Stability analysis of shear flows in a Hele–Shaw cell

A mathematical model describing motion of an inhomogeneous incompressible fluid in a Hele–Shaw cell is considered. Linear stability analysis of shear flow class is provided. The role of inertia, linear friction and impermeable boundaries in Kelvin–Helmholtz instability is studied. Hierarchy of simplified one-dimensional models of viscosity- and density-stratified flows is obtained in long-wave approximation. Interpretation of Saffman–Taylor instability is given in the framework of these models.