Finite-size effects on linear stability of pure-fluid convection

The linear stability of pure-fluid Rayleigh-Benard convection in a finite cell of arbitrary geometry can be formulated as a self-adjoint eigenvalue problem. This, when coupled with perturbation theory, allows one to deduce how the sidewalls affect its stability. In particular, it is shown that for almost all boundary conditions the difference between the onset Rayleigh number and its infinite-cell limit scales like L^-2 as the cell dimension L tends to infinity, and near the sidewall the temperature and velocity are of order L^-1 compared to their bulk values. The validity of replacing the true thermal boundary condition by a frequently used mathematically simpler homogeneous one is also demonstrated