Effects of geometrical imperfection at the onset of convection in a shallow two-dimensional cavity

A theoretical study is made of the effect of geometrical imperfections on the formation of weakly nonlinear convection in a shallow two-dimensional cavity uniformly heated from below. If the sidewalls of the cavity are not quite vertical then convective rolls first appear near these walls and subsequently spread inwards to the center of the cavity as the Rayleigh number is increased. If the horizontal surfaces are not quite parallel then the major effect is a lateral modulation of the rolls due to a combination of the misalignment of the horizontal surfaces and the presence of the sidewalls. As an interesting special case, the solution for a cavity of uniformly sloping base is presented and asymptotic methods are found to provide a remarkably accurate prediction for the critical Rayleigh number as a function of angle of inclination.