Instabilités et transition en fluides non newtoniens 
Nous nous intéressons à la transition vers la complexité spatiotemporelle et la turbulence dans des écoulements de fluides non newtoniens, plus spécifiquement rhéofluidifiants. Ces fluides, qui s'écoulent d'autant mieux que les contraintes appliquées augmentent, interviennent dans divers procédés industriels en génie pétrolier, en cimenterie, en agroalimentaire, en cosmétique, etc... Dans certains cas, le sang luimême peut être considéré comme un fluide rhéofluidifiant. La transition vers la turbulence dans ces fluides est difficile à étudier du fait de leur richesse de comportements, et parce que les équations modèles sont hautement non linéaires.
Nous étudions cette problématique expérimentalement, par exemple dans des écoulements en tuyau, où des phénomènes étranges peuvent se produire. Ainsi des états faiblement turbulents existent, qui n'ont pas d'équivalent en fluides newtoniens. Dans ces états faiblement turbulents l'écoulement axial moyen est fortement dissymétrique, le fluide s'écoulant rapidement d'un côté du tuyau, et plus lentement de l'autre côté, comme le montre la figure suivante (tirée de [12]):
Nous étudions aussi cette problématique théoriquement. Pour cela nous coopérons avec plusieurs chercheurs de par le monde :
Merci de considérer nos publications cidessous. Vous pouvez aussi observer ces animations réalisées à partir de calculs d'ondes non linéaires comme ceux de notre article [17] :
[1] 
Nonlinear
stability of Poiseuille flow of a Bingham fluid: Theoretical
results and comparison with phenomenological criteria,
C. Nouar and I. A. Frigaard,
J. Non Newt. Fluid Mech. 100, 127149 (2001).
We present new results on the nonlinear stability of Bingham fluid Poiseuille flows in pipes and plane channels. These results show that the critical Reynolds number for transition, Re_{c}, increases with Bingham number, B, at least as fast as Re_{c}B^{1/2} as B→∞. Estimates for the rate of increase are also provided. We compare these bounds and existing linear stability bounds with predictions from a series of phenomenological criteria for transition, as B→∞, concluding that only Hanks [AIChE J. 9 (1963) 306; 15 (1) (1963) 25] criteria can possibly be compatible with the theoretical criteria as B→∞. In the more practical range of application, 0≤B≤50, we show that there exists a large disparity between the different phenomenological criteria that have been proposed.

[2] 
On the
stability of Bingham fluid flow in an annular channel,
N. Kabouya and C. Nouar,
C. R. Méca. 331, 149156 (2003).
Linear stability of a fully developed Bingham fluid flow between two coaxial cylinders subject to infinitesimal axisymetric perturbations is investigated. The analysis leads to two uncoupled OrrSommerfeld equations with appropriate boundary conditions. The numerical solution is obtained using fourth order finite difference scheme. The computations were performed for various plug flow dimensions and radii ratios. Within the range of the parameters considered in this paper, the Poiseuille flow of Bingham fluid is found to be linearly stable.

[3] 
On
3Dimensional linear stability of Poiseuille flow of Bingham
fluids,
I. A. Frigaard and C. Nouar,
Phys. Fluids 15, 28432851 (2003).
Plane channel Poiseuille flow of a Bingham fluid is characterized by the Bingham number, B, which describes the ratio of yield and viscous stresses. Unlike purely viscous nonNewtonian fluids, which modify hydrodynamic stability studies only through the dissipation and the basic flow, inclusion of a yield stress additionally results in a modified domain and boundary conditions for the stability problem. We investigate the effects of increasing B on the stability of the flow, using eigenvalue bounds that incorporate these features. As B we show that threedimensional linear stability can be achieved for a Reynolds number bound of form Re = O(B^{3/4}), for all wavelengths. For long wavelengths this can be improved to Re = O(B), which compares well with computed linear stability results for twodimensional disturbances [J. Fluid Mech. 263, 133 (1994)]. It is also possible to find bounds of form Re = O(B^{1/2}), which derive from purely viscous dissipation acting over the reduced domain and are comparable with the nonlinear stability bounds in J. NonNewt. Fluid Mech. 100, 127 (2001). We also show that a Squirelike result can be derived for the plane channel flow. Namely, if the equivalent eigenvalue bounds for a Newtonian fluid yield a stability criterion, then the same stability criterion is valid for the Bingham fluid flow, but with reduced wavenumbers and Reynolds numbers. An application of these results is to bound the regions of parameter space in which computational methods need to be used.

[4] 
Nonlinear stability of a viscoplastically
lubricated viscous shear flow,
M. A. MoyersGonzalez, I. A. Frigaard and C. Nouar,
J. Fluid Mech. 506, 117146 (2004).
A common problem in multilayer shear flows, especially from the perspective of process engineering, is the occurrence of interfacial instabilities. Here we show how multilayer duct flows can in fact be made nonlinearly stable, by using a suitable lubricating fluid. First we show how interfacial instabilities may be eliminated through the introduction of a yield stress fluid as the lubricant and by preserving an unyielded layer adjacent to the interface. Second we show how to treat the nonlinear stability of a twolayer flow, allowing finite motion of the domains. We focus on the simplest practically interesting case of viscoplastically lubricated viscous shear flow: a coreannular pipe flow consisting of a central core of Newtonian fluid surrounded by a Bingham fluid. We demonstrate that this flow can be nonlinearly stable at significant Reynolds numbers and produce stability bounds. Our analysis can be straightforwardly generalized to other flows in this class.

[5] 
On the usage of viscosity regularisation methods
for viscoplastic fluid flow computation,
I. A. Frigaard and C. Nouar,
J. Non Newt. Fluid Mech. 127, 126 (2005).
Viscosity regularisation methods are probably the most popular current method for computing viscoplastic fluid flows. They are however generally used in an ad hoc manner. Here we examine convergence of regularised solutions to those of the corresponding exact models, in both mathematical and physical senses. Mathematically, the aim is to give practical guidance as to the order of error that one might expect for different regularisations and for different types of flow. Our theoretical results are illustrated with a number of computed example flows showing the orders of error predicted. Physically, the question is whether or not the regularised solutions behave in the same way as the exact solutions, qualitatively as well as quantitatively. We show that there are flows for which regularisation methods will generate their maximum errors, e.g. lubricationtype flows. In this context, we also consider the effects of regularisation on problems of hydrodynamic stability. For broad classes of problems, stability characteristics of the flow are incorrectly predicted by the use of viscosity regularisation methods.

[6] 
Laminar transitional and turbulent flow of yield
stress fluid in a pipe,
J. Peixinho, C. Nouar, C. Desaubry and B. Théron,
J. Non Newt. Fluid Mech. 128, 172184 (2005).
This paper presents an experimental study of the laminar, transitional and turbulent flows in a cylindrical pipe facility (5.5 m length and 30 mm inner diameter). Three fluids are used: a yield stress fluid (aqueous solution of 0.2% Carbopol), a shear thinning fluid (aqueous solution of 2% CMC) without yield stress and a Newtonian fluid (glucose syrup) as a reference fluid. Detailed rheological properties (simple shear viscosity and first normal stress difference) are presented. The flow is monitored using pressure and (laser Doppler) axial velocity measurements. The critical Reynolds numbers from which the experimental results depart from the laminar solution are determined and compared with phenomenological criteria. The results show that the yield stress contribute to stabilize the flow. Concerning the transition for a yield stress fluid it has been observed an increase of the root mean square (rms) of the axial velocity outside a region around the axis while it remains at a laminar level inside this region. Then, with increasing the Reynolds number, the fluctuations increase in the whole section because of the apparition of turbulent spots. The time trace of the turbulent spots are presented and compared for the different fluids. Finally, a description of the turbulent flow is presented and shows that the rms axial velocity profile for the Newtonian and nonNewtonian fluids are similar except in the vicinity of the wall where the turbulence intensity is larger for the nonNewtonian fluids.

[7] 
Observations of assymetrical flow behavior in transitional pipe
flow of yieldstress and other shearthinning liquids,
M.P. Escudier, R.J. Poole, F.
Presti, C. Dales, C. Nouar, C. Desaubry, L. Graham and L. Pullum,
J. Non Newt. Fluid Mech. 127, 143155 (2005).
The purpose of this brief paper is to report mean velocity profile data for fully developed pipe flow of a wide range of shearthinning liquids together with two Newtonian control liquids. Although most of the data reported are for the laminarturbulent transition regime, data are also included for laminar and turbulent flow. The experimental data were obtained in unrelated research programmes in UK, France and Australia, all using laser Doppler anemometry (LDA) as the measurement technique. In the majority of cases, axisymmetric flow is observed for the laminar and turbulent flow conditions, although asymmetry due to the Earth's rotation is evident for the laminar flow of a Newtonian fluid of low viscosity (i.e. low Ekman number). The key point, however, is that for certain fluids, both yieldstress and viscoelastic (all fluids in this study are shearthinning), asymmetry to varying degrees is apparent at all stages of transition from laminar to turbulent flow, i.e. from the first indications to almost fully developed turbulence. The fact that symmetrical velocity profiles are obtained for both laminar and turbulent flow of all the nonNewtonian fluids in all three laboratories leads to the conclusion that the asymmetry must be a consequence of a fluiddynamic mechanism, as yet not identified, rather than imperfections in the flow facilities.

[8] 
Linear stability involving the Bingham model when the
yield stress approaches zero,
C. Métivier, C. Nouar and J.P. Brancher,
Phys. Fluids 17, 104106 (2005).
The plane Poiseuille flow of Bingham fluid is characterized by a plug zone around the axis of the channel, where , the second invariant of the deviatoric stress tensor, is less than or equal to the yield stress _{0}. According to the Bingham model, the plug zone moves as a rigid body with a constant velocity. The dimension of the plug zone, scaled with the width of the channel, depends only on the Bingham number, B (the ratio of the yield stress to a nominal viscous stress). The linear stability analysis of this flow as well as the Rayleigh–Bénard Poiseuille flow is performed. The numerical results are discussed essentially for the case B1. By comparison with the Newtonian fluid (B=0), a discontinuous behavior of the critical conditions is observed. This discontinuity is a consequence of the linear stability analysis that allows the plug zone to remain intact.

[9] 
Modal and nonmodal linear stability of the
planeBinghamPoiseuille flow,
C. Nouar, N. Kabouya, J. Dusek and M. Mamou,
J. Fluid Mech. 577, 211239 (2007).
The receptivity problem of plane BinghamPoiseuille flow with respect to weak perturbations is addressed. The relevance of this study is highlighted by the linear stability analysis results (spectra and pseudospectra). The first part of the present paper thus deals with the classical normalmode approach in which the resulting eigenvalue problem is solved using the Chebychev collocation method. Within the range of parameters considered, the Poiseuille flow of Bingham fluid is found to be linearly stable. The second part investigates the most amplified perturbations using the nonmodal approach. At a very low Bingham number (B << 1), the optimal disturbance consists of almost streamwise vortices, whereas at moderate or large B the optimal disturbance becomes oblique. The evolution of the obliqueness as function of B is determined. The linear analysis presented also indicates, as a first stage of a theoretical investigation, the principal challenges of a more complete nonlinear study.

[10] 
Delaying transition to turbulence : Revisiting the
stability of shear thinning fluids,
C. Nouar, A. Bottaro and J. P. Brancher,
J. Fluid Mech. 592, 177194 (2007).
A viscosity stratification is considered as a possible mean to postpone the onset of transition to turbulence in channel flow. As a prototype problem, we focus on the linear stability of shearthinning fluids modelled by the Carreau rheological law. To assess whether there is stabilization and by how much, it is important both to account for a viscosity disturbance in the perturbation equations, and to employ an appropriate viscosity scale in the definition of the Reynolds number. Failure to do so can yield qualitatively and quantitatively incorrect conclusions. Results are obtained for both exponentially and algebraically growing disturbances, demonstrating that a viscous stratification is a viable approach to maintain laminarity.

[11] 
On linear stability of RayleighBénard Poiseuille
flow of viscoplastic fluids,
C. Métivier and C. Nouar,
Phys. Fluids 20, 104101 (2008).
The present paper deals with the onset of the twodimensional Rayleigh–Bénard convection for a plane channel flow of viscoplastic fluid. The influence of the yield stress on the instability and stability conditions characterized by the Rayleigh numbers denoted, respectively, Ra_{L} and Ra_{E} is investigated in the framework of linear analysis using modal and energetic approaches. The results show that the yield stress, represented by the Bingham number B, delays the onset of convection. For low values of the Reynolds number Re, the critical conditions Ra_{L} and Ra_{E} tend to be equal and the difference Ra_{L}−Ra_{E} increases with increasing Re, highlighting the nonnormality of the linear operator. For Re<1 and large B (BO(10)), it is shown that the critical Rayleigh number increases as B^{2} and the critical wave number evolves according to B^{1/4}.

[12] 
Transitional flow of a yieldstress fluid in a pipe.
Evidence of a robust coherent structure,
A. Esmael and C. Nouar,
Phys. Rev. E 77, 057302 (2008).
In two independent articles, Escudier and Presti [J. NonNewt. Fluid Mech. 62, 291 (1996)] and Peixinho et al. [J. NonNewt. Fluid Mech. 128, 172 (2005)] studied experimentally the flow structure of a yield stress fluid in a cylindrical pipe. It was observed that the mean, i.e., timeaveraged, velocity profiles were axisymmetric in the laminar and turbulent regimes, and presented an increasing asymmetry with increasing Reynolds number in the transitional regime. The present paper provides a threedimensional description of this asymmetry from axial velocity profiles measurements at three axial positions and different azimuthal positions. The observed transitional flow suggests the existence of a robust nonlinear coherent structure characterized by two weakly modulated counterrotating longitudinal vortices. This new state mediates the transition between laminar and turbulent flow.

[13] 
Nonlinear stability
of the Bingham RayleighBénard Poiseuille flow,
C. Métivier, I. A. Frigaard and C. Nouar,
J. NonNewt. Fluid Mech. 158, 127131 (2009).
A nonlinear stability analysis of the RayleighBénard Poiseuille flow is performed for a yield stress fluid. Because the topology of the yielded and unyielded regions in the perturbed flow is unknown, the energy method is used, combined with classical functional analytical inequalities. We determine the boundary of a region in the (Re,Ra)plane where the perturbation energy decreases monotonically with time. For increasing values of Reynolds numbers, we show that the energy bound for Ra varies like 1Re/Re_{EN}, where Re_{EN} is the energy stability limit of isothermal Poiseuille flow. It is also shown that Re_{EN} behaves asymptotically as 120 B^{1/2} for large B.

[14] 
Stability of plane CouettePoiseuille flow of shearthinning fluid,
C. Nouar and I. A. Frigaard,
Phys. Fluids 21, 064104 (2009).
A linear stability analysis of the combined plane Couette and Poiseuille flow of shearthinning fluid is investigated. The rheological behavior of the fluid is described using the Carreau model. The linearized stability equations and their boundary conditions result in an eigenvalue problem that is solved numerically using a Chebyshev collocation method. A parametric study is performed in order to assess the roles of viscosity stratification and the Couette component. First of all, it is shown that for shearthinning fluid, the critical Reynolds number for a twodimensional perturbation is less than for a three dimensional. Therefore, it is sufficient to deal only with a modified OrrSommerfeld equation for the normal velocity component. The influence of the velocity of the moving wall on the critical conditions is qualitatively similar to that for a Newtonian fluid. Concerning the effect of the shear thinning, the computational results indicate that this behavior leads to a decrease in the phase velocity of the traveling waves and an increase in stability, when an appropriate viscosity is used in the definition of the Reynolds number. Using a longwave version of the OrrSommerfeld equation, the cutoff velocity is derived. The mechanisms responsible for the changes in the flow stability are discussed in terms of the location of the critical layers, Reynolds stress distribution, and the exchange of energy between the base flow and the disturbance.

[15] 
Linear stability of the Rayleigh Bénard Poiseuille flow
for thermodependent viscoplastic fluids,
C. Métivier and C. Nouar,
J. NonNewt. Fluid Mech. 163, 18 (2009).
This work investigates the RayleighBénard Poiseuille flow of a Bingham fluid with temperaturedependent plastic viscosity according to an exponential law. In fully developed situation, the temperature profile is purely conductive and the axial velocity profile, determined numerically, is skewed toward the lower viscosity region. The linear stability analysis of this primary flow is performed, and the critical conditions above which the flow becomes unstable are determined. It is found that the critical conditions decrease with increasing temperature difference and a critical Rayleigh number scaling. It is shown that this destabilization is mainly due to the asymmetry of the basic flow. As well as the basic flow, the perturbed flow is also asymmetric. Indeed, the amplitude perturbation of the least stable mode is much higher in the yielded region having the largest width.

[16] 
Stability of the flow of a Bingham fluid in a channel:
eigenvalue sensitivity, minimal defects and scaling laws of transition,
C. Nouar and A. Bottaro,
J. Fluid Mech. 642, 349372 (2010).
It has been recently shown that the flow of a Bingham fluid in a channel is always linearly stable (Nouar et al., J. Fluid Mech. 577 211, 2007). To identify possible paths of transition we revisit the problem for the case in which the idealized base flow is slightly perturbed. No attempt is made to reproduce or model the perturbations arising in experimental environments  which may be due to the improper alignment of the channel walls or to imperfect inflow conditions  rather a general formulation is given which yields the transfer function (the sensitivity) for each eigenmode of the spectrum to arbitrary defects in the base flow. It is first established that such a function, for the case of the most sensitive eigenmode, displays a very weak selectivity to variations in the spanwise wavenumber of the disturbance mode. This justifies a further look into the class of spanwise homogeneous modes. A variational procedure is set up to identify the base flow defect of minimal norm capable of optimally destabilizing an otherwise stable flow; it is found that very weak defects are indeed capable to excite exponentially amplified streamwise travelling waves. The associated variations in viscosity are situated mostly near the critical layer of the inviscid problem. Neutrally stable conditions are found as function of the Reynolds number and the Bingham number, providing scalings of critical values with the amplitude of the defect consistent with previous experimental and numerical studies. Finally, a structured pseudospectrum analysis is performed; it is argued that such a class of pseudospectra provides information well suited to hydrodynamic stability purposes.

[17] 
PetrovGalerkin computation of nonlinear waves
in pipe flow of shearthinning fluids:
first theoretical evidences for a delayed transition,
N. Roland, E. Plaut and C. Nouar,
Computers & Fluids 39, 17331743 (2010).
A pseudospectral PetrovGalerkin code is developped in order to compute nonlinear traveling waves in pipe flow of shearthinning fluids. The framework is continuum mechanics and the rheological model used is the purely viscous Carreau model. The code is validated, and a study of its convergence properties is made. It is shown that exponential convergence is obtained, despite the highly nonlinear nature of the viscous diffusion terms. Physical computations show that, as compared with the case of a constantviscosity fluid, i.e., a Newtonian fluid, in the case of shearthinning fluids the critical Reynolds number of the saddlenode bifurcation where the waves with an azimuthal wavenumber m_{0}=3 appear increases significantly when the nonNewtonian effects come into play.

[18] 
Transitional flow of a nonNewtonian fluid in a pipe:
Experimental evidence of weak turbulence induced by shearthinning behavior,
A. Esmael, C. Nouar, A. Lefèvre and N. Kabouya,
Phys. Fluids 22, 101701 (2010).
The present letter is a thorough study of the flow regime where an asymmetry of the mean axial velocity profiles is observed for shearthinning fluids flow in a pipe. This study is based on a statistical analysis of the axial velocity fluctuations. It is shown that this flow regime exhibits features of a weak turbulence: chaotic in time and regular in space. More precisely, (i) power spectra of axial velocity fluctuations decay following a power law with an exponent very close to 3, (ii) largescale coherent structures are generated, and (iii) there is essentially no intermittency in this flow regime.

[19] 
Subcritical bifurcation of shearthinning plane
Poiseuille flows,
A. Chekila, C. Nouar, E. Plaut and A. Nemdili,
J. Fluid Mech. 686, 272298 (2011).
In a recent article (Nouar et al. 2007, ref. [10] of this page), a linear stability analysis of plane Poiseuille flow of shearthinning fluids has been performed. The authors concluded that the viscosity stratification delays the transition and that is important to account for the viscosity perturbation. The current paper focuses on the first principles understanding of the influence of the viscosity stratification and the nonlinear variation of the effective viscosity with the shear rate on the flow stability with respect to a finite amplitude perturbation. A weakly nonlinear analysis, using the amplitude expansion method is adopted as a first approach to study nonlinear effects. The bifurcation to twodimensional travelling waves is studied. For the numerical computations, the shearthinning behavior is described by 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 by the nonlinearity of the viscosity is smaller and in opposite phase with that generated by the quadratic nonlinear inertial terms and (iii) with increasing shearthinning effects, the bifurcation becomes highly subcritical. Consequently, the magnitude of the threshold amplitude of the perturbation, beyond which the flow is nonlinearly unstable, decreases. This result is confirmed by computing higher order Landau constants.

[20]  Pipe flow of shearthinning fluids,
S. N. LópezCarranza, M. Jenny and C. Nouar,
C. R. Mécanique 340, 602618 (2012).
Pipe flow of purely viscous shearthinning fluids is studied using numerical simulations. The rheological behavior is described by the Carreau model. The flow field is decomposed as a base flow and a disturbance. The perturbation equations are then solved using a pseudospectral PetrovGalerkin method. The time marching uses a fourthorder AdamsBashforth scheme. In the case of an infinitesimal perturbation, a threedimensional linear stability analysis is performed based on modal and nonmodal approaches. It is shown that pipe flow of shearthinning fluids is linearly stable and that for the range of rheological parameters considered, streamwiseindependent vortices are optimally amplified. Nonlinear computations are done for finite amplitude twodimensional disturbances, which consist of one pair of longitudinal rolls. The numerical results highlight a strong modification of the viscosity profile associated with the flow reorganization. For a given wall Reynolds number, shearthinning reduces the energy gain of the perturbation. This is due to a reduction of the exchange energy between the base flow and the perturbation. Besides this, viscous dissipation decreases with increasing shearthinning effects. 
[21]  TaylorCouette Instability in Anisotropic Clay Suspensions Measured Using SmallAngle Xray Scattering,
A. M. Philippe, C. Baravian, M. Jenny, F. Meneau and J. L. Michot,
Phys. Rev. Lett. 108, 254501 (2012).
In this Letter, we propose an original and novel experimental method to characterize both the onset and morphology of TaylorCouette instability occurring in a nonNewtonian cylindrical Couette flow. Using synchrotronbased rheological small angle Xray scattering experiments, we jointly investigate the shearthinning behavior of natural swelling clays suspensions and the associated anisotropy developing in such media. Combined with a linear stability analysis for power law fluids, a destabilizing effect is shown both numerically and experimentally and the vortices morphology is found to be dependent on the fluid index. Still, the strong destabilizing effect and large vortex size can not be assigned to shearthinning only, which clearly evidences the impact of medium anisotropy on TaylorCouette instability. 
[22]  Revisiting the stability of circular Couette flow of shearthinning fluids,
B. Alibenyahiaa, C. Lemaitre, C. Nouar and N. AitMessaoudene,
J. NonNewt. Fluid Mech. 183184, 3751 (2012).
Threedimensional linear stability analysis of Couette flow between two coaxial cylinders for shearthinning fluids with and without yield stress is performed. The outer cylinder is fixed and the inner one is rotated. Three rheological models are used: Bingham, Carreau and powerlaw models. Wide range of rheological, geometrical and dynamical parameters is explored. New data for the critical conditions are provided for Carreau fluid. In the axisymmetric case, it is shown that when the Reynolds number is defined using the innerwall shearviscosity, the shearthinning delays the appearance of Taylor vortices, for all the fluids considered. It is shown that this delay is due to reduction in the energy exchange between the base and the perturbation and not to the modification of the viscous dissipation. In the non axisymmetric case, contrary to Caton [1], we have not found any instability. 
[23]  Instability of streaks in pipe flow of shearthinning fluids,
S. N. LópezCarranza, M. Jenny and C. Nouar,
Phys. Rev. E 88, 023005 (2013).
This study is motivated by recent experimental results dealing with the transition to turbulence in a pipe flow of shearthinning fluids, where a streaky flow with an azimuthal wave number n=1 is observed in the transitional regime. Here, a linear stability analysis of pipe flow of shearthinning fluids modulated azimuthally by finite amplitude streaks is performed. The shearthinning behavior of the fluid is described by the Carreau model. The streaky base flows considered are obtained from twodimensional direct numerical simulation using finite amplitude longitudinal rolls as the initial condition and by extracting the velocity field at time t_{max}, where the amplitude of the streaks reaches its maximum, denoted by A_{max}. It is found that the amplitude A_{max} increases with increasing Reynolds number as well as with increasing amplitude E_{0} of the initial longitudinal rolls. For sufficiently large streaks amplitude, streamwise velocity profiles develop inflection points, leading to instabilities. Depending on the threshold amplitude A_{c}, two different modes may trigger the instability of the streaks. If A_{c} exceeds approximately 41.5% of the centerline velocity, the instability mode is located near the axis of the pipe, i.e., it is a “center mode.” For weaker amplitude A_{c}, the instability mode is located near the pipe wall, in the region of highest wall normal shear, i.e., it is a “wall mode.” The threshold amplitude A_{c} decreases with increasing shearthinning effects. The energy equation analysis indicates that (i) wall modes are driven mainly by the work of the Reynolds stress against the wall normal shear and (ii) for center modes, the contribution of the normal wall shear remains dominant; however, it is noted that the contribution of the Reynolds stress against the azimuthal shear increases with increasing shearthinning effects. 
[24]  Weakly nonlinear analysis of
RayleighBénard convection in
shearthinning fluids: nature of the
bifurcation and pattern selection,
M. Bouteraa, C. Nouar, E. Plaut, C. Métivier and A. Kalck,
J. Fluid Mech. 767, 696734 (2015).
A linear and weakly nonlinear analysis of natural convection of a layer of shearthinning fluids between two horizontal plates heated from below is performed. The objective is to examine the effects of the nonlinear variation of the viscosity with the shear rate on the nature of the bifurcation, the planform selection problem between rolls, squares and hexagons, and the consequences on the heat transfer coefficient. Navier slip boundary conditions are used at the top and bottom walls. The shearthinning behavior of the fluid is described by the Carreau model. By considering an infinitesimal perturbation, the critical conditions, corresponding to the onset of convection, are determined. At this stage, nonNewtonian effects do not play. The critical Rayleigh number decreases and the critical wave number increases when the slip increases. For a finite amplitude perturbation, nonlinear effects enter in the dynamic. Analysis of the saturation coefficients at cubic order in the amplitude equations shows that the nature of the bifurcation depends on the rheological properties, i.e. the fluid characteristic time and shearthinning index. For weakly shearthinning fluids, the bifurcation is supercritical and the heat transfer coefficient increases, as compared to the Newtonian case. When the shearthinning character is large enough, the bifurcation is subcritical, pointing out the destabilizing effect of the nonlinearities arising from the rheological law. Departing from the onset, the weakly nonlinear analysis is carried out up to fifth order in the amplitude expansion. The flow structure, the modification of the viscosity field and the Nusselt number are characterized. The competition between rolls, squares and hexagons is investigated. Unlike Albaalbaki & Khayat (2011), it is shown that only rolls are stable near onset. 
[25]  Numerical study of subcritical RayleighBénard convection
rolls in strongly shearthinning Carreau fluids,
M. Jenny, E. Plaut and A. Briard,
J. Non Newt. Fluid Mech. 219, 1934 (2015).
The RayleighBénard thermoconvection of Newtonian fluids has been extensively studied. The transition from the conductive, static state to thermoconvection flows corresponds in this case to a supercritical bifurcation. In shearthinning fluids, on the contrary, recent weakly nonlinear studies have shown that the transition may become subcritical. Using a custom numerical code developped with Freefem++ to compute bidimensionnal, fully nonlinear roll solutions in Carreau fluids, for a large range of rheological parameters, and more particularly for strongly shearthinning fluids, approaching powerlaw fluids, we confirm this result. A simple expression of the value of the Rayleigh number at which subcritical convection rolls appear is proposed. This law suggests to reconsider the choice of the reference viscosity for shearthinning fluids. Indeed, when the shearthinning effects increase, the critical Rayleigh number increases or decreases depending on the choice of the reference viscosity. The `neutral' or `effective' viscosity, which gives a constant value of the Rayleigh number at the onset of subcritical convection rolls, is close to the bulk average viscosity. In addition, a correlation is proposed to estimate the Nusselt number of subcritical rolls. 
[26]  Linear stability of TaylorCouette flow of shearthinning fluids:
modal and nonmodal approaches,
Y. Agbessi, B. Alibenyahia, C. Nouar, C. Lemaitre and L. Choplin,
J. Fluid Mech. 776, 354389 (2015).
In this paper, the response of circular Couette flow of shearthinning fluids between two infinitely long coaxial cylinders to weak disturbances is addressed. It is highlighted by transient growth analysis. Both powerlaw and Carreau models are used to describe the rheological behaviour of the fluid. The first part of the paper deals with the asymptotic longtime behaviour of threedimensional infinitesimal perturbations. Using the normalmode approach, an eigenvalue problem is derived and solved by means of the spectral collocation method. An extensive description and the classification of eigenspectra are presented. The influence of shearthinning effects on the critical Reynolds numbers as well as on the critical azimuthal and axial wavenumbers is analysed. It is shown that with a reference viscosity defined with the characteristic scales μ_{ref} = K (R_{1}Ω_{1}/d)^(n1) for a powerlaw fluid and μ_{ref} = μ_{0} for a Carreau fluid, the shearthinning character is destabilizing for counterrotating cylinders. Moreover, the axial wavenumber increases with Re_{2} and with shearthinning effects. The second part investigates the shorttime behaviour of the disturbance using the nonmodal approach. For the same inner and outer Reynolds numbers, the amplification of the kinetic energy perturbation becomes much more important with increasing shearthinning effects. Two different mechanisms are used to explain the transient growth, depending on whether or not there is a stratification of the angular momentum. On the Rayleigh line and for Newtonian fluids, the optimal perturbation is in the form of azimuthal streaks, which transform into Taylor vortices through the antiliftup mechanism. In the other cases, the optimal perturbation is initially oriented against the base flow, then it tilts to align with the base flow at optimal time. The scaling laws for the optimal energy amplification proposed in the literature for Newtonian fluids are extended to shearthinning fluids. 
[27]  Weakly nonlinear analysis of RayleighBénard convection in a nonNewtonian fluid between plates of finite conductivity: Influence of shearthinning effects,
M. Bouteraa and C. Nouar,
Phys. Rev. E 92, 063017 (2015).
Finiteamplitude thermal convection in a shearthinning fluid layer between two horizontal plates of finite thermal conductivity is considered. Weakly nonlinear analysis is adopted as a first approach to investigate nonlinear effects. The rheological behavior of the fluid is described by the Carreau model. As a first step, the critical conditions for the onset of convection are computed as a function of the ratio ξ of the thermal conductivity of the plates to the thermal conductivity of the fluid. In agreement with the literature, the critical Rayleigh number Ra_{c} and the critical wave number k_{c} decrease from 1708 to 720 and from 3.11 to 0, when ξ decreases from infinity to zero. In the second step, the critical value α_{c} of the shearthinning degree above which the bifurcation becomes subcritical is determined. It is shown that α_{c} increases with decreasing ξ. The stability of rolls and squares is then investigated as a function of ξ and the rheological parameters. The limit value ξ_{c}, below which squares are stable, decreases with increasing shearthinning effects. This is related to the fact that shearthinning effects increase the nonlinear interactions between sets of rolls that constitute the square patterns. For a significant deviation from the critical conditions, nonlinear convection terms and nonlinear viscous terms become stronger, leading to a further diminution of ξ_{c}. The dependency of the heat transfer on ξ and the rheological parameters is reported. It is consistent with the maximum heat transfer principle. Finally, the flow structure and the viscosity field are represented for weakly and highly conducting plates. 
[28]  Natural convection in shearthinning fluids:
Experimental investigations by MRI,
M. Darbouli, C. Métivier, S. Leclerc, C. Nouar,
M. Bouteraa and D. Stemmelen,
Int. J. Heat & Mass Transfer 95, 742754 (2016).
An experimental investigation of the RayleighBénard convection in shearthinning fluids using MRI technics is presented. The experimental setup consists on a cylindrical cavity defined by a finite aspect ratio A = D/d = 6. Qualitative and quantitative results are provided. Flow structure is determined from velocity mapping for a Newtonian fluid, Glycerol and for shearthinning fluids, Xanthan gum aqueous solutions with weight concentrations ranging from 0.1% to 0.2%. In the case of the Glycerol and the Xanthan solution at 0.1%, one recovers similar results in terms of criticality with Ra_{c} ≃ 1800 and patterns since the convection is characterized by rolls. When the Xanthan concentration is increased, the critical Rayleigh number is not modified, however the onset occurs with hexagonal pattern. Because the critical temperature differences increase with the concentrations due to an increase in viscosity, one can think that hexagonal patterns are due to variations of physical properties with temperature (non OberbeckBoussinesq effects). Similarities with some results obtained in the Newtonian case are highlighted. We have observed a transition from hexagonal patterns to rolls by increasing the Rayleigh number. This pattern transition is characterized by a discrepancy in the maximal velocity values. By using shearthinning fluids, results show an increase in the intensity of convection compared with the Newtonian case. 
[29]  Nonlinear waves with a threefold rotational
symmetry in pipe flow:
influence of a strongly shearthinning rheology,
E. Plaut, N. Roland and C. Nouar,
resubmitted to the
J. Fluid Mech. in December 2016.
In order to model the transition to turbulence in pipe flow of nonNewtonian fluids, the influence of a strongly shearthinning rheology on traveling waves with a threefold rotational symmetry is analyzed. The rheological model is Carreau's law. Besides the shearthinning index n_{C}, the dimensionless characteristic time λ of the fluid is considered as the main nonNewtonian control parameter. If λ=0, the fluid is Newtonian. In the relevant limit λ → +∞, the fluid approaches a powerlaw behaviour. The laminar base flows are first characterized. To compute the nonlinear waves, a PetrovGalerkin code is used, with continuation methods, starting from the Newtonian case, i.e., the solutions of Faisst & Eckhardt (2003), Wedin & Kerswell (2004). The axial wavenumber is optimized and the critical waves appearing at minimal values of the Reynolds number Re_{w} based on the mean velocity and wall viscosity are characterized. As λ increases, the saddlenode bifurcation, at which the critical wave appears, corresponds to a constant value of the Reynolds number based on the mean velocity and viscosity. This viscosity, close to the one of the laminar flow, can be estimated analytically. Therefore the experimentally relevant critical Reynolds number Re_{wc} can also be estimated analytically. This Reynolds number may be viewed as a lower estimate of the Reynolds number for the transition to developed turbulence. This demonstrates a quantified stabilizing effect of the shearthinning rheology. Finally, the increase of the pressure gradient in waves, as compared to the one in the laminar flow with the same mass flux, is calculated, and a kind of `drag reduction effect' is found. 
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