Effect of inertia on the Marangoni instability of two-layer channel flow, Part I: numerical simulations

A numerical method is developed for simulating the flow of two superposed liquid layers in a two-dimensional channel confined between two parallel plane walls, in the presence of an insoluble surfactant. The algorithm combines Peskin's immersed-interface method with the diffuse-interface approximation, wherein the step discontinuity in the fluid properties is replaced by a transition zone defined in terms of a mollifying function. A finite-difference method is implemented for integrating the generalized Navier-Stokes equation incorporating the jump in the interfacial traction, and a. finite-volume method is implemented for solving the surfactant transport equation over the evolving interface. The accuracy of the overall scheme is confirmed by successfully comparing the numerical results with the predictions of linear stability analysis and numerical simulations based on a boundary-element method for Stokes flow. Results for selected case studies suggest that inertial effects have a mild effect on the growth rate of the surfactant-induced Marangoni instability.