Effect of the density ratio on the oscillatory Saffman–Taylor instability in vertical conical Hele–Shaw cell

Abstract

The stability of an oscillating interface between two immiscible liquids with a high viscosity contrast in an axisymmetric conical Hele–Shaw cell in dependence on the density ratio is studied experimentally. The symmetry axis of the cell is vertical, while the tangential component of the gravy acts on the axisymmetric interface. The contact line is almost motionless, while the low-viscosity liquid penetrates a high-viscosity one in the form of an axisymmetric “tongue” in the course of an oscillating cycle. The increase in the oscillation amplitude leads to the development of azimuthal patterns (fingers) at the interface. Fingers of a low-viscosity liquid appear when a viscous liquid is being squeezed out of the cell and reach their maximum length at the maximum displacement of the interface. Then, the fingers decrease and are replaced by small depressions penetrating into the low-viscosity liquid in the phase of maximum boundary displacement toward the low-viscosity liquid. The increase in the density ratio of liquids has a stabilizing effect on the interface: The instability threshold is shifted to the higher oscillation amplitudes. Also, the stability threshold is independent of whether the high-viscosity or low-viscosity liquid is denser than the other one. We propose a new dimensionless parameter that controls the stability of the interface—the multiplication of the square root of the capillary number and the dimensionless amplitude of interface oscillations. It is revealed that the critical value of the stability parameter increases linearly with an increase in the density ratio of liquids.