Boundary conditions on the envelope function of convective rolls close to onset

Boundary conditions are derived for the envelope function of convective rolls approaching a rigid sidewall at an arbitrary orientation. This generalizes previous results for parallel and perpendicular rolls. It provides the first step in a study of the convective pattern to be expected close to onset in Rayleigh–Benard cells of any cross section, and allows consideration of more complicated patterns in rectangular or circular cross sections.