Magnetohydrodynamic instability in a partially heated vertical channel

Abstract

The linear stability and the nonlinear behavior of a two-dimensional magnetohydrodynamic (MHD) opposing mixed convection flow of an electrically conducting fluid mixture in a partially and symmetrically heated vertical channel of finite length under an applied transverse magnetic field is studied using numerically generated perturbed functions. The problem depends on the following dimensionless parameters of the fluid mixture: the flow Reynolds number ($Re=100$), the Prandtl number ($Pr=7$), the Richardson number ($Ri=7$), and the Hartmann ($Ha$) number, together with geometrical parameters of the vertical channel. The nonlinear behavior is studied by solving numerically the full nonlinear equations and employing a temporal asymmetric perturbation of the $Ha$ number. The nonlinear stability results show that for relatively large values of the $Ha$ number, the flow is stable and the evolution of the heat transfer response is symmetric. For decreasing values of the $Ha$ number, for a critical value of $Ha=4$, symmetry breaks and a stable nonsymmetric flow and heat transfer response is reached. Our findings reveal the existence of a hysteresis loop describing the nonlinear behavior for the resulting evolution of the overall Nusselt numbers at different $Ha$ numbers. A linear stability analysis using a symmetrical non-parallel thermal base flow has also been performed for the same parameter values. The symmetric flow system shows instability for a critical value of $Ha=3.68$, where the vertical separation of the two vortical structures oscillates with a fixed dimensionless frequency of $St=0.011$. The results show that the nonlinear behavior using the full nonlinear equations reveals hidden instability for the linear analysis. Furthermore, we demonstrate that for the chosen set of parameters and at sufficient high values of the $Ha$ number, the complex interactions related to the effects of shear, opposing buoyancy, and magnetic damping can be effectively used to stabilize the flow.