The effect of distant sidewalls on the transition to finite amplitude Benard convection

The theory of the nonlinear development of Benard convection in an infinite fluid layer confined between horizontal boundaries predicts that the amplitude of the motion undergoes a bifurcation as the Rayleigh number passes through the critical value for instability predicted by linear theory. Segel (1969) has shown that this is also the case if the flow is confined laterally by rigid perfectly insulating sidewalls. In the present paper it is shown that if there is a small heat transfer through these walls (so that the boundary conditions there are inconsistent with a state of no motion) the bifurcation is in general replaced by a smooth transition to finite amplitude convection. This effect also ensures that the motion is stable. Although an idealized theoretical model is assumed in which the flow is two-dimensional and stress-free at the horizontal boundaries, the results apply qualitatively to more realistic models.