One-dimensional mixing layer model for a shear Hele-Shaw flow

A shear flow of a viscosity-stratified fluid in a Hele-Shaw cell is considered. The long-wave approximation is applied to the governing equations. To describe the evolution of the mixing layer, a special flow with a three-layered structure is considered. A one-dimensional model is derived by averaging the motion equations over the cell width, taking into account the flow structure. For a stationary flow, solutions of motion equations are constructed. The influence of viscosity on the mixing layer evolution is investigated by performing a numerical experiment for a flow with different viscosities in the layers and for a flow with always zero viscosity. It is shown that viscosity has a significant influence on the flow evolution.