Instabilities and transition in non-Newtonian fluids

M. Jenny, C. Metivier, C. Nouar & E. Plaut


      We are interested in the transition to spatio-temporal complexity and turbulence in flows of non-Newtonian fluids, more specifically of shear-thinning fluids. Such fluids, which flow more and more easily when the applied stresses increase, are often encountered in Oil, Ciment or Food Industry; in some cases, Blood can also be considered as a shear-thinning fluid. The transition to turbulence in such fluids is difficult to study because of the richness of behaviour of those fluids, and because the model equations are highly nonlinear.

We study this topics experimentally, for instance in pipe flow where strange phenomena can occur. Indeed it is found that weakly turbulent states exist which have no counterpart in newtonian fluids. In those weakly turbulent states the axial mean flow is highly unsymmetric, the fluid flowing rapidly in one side of the pipe, slowly in the other half, as shown in the following figure (taken from [12]):

We also study this topics theoretically. For this purpose we cooperate with several researchers all over the world:

Please have a look at our papers below. You might also like to observe these animations, realized with the code presented in our 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, 127-149 (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, Rec, increases with Bingham number, B, at least as fast as Recnot, vert, similarB1/2 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, 149-156 (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 Orr-Sommerfeld 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 3-Dimensional linear stability of Poiseuille flow of Bingham fluids, I. A. Frigaard and C. Nouar, Phys. Fluids 15, 2843-2851 (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 non-Newtonian 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-->[infinity] we show that three-dimensional linear stability can be achieved for a Reynolds number bound of form Re = O(B3/4), for all wavelengths. For long wavelengths this can be improved to Re = O(B), which compares well with computed linear stability results for two-dimensional disturbances [J. Fluid Mech. 263, 133 (1994)]. It is also possible to find bounds of form Re = O(B1/2), which derive from purely viscous dissipation acting over the reduced domain and are comparable with the nonlinear stability bounds in J. Non-Newt. Fluid Mech. 100, 127 (2001). We also show that a Squire-like 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 visco-plastically lubricated viscous shear flow, M. A. Moyers-Gonzalez, I. A. Frigaard and C. Nouar, J. Fluid Mech. 506, 117-146 (2004).

A common problem in multi-layer shear flows, especially from the perspective of process engineering, is the occurrence of interfacial instabilities. Here we show how multi-layer 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 two-layer flow, allowing finite motion of the domains. We focus on the simplest practically interesting case of visco-plastically lubricated viscous shear flow: a core-annular 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 visco-plastic fluid flow computation, I. A. Frigaard and C. Nouar, J. Non Newt. Fluid Mech. 127, 1-26 (2005).

Viscosity regularisation methods are probably the most popular current method for computing visco-plastic 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. lubrication-type 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, 172-184 (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 non-Newtonian fluids are similar except in the vicinity of the wall where the turbulence intensity is larger for the non-Newtonian fluids.

[7] Observations of assymetrical flow behavior in transitional pipe flow of yield-stress and other shear-thinning 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, 143-155 (2005).

The purpose of this brief paper is to report mean velocity profile data for fully developed pipe flow of a wide range of shear-thinning liquids together with two Newtonian control liquids. Although most of the data reported are for the laminar-turbulent 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 yield-stress and viscoelastic (all fluids in this study are shear-thinning), 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 non-Newtonian fluids in all three laboratories leads to the conclusion that the asymmetry must be a consequence of a fluid-dynamic 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 tau, the second invariant of the deviatoric stress tensor, is less than or equal to the yield stress tau0. 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 B<<1. 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 non-modal linear stability of the plane-Bingham-Poiseuille flow, C. Nouar, N. Kabouya, J. Dusek and M. Mamou, J. Fluid Mech. 577, 211-239 (2007).

The receptivity problem of plane Bingham-Poiseuille 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 normal-mode 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 non-modal 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, 177-194 (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 shear-thinning 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 Rayleigh-Bénard Poiseuille flow of viscoplastic fluids, C. Métivier and C. Nouar, Phys. Fluids 20, 104101 (2008).

The present paper deals with the onset of the two-dimensional 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, RaL and RaE 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 RaL and RaE tend to be equal and the difference RaL−RaE increases with increasing Re, highlighting the non-normality of the linear operator. For Re<1 and large B (B>=O(10)), it is shown that the critical Rayleigh number increases as B2 and the critical wave number evolves according to B1/4.

[12] Transitional flow of a yield-stress 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. Non-Newt. Fluid Mech. 62, 291 (1996)] and Peixinho et al. [J. Non-Newt. 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., time-averaged, 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 three-dimensional 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 counter-rotating longitudinal vortices. This new state mediates the transition between laminar and turbulent flow.

[13] Nonlinear stability of the Bingham Rayleigh-Bénard Poiseuille flow, C. Métivier, I. A. Frigaard and C. Nouar, J. Non-Newt. Fluid Mech. 158, 127-131 (2009).

A nonlinear stability analysis of the Rayleigh-Bénard 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 1-Re/ReEN, where ReEN is the energy stability limit of isothermal Poiseuille flow. It is also shown that ReEN behaves asymptotically as 120 B1/2 for large B.

[14] Stability of plane Couette-Poiseuille flow of shear-thinning 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 shear-thinning 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 shear-thinning fluid, the critical Reynolds number for a two-dimensional perturbation is less than for a three dimensional. Therefore, it is sufficient to deal only with a modified Orr-Sommerfeld 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 long-wave version of the Orr-Sommerfeld 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. Non-Newt. Fluid Mech. 163, 1-8 (2009).

This work investigates the Rayleigh-Bénard Poiseuille flow of a Bingham fluid with temperature-dependent 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, 349-372 (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] Petrov-Galerkin computation of nonlinear waves in pipe flow of shear-thinning fluids: first theoretical evidences for a delayed transition, N. Roland, E. Plaut and C. Nouar, Computers & Fluids 39, 1733-1743 (2010).

A pseudospectral Petrov-Galerkin code is developped in order to compute nonlinear traveling waves in pipe flow of shear-thinning 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 constant-viscosity fluid, i.e., a Newtonian fluid, in the case of shear-thinning fluids the critical Reynolds number of the saddle-node bifurcation where the waves with an azimuthal wavenumber m0=3 appear increases significantly when the non-Newtonian effects come into play.

[18] Transitional flow of a non-Newtonian fluid in a pipe: Experimental evidence of weak turbulence induced by shear-thinning 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 shear-thinning 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) large-scale coherent structures are generated, and (iii) there is essentially no intermittency in this flow regime.

[19] Subcritical bifurcation of shear-thinning plane Poiseuille flows, A. Chekila, C. Nouar, E. Plaut and A. Nemdili, J. Fluid Mech. 686, 272-298 (2011).

In a recent article (Nouar et al. 2007, ref. [10] of this page), a linear stability analysis of plane Poiseuille flow of shear-thinning 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 two-dimensional travelling waves is studied. For the numerical computations, the shear-thinning 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 shear-thinning 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 shear-thinning fluids, S. N. López-Carranza, M. Jenny and C. Nouar, C. R. Mécanique 340, 602-618 (2012).

Pipe flow of purely viscous shear-thinning 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 pseudo-spectral Petrov-Galerkin method. The time marching uses a fourth-order Adams-Bashforth scheme. In the case of an infinitesimal perturbation, a three-dimensional linear stability analysis is performed based on modal and non-modal approaches. It is shown that pipe flow of shear-thinning fluids is linearly stable and that for the range of rheological parameters considered, streamwise-independent vortices are optimally amplified. Nonlinear computations are done for finite amplitude two-dimensional 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, shear-thinning 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 shear-thinning effects.

[21] Taylor-Couette Instability in Anisotropic Clay Suspensions Measured Using Small-Angle X-ray 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 Taylor-Couette instability occurring in a non-Newtonian cylindrical Couette flow. Using synchrotron-based rheological small angle x-ray scattering experiments, we jointly investigate the shear-thinning 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 shear-thinning only, which clearly evidences the impact of medium anisotropy on Taylor-Couette instability.

[22] Revisiting the stability of circular Couette flow of shear-thinning fluids, B. Alibenyahiaa, C. Lemaitre, C. Nouar and N. Ait-Messaoudene, J. Non-Newt. Fluid Mech. 183-184, 37-51 (2012).

Three-dimensional linear stability analysis of Couette flow between two coaxial cylinders for shear-thinning 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 power-law 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 inner-wall shear-viscosity, the shear-thinning 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 shear-thinning fluids, S. N. López-Carranza, 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 shear-thinning 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 shear-thinning fluids modulated azimuthally by finite amplitude streaks is performed. The shear-thinning behavior of the fluid is described by the Carreau model. The streaky base flows considered are obtained from two-dimensional direct numerical simulation using finite amplitude longitudinal rolls as the initial condition and by extracting the velocity field at time tmax, where the amplitude of the streaks reaches its maximum, denoted by Amax. It is found that the amplitude Amax increases with increasing Reynolds number as well as with increasing amplitude E0 of the initial longitudinal rolls. For sufficiently large streaks amplitude, streamwise velocity profiles develop inflection points, leading to instabilities. Depending on the threshold amplitude Ac, two different modes may trigger the instability of the streaks. If Ac 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 Ac, 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 Ac decreases with increasing shear-thinning 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 shear-thinning effects.

[24] Weakly nonlinear analysis of Rayleigh-Bénard convection in shear-thinning fluids: nature of the bifurcation and pattern selection, M. Bouteraa, C. Nouar, E. Plaut, C. Métivier and A. Kalck, J. Fluid Mech. 767, 696-734 (2015).

A linear and weakly nonlinear analysis of natural convection of a layer of shear-thinning 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 shear-thinning 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, non-Newtonian 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 shear-thinning index. For weakly shear-thinning fluids, the bifurcation is supercritical and the heat transfer coefficient increases, as compared to the Newtonian case. When the shear-thinning 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 Rayleigh-Bénard convection rolls in strongly shear-thinning Carreau fluids, M. Jenny, E. Plaut and A. Briard, J. Non Newt. Fluid Mech. 219, 19-34 (2015).

The Rayleigh-Bénard 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 shear-thinning 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 shear-thinning fluids, approaching power-law 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 shear-thinning fluids. Indeed, when the shear-thinning 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 Taylor-Couette flow of shear-thinning fluids: modal and non-modal approaches, Y. Agbessi, B. Alibenyahia, C. Nouar, C. Lemaitre and L. Choplin, J. Fluid Mech. 776, 354-389 (2015).

In this paper, the response of circular Couette flow of shear-thinning fluids between two infinitely long coaxial cylinders to weak disturbances is addressed. It is highlighted by transient growth analysis. Both power-law and Carreau models are used to describe the rheological behaviour of the fluid. The first part of the paper deals with the asymptotic long-time behaviour of three-dimensional infinitesimal perturbations. Using the normal-mode 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 shear-thinning 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 μref = K (R1Ω1/d)^(n-1) for a power-law fluid and μref = μ0 for a Carreau fluid, the shear-thinning character is destabilizing for counter-rotating cylinders. Moreover, the axial wavenumber increases with Re2 and with shear-thinning effects. The second part investigates the short-time behaviour of the disturbance using the non-modal approach. For the same inner and outer Reynolds numbers, the amplification of the kinetic energy perturbation becomes much more important with increasing shear-thinning 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 anti-lift-up 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 shear-thinning fluids.

[27] Weakly nonlinear analysis of Rayleigh-Bénard convection in a non-Newtonian fluid between plates of finite conductivity: Influence of shear-thinning effects, M. Bouteraa and C. Nouar, Phys. Rev. E 92, 063017 (2015).

Finite-amplitude thermal convection in a shear-thinning 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 Rac and the critical wave number kc 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 αc of the shear-thinning degree above which the bifurcation becomes subcritical is determined. It is shown that αc 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 ξc, below which squares are stable, decreases with increasing shear-thinning effects. This is related to the fact that shear-thinning 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 ξc. 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 shear-thinning fluids: Experimental investigations by MRI, M. Darbouli, C. Métivier, S. Leclerc, C. Nouar, M. Bouteraa and D. Stemmelen, Int. J. Heat & Mass Transfer 95, 742-754 (2016).

An experimental investigation of the Rayleigh-Bénard convection in shear-thinning 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 shear-thinning 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 Rac ≃ 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 Oberbeck-Boussinesq 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 shear-thinning fluids, results show an increase in the intensity of convection compared with the Newtonian case.

[29] Taylor-Couette instability in thixotropic yield stress fluids, M. Jenny, S. Kiesgen de Richter, N. Louvet, S. Skali-Lami and Y. Dossmann, Phys. Rev. Fluids 2, 023302 (2017).

We consider the flow of thixotropic yield stress fluids between two concentric cylinders. To account for the fluid thixotropy, we use Houška's model [Houška, Ph.D. thesis, Czech Technical University, Prague, 1981] with a single structural parameter driven by a kinetic equation. Because of the yield stress and the geometric inhomogeneity of the stress, only a part of the material in the gap may flow. Depending on the breakdown rate of the structural parameter, the constitutive relation can lead to a nonmonotonic flow curve. This nonmonotonic behavior is known to induce a discontinuity in the slope of the velocity profile within the flowing material, called shear banding. Thus, for fragile structures, a shear-banded flow characterized by a very sharp transition between the flowing and the static regions may be observed. For stronger structures, the discontinuity disappears and a smooth transition between the flowing and the static regions is observed. The consequences of the thixotropy on the linear stability of the azimuthal flow are studied in a large range of parameters. Although the thixotropy allows shear banding in the base flow, it does not modify fundamentally the linear stability of the Couette flow compared to a simple yield stress fluid. The apparent shear-thinning behavior depends on the thixotropic parameters of the fluid and the results about the onset of the Taylor vortices in shear-thinning fluids are retrieved. Nevertheless, the shear banding modifies the stratification of the viscosity in the flowing zone such that the critical conditions are mainly driven by the width of the flowing region.

[30] Nonlinear waves with a threefold rotational symmetry in pipe flow: influence of a strongly shear-thinning rheology, E. Plaut, N. Roland and C. Nouar, J. Fluid Mech. 818, 595-622 (2017).

In order to model the transition to turbulence in pipe flow of non-Newtonian fluids, the influence of a strongly shear-thinning rheology on the traveling waves with a threefold rotational symmetry of Faisst & Eckhardt (2003) ; Wedin & Kerswell (2004) is analyzed. The rheological model is Carreau's law. Besides the shear-thinning index nC, the dimensionless characteristic time λ of the fluid is considered as the main non-Newtonian control parameter. If λ = 0, the fluid is Newtonian. In the relevant limit λ → +∞, the fluid approaches a power-law behaviour. The laminar base flows are first characterized. To compute the nonlinear waves, a Petrov-Galerkin code is used, with continuation methods, starting from the Newtonian case. The axial wavenumber is optimized and the critical waves appearing at minimal values of the Reynolds number Rew based on the mean velocity and wall viscosity are characterized. As λ increases, these correspond 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 Rewc 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 shear-thinning 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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