Low viscosity contrast fingering in a rotating Hele-Shaw cell

Abstract

We study the fingering instability of a circular interface between two immiscible liquids in a radial Hele-Shaw cell. The cell rotates around its vertical symmetry axis, and the instability is driven by the density difference between the two fluids. This kind of driving allows studying the interfacial dynamics in the particularly interesting case of an interface separating two liquids of comparable viscosity. An accurate experimental study of the number of fingers emerging from the instability reveals a slight but systematic dependence of the linear dispersion relation on the gap spacing. We show that this result is related to a modification of the interface boundary condition which incorporates stresses originated from normal velocity gradients. The early nonlinear regime shows nearly no competition between the outgrowing fingers, characteristic of low viscosity contrast flows. We perform experiments in a wide range of experimental parameters, under conditions of mass conservation ͑no injection͒, and characterize the resulting patterns by data collapses of two characteristic lengths: the radius of gyration of the pattern and the interface stretching. Deep in the nonlinear regime, the fingers which grow radially outwards stretch and become gradually thinner, to a point that the fingers pinch and emit drops. We show that the amount of liquid emitted in the first generation of drops is a constant independent of the experimental parameters. Further on there is a sharp reduction of the amount of liquid centrifugated, punctuated by periods of no observable centrifugation.

Numerical simulations of interfacial instabilities on a rotating miscible droplet in a time‐dependent gap Hele–Shaw cell with significant Coriolis effects

“…These phenomena can also be observed by contours of vorticity, depicted in Figure 2(c). The new layer presents outgoing ÿngers, which develop a peculiar fork-like shape, which is also identiÿed in immiscible situation [7], at t = 20. Besides, compared to the reference case, the ÿngering pattern clearly shows a rotational trend because of the stronger Coriolis e ects induced by the faster rotating speed.…”

Fingering patterns on an expanding miscible drop in a rotating Hele‐Shaw cell

“…Based on the Hele-Shaw theory, Alvarez-Lacalle et al [2] investigated the interfacial instabilities of immiscible fluids in a rotating cell both experimentally and numerically, in which Coriolis forces are neglected. Finger pinch-off and droplet emission are detected at the low viscosity contrast and the low surface tension in their experiments.…”

Interfacial instability in a time-periodic rotating Hele-Shaw Cell

“…Several theoretical and experimental works have been carried out to study the interfacial instability in an annular Hele-Shaw cell [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. These works have been extended to include effects arising from a constant rotation rate of the system and have mainly focused on the study of the linear growth rate, also, on the understanding fingering mechanisms of the interface when one of two immiscible fluids is accelerated toward the other under centrifugal forces.…”

“…In his work, the coriolis force is included in an ad hoc manner and it was shown that circular drop is unstable both to a translation, and depending on the rotation rate, to a number of fingering modes. Later on, the linear stability analysis originally performed in [2] has been extended to an arbitrary density and viscosity contrast by carrillo et al, [3][4][5], and to a low viscosity contrast flows by Alvarez-Lacalle et al [8]. In [3], Carrillo et al have studied, both theoretically and experimentally, the behavior of the interface between two immiscible fluids in a rotating Hele-Shaw cell when the inner fluid is injected, with constant injection rate, through a hole at the rotation axis.…”

“…Using a linear stability analysis and neglecting coriolis forces, he has determined the growth rate when both the centrifugal forces and magnetic forces are induced and has shown that the magnetic forces act to stabilize the fluid interface. Thereafter, Alvarez-Lacalle et al, [8], have studied the dynamics and morphology of the viscous fingering patterns formed at the circular interface between two immiscible fluids in a rotating Hele-Shaw cell in the low-viscosity contrast limit. These authors have found that, in addition to viscosity contrast and surface tension, the gap spacing of the cell also plays an important role in the linear regime to characterize finger competition processes.…”