Steady Boussinesq convection: Parametric analyticity and computation

Steady solutions to the Navier–Stokes equations with internal temperature forcing are considered. The equations are solved in two dimensions using the Boussinesq approximation to couple temperature and density fluctuations. A perturbative Stokes expansion is used to prove that that steady flow variables are parametrically analytic in the size of the forcing. The Stokes expansion is complemented with analytic continuation, via functional Padé approximation. The zeros of the denominator polynomials in the Padé approximants are observed to agree with a numerical prediction for the location of singularities of the steady flow solutions. The Padé representations not only prove to be good approximations to the true flow solutions for moderate intensity forcing, but are also used to initialize a Newton solver to compute large amplitude solutions. The composite procedure is used to compute steady flow solutions with forcing several orders of magnitude larger than the fixed-point method developed in previous work.