Rayleigh-Benard convection: the container shape matters

We derive that the critical Rayleigh number for the onset of convection in right cylindrical cells with no-slip boundaries, of small diameter-to-height aspect ratios Γ, for any shape of plates, grows as ∼(1+A/Γ²)(1+B/Γ²), where A and B are determined by the cell shape and boundary conditions for the velocity and temperature. Under the assumption that in the expansions of the reduced temperature and velocity by the onset of convection in terms of the eigenfunctions of the Laplace operator, the contributions of the constant-sign eigenfunctions, both in the vertical and at least in one horizontal direction, vanish, we derive precise estimates of the critical Rayleigh number and determine the values of A and B for cylindrical and box-shaped cells. We compare our results with those from the linear stability analysis. Furthermore, we derive the relevant length scale in Rayleigh-Benard convection, which tends to the cell height for Γ→∞ and to the cell diameter for Γ→0. Finally, we discuss the optimum of the cell shape and show further applications of the developed ansatz. You are welcome to read the paper in Phys. Rev. Fluids (2021) with the same author and title.