | Energy transient growth in the Taylor-Couette problem[END_REF] and [START_REF] Maretzke | Transient growth in linearly stable Taylor-Couette flows[END_REF]. Very good agreement is observed. The conditions of [START_REF] Hristova | Transient growth in Taylor-Couette flow[END_REF] given in their table II were also simulated. A relative discrepancy of less than 1% was obtained.

As a reference case, we further studied the transient growth for a Newtonian fluid. Figure 14 shows contours of the optimal growth for a Newtonian fluid in the plane (Re 1 , Re 2 ). The corresponding axial and azimuthal wavenumbers are represented in figures 15 and 16. It is interesting to observe that in the counter-rotating regime, the optimal azimuthal wavenumber varies little (10 < m opt < 13) and is much higher than the critical wavenumber m c . Concerning k opt , it is smaller than the critical axial wavenumber k c . For a shearthinning fluid, the contours of optimal energy amplification are displayed in figure 17.

The amplification appears to be much larger than for a Newtonian fluid. In table 2, the optimal growth conditions G opt , k opt , m opt and t opt are compared for Newtonian, powerlaw and Carreau fluids at given Re 1 and Re 2 for thin and wide gaps. It is observed that the optimal amplification is highest for Carreau fluid and at thin gap. The azimuthal and axial optimal wavenumbers are practically similar for Newtonian and shear-thinning fluids. The optimal time is greater for power-law and even greater for Carreau fluids.

These results may be presented in a concise manner by using the shear Reynolds number Re, and the rotation number R Ω . This representation is more relevant in the determination of scaling laws for the optimal energy amplification [START_REF] Maretzke | Transient growth in linearly stable Taylor-Couette flows[END_REF]. Figure 18 shows the characteristics of the optimal perturbation as a function of Re for different values of R Ω . We recover the scaling G opt ∝ Re 2/3 for sufficiently large Re as indicated by [START_REF] Maretzke | Transient growth in linearly stable Taylor-Couette flows[END_REF]. The optimal azimuthal wavenumber varies slightly with R Ω . At The optimal time t opt decreases asymptotically as t opt ∝ Re -2/3 .

For the particular case of R Ω = -1, i.e. along the Rayleigh line, the optimal perturbation is axisymmetric with k opt ≈ π as shown by figure 19. The optimal amplification varies as, G opt ∝ Re 2 (figure 20). The optimal time at which G opt is reached decreases very slightly with Re.

The characteristics of optimal perturbations were also determined for power-law and Carreau fluids as shown in figures 21 and 22. It is interesting to note that we recover the same scaling as for a Newtonian fluid. Indeed, at large Re, G opt ∝ Re 2/3 ; t opt ∝ Re -2/3 ; 10 m opt 13 and 1 k opt < 2 much lower than k c . Comparison with the Newtonian case indicates that for given Re and R Ω , G opt increases with increasing shear-thinning degree as shown in figure 23. This is related to the increase of the non-normality of the linear operator emphasized in §??. The approximate scaling is found:

G opt G opt n=1 ∝ exp(-n α ) (7.9)
where G opt n=1 is the optimal energy growth for a Newtonian fluid and α = 1/3.

Evolution of the optimal perturbation: transient growth mechanism

In this section, the optimal perturbation and its time evolution are described for different values of Re 1 and Re 2 in the linearly stable domain.

Newtonian fluids

For Newtonian fluids, numerical results show that the Rayleigh line should be distinguished from the rest of the domain. The contours of the azimuthal vorticity and azimuthal velocity of the optimal perturbation at initial time t = 0 and optimal time t = t opt are presented in figure 24 . It appears that the optimal perturbation is a superposition along the axial direction of high and low azimuthal velocity streaks. Figure 25 shows the contours of the local kinetic energy associated to the azimuthal, radial and axial components of the velocity perturbation. At initial time more than 99% of the kinetic energy of the perturbation is concentrated in the azimuthal component. At optimal time t = t opt , the kinetic energy is merely concentrated in the radial and axial components. This is also illustrated by figure 26, where we have represented the spatial average of the kinetic energy contained in a volume

V = 2π 2 R 2 2 -R 2 1 /k, E r = 1 V ũũ * ; E θ = 1 V ṽṽ * ; E z = 1 V w w * (7.10) with . = R2 R1 2π 0 2π k 0 real(.)r dr dθ dz . (7.11)
The optimal perturbation which is initially in the form of azimuthal streaks transforms progressively through the anti-lift up mechanism [START_REF] Antkowiak | On vortex rings around vortices: an optimal mechanism[END_REF]) into azimuthal rolls. This is illustrated by the perturbation equation written for the azimuthal vorticity ω ′ θ = ∂u ′ /∂z -∂w ′ /∂r.

In the inviscid limit, its evolution is governed by

∂ω ′ θ ∂t = -2Re 1 Ω b ∂v ′ ∂z (7.12)
where Ω b = V b /r is the basic angular velocity. It clearly appears that ω ′ θ is driven by the axial gradient of the azimuthal streaks ∂v ′ ∂z . According to [START_REF] Antkowiak | On vortex rings around vortices: an optimal mechanism[END_REF], the fluid particles with positive azimuthal streaks undergo an enhanced centrifugal force that will not be balanced by the pressure gradient, and will thus be pushed towards the outer wall. On the contrary, the particles with negative azimuthal streaks will be pushed against the inner wall. This process will initiate the formation of azimuthal rolls. of plane (Re 1 , Re 2 ) is qualitatively similar to that obtained for a Newtonian fluid outside the Rayleigh line. Further computations show that the evolution of the optimal perturbation with time for shear-thinning fluids with η = 0.881 and η = 0.4 are similar to the Newtonian one. However, the optimal perturbation is merely concentrated near the inner wall and more pronounced for large gap η = 0.4. Furthermore, the transient growth is reduced with decreasing η.

Discussion

3D optimal perturbations are initially oriented in the direction opposite to the base flow and aligned with it at optimal time, e.g. figure 27(c) and figure 27(c ′ ). The same evolution was observed for other shear flows such as the boundary-layer flow (see [START_REF] Cherubini | Optimal wave packets in a boundary layer and initial phases of a turbulent spot[END_REF]; [START_REF] Monokrousos | Global three-dimensional optimal disturbances in the blasius boundary -layer flow using time-steppers[END_REF] and references therein). This is in contrast with 2D optimal perturbations which are oriented transversally to the main shear at the optimal time, as shown by [START_REF] Butler | Three -dimensional optimal perturbations in viscous shear flow[END_REF] quence, the velocity along the contour has to increase to conserve the initial circulation.

In 3D situation, another mechanism, absent in 2D, is active. Indeed, the basic shear vorticity, pointing in the spanwise (here axial) direction, may be stretched due to variations of areas in the shear plane as it is was shown by [START_REF] Vitoshkin | On the role of vortex stretching in energy optimal growth of three -dimensional perturbations on plane parallel shear flows[END_REF] in plane shear flows. These authors pointed out that the interplay between the spanwise vorticity q and the planar divergence d fields play a dominant role in energy growth. We extended their analysis to the case of Taylor-Couette flow and found that there is (as in the plane shear flow) a mutual growth of d and q and the maximum is reached at t = t opt , i.e.

when the perturbation is tilted with the basic shear. Following [START_REF] Vitoshkin | On the role of vortex stretching in energy optimal growth of three -dimensional perturbations on plane parallel shear flows[END_REF], the role played by the interaction between d and q is highlighted in the analysis of the energy growth via the Reynolds stress term. For this, a 2D Helmholtz decomposition is performed ũ = u d + u q with u d = ∇(φ) and u q = -rot(ψ e z ) (7.13)

where φ and ψ are scalar functions, such that the planar divergence d = ∂u ′ /∂r + u ′ /r + (1/r) ∂v ′ /∂θ and the axial vorticity q

= (1/r)∂(rv ′ )/∂r -(1/r)∂u ′ /∂θ satisfy d = ∆ H (φ) and q = ∆ H (ψ) , (7.14)
where ∆ H = ∂ 2 /∂r 2 + (1/r)∂/∂r + (1/r 2 )∂/∂θ 2 is the 2D Laplacian on the shear plane.

At the inviscid limit the spatially averaged energy equation is

∂ E ∂t = -Re 1 γb rθ ũṽ * = -Re 1 γb rθ (u d + u q ) v * d + v * q = -Re 1 γb rθ u d v * d + γb rθ u d v * q + γb rθ u q v * d + γb rθ u q v * q . (7.15) with E = |ũ| 2 2 , u d = (u d , v d , w d ) and u q = (u q , v q , w q ).
The first term in the R.H.S of (7.15) is the contribution of the divergent field. The second and the third terms, i.e. mixed q -d terms translated the interplay between the planar divergence and the axial vorticity. The fourth term represents the 2D Orr mechanism.

The contribution of the different terms as a function of time is shown in figure 28(a) for a Newtonian fluid and figure 28(b) for a power-law fluid. Qualitatively, evolutions of the different Reynolds stress components are quite similar, however the amplitudes reached are higher for power-law fluids. For both fluids, the energy growth is initially dominated by the rotational term -γb rθ u q v * q , i.e. the 2D Orr mechanism. From t ≈ t opt /2, the vortices are tilted with the shear, -γb rθ u q v * q becomes negative, i.e. Orr mechanism contributes to the decay of the energy. The contribution of the planar divergence term is positive but very small. The contribution of the mixed term d -q is positive and are at the origin of the increase of E when the vortices align with the base shear.

Conclusion

In this paper, we performed a three-dimensional linear stability analysis of a circular Couette flow of shear-thinning fluids in a thin (η = 0.881) and a wide (η = 0.4) gaps between two infinitely coaxial cylinders. Long and short times evolutions of the disturbances is investigated. Two rheological models were adopted, power-law and Carreau models, to describe the shear-thinning behavior of the fluids.

Firstly, we considered the asymptotic evolution at long times. Using a normal-mode approach, an eigenvalue problem was derived and solved with a Chebyshev spectral collocation method. The results show that for a given Re 2 , the shear-thinning character leads to a decrease of the inner critical Reynolds number Re 1c except in the vicinity of Re 2 = 0 and for a co-rotating regime with a small gap. This reduction is due to an enhancement of the energy exchange between the perturbation and the base flow, which is probably a consequence of the increase of the shear rate at the inner cylinder wall.

For counter-rotating cylinders, the threshold of the outer Reynolds number, |Re 2 | at which the critical modes switches from axisymmetric Taylor vortices to spiral vortices increases significantly with increasing shear-thinning effects. The axial wavelength of these spiral vortices is strongly reduced with increasing shear-thinning effects, because of the reduction of the "effective gap" where the Rayleigh's instability criterion holds.

Secondly, we studied the transient growth of the perturbation kinetic energy that occurs at short times due to the non-normality of the operators involved in the linearized perturbation equations. The evidence of non-normality was highlighted by plotting εpseudospectra. It was shown that the sensitivity of eigenvalues increases with more pronounced shear-thinning character. This effect is confirmed by the strong increase, up to one order of magnitude of the perturbation energy amplification compared to the Newtonian fluid for given Re 1 and Re 2 .

As far as the structure of the optimal perturbation is concerned, we showed that at large shear Reynolds numbers and for a thin gap, it exhibits a large azimuthal wavenumber m opt ≈ 12 and a small axial wavenumber 1 k opt 2. Whereas for a wide gap (η = 0.4), m opt = 2 and k opt ∝ 1. It appears that shear-thinning effects have little influence on k opt and m opt .

-In the particular case of a Newtonian fluid with (Re 1 , Re 2 ) on the Rayleigh line, the angular momentum is constant and the optimal perturbation consists of a superposition along the axial direction of high and low azimuthal velocity streaks which evolve towards

Taylor vortices via the anti-lift-up mechanism. The optimal energy growth behaves as

G opt ∝ Re 2 .
-Outside the Rayleigh line, the optimal perturbation, initially oriented against the base shear, tilts until it almost aligns with the shear at t = t opt . The amplification of the 3D optimal perturbation is analyzed in terms of interplay between the planar divergence and the axial vorticity as suggested by [START_REF] Vitoshkin | On the role of vortex stretching in energy optimal growth of three -dimensional perturbations on plane parallel shear flows[END_REF]. The decomposition of the Reynolds stress in terms of the various contributions from the divergent and the rotational fields indicates that the energy growth is dominated by the mixed terms -(u d v q +u q v d ) γb In this case, q = (ũ, ṽ) T and the operators M and L are expressed below

M =       DD * -k 2 i m r D - i m k 2 r D * 1 + m 2 k 2 r 2       (A 2) L = L I + L V 1 + L V 2 + L V 3 (A 3) L I = Re 1       i m r -V b D + D V b D * + k 2 V b -2 k 2 V b r + m 2 r 2 -2 V b r + D V b + V b D -D * V b - m 2 k 2 r 2 V b D * -i m r 1 + m 2 k 2 r 2 V b       (A 4) L V 1 = µ b       (DD * -k 2 ) 2 i m 2 k 2 r 2 + D ∆ 1 r ∆ -2D r + 1 r 2 i m r - ∆D * k 2 + 2 r DD * -k 2 + m 2 k 2 r 2 ∆ -2D r + 1 r 2 -k 2       (A 5) L V 2 = Dµ b     D 2 D * + ∆D * -k 2 D i m r ∆ + D 2 -4D r + 3 r 2 - i m k 2 r D D * 1 + m 2 k 2 r 2 D     + D 2 µ b     k 2 + D D * i m r D 0 0     (A 6) L V 3 = µ t -µ b       k 2 m 2 r 2 -i k 2 m D r i m D * r D + 2 r D       + D µ t -µ b       0 0 i m r D       (A 7)
A.2. (ũ, w) formulation

In this case, q = (ũ, w) T and the operators M and L are expressed below The eigenvalue problem reads

M =       D 2 * + 1 r D * -m 2 r 2 i k D + 2 i k r k D * i k 2 + i m 2 r 2       (A 8) L = L I + L V 1 + L V 2 + L V 3 (A 9) L I = Re 1     i m r 2 2 D V b + V b D -3 V b D * + m 2 V b r 3 k m V b r 2 i k m r D V b -V b D m r m 2 V b r 2 + k 2 V b     + Re     i m r -DV b r + D 2 V b -V b DD * k m r V b D + DV b 0 0     (A 10) L V 1 = µ b     -2 m 2 r 4 i k 2 r + D D * D -k 2 + 2D * D r -4 r + D m 2 r 2 k ∆ + 2D r D * i k 2 ∆ + 2 D r     + µ b     2 r + D ∆ D * + 2 r D 2 + D r D * -m 2 r 2 ∆ -m 2
Lq = σ M q (B 1)
where q = (u, v) T or (u, w) T depending on the formulation adopted.

For a Newtonian fluid, on the Rayleigh line, the angular momentum of the base flow is constant d dr (rV b ) 2 = 0, thus

V b = a r , ( B 2)
where a is a constant scalar and

D * V b = DV b + V b r = 0 . (B 3)
As stressed by [START_REF] Gebhardt | The Taylor -Couette eigenvalue problem with independently rotating cylinders[END_REF], two types of modes exist. The first type corresponds to v = 0 and the second to v = 0.

-If v = 0 then the v equation of the (u, v) formulation is considered (second line of system (B 1)),

σ v = µ b DD * v -k 2 v (B 4)
This equation is multiplied by the complex conjugate of v, v * and integrated over volume is required to extend the pseudospectrum into the unstable half plane. For axisymmetric

V = 2π 2 R 2 2 -R 2 1 /k σ R2 R1 2π

1. 1 .

 1 Review on Taylor-Couette flow in shear-thinning fluids Circular Couette flow of shear-thinning fluids are mainly characterized by viscosity stratification in the annular space. The degree of viscosity stratification defined by |dµ b /dr|

  Figure 1 displays some particular values of R Ω in the usual parameter space (Re 1 , Re 2 ) for a Newtonian fluid and for a power-law fluid with n = 0.3. Shear-thinning effects introduce a slight deviation of constant R Ω lines. For Carreau fluids, r, Re, and R Ω are evaluated numerically. The constant R Ω lines are

Figure 1 .

 1 Figure 1. Parameters of Dubrulle et al. (2005) represented in the (Re1, Re2) plane at η = 0.881. Contours of rotation number RΩ for (a) Newtonian fluid and (b) power-law fluid with n = 0.3. The arrow indicates that Re increases along the constant RΩ line.

Figure 2 .Figure 3 .

 23 Figure 2. Base flow for power-law fluids in the case of counter-rotating cylinders with η = 0.4, Re1 = 300 and Re2 = -3000. (a) Velocity profiles and (b) viscosity profiles for (1) n = 1 (Newtonian fluid) (2) n = 0.5 and (3) n = 0.3. (-+ -) Carreau fluid with n = 0.5 and λ = 10.

Figure 4 .

 4 Figure 4. Rayleigh line for a power-law fluid with (a) η = 0.881 and (b) η = 0.4: (1) Newtonian fluid n = 1, (2) n = 0.5, (3) n = 0.2 and (4) n = 0.1. (---) solid rotation line. in the (Re 1 , Re 2 ) plane by the Rayleigh line, on which ∃ r such that d dr (L 2 ) = 0. For a Newtonian fluid, the Rayleigh line equation reads Re 1 = Re 2 η . (4.1)

  For a shear-Y. Agbessi, B. Alibenyahia, C. Nouar, C. Lemaitre and L. Choplin thinning fluid, the calculations show that the Rayleigh line deviates from R Ω = -1.

  .3) Similar results are observed for a wide gap, figure 6(b).

Figure 5 .Figure 6 .Figure 7 .Figure 8 .Figure 9 .

 56789 Figure 5. Eigenspectra for a Newtonian fluid undergoing an axisymmetric perturbation m = 0 for η = 0.881, k = π, Re1 = 400, and varying Re2. (a) Re2 = 352.4 (on the Rayleigh line) and (b) Re2 = 0.

Figure 10 .Figure 11 .

 1011 Figure 10. Critical conditions of instability for a power-law fluid with two radius ratios, η = 0.881 (figures on the left) and η = 0.4 (figures on the right). Effect of the shear thinning index. (a) and (b) Marginal stability curves in plane (Re1, Re2), (c) and (d) critical azimuthal wavenumbers, (e) and (f) critical axial wavenumbers. (1) Newtonian fluid, n = 1, (2) n = 0.7, (3) n = 0.5 and (4) n = 0.3.

Figure 12 .Figure 13 .

 1213 Figure12. Radii at which functions (•) f (r) and (△) g1(r) + g2(r) defined by equations (6.5) -(6.7), are maximal, for a power-law fluid with n = 0.5 and η = 0.881.

Figure 14 .

 14 Figure 14. Contours of optimal energy growth G opt in the plane (Re1, Re2) for a Newtonian fluid with η = 0.881: (a) counter-rotating cylinders, (b) co-rotating cylinders. The thick line is the marginal stability curve.

(Figure 15 .Figure 16 .

 1516 Figure 15. Contours of optimal wavenumbers in the (Re1, Re2) plane for a Newtonian fluid flowing between counter-rotating cylinders with η = 0.881. (a) Optimal azimuthal wavenumber m opt and (b) optimal axial wavenumber k opt . The thick line is the marginal stability curve.

Figure 17 .

 17 Figure 17. Contours of optimal energy growth G opt in the plane (Re1, Re2) for a power-law fluid with n = 0.5 and η = 0.881 : (a) counter-rotating cylinders, (b) co-rotating cylinders. The thick line is the marginal stability curve.

Figure 18 .Figure 19 .

 1819 Figure 18. Optimal energy growth and characteristics of the optimal perturbation as a function of the shear Reynolds number Re for different values of the rotation number RΩ. Newtonian fluid with η = 0.881. (a) Optimal energy growth G opt , (b) optimal axial wavenumber k opt , (c) optimal azimuthal wavenumber m opt and (d) optimal time t opt .

Figure 20 .

 20 Figure 20. (a) Optimal energy growth and (b) optimal time versus shear Reynolds number along the Rayleigh line (RΩ = -1) for a Newtonian fluid with η = 0.881.η = 0.881, figure18(c) shows again that asymptotically 10 m opt 13. In this range of m opt , the kinetic energy amplification does not change significantly. Concerning the axial wavenumber, it is observed that it is much lower than the linear critical wavenumber k c , figure 18(b). The optimal time t opt decreases asymptotically as t opt ∝ Re -2/3 .

Figure 21 .

 21 Figure 21. Optimal energy growth and characteristics of the optimal perturbation as a function of the shear Reynolds number Re for different values of the rotation number RΩ. Power-law fluid with n = 0.5 and η = 0.881. (a) Optimal energy growth G opt , (b) optimal axial wavenumber k opt , (c) optimal azimuthal wavenumber m opt and (d) optimal time t opt .

Figure 22 .Figure 23 .

 2223 Figure 22. Optimal energy growth and characteristics of the optimal perturbation as a function of the shear Reynolds number for different values of the rotation number RΩ. Carreau fluid with n = 0.5, λ = 10 and η = 0.881. (a) Optimal energy growth G opt , (b) optimal axial wavenumber k opt , (c) optimal azimuthal wavenumber m opt and (d) optimal time t opt .

Figure 24 .

 24 Figure 24. Optimal perturbation represented in the plane (r, z) for a Newtonian fluid along the Rayleigh line, η = 0.881, Re1 = 1135 and Re2 = 1000, with m opt = 0, k opt = 3.32 and G opt = 5.58. Contours of the azimuthal vorticity component at (a) t = 0 and (b) optimal time t = t opt . Contours of the azimuthal velocity component at (c) t = 0, vmax = 0.52 and (d) optimal time t = t opt , vmax = 0.1; the contours are normalized by the maximum velocity at time t and shown by steps of 0.2. The solid lines represent positive values and the dotted lines represent negative values.

Figure 25 .Figure 26 .

 2526 figure 27(a) in plane (r, z) and in figure27(a') in plane (r, θ). These contours will tilt progressively until they almost align with the shear at t = t opt as shown in figures 27(c) and 27(c').

Figure 27 .Figure 28 .

 2728 Figure 27. Optimal perturbation for a Newtonian fluid for η = 0.881, Re1 = 101, Re2 = -1061.2 with m opt = 13, k opt = 2.176. Left: contours of the kinetic energy in the plane (r, z) at (a) initial time t = 0 , (b) t ≈ t opt /2 and (c) optimal time t = t opt . Right: contours of the kinetic energy in the plane (r, θ) at (a ′ ) initial time t = 0 , (b ′ ) t ≈ t opt /2 and (c ′ ) optimal time t = t opt . The quantity d = ∂u ′ /∂r + u ′ /r + (1/r) ∂v ′ /∂θ is the 2D divergence on the shear plane.

  rθ which overwhelms the decaying effect of the Orr-mechanism. The numerical results show that the optimal energy growth behaves as G opt ∝ Re 2/3 in agreement with[START_REF] Maretzke | Transient growth in linearly stable Taylor-Couette flows[END_REF]. For shear-thinning fluids; the Rayleigh line should not be distinguished from other situations, the tilting mechanism, described above, applies and the numerical results show that G opt = G opt n=1 × exp(-n 1/3 ). The transient energy growth is enhanced with increasing shear-thinning effects. Thus, shear-thinning promotes transition to turbulence by the linear superposition of eigenmodes. This tendency was observed in a plane Couette flow (Liu & Liu 2011) and a Hagen-Poiseuille flow (Liu & Liu 2012) of power-law fluids involving another transient growth mechanism.Appendix A. Operators involved in the linearized perturbation equations

  Demonstration that the eigenvalues for a Newtonian fluid on the Rayleigh line are real

DDFigure 32 .

 32 Figure 32. Pseudospectra for shear-thinning fluids with Re1 = 101, Re2 = -1061.2 and η = 0.881. (a) Power-law fluid with n = 0.5, k = 1.647, m = 11; (b) Carreau fluid with n = 0.5, λ = 10, k = 1.842, m = 12.

Figure 31

 31 Figure 31(a) displays the spectral portrait of operator L for a Newtonian fluid, on the Rayleigh line with k = 3.32, m = 0, Re 1 = 6.30×10 3 , Re 2 = 5.55×10 3 . The iso-contours labeled p delineate the boundaries in the complex plane of the ε-pseudospectrum with ε = 10 -p . Non-normality effects are more pronounced for eigenvalues near the origin. For the given values of Re 1 and Re 2 , a perturbation of O 10 -3 allows the pseudospectrum to protrude into the unstable region. Figure 31(b) shows the spectral portrait of the Taylor-Couette flow outside the Rayleigh line. Non-normal effects are more pronounced at the intersection of the three branches of the spectrum. A perturbation of O 10 -0.7

  Reynolds numbers Re 1 and Re 2 . In order to separate shear and centrifugal effects,

	Dubrulle et al. (2005) introduced another parametrization replacing Re 1 and Re 2 by
	a shear Reynolds number Re, ratio between the shear time scale | γb rθ ( r)| -1 and the vis-
	cous time scale ρ d2 /μ and a rotation number R Ω , ratio between shear time scale and a
	rotation time scale:

Table 1 .

 1 Comparison of the present results with those of[START_REF] Meseguer | Energy transient growth in the Taylor-Couette problem[END_REF] andMaretzke et al.

		Meseguer (2002)	Maretzke et al. (2014)	Present work (N = 40)
	Re1 Re2	k opt m opt G opt	k opt m opt G opt	k opt m opt G opt
	591 -2588	1.994 10 71.36	1.997 10 71.58	2.015 10 71.58
	523 -2975	1.996 11 71.58	1.998 11 71.81	1.985 11 71.80
	473 -3213	1.920 11 71.64	1.922 11 71.87	1.924 11 71.87
	405 -3510	1.839 11 71.75	1.841 11 71.99	1.833 11 71.99

We adopt here a definition of the Reynolds numbers based on the viscosity evaluated at the inner wall, μb ( R1 )

Figure 29 shows marginal stability curves in terms of Re 1w and Re 2w ,

It appears that the shear-thinning behavior has a destabilizing effect for a thin gap η = 0.881 when both cylinders are counter-rotating. However, for a large gap η = 0.4, stabilization is observed. The opposite effect is found by using as scale the viscosity defined by (2.9). The selection of the viscosity scale may be considered to be simply a matter of choice, however the conclusions that one reaches by comparing different shearthinning fluids among themselves and against Newtonian fluids can be radically different from one choice to the other.

-


G I Bara
email: fibara@ensem.inpl-nancy.fr

J Daafouz
email: jdaafouz@ensem.inpl-nancy.fr

F Kratz
email: fkratz@ensem.inpl-nancy.fr

J Ragot
email: jragotg@ensem.inpl-nancy.fr

OBSERVERS INTERPOLATION FOR STATE RECONMSTRUCTION OF DISCRETE TIME LPV SYSTEMS: A LMI APPROACH

Keywords: LPV systems, LPV Observers, Interpolation, Quadratic stability. 1.INTRODUCTION

This paper invistigates a state observation design problem for discrete time linear parameter varying (LPV) systems. The main contribution of this paper consists in providing an interpolation scheme to build the LPV observer. We show that an appropriate choice of the interpolation functions allowto use available quadratic stability conditions to design an LPV observer.

Among recent advance in control theory, parameter varying modeling appeared to be well suited in describing the behaviour of a large class of physical systems, see 1] and references therein. Linear Parameter Varying systems theory has been motivated bythe gain scheduling approach which attempts to provide a systematic methodology to design parameter-dependent control laws that guarantee stability and performances speci cations 2], 3], ... In this paper we are interested in state reconstruction of discrete time LPV systems. This problem can be investigated using di erent approaches. One can seek to develop a switching strategy which consists in a collection of linear time invariant (LTI) observers that are scheduled based on the measurement of the parameters. The main di culty consists in the choice of a switching strategy that ensures at least the stability property. One can also choose to avoid switching di culties and look for methods that guarantee smooth transition between several observers using, for example, interpolation techniques... Here, we propose a solution to state reconstruction of discrete time LPV systems. The state observer is also parameter varying and is obtained using interpolation of a collection of LTI observers.

In 4], a linear interpolation technique is used to design state feedback observer-based controllers for LPV systems. In the case of a scalar scheduling variable, a stability preserving interpolation in terms of frozen values of the scheduling variable is proposed. An application of interpolation based observer to the design of a gain scheduled robust observerbased controller is also proposed in 5]. These contributions deal with continuous time LPV systems and the used techniques are di erent from the one developped in this paper.

The paper is organized as follows. Section 2 gives the problem statement and develops a solution for the interpolation based LPV observation problem. Conditions allowing the design of the parameter varying observer are given in terms of LMI conditions. Di erenterror convergence criteria are discussed. Section 3isdevoted to a numerical example.

2.PROBLEM FORMULATION

We consider LPV systems given by:

x(k +1) = A( (k))x(k)+Bu(k) y(k) = Cx(k) (1)
where x(k) 2 R n is the state vector, u(k) 2 R m is the control vector and y(k) 2 R p is the measured output vector. The dynamical matrix depends on a time varying parameter and is given by:

A( (k)) = A 0 + (k) 1 (k) 2 1 2 ] (2)
We assume that is bounded and real-time measurable. We look for a parameter varying observer with the following structure

x(k +1) = A( (k))x + Bu+ L( (k))(y(k) ; ŷ(k)) ŷ(k) = Cx(k) (3)
The parameter varying gain L( (k)) is obtained by interpolation of an o -line computed constant gains, namely those corresponding to the observers estimating the state of the original system when (k)= 1 and (k)= 2 :

x(k +1) = A i x(k)+Bu(k) y(k) = Cx(k) (4)
with

A i = A( i ) i =1 2
The problem to be dealt with in this paper can be formulated as follows: Find a collection of linear time invariant observers built for the extremal values of the time varying parameter (k), such that an interpolation procedure leads to a parameter varying observer for the original LPV system.

Let L i be the constant gains corresponding to the observers estimating the state of the original system for the extremal values (4).

x(k +1) = A i x(k)+Bu(k)+ L i (y(k) ; ŷ(k)) (5) ŷ(k) = Cx(k) The parameter varying gain matrix L( (k)) is obtained byaninterpolation of the gains L i L( (k)) = 1 ( (k))L 1 + 2 ( (k))L 2 (6)
where the interpolation functions 1 ( (k)) and 2 ( (k)) allow a smooth transition between the extremal observers according to actual value of (k). Wechoose linear interpolation functions as depicted in gure 1.

In this case, the functions 1 ( (k)) and 2 ( (k)) are given by:

1 ( (k)) = (k) ; 1 1 ; 2 2 =1; 1 (7) Using the LPV observer (3), the dynamic of the observation error e(k)=x(k) ; x(k) is gi- ven by: e(k +1)= ; A( (k)) ; L( (k))C e(k)
which can be rewritten, using [START_REF] Bernussou | new discrete-time robust stability condition[END_REF], as :

e(k +1)= ; 1 ( (k)) Ã1 + 2 ( (k)) Ã2 e(k) (8) with Ã1 = A 1 ; L 1 C and Ã2 = A 2 ; L 2 C and i ( (k)) > 0 X i i ( (k)) = 1 i =1 2 (9)
The interpolated parameter varying observer determination reduces to the computation of the constantgainsL i ,such that the polytopic uncertain system (8) is asymptotically stable. As the parameter (k)istimevarying, and no limitation has been considered on the rate of variation, we look for a single Lyapunov function to prove stability. A new discrete time Lyapunov condition developped in 6] is used. The following Theorem recalls such a condition.

Theorem 1 6] The following conditions are equivalent i) There exists a symmetric matrix P > 0 such that A 0 PA; P < 0 (10) ii) There exist a symmetric matrix P > and a matrix G such that

;P A 0 G 0 GA P ; (G + G 0 ) < 0 (11)
As stated in 6], due to the presence of an extra degree of freedom, namely the matrix G, results based on the evaluation of condition ii) includes as a particular case those based on the classical quadratic stability coniditon i). The following proposition makes use of this new condition to provide the gains matrices L i .

Proposition 1 Suppose that there exist matrices P = P 0 , G and R i , i =1 2, such that ;P A 0 i G 0 ; C 0 R 0 i GA i ; R i C P ; (G + G 0 ) < 0 [START_REF] Leith | \Survey of gain scheduling analysis and design[END_REF] then the extremal LTI observers are given by [START_REF] Hyde | \The application of scheduled h 1 controllers to a vstol aircraft[END_REF] with L i = G ;1 R i . Moreover, the global error dynamic (8) is asymptotically stable and the LPV observer is given by ( 3) with ( 6) and [START_REF] Bernussou | new discrete-time robust stability condition[END_REF].

Proof: If condition ( 12) is satis ed then it is quite easy to check that (A i ; L i C) 0 P(A i ; L i C) ; P < 0 i =1 2 which means that the error dynamic is asymptotically stable for the extremal values of (k). In this case, the corresponding observers are given by [START_REF] Hyde | \The application of scheduled h 1 controllers to a vstol aircraft[END_REF]. Since the lyapunov matrix P is common and according to (9), the global error dynamic (8) is asymptotically stable. Hence, the LPV observer is given by ( 3)with ( 6) and [START_REF] Bernussou | new discrete-time robust stability condition[END_REF].

The error dynamics obtained using a parameter varying observer built as indicated previously converges to the origin asymptotically.

One can also look for a faster convergence rate. Depending on the nature of variation of the parameter (k), one can look for di erent kind of performance criteria.

2.1.Slow parameter variations

In the case of slow parameter variations such that the system can be considered as time invariant over time intervals, a pole placement constraint can be imposed to improve the convergence of the observation error. One can look for a pole placement of the dynamical error observation matrix in a disk D( r) centered at ( 0) and with radius r such that j j + r < 1. Such a region allows to specify good transient behaviour by mean of managing speed and damping indexes 7], 8]...

Introduce the following change of variables:

A ir = A i ; I r C r = C r (13)
The problem of interpolated parameter varying state observer synthesis reduces to nd the gains L i such that the poles of the dynamical error matrix (A( ) ; L( )C) belong to a speci ed disk D( r). This problem is equivalent to ensure that the poles of the modi ed slowly varying parameter matrix (A( ) ; L( )C ; I) r belong to the unit disk C(0 1). Using the change of variables (13), wehave the following proposition.

Proposition 2 Suppose that there exist matrices P = P 0 , G and R i , i =1 2, such that

;P A 0 ir G 0 ; C 0 r R 0 i GA ir ; R i C r P ; (G + G 0 ) < 0 (14)
then the extremal LTI observers are given by [START_REF] Hyde | \The application of scheduled h 1 controllers to a vstol aircraft[END_REF] with L i = G ;1 R i . Moreover, the parameter varying observer is given by ( 3) with ( 6) and ( 7) and the poles of the error dynamical matrix belong to the speci ed disk D( r).

2.2.Fast parameter variations

If the parameter variations are such that one can not aproximate the system behaviour by a piece-wise LTI system behaviour, one can also look for a faster convergence rate using the decay rate criteria 9]. In the discrete time case, the decay rate is de ned as the larget scalar 1such that

lim k!1 k ke k k =0 (15)
holds for all trajectories e(k). The asymptotic convergence corresponds to =1. We can use the quadratic Lyapunov function

V (e(k)) = e 0 (k)Pe(k)
to establish a lower bound on the decay rate . The LTI observer gains L i are then designed using the following proposition.

Proposition 3 Assume that the following convex optimisation problem Min s.t

; P A 0 i G 0 ; C 0 R 0 i GA i ; R i C P ; (G + G 0 ) < 0 0 < <1 (16)
admits a solution = with the correponding matrices P = P 0 , G = G and R i , then the interpolated parameter varying observer obtained with L i = G ;1 R i leads to an error convergence to the origin with a decay rate at least equal to ; 1 2 .

Proof: If (16) admits a solution then (A( (k)) ; L( (k))C) 0 P( ) ; P < 0 which means that V (e(k +1))< V(e(k)) 0 < <1 (17

)
holds for all trajectories e(k). Hence, V (e(k)) < k V (e(0)) or equivalently ke(k)k < k 2 :M with M > 0 a scalar given by M = kP 1 2 kkP ; 1 2 kke(0)k

3.ILLUSTRATIVE EXAMPLE

We consider a LPV system given by ( 1 The corresponding LPV observer has been simulated with an initial error e(0) = 10 ;5 1 0 , a control u = 0 and a fast parameter variation as shown in gure 2. The obtained behaviour is reported in gure 4.

Figure 3 shows the interpolation functions 1 ( (k)) and 2 ( (k)).

The same example has been considered for slowvariation of the parameter . To guarantee a good convergence rate of the observation error, a disk centered at 0 with radius 0.5 has been speci ed. Solving the LMIs given in Proposition 2, we get the following result Figure 5 shows the pole location of the error dynamic as well as the behaviour obtained with the proposed LPV observer. The simulation has been performed for 60 values of with the same initial error and control value as previously.

4.CONCLUSION

An interpolation based method has been proposed to design a parameter varying observer. It is based on a new LMI condition with an extra degree of freedom. The choice of linear interpolation functions has been made for simplicity reasons. Polynomial inteprolation functions has also been investigated but the improvement of the observer behaviour is not satisfactory regarding to the complexityofthe mathematical developpments. Only the scalar parameter case has been considered in this paper. Extending the proposed results to the vector case as well as reducing the conservatism of the our approach will be addressed in a near future. 8] G. Garcia, J. Daafouz, and J.

  Figure 1: Interpolation functions

Figure 3 :Figure 5 :

 35 Figure 2: Parameter variation

  Bernussou, \Output feedback disk pole assignment for systems with positive real uncertainty", IEEE Tran. on Auto. Cont.,vol. AC-41, pp. 1385{1391, 1996. 9] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequality in Systems and Control Theory, Studies in Applied Mathematics, 1994.