Bifurcations and stability of steady regimes of convective flows in an inclined rectangular cavity

The effects of the Grashof number Gr and the angle of inclination α of the cavity on the structure and stability of steady convective flows in a rectangle with free boundaries are studied. Between the two opposite isothermal sides, horizontal at α=0°, the temperature difference is maintained, and the other sides are thermally insulated. The flow of the fluid is assumed to be flat and described by the equations of thermal convection in the Boussinesq approximation. In the case of heating, explicit analytical expressions for increments and critical Grashof numbers of small perturbations of mechanical equilibrium are obtained strictly from below. By solving the multidimensional Newton method of a system of algebraic equations obtained by discretizing the equations of thermal convection on a rectangular grid, stationary regimes are determined. To investigate the stationary modes found for stability with respect to small perturbations, we find the values of the parameters for which the Jacobian of the system for the amplitudes of small perturbations is zero. For α=0° increasing the Grashof number from the state of mechanical equilibrium, as a result of three successive fork bifurcations, two stable single-shaft bunches and two unstable two- and three-shaft stationary regimes branch off. Each two-shaft mode as a result of a fork bifurcation breaks up into two unstable modes and one stable mode. All these fork bifurcations, except for the second equilibrium bifurcation, are structurally unstable: they decay with a small change in the angle of inclination of the cavity. The evolution of the flow structure is traced as the angle of inclination of the rectangle and Grashof number change. Areas are selected in the parameter plane (α, Gr), where one, three, five, seven, nine or eleven stationary modes exist for each set of parameters.