Subcritical bifurcation of shear-thinning. plane Poiseuille flows

Transcription

1 Under consideration for pulication in J. Fluid Mech. 1 Sucritical ifurcation of shear-thinning plane Poiseuille flows A. CHEKILA 1,2, C. NOUAR 2, E. PLAUT 2 AND A. NEMDILI 1 1 Laoratoire de Rhéologie, Faculté de Génie Mécanique, Université des Sciences et de la Technologie d Oran, Algérie 2 LEMTA, UMR 7563 (CNRS- Nancy Université), 2 Avenue de la Forêt de Haye, BP 16, 5454 Vandoeuvre Cedex, France (Received?? and in revised form??) In a recent article (Nouar et al. 27), a linear staility analysis of plane Poiseuille flow of shear-thinning fluids has een performed. The authors concluded that the viscosity stratification delays the transition and that is important to account for the viscosity perturation. The current paper focuses on the first principles understanding of the influence of the viscosity stratification and the nonlinear variation of the effective viscosity µ with the shear rate γ on the flow staility with respect to a finite amplitude perturation. A weakly nonlinear analysis, using the amplitude expansion method is adopted as a first approach to study nonlinear effects. The ifurcation to two-dimensional travelling waves is studied. For the numerical computations, the shear-thinning ehavior is descried y the Carreau model. The rheological parameters are varied in a wide range. The results indicate that (i) the nonlinearity of the viscous terms tends to reduce the viscous dissipation and to accelerate the flow, (ii) the harmonic generated y the nonlinearity µ( γ) is smaller and in opposite phase with that generated y the quadratic nonlinear inertial terms and (iii) with increasing shear-thinning effects, the ifurcation ecomes highly sucritical. Consequently, the magnitude of the threshold amplitude of the perturation,

2 2 A. Chekila, C. Nouar, E. Plaut and A. Nemdili eyond which the flow is nonlinearly unstale, decreases. This result is confirmed y computing higher order Landau constants. 1. Introduction Non-Newtonian fluid flows occur in many industrial processes. The control of these processes requires knowledge of the flow structure and in particular of the conditions for staility and transition to turulence. Most non-newtonian fluids have two common properties: viscoelasticity and shear-thinning. This theoretical work is related to the shear flow staility of purely viscous shear-thinning fluids, i.e., fluids without elastic response and for which the effective viscosity ˆµ decreases with increasing shear rate. Flows of these fluids are mainly characterized y a viscosity stratification in the direction normal to the wall. It has een shown y several authors that the staility characteristics of parallel shear flows can e significantly modified y such a viscosity stratification, which can also e otained when the viscosity ˆµ depends on an intensive quantity oeying an advectiondiffusion equation. Wall & Wilson (1996) considered the staility of plane channel flow of a Newtonian fluid with temperature-dependent viscosity, the walls eing maintained at different temperatures. Four different viscosity models were considered. They found that a non-uniform increase of the viscosity in the channel always stailizes the flow, whereas a non-uniform decrease in the channel may either destailize or stailize the flow. These results were explained in terms of three physical effects, namely ulk effect due to uniform increase or decrease of viscosity, a velocity-profile shape effect as the asic velocity profile ecomes non-symmetrical and a thin-layer effect when a thin layer of lower or higher viscosity develops adjacent to a channel wall. The influence of heating on the staility of channel flow was also recently addressed y Sameen & Govindarajan

3 Sucritical ifurcation of shear-thinning channel flows 3 (27). They showed that a decrease in viscosity towards the wall stailizes the flow. The effect of viscosity stratification on the staility of plane channel flow with respect to infinitesimal perturations has een also analyzed y Ranganathan & Govindarajan (21), Govindarajan (22), Govindarajan et al. (23) and Chikkadi et al. (25). They showed that any viscosity profile in the critical layer, in which the viscosity decreases towards the wall, delays significantly the onset of two-dimensional modes. This effect was related to a reduced energy intake from the mean flow to the fluctuations. The energy dissipation responds less drastically to changes in viscosity. It was then argued that similar physics should hold in the case of turulent drag reduction y polymers additive, when the production layer overlaps the viscosity-stratified layer. Recently, Nouar et al. (27) investigated the linear staility of viscously stratified channel flow, focusing on shear-thinning fluids modeled y the Carreau rheological law. The degree of stailization oserved is more modest than the prediction of Chikkadi et al. (25). The disagreement stems from the neglect of the viscosity perturation in Chikkadi et al. (25), Govindarajan et al. (21), Govindarajan et al. (23). When it is taken into account the fluctuating shear-stress tensor ecomes anisotropic (Nouar et al. 27). Indeed, denoting y x and y the streamwise and wall-normal directions, the xy-component of the viscous stress tensor perturation implies the tangent viscosity, which is smaller than the effective viscosity. The other components of the viscous stress tensor perturation are proportional to the effective viscosity (see equations 2.24, 2.25 and 2.3 elow). In the studies descried aove, the influence of the viscosity stratification on the staility of plane channel flow has een descried in the framework of linear staility analysis. It is natural to inquire if the stailizing effects oserved in the linear regime exist also in the nonlinear regime. Here, we seek a first-principles understanding of the influence of the viscosity stratification and the nonlinear variation of the effective viscosity ˆµ with

4 4 A. Chekila, C. Nouar, E. Plaut and A. Nemdili the shear rate ˆ γ on the flow staility in the presence of finite amplitude disturances. A weakly nonlinear analysis is used as a first approach to take into account nonlinear effects. These effects will e analyzed through (i) the nature of the ifurcation, (ii) the modification of the ase flow, (iii) the generated harmonic and (iv) the threshold amplitude of the perturation which limits the asin of attraction of the laminar parallel flow. To our knowledge, the weakly nonlinear staility analysis of shear-thinning fluid flow in a plane channel has not een performed so far. For such non Newtonian fluids, the viscous term in the momentum equation, which is linear (proportional to the Laplacian of the velocity) in Newtonian fluids, ecomes highly nonlinear. In fact, Govindarajan et al. (23) studied the effect of viscosity stratification on the development of secondary instailities, i.e. three dimensional linear instailities of finite amplitude Tollmien-Schlichting waves. The authors found a strong stailization. The secondary instailities modes are slaved y the primary mode and are rapidly damped. In this study, the authors did not include the viscosity fluctuation in the staility equations. This can e justified if an infinite scalar coefficient diffusion is considered for the perturation. This assumption is, however, incompatile with a laminar steady viscosity profile (Ern et al. 23). Taking into account the viscosity perturation may modify the conclusions drawn y the authors. Here, we consider only the case of shear-thinning ehavior, i.e. the case where ˆµ decreases as the shear rate increases. Particulate dispersions, polymer colloids and polymer solutions can display this ehavior aove a certain concentration threshold. For polymer solutions, the shear-thinning phenomenon arises ecause the dissolved macromolecules form aggregates via hydrogen onds and polymer chain entanglement, which are progressively disrupted under the influence of increasing shear rate (Phillips & Williams 2). In addition, the individual polymer network may align in the direction of the flow, reducing further the energy dissipated under shear. For suspension of particles, the mechanism

5 Sucritical ifurcation of shear-thinning channel flows 5 of shear-thinning is descried for instance in Quemada (1978). The choice of a suitale constitutive model is not crucial at this stage: all the analytical relations are given for a general nonlinear purely viscous fluid. For the numerical applications, we adopt the Carreau (1972) model. This model has een chosen ecause it has a sound theoretical asis and is frequently adopted to descrie the rheological ehavior of shear-thinning fluids. Staility analysis data are also availale in the literature for this fluid model. This article is organized as follows. In 2 we formulate the physical prolem, state the governing equations and define the dimensionless parameters. The velocity and viscosity profiles of the ase state are discussed and the disturance equations are derived. Susequently, the linearization of the disturance equations and the eigenvalue prolem derivation for the linear staility analysis are presented in 3. In 4, the weakly nonlinear scheme is descried in detail as well as the method of determination of the Landau constant. Section 5 presents and discusses the numerical results focusing on the influence of the shear-thinning ehavior of the fluid. Finally, 6 summarizes the salient conclusions of the present study. 2. Poiseuille flows of shear-thinning fluids 2.1. Governing equations - Dimensionless parameters We consider the flow of a shear-thinning incompressile fluid etween two plates at ŷ = ±ĥ. A constant pressure gradient ˆP ˆx is imposed in the streamwise direction e x. The wall-normal direction is defined y the unit vector e y. Here and in what follows, the quantities with hat (ˆ.) are dimensional. Distances are scaled with ĥ, velocity with the maximal velocity Û of the ase flow, time with ĥ, pressure and stresses with Û ˆρÛ2.

6 6 A. Chekila, C. Nouar, E. Plaut and A. Nemdili Using these scales, the governing equations in dimensionless form are:.u =, (2.1) U t + (U. ) U = P +.τ, (2.2) where U = Ue x + V e y denotes the fluid velocity, P the pressure (incorporating gravity effects) and τ the deviatoric stress tensor. The fluid is supposed to e purely viscous, i.e., its viscosity depends only on the shear rate. The constitutive equation reads: τ = 1 Re µ(γ) γ with γ = U + ( U)T. (2.3) Here, Re is the Reynolds numer defined as: γ is the rate-of-strain tensor and Γ its second invariant: Re = ˆρÛ ĥ ˆµ, (2.4) Γ = 1 2 γ ij γ ij. (2.5) For the numerical applications, the viscosity ˆµ is given y the Carreau model (Bird et al. 1987), ˆµ ˆµ ( = 1 + ˆµ ˆµ ˆλ ) nc ˆΓ, (2.6) where ˆµ and ˆµ are the viscosities at low and high shear rate, n c < 1 the shearthinning index, ˆλ the characteristic time of the fluid. The location of the transition from the Newtonian plateau to the shear-thinning regime is determined y ˆλ, since 1/ˆλ defines the characteristic shear rate for the onset of shear-thinning. Increasing ˆλ reduces the Newtonian plateau to lower shear rates. The infinite-shear-rate viscosity is generally associated with a reakdown of the fluid, and is frequently significantly smaller (1 3 to 1 4 times smaller) than ˆµ, see Bird et al. (1987) and Tanner (2). The ratio ˆµ /ˆµ

7 Sucritical ifurcation of shear-thinning channel flows 7 will e thus neglected in the following. The dimensionless effective viscosity is then µ = ˆµˆµ = ( 1 + λ 2 Γ ) nc 1 2 with λ = ˆλ ĥ/û = ˆλ ˆρ ĥ2 /ˆµ Re = Λ Re, (2.7) where 1/λ is the dimensionless characteristic shear rate for the onset of shear-thinning and Λ the ratio of the characteristic time of the fluid to the viscous diffusion time. This ratio Λ is fixed for a given fluid and flow geometry. The Newtonian ehavior, ˆµ = ˆµ, is otained y setting n c = 1 or ˆλ =. If ˆΓ >> 1/ˆλ 2, the Carreau model reduces to the power-law model ˆµ = ˆKˆΓ nc 1 2 with a consistency ˆK = ˆµ ˆλn c Base flows The ase flow is parallel and steady: U = U (y)e x and P = P (x). The momentum equation reduces to with = dp dx + 1 Re µ = ( 1 + λ 2 ( du dy ( ) d du µ dy dy ) 2 ) nc 1 2. (2.8) The suscript refers to the ase flow. Numerical solutions of (2.8) are required, since this equation is nonlinear. An iterative spectral method is used for this purpose. Examples of the asic velocity, U (y), and the viscosity profile, µ (y), are given in figures 1 and 2 for various values of the parameters λ and n c. As expected, the velocity profiles flatten with increasing shear-thinning and the wall axial velocity gradients increase therey reducing the wall shear-viscosity. The figure 2 shows that oth decreasing n c and increasing λ have the effect of increasing shear-thinning in the fluid, ut in two distinct ways. For fixed n c, increasing λ to large values drives an increase of the viscosity-gradient dµ /dy near the axis, and its decrease near the wall (figure 2a). Vice-versa, when λ is fixed and n c is decreased (figure 2). Because of the crucial role of the viscosity stratification

8 8 A. Chekila, C. Nouar, E. Plaut and A. Nemdili (3).8 (4).6.6 U U (a) y () y Figure 1. Basic velocity profiles: (a) n c =.5 and different values of λ: λ = Newtonian; (2) λ = 1; (3) λ = 1. () λ = 1 and different values of n c: n c = 1 Newtonian; (2) n c =.7; (3) n c =.5; (4) n c =.3. in the critical layer, in the linear staility analysis, we have represented in figure 3 the wall viscosity-gradient dµ /dy w as function of the rheological parameters n c and λ. From λ =, i.e., Newtonian case, dµ /dy w increases sharply, reaches a maximum at λ 1 and then decreases with increasing λ. It can e shown using (2.7) that when λ >> 1, (dµ /dy) w (n c 1) λ nc 1 (Γ ) nc 1 w (dγ /dy) w, where (Γ ) w and (dγ /dy) w can e calculated analytically using power-law model. Thus, dµ /dy w decreases as λ nc 1. However, one should take care of the fact that (µ ) w also decreases with increasing λ. ( ) 1 It appears therefore more appropriate to consider the ratio dµ dy as function of the rheological parameters. This ratio is called here the degree of viscosity stratification and denoted Sv. Figure 3() shows that (Sv) w increases with increasing shear-thinning effects. For λ >> 1, (Sv) w saturates to a constant value identical to that calculated for a power-law fluid, i.e., (1 n c )/n c. µ

9 µ Sucritical ifurcation of shear-thinning channel flows 9 (5) (4) (3) (2) (a) y µ (4) (2) (3) () y Figure 2. Basic viscosity profiles: (a) n c =.5 and different values of λ: λ = Newtonian; (2) λ = 1; (3) λ = 1; (4) λ = 5; (5) λ = 1. () λ = 1 and different values of n c: n c = 1 Newtonian; (2) n c =.7; (3) n c =.5; (4) n c = n c = n =.3 c d µ / d y w.6.4 n c =.5 Sv n =.4 c n =.5 c.2 n c =.7.5 n =.6 c n =.7 c (a) λ () λ Figure 3. (a) Wall viscosity-gradient as function of the rheological parameters. () Degree of viscosity stratification Sv at the wall as function of the rheological parameters Disturance equations The velocity U and pressure P of the distured flow are split into the asic field (with suscript ) and the disturance: U = U + u, P = P + p. (2.9)

10 1 A. Chekila, C. Nouar, E. Plaut and A. Nemdili Here, we consider two-dimensional disturances in the (x, y)-plane added to the asic field. It is worthy to note that there is no equivalent of Squire s theorem for nonlinear viscous fluids. However, the numerical tests performed in Nouar & Frigaard (29), for a large range of rheological parameters and of streamwise and spanwise wave numers, show that the lowest critical Reynolds numer is otained for spanwise homogeneous perturation. This result could e anticipated since the viscosity felt y the perturation in two-dimensional situation (tangent viscosity) is less than that in three-dimensional situation, as it also indicated in equation (2.25) elow. We introduce the streamfunction for the disturance, ψ(x,y;t), such that u = ψ y, v = ψ x. (2.1) The equation governing the stream function of an unsteady disturance is otained y cross-differentiating x and y momentum equations and eliminating the pressure. This yields the vorticity equation, t ψ = ( D 2 U U ) ψ x ( 2 + y 2 2 x 2 + J(ψ, ψ) + 2 x y [τ xx (Ψ + ψ) τ yy (Ψ + ψ)] ) τ xy (Ψ + ψ) (2.11) where D d/dy, Ψ is the stream function associated with the ase flow U = DΨ, J (f,g) is the Jacoian defined y ( f/ x) ( g/ y) ( f/ y) ( g/ x), 2 / x / y 2 and τ ij (Ψ + ψ) = µ(ψ + ψ) γ ij (Ψ + ψ). No-slip oundary conditions are imposed at the walls: ψ x = ψ = at y = ±1. (2.12) y For a small amplitude disturance, the viscosity of the pertured flow can e expanded around the ase flow as: µ(ψ + ψ) = µ + µ 1 (ψ) + µ 2 (ψ, ψ) + µ 3 (ψ, ψ, ψ) +..., (2.13)

11 where Sucritical ifurcation of shear-thinning channel flows 11 µ 1 (ψ) = µ γ ij γ ij (ψ), (2.14) µ 2 (ψ,ψ) = 1 2 µ 2 γ ij γ kl γ ij (ψ) γ kl (ψ), (2.15) µ 3 (ψ, ψ, ψ) = 1 3 µ 6 γ ij γ kl γ pq γ ij (ψ) γ kl (ψ) γ pq (ψ). (2.16) The deviatoric stresses in the distured flow can also e written as τ ij (Ψ + ψ) = τ ij (Ψ ) + τ 1,ij (ψ) + τ 2,ij (ψ, ψ) + τ 3,ij (ψ, ψ, ψ) +..., (2.17) with τ 1,ij (ψ) = 1 Re [µ 1 (ψ) γ ij (Ψ ) + µ γ ij (ψ) ], (2.18) τ 2,ij (ψ,ψ) = 1 Re [µ 2 (ψ,ψ) γ ij (Ψ ) + µ 1 (ψ) γ ij (ψ) ], (2.19) τ 3,ij (ψ,ψ,ψ) = 1 Re [µ 3 (ψ,ψ,ψ) γ ij (Ψ ) + µ 2 (ψ,ψ) γ ij (ψ) ]. (2.2) In the following, A 1 (ψ), A 2 (ψ,ψ) and A 3 (ψ,ψ,ψ) where A stands for τ ij or µ, will e denoted respectively A 1, A 2 and A 3. In the ase flow γ ij = γ ij (Ψ ) = if ij xy,yx and γ ij = DU if ij = xy,yx. Setting Γ = (DU ) 2 and Γ 2 = (1/2) γ ij (ψ) γ ij (ψ), the expressions of µ 1,µ 2 and µ 3 can e simplified, µ 1 = 2 µ Γ γ xy γ xy (ψ), (2.21) µ 2 = µ Γ Γ µ Γ 2 Γ γ xy 2 (ψ), (2.22) µ 3 = 2 2 µ Γ 2 γ xy γ xy (ψ) Γ µ ) 3 3 Γ 3 ( γ xy γ 3 xy (ψ). (2.23)

12 12 A. Chekila, C. Nouar, E. Plaut and A. Nemdili Replacing µ 1, µ 2 and µ 3 y their corresponding expressions (2.21)-(2.23) into eqs. (2.18)- (2.2) we otain τ 1,ij = 1 Re µ γ ij (ψ) if ij xy, yx, (2.24) τ 1,xy = τ 1,yx = 1 [ ] µ µ + 2Γ Re Γ γ xy (ψ) = 1 Re µ t γ xy (ψ), (2.25) τ 2,ij = 1 Re µ 1 γ ij (ψ) if ij xy, yx, (2.26) τ 2,xy = 1 Re [µ 1 γ xy (ψ) + µ 2 γ xy (Ψ )], (2.27) τ 3,ij = 1 Re µ 2 γ ij (ψ) if ij xy,yx, (2.28) τ 3,xy = 1 Re [µ 2 γ xy (ψ) + µ 3 γ xy (Ψ )]. (2.29) In equation (2.25), µ µ t = µ + 2Γ Γ (2.3) is the tangent viscosity, which can e also defined y µ t = ( τ xy / γ xy ). For shearthinning fluid, dµ /dγ < and µ t < µ. Finally, the disturance equation (2.11) can e rewritten ψ = L(ψ) + N (ψ). (2.31) t The linear and nonlinear terms of (2.11) are gathered in L(ψ) and N (ψ) respectively, where we separate the contriutions of the inertial and viscous terms as follows: L = L I + L V 1 + L V 2 + L V 3, (2.32) N = N I + N V quad + N V cu +..., (2.33)

13 with Sucritical ifurcation of shear-thinning channel flows 13 L I (ψ) = ( D 2 U U ) ψ x, (2.34) Re L V 1 (ψ) = µ 2 (ψ), (2.35) Re L V 2 (ψ) = ( [ D 2 ) µ γxy (ψ) + Dµ 2 γ xy y (ψ) + γ xx x (ψ) γ ] yy x (ψ), (2.36) ( ) 2 Re L V 3 (ψ) = y 2 2 x 2 [(µ t µ ) γ xy (ψ)], (2.37) N I (ψ,ψ) = J (ψ, ψ), (2.38) 2 Re N V quad (ψ,ψ) = x y [µ 1 ( γ xx (ψ) γ yy (ψ) ) ] ( ) 2 + y 2 2 x 2 [µ 1 γ xy (ψ) + µ 2 γ xy (Ψ ) ], (2.39) 2 Re N V cu (ψ,ψ,ψ) = x y [µ 2 ( γ xx (ψ) γ yy (ψ) ) ] ( ) 2 + y 2 2 x 2 [µ 2 γ xy (ψ) + µ 3 γ xy (Ψ ) ]. (2.4) The suscripts I and V refer to inertial and viscous terms. L V 1 is a Newtonian-like viscous term, L V 2 and L V 3 reflect respectively the spatial stratification of the viscosity and the anisotropy of the deviatoric shear-stress tensor perturation. N I is the nonlinear inertial term. N V quad and N V cu gather the quadratic and cuic terms which arise from the nonlinear variation of the effective viscosity with the shear rate. Higher order expansions, up to seventh order, are given in Appendix C 3. Linear staility analysis 3.1. Direct prolem To solve the linearized version of the disturance equation (2.31), we use the normal mode analysis, assuming that ψ(x,y;t) = f 1,1 (y)exp [iα (x ct)] (3.1)

14 14 A. Chekila, C. Nouar, E. Plaut and A. Nemdili with α the axial wave numer and c = c r + ic i the complex wave speed. Its real part c r is the phase velocity and the imaginary part yields the growth rate α c i. It can e shown that f 1,1 (y) satisfies the modified Orr-Sommerfeld equation L 1 f 1,1 = (3.2) with L 1 iαcs 1 + iα [ U S 1 D 2 ] 1 [ U µ S (Dµ ) S 1 D + ( D 2 ) ] µ G1 Re 1 Re G 1 [(µ t µ ) G 1 ]. (3.3) The operators S n and G n are defined y S n D 2 n 2 α 2 and G n D 2 + n 2 α 2, n 1. (3.4) The oundary conditions are f 1,1 = Df 1,1 = at y = ±1. (3.5) The eigenvalue prolem (3.2)-(3.5) is invariant with respect to the symmetry y y, therefore the eigenmodes are either odd or even under y y. We consider only the eigenmodes which are unstale at the lowest Reynolds numer, which appear to e the even modes as in the Newtonian case (Drazin & Reid 1995). The oundary conditions at y = 1 can therefore e replaced y the parity conditions: Df 1,1 = D 3 f 1,1 = at y =. (3.6) Since any multiple of an eigenfunction is also a solution of (3.2), f 1,1 can e normalized according to f 1,1 = 1 at y =, (3.7) which fixes amplitude and phase of the wave (3.1). The eigenvalue prolem (3.2) with its oundary conditions is solved using Cheychev collocation method, see e.g. Schmid &

15 Sucritical ifurcation of shear-thinning channel flows 15 Henningson (21). The characteristics of the destailizing waves and the critical parameters α c,c c and Re c (critical wavenumer, critical phase velocity and critical Reynolds numer) will e given in 5.1, after the presentation of the weakly nonlinear analysis Adjoint prolem The adjoint operator of L 1, equation (3.2), is defined with the hermitian inner product, (f,g) = 1 f(y)g (y)dy, etween two functions f(y) and g(y) of H 2 ([,1]). The homogeneous adjoint prolem associated to the equation (3.2) is given as L 1 f 1,1 =, (3.8) with L 1 iαc S 1 iα [U S (DU ) D] 1 [ µ S (Dµ ) S 1 D + ( D 2 ) ] µ G1 Re 1 Re G 1 [(µ t µ ) G 1 ] (3.9) and Df 1,1 = D3 f 1,1 = at y = ; f 1,1 = Df 1,1 = at y = 1. (3.1) 4. Weakly nonlinear analysis 4.1. General asymptotic expansion - Landau equation Once the critical parameters are known, in order to study the saturation of the ifurcation to the waves, a weakly nonlinear analysis is used. Due to the nonlinear terms in the disturance equation (2.31), it is clear that a nonlinear solution which contains the fundamental mode in exp [±iα c (x c c t)] must also contain harmonic modes in exp [niα c (x c c t)] for all n integers. Therefore, it seems natural to represent the nonlinear disturance as the

16 16 A. Chekila, C. Nouar, E. Plaut and A. Nemdili Fourier series ψ (x, y, t) = φ (y,t) + + n= n φ n (y,t) exp [niα c (x c c t)], (4.1) where we have separated out the mean part φ (y,t), i.e. the mean flow distortion. Because ψ is real, we have that φ n = φ n, where the star denotes complex conjugation. Sustituting (4.1) into (2.31) comined with (2.32)-(2.4) and (2.21)-(2.23) and separating out the coefficients of like exponentials, we otain an infinite set of coupled nonlinear partial differential equations for the Fourier components φ n. The nonlinearity and coupling of this infinite set makes its solution difficult. However, if the amplitude A of the fundamental wave is small, the Fourier components φ n can e sought using a perturation method expanding around the solution of the linear prolem. It is clear that if the linear solution is O(A), the leading term of φ 2 is O(A 2 ) ecause of the interaction of the fundamental with itself. The same reasoning applied for higher harmonics indicate that φ n can e written as φ n (y, t) = A n (t)f n (y,t) if n > and φ (y, t) = A 2 f (y,t), (4.2) with f n (y,t) = f (y,t) = + m=1 + m=1 A 2(m 1) f n,n+2(m 1) (y) if n (4.3) A 2(m 1) f,2m (y), (4.4) see e.g. Watson (196); Stuart (196); Reynolds & Potter (1967); Herert (1983). In the aove equations, A = A(t) is the complex amplitude of the fundamental, descried as a general function of time and considered as a small parameter. The suscripts to function f k,l means that it has an harmonic index k and asymptotic order l. The time evolution of the disturance amplitude A is given y the Stuart-Landau equa-

17 tion: Sucritical ifurcation of shear-thinning channel flows 17 da + dt = iα (c c c) A + g j A 2j A, (4.5) j=1 where c is the complex phase velocity evaluated in the vicinity of the critical conditions c = c(α c,re) and g j is jth Landau coefficient. This is in fact a rigorous series expansion in powers of A at the onset, i.e., for c = c c, ut it is customary to truncate the series at a finite order, and to use the Landau equation (4.5) at a finite distance of the onset. As A is assumed complex, the real amplitude is given y the the modulus of the amplitude A which is determined y: d A 2 dt = 2α c i A j=1 g jr A 2(j+1), (4.6) where the suscript jr in g jr means the real part of g j. For linearly unstale flows, the nature of the ifurcation is determined g 1r. If g 1r is positive, then the instaility is sucritical, while if g 1r is negative, the instaility is supercritical. The imaginary part g 1i gives the first order correction to the frequency of the asic wave due to nonlinear effects. In the system of equations otained after sustituting (4.1) into (2.31) and separating terms of like exponential, the Fourier components φ n are replaced y their expression (4.2) comined with (4.3)-(4.5). Equating the same powers A, a hierarchy of ordinary differential equations for the functions f k,l is otained. These are solved sequentially eginning from k = 1,l = 1. The prolem k = 1, l = 1 is the linear prolem (3.2) which gives the critical point around which the harmonic-amplitude expansion is carried out. The prolem k =, l = 2 yields the first correction to the mean flow. The prolem k = 2, l = 2 yields the first harmonic of the fundamental mode. The prolem k = 1, l = 3 yields the coefficient g 1 of feedack on the fundamental mode. In order to avoid overloading the text, the linear (L k ), ilinear (N I and N V quad ) and

18 18 A. Chekila, C. Nouar, E. Plaut and A. Nemdili trilinear forms (N V cu ), involved in the ordinary differential equations for the functions f k,l, are given in Appendix A Modification of the mean flow The interaction of the fundamental mode Af 1,1 with its complex conjugate produces a mean flow correction proportional to A 2, i.e., A 2 u,2, where u,2 = Df,2. Actually, this modification of the mean flow cannot e calculated using the vorticity equation (2.31), since, in this x-independent case, the linear operator implied is not invertile. Instead, one has to consider the streamwise mean momentum equation. In the situation of a fixed axial pressure gradient, it reduces to: where. x = α 2π 2π/α y (uv) = x y τ 1,xy + x y τ 2,xy, (4.7) x (.)dx. Using (2.25),(2.27) and (2.22), it can e shown after some algera, that u 2 satisfies the following equation: 1 Re D (µ t Du,2 ) = iαd ( f 1,1 Df1,1 f1,1df ) 1,1 2 [ Re D γ xy µ ( Γ 3 G 1 f 1, α 2 Df 1,1 2) ] 4 [ Re D γ xy 2 ] µ Γ Γ 2 G 1 f 1,1 2. (4.8) with the oundary and parity conditions u,2 = at y = 1 and Du,2 = at y = (4.9) 4.3. Generation of the first harmonic of the fundamental mode The interaction of the fundamental with itself, through the quadratic nonlinear terms of the perturations equation, produces first harmonic term, f 2,2 A 2 E 2. At order k = 2, l = 2, the perturation equation (2.31) comined with (4.2)-(4.5) reduces to: L 2 f 2,2 = N I (f 1,1,f 1,1 ) + N V quad (f 1,1,f 1,1 ). (4.1)

19 Sucritical ifurcation of shear-thinning channel flows 19 The oundary and parity conditions read f 2,2 = Df 2,2 = at y = 1 and f 2,2 = D 2 f 2,2 = at y =. (4.11) 4.4. Calculation of the cuic Landau constant The cuic Landau constant g 1 appears in the prolem k = 1, l = 3. This prolem represents the feedack at order O(A 3 ) on the fundamental mode through the nonlinear interactions of the fundamental with the first harmonic and with the modification of the mean flow. It reads: L 1 f 1,3 = g 1 S 1 f 1,1 + N I (f,2 f 1,1 ) + N V quad (f,2 f 1,1 ) + N I (f 2,2 f 1,1 ) + N V quad (f 2,2 f 1,1 ) + N V cu (f 1,1,f 1,1 f 1,1 ), (4.12) where f k,l = fk,l. In the critical conditions, equation (4.12) has a non trivial solution if the Fredholm solvaility condition is satisfied, i.e., orthogonality of the inhomogeneous part of the equation (4.12) to the null-space of the adjoint operator of L 1, see 3.2. The cuic Landau coefficient is then readily otained, g 1 = g I 1 + g V 1 = ( g I 1 + g I 12) + ( g V 1 + g V 12 + g V 1 11), (4.13) with ( ) ( ) N I (f,2 f 1,1 ),f g1 I 1,1 N V quad (f,2 f 1,1 ),f = ( ), g V 1,1 S 1 f 1,1, f 1 = ( ), (4.14) 1,1 S 1 f 1,1, f 1,1 ( ) ( ) N I (f 2,2, f 1,1 ),f g12 I 1,1 N V quad (f 2,2 f 1,1 ),f = ( ), g V 1,1 S 1 f 1,1, f 12 = ( ), (4.15) 1,1 S 1 f 1,1, f 1,1 ( ) N V cu (f 1,1,f 1,1 f 1,1 ),f g1 11 V 1,1 = ( ), (4.16) S 1 f 1,1, f 1,1

20 2 A. Chekila, C. Nouar, E. Plaut and A. Nemdili where g1 I and g1 V are the feedack of the mean flow correction onto the wave through the nonlinear inertial and nonlinear viscous terms, g12 I is the feedack of the harmonic onto the wave, etc Results and discussion 5.1. Linear prolem A spectral collocation method ased on Cheyshev polynomials is applied to determine the eigenvalues c and the corresponding eigenfunctions. The differential equation (3.2) comined with (3.3) is discretized on the Gauss-Loatto grid. The resulting generalized eigenvalue prolem is solved using the QZ algorithm with Matla. Spectra with increasing collocation points (N = 5, 1, 15, 2, 3) were compared to determine the adequate numer (N + 1) of Cheyshev polynomials. It is found that, in order to keep the same resolution accuracy (within five digits) of the least stale mode for all the situations considered here, it is necessary to increase N as the shear-thinning ehavior of the fluid ecomes more important. For instance, N = 2 for a Carreau fluid with n c =.1 and λ = 1, whilst N = 5 in the Newtonian case. Marginal staility curves and the associated critical conditions are determined for different rheological parameters (n c,λ). To determine the critical conditions, for a given rheological parameters, α and Re are varied until c i 1 5. Numerical results have een already given in (Nouar et al. 27). When the Reynolds numer is defined with the wall-shear viscosity Re cw = Re c /(µ ) w, (5.1) the shear-thinning delays the linear instaility as shown in Tale 1. The results of Nouar et al. (27) are supplemented here y additional data for large values of λ and another shear-thinning index n c. In figure 4, the critical Reynolds numer Re cw is depicted vs

21 Sucritical ifurcation of shear-thinning channel flows 21 n c λ Re cw α c c c Tale 1. Critical Reynolds numer Re cw, critical wavenumer α c and critical wave speed c c for various rheological parameters. λ for different values of n c. The stailizing effect of the shear-thinning is clearly illustrated. One has also to note, that the evolution of Re cw v.s. the rheological parameters is correlated to that of the degree of viscosity stratification at the wall, i.e., 1 µ Dµ at y = 1 (see figure 3). The evolution of the critical wavenumer with the shear-thinning ehavior has een descried in (Nouar et al. 27). At large λ and when n c is decreased, the waves have a shorter wavelength, and also a lower phase speed (Tale 1). Remark Note that, if the perturation of the viscosity is not taken into account, i.e., one sets µ t = µ in (3.3), the stailizing effect is much stronger. The critical Reynolds numer is 2 to 3 times higher. Some characteristics of the critical wave are shown in figure 5 for Newtonian and Carreau fluids. For such fluids, the streamlines are more concentrated near the wall, indicating the presence of steep velocity gradient and a large unaffected core region.

22 22 A. Chekila, C. Nouar, E. Plaut and A. Nemdili 1 4 Re cw n =.3 c n =.4 c n =.5 c n =.6 c 1 n c =.7 n c = (a) 1 λ δ c () (2) (3) n c Figure 4. (a) Critical Reynolds numer defined with the wall shear viscosity as function of the dimensionless characteristic time λ of the fluid for different values of n c. () Thickness of the critical layer as function of n c for different λ: λ =.5; (2) λ = 1; (3) λ = 1. The slope of the separatrices (lines where the stream function vanishes) near the wall is related to the energy exchange etween the mean flow and the disturance. It has een shown (Plaut et al. 28) that for the disturance to extract energy from the asic flow through the action of the Reynolds stress, the separatrices must slope against the mean flow. The thickness δ c of the critical layer, where the exchange of energy etween the perturation and the ase flow takes place, is represented as function of n c for different λ, in figure 4(). The critical layer ecomes thinner when the shear-thinning effects increase. The structure of the critical eigenfunctions is depicted in figure 6 for λ = 1 and different values of n c and in figure 7 for n c =.5 and different λ. Concerning the critical eigenfunctions of the adjoint linear operator, they are given in Appendix B for various values of the shear-thinning index.

23 Sucritical ifurcation of shear-thinning channel flows y.5 y (a) x () x Figure 5. Iso-values of the streamfunction associated with the critical waves. (a) Newtonian fluid; () Carreau fluid with λ = 1 and n c =.3: Continuous lines for positive values of ψ:.1 near the walls then with a step of.1 until.9. Dotted lines for negative values of ψ:.1 near the walls then with a step of.1 until.9. The thick lines are the separatrices where the stream-function vanishes..5 1 (4) (4).8 (2) (3).5 f 1,1 r.6 f 1,1 i (a) y () y Figure 6. (a) Real part, () imaginary part of the eigenfunctions associated to the critical mode for λ = 1 and different values of the shear-thinning index n c: n c = 1, Newtonian; (2) n c =.7; (3) n c =.5; (4) n c = Energy equation To analyze the importance of the viscosity perturation on the critical Reynolds numer, it is useful to consider the Reynolds-Orr energy equation truncated at order A 2, 1 d u 2 2 dt 1 + v1 2 xy = + 1 Re 1 Re µ du (v 1 ) : (v 1 ) xy u 1 v 1 dy (µ µ t ) ( γ xy ) 2 xy xy (5.2)

24 24 A. Chekila, C. Nouar, E. Plaut and A. Nemdili.5 1 (3) (2).8.5 (3) (2) f 1,1 r.6 f 1,1 i (a) y () y Figure 7. (a) Real part, () imaginary part of the eigenfunctions associated to the critical mode for n c =.5 and different values of λ: λ =, Newtonian; (2) λ = 1; (3) λ = 1. where. xy = 2π/α 1 (.)dydx, u 1 and v 1 are the streamwise and the wall-normal velocities of the fundamental mode. The first term in the right-hand-side, due to dissipation, is always negative. The second term is positive, since the waves extract energy from the asic shear flow, near the wall where the separatrices slope. The third term arises from the viscosity perturation. It is definite positive, ecause µ > µ t for shear-thinning fluids. It traduces a reduction of the viscous dissipation and may e viewed as an energy source term. It increases when the difference etween the effective and the tangent viscosities, i.e. increases, with increasing shear-thinning effects. The onset of instaility is then found earlier than in the case where the viscosity perturation is not taken into account Modification of the mean flow As for the linear prolem, equation (4.8) with the associated oundary conditions (4.9) is solved numerically using a spectral collocation method ased on Cheyshev-polynomials. Figure 8 shows the modification of the mean flow at order A 2 for two sets of rheological parameters, λ = 1 and different values of n c and n c =.5 and different values of

25 1 Sucritical ifurcation of shear-thinning channel flows u,2 5 (3) (2) u,2 1 2 (2) (4) 1 u,2 / 2 (5) (a) y () y (3) (4) Figure 8. Modification of the ase flow. (a) λ = 1 and n c = 1 Newtonian; (2) n c =.7; (3) n c =.5; (4) n c =.3; (5) n c =.1, for this case, we have represented u,2/2 rather than u,2. () n c =.5 and λ = Newtonian; (2) λ = 1; (3) λ = 1; (4) λ = 1. λ. By comparison with the Newtonian case (curve 1), a remarkale difference is that the fluid can e accelerated near the wall. It is always significantly decelerated in the central zone. With increasing shear-thinning effects, the acceleration ecomes stronger and more confined near the wall. Canceling artificially the nonlinear viscous terms in (4.8) allows to highlight the contriution of the inertial terms on the modification of the mean flow and vice-versa, to highlight the contriution of the nonlinear viscous terms. The results, shown in figure 9, prove that the interaction of the fundamental with its complex conjugate through nonlinear viscous terms, accelerates the fluid in all the width etween the two plates (curves 4 in figure 9). In contrast, the inertial terms reduce the flow rate as for a Newtonian fluid (curves 3 in figure 9). In order to interpret these results, equation (4.7) is integrated once, and ecause of the symmetry of u 2 and f 1,1 under y y, we otain 1 Re µ t Du,2 = u 1 v 1 x τ 2,xy x. (5.3)

26 26 A. Chekila, C. Nouar, E. Plaut and A. Nemdili u, viscous terms inertial terms (a) y (4) (2) (3) u, () viscous terms inertial terms (4) (2) (3) y Figure 9. Modification of the ase flow: (a) n c =.5 and λ = 1; () n c =.3 and λ = 1. Reference curve, Newtonian fluid; (2) Carreau fluid; (3) Carreau fluid with the nonlinear inertial terms only; (4) Carreau fluid with the nonlinear viscous terms only. The first term on the right-hand of the aove equation can e related to the slope x s of the separatrices represented in figure 5 via the relation u 1 v 1 x = v1 2 x x s (Plaut et al. 28). The numerical results show that u 1 v 1 x is of the sign of du /dy, and increases in magnitude, in the critical layer, with increasing shear-thinning effects. Therefore, the strong deceleration of the flow illustrated y the curves 3 in figures 9(a) and 9() can e explained y the increase of the Reynolds-stress. The numerical results show also that τ 2,xy x, the viscous shear-stress arising from the interaction of the fundamental with its complex conjugate, is positive and increases with increasing the shear-thinning effects. This can explain the acceleration of the flow, induced y the nonlinear viscous terms, illustrated y the curves 4 in figures 9(a) and 9(). The term on left-hand side of (5.3), i.e., (1/Re)µ t (du 2 /dy) = τ 1,xy involves the tangent viscosity µ t < µ which amplifies the accelerations and decelerations of the flow descried aove. Alternatively, fixed flow rate conditions can e imposed. In this case, a correction P,2 x to the pressure field is introduced at order A 2 such that the modified mean-flow correction

27 ũ,2 satisfies Sucritical ifurcation of shear-thinning channel flows 27 ũ,2 (y) y =. (5.4) The streamwise component of the mean momentum equation at order A 2 reads d dy (uv),2 = d P,2 dx + 1 Re which, y comparison with (4.8) gives d dy [ ] dũ,2 µ t + d dy dy (τ 2xy),2 (5.5) ũ,2 (y) = u,2 (y) Re d P,2 dx 1 y ζ µ t dζ with d P,2 dx = 1 Re 1 1 u,2(y)dy 1 y (ζ/µ t)dζdy (5.6) All the integrals are evaluated numerically using Clenshaw and Curtis method. The mean flow correction ũ,2 is shown in figure 1 for fixed λ and different values of n c. With increasing shear-thinning effects, the flow is accelerated in a thin region adjacent to the wall, followed y a deceleration and then a slight acceleration in a large region around the axis. The fact that the mean pressure gradient d P,2 /dx is negative as it is shown in the figure 1() signifies that at fixed flow rate, the transition to generalized Tollmien-Schlichting waves is accompanied y an increase of the head loss. The figure 1() suggests also that there is an optimal values of the rheological parameters for which the increase of the head loss is minimum and lower than in the Newtonian case First harmonic of the fundamental Equation (4.1) with the associated oundary conditions (4.11) is solved using the same numerical approach as in 5.3. The first harmonic of the fundamental which has half its wavelength is displayed in figure 11 for a Carreau fluid with λ = 1 and n c =.3. The real and the imaginary parts of f 2,2 are shown in figure 12 for λ = 1 and different values of n c. It is oserved that shear-thinning induces a significant amplification of f 2,2. To determine the contriution of the inertial terms to the generation of the first harmonic, the nonlinear viscous terms in the r.h.s. of equation (4.1) are canceled artificially, and

28 28 A. Chekila, C. Nouar, E. Plaut and A. Nemdili u ~,2 2 4 (2) (3) 1 2 ( dp ~ 2 / d x ) (4) y (a) () n c Figure 1. (a) Modification of the ase flow for fixed flow rate at λ = 1 and different n c: Reference curve otained for Newtonian fluid and represented y the dashed line; (2) n c =.7; (3) n c =.5; (4) n c =.3. () Correction to the mean pressure gradient versus n c at fixed flow rate and λ = 1. vice-versa to otain the contriution of the quadratic nonlinearity arising from the viscosity. The results are shown in figure 13. It appears that the first harmonic generated y the nonlinear viscous terms is smaller and in opposite phase with that generated y the quadratic nonlinear inertial terms. This reveals proaly, that the mechanism of energy exchange etween the fundamental and its harmonic via the nonlinear inertial terms is different compared to that via the nonlinear viscous terms Cuic Landau constant The cuic coefficient g 1, i.e., the first Landau constant in (4.5) is determined for different critical sets (n c,λ,re,α). The integrals in (4.14)-(4.16) are evaluated numerically y means of Clenshaw and Curtis Method. In figure 14(a), we plot g 1r = Re(g 1 ) in terms of λ for fixed values of n c. As one could expect, the sign of g 1r is found positive indicating a sucritical ifurcation. For fixed n c, g 1r increases as the dimensionless characteristic time λ increases and tends towards an asymptote. In the same way, for a given λ, g 1r

29 Sucritical ifurcation of shear-thinning channel flows y x.9 Figure 11. Iso-values of Re ( f 2,2E 2) at the critical conditions and t =, for a Carreau-fluid with λ = 1 and n c =.3. Continuous and dotted lines correspond respectively to positive and negative values of Re ( f 2,2E 2). They are normalized y the maximum value and displayed y step of.1. f 2,2 r (4) (3) (2) (a) y f 2,2 i () y (2) (3) (4) Figure 12. Real (a) and imaginary () parts of the first harmonic of the critical mode for λ = 1 and different values of the shear-thinning index n c: n c = 1 Newtonian case; (2) n c =.7; (3) n c =.5; (4) n c =.3. increases strongly as n c decreases. Thus shear-thinning effects, while tending to promote staility on a linear asis, also tend to augment the sucritical character of the ifurcation. Contriutions of the different terms g I 1, g I 12,g V 1,... (see equation 4.13) that control

30 3 A. Chekila, C. Nouar, E. Plaut and A. Nemdili (3) (2) (3) 4 f 2,2 r 3 2 f 2,2 i 2 4 (2) 1 (4) (a) y () y (4) Figure 13. Real (a) and imaginary () parts of the first harmonic of the fundamental mode at the critical condition for Carreau fluid with λ = 1 and n c =.3. n c = 1 Newtonian case as a reference curve; (2) Carreau fluid ; (3) Carreau fluid: contriution of the inertial terms; (4) Carreau fluid: contriution of the nonlinearities arising from the viscous terms. the value of g 1r are given in tales 2 and 3 for fixed pressure gradient and fixed flow rate respectively. The data show that with increasing shear-thinning effects, the feedack of the harmonic onto the waves plays an important role in the sucritical ifurcation, whilst in the Newtonian case g12 I is negative and the sucritical nature of the ifurcation is due to the feedack of the mean flow correction onto the wave (Reynolds & Potter 1967; Plaut et al. 28). If the nonlinear viscous terms are canceled artificially in the disturance equation (2.31), higher values of g 1r are found. This can e assessed y comparing curve (2) otained without nonlinear viscous terms and curve (3) in figure 14(). For λ 1, the relative difference is aout 2%. In contrast, if the nonlinear inertial terms are canceled in the disturance equation (2.31), g 1r is negative and the ifurcation ecomes supercritical (see the curve 4 of figure 14. Hence, for shear-thinning fluids, the nonlinear variation of µ with the shear-rate favors a supercritical ifurcation.

31 Sucritical ifurcation of shear-thinning channel flows 31 g 1 r (a) 1 λ (5) (4) (3) (2) g 1r (2) nonlinear inertial terms nonlinear viscous terms g 1r 1 1 () λ (3) (4) Figure 14. (a) Real part of the Landau constant as function of λ and different values of n c, under the fixed pressure-gradient condition. Newtonian case n c = 1; (2) n c =.7; (3) n c =.5; (4) n c =.3; (5) n c =.2. () Real part of the Landau constant versus λ for a given shear-thinning index n c =.3. Newtonian fluid, (2) Carreau fluid, (3) Only nonlinear inertial terms are retained and (4) Only nonlinear viscous terms are retained; in this case the opposite of g 1r is shown n c g 1r g I 1 g V 1 g I 12 g V 12 g V Tale 2. Fixed pressure gradient condition. Real part of the first Landau constant and contriution of the nonlinear inertial and nonlinear viscous terms, for different values of the shear-thinning index at λ = 1. For the Newtonian case, the value of g 1r is in agreement with that given y Reynolds & Potter (1967).

32 32 A. Chekila, C. Nouar, E. Plaut and A. Nemdili n c g 1r g I 1 g V 1 g I 12 g V 12 g V Tale 3. Same as tale 2 ut for fixed flow rate condition. For the Newtonian case, the value of g 1r is in agreement with that given y Fujimura (1989) Threshold amplitude Beside the Landau coefficient, the threshold amplitude, A c, which limits the asin of attraction of undistured parallel flow, is another important quantity in the nonlinear staility analysis. It is otained y setting da/dt = in (4.5): the linear growth rate and its nonlinear correction alance. In the neighorhood of the critical conditions such that (Re c Re) /Re c = ǫ << 1, using Taylor expansion, αc i can e written as α c i = ǫ/τ + O(ǫ 2 ), where the τ is a characteristic time. Hence, to lowest order in ǫ, the threshold amplitude is ǫ A c =. (5.7) τ g 1r One has to note that (dc i /dǫ) ǫ= has to e evaluated for a given fluid and flow geometry that is for a constant Λ = λ/[re c (1 ǫ)]. Figure 15(a) shows the threshold amplitude in the form A c / ǫ as a function of the shear-thinning index n c. The curve illustrates that the flow ecomes much more sensitive to small disturances since the threshold

33 Sucritical ifurcation of shear-thinning channel flows A c / ε 1/2 1 2 Threshold energy, ζ / ε (a) nc 1 5 () nc Figure 15. Fixed pressure gradient condition. (a) Scaled threshold amplitude, eyond which the flow is nonlinearly unstale, as function of n c at λ = 1. () Threshold kinetic energy as a function of the shear-thinning index, n c at λ = 1. amplitude significantly decreases as the shear-thinning effects increase, i.e., the flow ecomes relatively more nonlinearly unstale. The numerical results indicate also that the characteristic time τ decreases. For instance, for a Newtonian fluid, we have τ = 13.1, in agreement with Herert (198), compared to otained for a Carreau fluid with n c =.2 and λ = 1. One has to note that the numerical values of the Landau constant and hence the values of A c depend upon the normalization condition used for the eigenfunctions 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 threshold kinetic energy required for a finite amplitude instaility, defined y 1 ζ = 2 A c 2 [ Df1,1 2 + αc 2 f 1,1 2] dy. (5.8) Figure 15() plots the threshold kinetic energy in the form ζ/ǫ as a function of the shearthinning index n c, for λ = 1. As it can e oserved, the threshold kinetic energy decreases significantly as n c decreases, highlighting the destailizing effect of shear-thinning for finite amplitude perturations.

34 34 A. Chekila, C. Nouar, E. Plaut and A. Nemdili 5.7. Validation y computing higher-order Landau constants Figure 15 indicates that the threshold amplitude eyond which the flow is nonlinearly unstale decreases with increasing shear-thinning effects. This result was otained y truncating the series (4.5) to the first Landau constant, at cuic order in A. For a significant deviation from the critical condition, terms of higher order ecome large and should e taken into account. Weakly nonlinear expansion was then carried out up to seventh order in amplitude under fixed flow rate condition. For the sake of clarity, calculation details are not given in the text, (see Appendix C). As in the Newtonian case, for shear-thinning fluids, the real part of Landau constants are of alternating sign and increase very fast. This increase is stronger with increasing shear-thinning effects as is shown y the data in tale 4. Figure 16 shows the evolution of the threshold amplitude A c versus the relative departure from the critical conditions, ǫ = (Re c Re)/Re c, for different values of the shear-thinning index. The results are presented at cuic, fifth and seventh order in the amplitude expansion. At n c =.3, the three curves (cuic, fifth and seventh order) are not distinguishale within plotting accuracy. At ǫ =.5, the relative difference in A c etween the fifth and the seventh order is.6% for n c =.3 and 4.6% for a Newtonian fluid. The threshold amplitude decreases with increasing shear-thinning effects. This result is valid at least for a reasonale relative departure from the critical conditions. 6. Conclusion The present work focuses on the first principles understanding of the influence of shear-thinning effects on the flow staility with respect to finite amplitude perturations. The fluid is supposed purely viscous. Hence, in presence of a perturation u, it is assumed that the viscosity instantaneously adjusts the shear rate of the pertured flow