Linear onset of convective instability for Rayleigh-Bénard-Couette flows of viscoelastic fluids

Abstract

A linear stability analysis of a plane Couette flow in the presence of an unstable vertical temperature gradient is investigated numerically for viscoelastic fluids using the Oldroyd-B model. Three main configurations are considered. In the first configuration, named case (i), both walls move with the same velocity, resulting in a shear free flow. In the second configuration, named case (ii), the lower wall is kept fixed while the upper wall moves with a constant velocity parallel to the x axis, i.e there is a non-zero net flow. In the third configuration, named case (iii), the horizontal plates move in opposite directions but with the same velocity magnitude, i.e. there is a zero net flow. The effects of the Reynolds number Re, relaxation time or elasticity number λ1, ratio Γ between retardation and relaxation times and the ratio VR between lower and upper plate velocities on the critical Rayleigh numbers, critical frequencies and convection cell sizes are examined. When Newtonian fluids are considered, the critical Rayleigh number for the appearance of longitudinal rolls (LRs), which are rolls with axes parallel to the x axis, is independent of Re. The same cannot be said about viscoelastic fluids. There is a range of λ1 and Γ values, which defines a strongly elastic regime, where the system experiences a Hopf bifurcation giving rise to oscillatory LRs. On the other hand, in the parametric region outside of this range, which defines a weakly elastic regime, LRs are stationary. Since there is a Squire transformation for this problem, the linear characteristics of rolls with arbitrary orientation can be deduced from those determined for transverse rolls (TRs), which are rolls with axes perpendicular to the x axis. In the weakly elastic regime, TRs are stabilized by an increasing Re. This effect is stronger in case (ii) than in case (iii). Furthermore, TRs are stationary in case (iii) but oscillatory in case (ii), since the net flow breaks the reflexion symmetry x→(−x) and selects a travelling wave propagating in the main flow direction. Finally, elasticity has a stabilizing effect whereas concentrated polymeric solutions are less stable than diluted ones. On the other hand, in the strongly elastic regime, the constant shear flow promotes the simultaneous appearance of two travelling waves (TW). Because of the mirror symmetry in case (iii), these TWs propagate in opposite directions but with the same phase velocity magnitude. In case (ii), the nonzero net flow induces different values for the TW phase velocities. Finally, λ1 and Γ have opposite effects. Although increasing λ1 (Γ) destabilizes (stabilizes) TWs for small Re, it stabilizes (destabilizes) them for moderate and high values of Re.