Weakly nonlinear dynamics of thermoconvective instability involving viscoplastic fluids

Abstract

In this paper, a weakly nonlinear stability of viscoplastic fluid flow is performed. The system consists of a plane Rayleigh–Bénard–Poiseuille (RBP) flow of a Bingham fluid. The basic flow is characterized by a central plug zone, of 2yb width, in which the stresses are smaller than the Bingham number B, the dimensionless yield stress. The Bingham model assumes that inside this zone the material moves as a rigid solid, and that outside this zone it behaves as a viscous fluid. The aim of this study is to investigate the influence of the yield stress on the instability conditions. The linear stability analysis is performed using a modal method and provides critical values of Rayleigh and wavenumbers, from which the system becomes unstable. The critical mode, i.e. the least stable mode, is also determined. This mode, also called the fundamental mode, creates perturbation harmonics which cannot be neglected above criticality. The weakly nonlinear analysis is performed for small-amplitude perturbations. In this study, the quadratic modes of the perturbation are determined. Results indicate that the nonlinear modes perturbation can attain high maximal values, which is the consequence of the high variations of viscosity in the flow. The characterization of the complex Landau equation sheds light on a transition in terms of the bifurcation nature above a critical Péclet number Pec = O(1). Below Pec, it is found that a supercritical equilibrium state could exist, such as in the Newtonian case, while above Pec, the bifurcation becomes subcritical. One observes a sharp transition from supercritical to subcritical bifurcation as the Péclet value is increased. A dependence of Pec on the yield stress is highlighted since the subcritical bifurcation is first observed for weak values of yb (yb < O(10−1)). For this range of values, the transition is mainly due to the presence of the unyielded region via non-homogeneous boundary conditions at the yield surfaces. Then for yb > O(10−1), the change of the bifurcation nature is due to the variations of the effective viscosity in the unyielded regions.