Saffman – Taylor instability in poorly miscible viscoelastic flows

A renowned problem of a viscous fluid displacement by a less viscous one from a Hele-Shaw cell was considered. Both fluids exhibited viscoelastic Maxwell rheology with upper convective derivative. A unified approach, which is independent of particular rheology, was applied to derive averaged two-dimensional equations of motion (so-called Hele-Shaw models). The equations were based on Reynolds class averaging procedure. Linear stability analysis was performed under these new governing equations with a special set of boundary conditions for the case of viscoelastic fluids. Dispersion curves showed that, in contrast to the purely Newtonian case, two regimes of disturbance growth were possible: viscous and elastic. We studied the influence of the main dimensionless parameters, in particular, two Deborah numbers for a displacing and a displaced fluid, and the viscosity ratio, on the growth of small disturbances on the interface. In accordance with previous theoretical studies, in the elastic regime there is a disturbance with an infinite growth rate.