Elastic fingering in a rotating Hele-Shaw cell

TL;DR

This paper investigates the steady-state fingering instability of elastic membranes in a rotating Hele-Shaw cell, analyzing how membrane tension, elasticity, and rotation influence pattern formation and bifurcations.

Contribution

It provides a comprehensive numerical analysis of nonlinear finger states, secondary modes, and bifurcation structures considering both inextensible and extensible membranes.

Findings

Centrifugal Rayleigh-Taylor-like instability occurs when inner fluid density exceeds outer.

Membrane tension and elasticity significantly influence finger scale and bifurcation behavior.

System scale ratio determines bifurcation order and pattern profiles.

Abstract

We consider the steady-state fingering instability of an elastic membrane separating two fluids of different density under external pressure in a rotating Hele-Shaw cell. Both inextensible and highly extensible membranes are considered, and the role of membrane tension is detailed in each case. Both systems exhibit a centrifugally-driven Rayleigh-Taylor--like instability when the density of the inner fluid exceeds that of the outer one, and this instability competes with the restoring forces arising from curvature and tension, thereby setting the finger scale. Numerical continuation is used to compute not only strongly nonlinear primary finger states up to the point of self-contact but also secondary branches of mixed modes and circumferentially localized folds as a function of the rotation rate and the externally imposed pressure. Both reflection-symmetric and symmetry-broken chiral states are computed. The results are presented in the form of bifurcation diagrams. The ratio of system scale to the natural length scale is found to determine the ordering of the primary bifurcations from the unperturbed circle state as well as the solution profiles and onset of secondary bifurcations.