Shear Hele-Shaw flows of a weakly compressible fluid

We derive the integro-differential system of equations describing shear flows of a weakly compressible multicomponent fluid in a Hele-Shaw cell. The propagation velocities of nonlinear perturbations in the fluid are determined, and the characteristic form of the system is obtained. We formulate conditions for the hyperbolicity of the model that are necessary for a correct statement of the Cauchy problem. One-dimensional multilayered flows are considered, and the interpretation of the Saffman-Taylor instability in the framework of the two-layered fluid scheme is given.