Onset of Double-Diffusive Convection in a Rectangular Cavity and Its Generation Mechanism

Two-dimensional double diffusive convection in a binary fluid mixture filled in a container with a rectangular cross section is investigated by linear stability analyses, numerical simulations and numerical calculations of steady solutions in the present paper. We mainly consider an ethanol-water mixture as the binary fluid, in which heat and ethanol diffuse in different time scales affecting the fluid motion through buoyancy force and the Soret effect. The bottom of the cavity is kept at a higher temperature than the top, and the side boundary walls are assumed to be perfectly insulating. The impermeability condition of mass is applied on all the boundaries. We obtain the critical condition for the onset of double diffusive convection, and examine the flow field at the criticality. It is found that the most unstable mode of disturbance is oscillatory at the criticality for negative values of the separation number, though it is a steady mode of disturbance for positive or null values of the separation number. We discuss the driving mechanism of the steady and oscillatory convections by evaluating torques exerted on the fluid due to the buoyancy force, the pressure and the viscosity separately in each. We find in numerical simulations that the convection, even if it is oscillatory initially, always attains a steady state in due course in the case of a container with a square cross section. The bifurcation diagram of the steady convection is obtained numerically and the relation between the steady convection and the oscillatory mode of disturbance arising due to the linear instability is briefly discussed.