Viscous fingering in poorly miscible power-law fluids

A renowned problem of a viscous fluid displacement by a less viscous one from a Hele–Shaw cell is considered. Both fluids exhibit non-Newtonian properties: a power-law viscosity dependence on strain rates (Ostwald–de Waele rheology). A unified approach independent of particular rheology is applied to derive averaged two-dimensional equations of motion (so-called Hele–Shaw models). The equations are based on Reynolds class averaging procedure. Under these governing equations, linear stability analysis of the radial interface is conducted with a new key idea—possibility of characteristic size selection even in the absence of stabilizing factors such as surface tension and molecular diffusion. For proving this, proper boundary conditions are set on the interface, namely, the equality of full normal stresses including viscous ones, instead of the simple equality of pressures.