Finger instability of oscillating liquid–liquid interface in radial Hele-Shaw cell

Abstract

The dynamics of the interface between two immiscible liquids with a high viscosity contrast is studied experimentally when the liquids are pumped through a radial Hele-Shaw cell. Two cases are considered: a monotonous radial displacement of the viscous fluid, when the classical Saffman–Taylor instability develops, and an oscillatory interface motion due to harmonic flowrate modulation in the absence of the average displacement flow. At small amplitudes of flowrate modulation, the interface performs axisymmetric radial oscillations, maintaining the ring shape during the entire period, while with an increase in the amplitude, it loses stability in a threshold manner. In the phase of fluid displacement, finger instability develops at the interface in the form of an azimuthally periodic structure during a fraction of the period. Fingers reach the greatest length in the phase of maximum fluid displacement, while in the contraction phase (maximum displacement toward the cell center), the interface restores its concentric shape. The threshold for the occurrence of finger instability is determined by the relative amplitude of interface oscillations and under conditions of high contrast of viscosities (one liquid oscillates following the “viscous” law and the other obeys the “inviscid” law) coincides at different oscillation frequencies and different average radii of the interface. The discovered type of instability is new and is studied for the first time. A comparison of the wavelengths of the pulsating fingers with the well-known case of continuous displacement of a viscous fluid in a Hele-Shaw cell indicates that the Saffman–Taylor instability mechanism underlies the observed phenomenon.