Parallel flow in Hele-Shaw cells

Abstract

We consider the parallel flow of two immiscible fluids in a Hele-Shaw cell. The evolution of disturbances on the fluid interfaces is studied both theoretically and experimentally in the large-capillary-number limit. It is shown that such interfaces support wave motion, the amplitude of which for long waves is governed by a set of KdV and Airy equations. The waves are dispersive provided that the fluids have unequal viscosities and that the space occupied by the inner fluid does not pertain to the Saffman-Taylor conditions (symmetric interfaces with half-width spacing). Experiments conducted in a long and narrow Hele-Shaw cell appear to validate the theory in both the symmetric and the non-symmetric cases.

Shear instability of two-fluid parallel flow in a Hele–Shaw cell

“…In the first section, we recall the classical results of the stability analysis of this flow governed by the Euler equation, known as the Kelvin-Helmholtz theory. In the second section, we present the linear stability analysis given by the Darcy equation following the work of Zeybek and Yortsos 16,17 but adding the effect of gravity. At last, in the third section, we present a new equation taking into account some terms of the two previous equations and we perform the linear stability analysis of this new equation.…”

“…Assuming the incompressibility of the two fluids leads to Laplace equation for the pressure: ⌬pϭ0. We then followed the linear stability analysis performed by Zeybek and Yortsos 16,17 in the case of parallel flow in Hele-Shaw cell but adding the effect of gravity. Assuming small perturbations of the form p(y)exp͓ik(xϪct)͔ for the pressure superimposed to the basic stationary and unidirectional flow along x direction and imposing the usual continuity of displacement at the interface yϭ0, the jump condition for the pressure at the interface due to surface tension and the exponential decays at yϭϮϱ leads to the following dispersion relation:…”

Capillary-driven instability of immiscible fluid interfaces flowing in parallel in porous media

“…However, at larger flow rates, where viscous forces dominate, shear-driven Kelvin-Helmholtz type instabilities are believed to occur [9,10,11]. Both theoretical and experimental work has been invested in studying the Kelvin-Helmholtz instability in vertical Hele-Shaw cells * thomas@numericalrocks.com † Alex.Hansen@ntnu.no [3,12,13], as it provides a model for parallel flow in porous media in the viscous regime.…”

Parallel flow in Hele-Shaw cells with ferrofluids

“…Recent experimental and theoretical studies [3][4][5] examined the dynamics of fluid interfaces under parallel flow in Hele-Shaw cells. Zeybek and Yortsos [3,4] studied, both theoretically and experimentally, parallel flow in a horizontal Hele-Shaw cell in the large capillary number limit. For finite capillary number and wavelength, linear stability analysis indicates that small perturbations decay, but the rate of decay vanished in the limit of large capillary numbers and large wavelength.…”

“…In the following, we discuss the action of the applied magnetic field on the solitons that appear in parallel flow in Hele-Shaw cells. To treat the problem rigorously would require reproducing the analysis of Zeybek and Yortsos [3,4] Take the generic form of a KdV soliton,…”