Analytical approximations of laminar flow states above a curved wall heated from below

An analytical framework is developed for estimating wall-bounded plane laminar flow states above curved walls subject to external shear and heating. The framework is applicable to arbitrary lower-wall geometries with arbitrary temperature prescriptions. The study relies on perturbation expansions of flow states with respect to the inverse of the dimensionless channel wavelength. Exact closed-form solutions are obtained for systems of perturbation equations up to the third order, which have the freedom to allow for any flow constraint. As an application of the analytical framework, we examine flow states above two specific instances of wall geometries with long channel wavelengths. The approximated flow states shed insights into morphologies of laminar convection states at low shear rates (i.e. low Reynolds numbers).