Roll waves structure in two-layer Hele-Shaw flows

•The mathematical model of the two-layer fluid motion in a Hele-Shaw cell motion is proposed.•Conditions of roll waves existence are derived.•Numerical construction of the roll waves on the interface is provided.•The class of exact solutions corresponding to the roll waves regime is obtained.

The mathematical model of inhomogeneous fluid motion in a Hele-Shaw cell is proposed. Based on this model the equations for describing two-layer flows and development of roll waves at the interface are derived. Conditions of roll waves existence are formulated in terms of Whitham criterion. Numerical calculations of the interface position are provided. It is shown that small perturbations of the interface in the inlet section of the channel lead to the roll waves for certain parameters of the flow. Two-parametric class of exact solutions corresponding to the roll waves regime is obtained. Diagrams of critical depths of roll waves development are constructed.