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

Abstract

Abstract. The effect of time-periodic angular velocity on the interfacial instability of two immiscible, viscous fluids of different densities and confined in an annular Hele-Shaw cell is investigated. An inviscid linear stability analysis of the viscous and time dependent basic flow leads to a periodic Mathieu oscillator describing the evolution of the interfacial amplitude. We show that the relevant parameters that control the interface are the Bond number, viscosity ratio, Atwood number and the frequency number.

Control of the Interfacial Instabilities in a circular Hele-Shaw cell oscillating with a periodic angular velocity

“…Figures 2 (a) and (b) depict the evolution of the Bond number, B o , versus the wave number, n, for the waterair interface (γ = 0.072 N/m and A ρ = −0.9) and for different values of the dimensional frequency of the oscillations ω * . It is worth noting from results that, the interface is unstable to the parametric instability, unlike in [7][8] where the interface can be linearly unstable to the parametric instability and and to the Rayleigh-Taylor instability due to the centrifugal forces via the constant term of rotation. By inspecting this figures, two important main conclusions can be drawn.…”

“…It was shown that the Coriolis forces does not alter the stability of a dis-turbance with a particular wave number but reduces the maximum growth rate. Recently and in contrast to the majority of studies carried out in the rotating Hele-Shaw cell setup, the effect of time dependent angular velocity on the dynamics of such confined flows is only started to be investigated in the literature by Bouchgl et al [7] and Anjos et al [8]. In [7], the authors present a mathematical formulation of the physical problem based on the inviscid linear stability analysis.…”

“…Recently and in contrast to the majority of studies carried out in the rotating Hele-Shaw cell setup, the effect of time dependent angular velocity on the dynamics of such confined flows is only started to be investigated in the literature by Bouchgl et al [7] and Anjos et al [8]. In [7], the authors present a mathematical formulation of the physical problem based on the inviscid linear stability analysis. Hereafter, Anjos et al [8] have focused on examining how the time-dependent angular velocity and the inertia affect the finger competition events that traditionally arise in rotating Hele-Shaw flows.…”

“…Hereafter, Anjos et al [8] have focused on examining how the time-dependent angular velocity and the inertia affect the finger competition events that traditionally arise in rotating Hele-Shaw flows. Following the study carried out on the modulation of rotation [7][8], we are interested in the present work to a zero mean modulation of the cell. In this context, we investigate the effect of frequency on the threshold of instability for the initially unstable configuration, where the inner fluid and the outer one are considered respectively heavy and light.…”

Dynamics of liquids with high viscosity contrast in unevenly rotating Hele-Shaw cell

“…The number of theoretical and experimental works devoted to the morphological instability of the interface in the Hele-Shaw cell during rotation is large; a review of those can be found in [4], which theoretically explored how the time-dependent angular velocity and inertia affect the fingers competition arising in rotating Hele-Shaw flows. In theoretical work [5], it is noted that the time-periodic angular velocity affects the threshold of the instability, tending to stabilize interfacial disturbances. Of no less interest is the effect of rotation velocity modulation on the interface when a denser liquid occupies the stable position in the centrifugal field, i.e.…”