A 2D Model for Heat Transport in a Hele–Shaw Geometry

This paper is devoted to the theoretical and numerical analysis of the heat transport problem through a viscous and incompressible fluid in a Hele–Shaw geometry. This model corresponds to a bi-dimensional system derived from the 3D-Navier–Stokes equations coupled with an advection-diffusion equation for the heat transport. We analyze the existence of global solutions and construct a numerical scheme, based on finite element approximations in space and finite differences in time. We prove the well-posedness of this numerical scheme and develop the corresponding convergence analysis. The numerical results show the instability of the convective motion, leading to the development of thermal plumes enhancing the heat transport. In addition, our numerical results validate the relation between the time-averaged Nusselt and Rayleigh numbers at the high-Rayleigh regime, as investigated numerically in Letelier et al. (J Fluid Mech 864:746–767, 2019).