Dynamic evolution of fingering patterns in a lifted Hele–Shaw cell

Abstract

We present a study on pattern formation in a Newtonian liquid during lifting of a circular Hele–Shaw cell. When a confined layer of oil is subject to such a stretch flow, air penetrates into the liquid from the sides and a fingering instability, a variant of the classical Saffman–Taylor instability, evolves. This setting has the particularity that the finger growth takes place in a conserved volume of liquid and that the dimensionless surface tension, the control parameter which governs the Saffman–Taylor instability, is changing with time. This leads to a constantly evolving pattern, which we investigate with regard to number of fingers and finger amplitude. We distinguish in the pattern at each instant growing fingers and stagnant fingers. Systematically varying the properties of the viscous oil and the geometry of the Hele–Shaw cell, we show that the number of growing fingers is at each moment well described by a simple approach based on linear stability analysis and depends only on the dimensionless surface tension. In contrast, the finger amplitude and consequently the total number of fingers (growing and stagnant fingers) depend also on the cell confinement. We demonstrate that the finger amplitude has a distinct influence on the debonding force. Higher finger amplitude and number of fingers lead to lower forces.