Linear stability of the Rayleigh–Bénard Poiseuille flow for thermodependent viscoplastic fluids

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Abstract

This work investigates the Rayleigh–Bénard Poiseuille flow of a Bingham fluid with temperature-dependent plastic viscosity according to the model μˆp=aexp(bˆTˆ). 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 |k|=|bˆ|δTˆ and that the critical Rayleigh number scales as exp(0.8|k|). 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.

Introduction

Viscoplastic fluids associated with shear flows in the presence of heat transfer are commonly found in industrial processes, included those involved in the oil industry, cosmetics and food processing, as well as in natural settings such as geophysics or biology. These fluids are characterized by a solid-like behavior at low stresses and a fluid-like behavior above the yield stress. The Bingham model is frequently used to account for this behavior. Although simple, it contains all of the features of a viscoplastic fluid. For unidirectional shear flow with velocity Û(ŷ), the relationship between the yield stress τˆxy and the velocity gradient dÛ/dŷ is:τˆxy=sgndÛdŷτˆy+μpdÛdŷ|τˆxy|τˆy,dÛdŷ=0|τˆxy|<τˆy,where τˆy is the yield stress and μˆp is the plastic viscosity. Forrest and Wilkinson show in [1] that the temperature dependence of the yield stress is weak, and thus negligible compared to the temperature dependence of the plastic viscosity. The explanation for such weak thermodependence is that the yield stress is mainly dependent upon a mechanical locking of the fluid, which is essentially temperature independent.

Forced convection in thermodependent yield stress fluids has been considered by several authors, see for instance [2], [3], [4]. The aim of these studies was to provide a relationship between the heat transfer coefficient and the rheological properties of the fluid. The case of mixed convection was studied by Patel and Ingham [5] for a vertical plane channel. The dynamical and thermal fields are assumed fully developed. Different configurations of the flows were determined according to the ratio of the Grashof number to the Reynolds number. The characteristics of the thermal convection for a thermodependent yield stress fluids in a horizontal circular pipe were investigated by Nouar [4]. Assuming that free convection is weak compared to the forced convection, an asymptotic approach is developed.

Here, we consider the situation of an axial flow of Bingham plastic fluid in a horizontal plane channel, where the lower plate is maintained at higher temperature than the upper one. This mixed convection problem has two sources of potential instability. The first one comes from the asymmetrical heating effect which implies an unstable stratification density and leads, at low flow rate, to a buoyancy-driven instability. The second source of instability is the shear-driven one which occurs at high flow rate and leads to the Tollmien–Schlichting instabilities.

At low flow rate, the first source of instability is considered and the problem is termed as the Rayleigh–Bénard Poiseuille (RBP) problem. Its convective pattern depends on the temperature gradient and on the axial flow rate due to forced convection. Numerous papers on the RBP problem for Newtonian fluids exist, a review can be found in [6], [7]. Globally two convective modes can be observed: (i) longitudinal rolls which consist of helicoidal thermoconvective rolls aligned with the flow direction; and (ii) transverse rolls which correspond to travelling convection rolls with axis perpendicular to the flow direction. They appear as a chain of multiple counter-rotating rolls moved by the main flow rate. These two kinds of convective patterns may appear as an absolute or convective instability (see [8] for instance). Compared to the Newtonian case, very little is known for the non-Newtonian fluids. Recently, the linear stability analysis for the RBP flow involving viscosplastic fluids was performed by Métivier and Nouar [9] for a non-thermodependent Bingham model. The base flow is mainly characterized by a central plug zone moving as a rigid solid and a non-linear variation of the effective viscosity between the wall and the yield surface, where it is infinite. In the frame of linear stability analysis, it is assumed implicitly that the perturbation is infinitesimal compared to all scales of the basic flow. The yield surfaces are linearly perturbed and the plug zone remains intact. The influence of the yield stress on critical conditions arises from the variation of the mean viscosity, the viscosity stratification and the modification of the yielded zone where the Rayleigh–Bénard rolls develop. The authors show and characterize the stabilizing effect of the yield stress on the critical conditions. It is shown that the onset of instability is essentially governed by the presence of the plug zone and by the increase in mean viscosity.

In this article, we propose to extend the work done in [9] to a more realistic situation, where the effective viscosity depends on temperature. To the best of our knowledge, the effect of the viscosity thermodependence on linear stability of shear flows coupled to thermal gradients was only considered for Newtonian fluids for high Reynolds numbers, i.e. shear-driven instability, see Wall and Wilson [10], [11] and Sameen and Govindarajan [12]. Our study starts with the determination of the plane Bingham RBP fully developed flow in Section 2. The linear temporal stability analysis is presented in Section 3. The analysis focuses on the 2D situation, where the convective mode consists of transverse rolls. The critical conditions are determined numerically in Section 4 and the influence of the thermal viscosity stratification is sorted.

Section snippets

Governing equations

We consider the Poiseuille flow under an imposed axial pressure gradient of a thermodependent yield stress fluid in a horizontal plane channel. The upper and lower walls are at constant temperatures, Tˆ0δTˆ/2 and Tˆ0+δTˆ/2, respectively. The hat notation is used for all dimensional variables. The Rayleigh–Bénard configuration corresponds to the case where the lower plate is heated and the upper plate is cooled. Hence, the temperature difference δTˆ is set positive.

The dimensionless problem is

Results and discussion

For k=0, it is shown in [9] that an increase in the Bingham number stabilizes the flow compared to the Newtonian case. This stabilization is essentially due to the presence of the plug zone and the increase in effective viscosity. For non-zero values of k, the ratios Rac/(Rac(k=0)) and αc/(αc(k=0)) permit to characterize the only effect of the thermodependent plastic viscosity. In this respect, Fig. 6 presents the evolution of these ratios as a function of the Pearson number. Results show that

Conclusion

The stability of the plane Bingham Rayleigh–Bénard Poiseuille flow has been studied with the inclusion of the plastic viscosity dependence on temperature. Small values of Pearson number (|k|<O(1)) have been considered in this study. Similar to the non-thermodependent case (k=0), the fully developed flow is characterized by the presence of the two phases, i.e. the gel-like and fluid-like ones. However, the non-zero values of the Pearson number lead to an asymmetrical flow for which the unyielded

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