Stability of the Thermal Convection in a Tilted Square Cavity

The stability of the two-dimensional thermal convection in a square cavity heated from one side is investigated by restricting two-dimensional disturbances. The four sides of the cavity are assumed to be rigid and perfectly thermal conducting. The fluid is in a motionless state when the cavity is placed horizontally with the hotter side down. A thermal convection occurs as a pitchfork bifurcation from the motionless state due to the instability if the temperature difference exceeds a critical value in that case, but it always occurs even if the temperature difference is very small when the cavity is tilted and the bifurcation is the imperfect pitchfork bifurcation. The equilibrium state of the convection is calculated numerically in the range of 0° - 90° for the angle of the tilt. The stability of the equilibrium state is also investigated. It is shown that the instability occurs as a Hopf bifurcation or a pitchfork bifurcation depending on the angle of the tilt.