Marangoni flow traveling with reaction fronts: Eikonal approximation

Chemical reaction fronts traveling in liquids generate gradients of surface tension leading to fluid motion. This surface tension driven flow, known as Marangoni flow, modifies the shape and the speed of the reaction front. We model the front propagation using the Eikonal relation between curvature and normal speed of the front, resulting in a front evolution equation that couples to the fluid velocity. The sharp discontinuity between the reactants and products leads to a surface tension gradient proportional to a delta function. The Stokes equations with the surface tension gradient as part of the boundary conditions provide the corresponding fluid velocity field. Considering stress free boundaries at the bottom of the liquid layer, we find an analytical solution for the fluid vorticity leading to the velocity field. Solving numerically the appropriate no-slip boundary condition, we gain insights into the role of the boundary condition at the bottom layer. We compare our results with results from two other models for front propagation: the deterministic Kardar-Parisi-Zhang equation and a reaction-diffusion equation with cubic autocatalysis, finding good agreement for small differences in surface tension.