The primary and inverse instabilities of directional viscous fingering

Abstract

Consider two infinitely long cylinders of different radii with one inside the other but off-centred. The gap between the two cylinders is partially filled with a viscous fluid. As the cylinders rotate with independent velocities U1 and U2, a thin liquid film coats each of their surfaces all the way around except in the region where the viscous fluid completely fills the gap. Interface conditions that connect solutions of averaged equations in the viscous fluid region with solutions in the thin film region are derived. For the two-interface problem analysed here, two types of instabilities occur depending on the amount of viscous fluid between the cylinders. For large fluid volume, the primary supercritical instability occurs when the front interface becomes unstable as the cylinder velocities are increased. For small fluid volume, the back interface passes through the region where the gap width is a minimum to the same side as the front interface. Steady state solutions with straight interface edges exhibit a turning point with respect to the cylinder velocities. The back interface becomes unstable at the turning point; this inverse instability is subcritical.