THE PRIMARY AND INVERSE INSTABILITIES OF DIRECTIONAL VISCOUS FINGERING

Consider two infinitely long cylinders of different radii with one ins ide the other but off-centred. The gap between the two cylinders is pa rtially filled with a viscous fluid. As the cylinders rotate with inde pendent velocities U-1 and U-2, it thin liquid film coats each of thei r surfaces all the way around except in the region where the viscous f luid completely fills the gap. Interface conditions that connect solut ions of averaged equations in the viscous fluid region with solutions in the thin film region are derived. For the two-interface problem ana lysed here, two types of instabilities occur depending on the amount o f viscous fluid between the cylinders. For large fluid volume, the pri mary supercritical instability occurs when the front interface becomes unstable as the cylinder velocities are increased. For small fluid vo lume, the back interface passes through the region where the gap width is a minimum to the same side as the front interface. Steady state so lutions with straight interface edges exhibit a turning point with res pect to the cylinder velocities. The back interface becomes unstable a t the turning point; this inverse instability is subcritical.