P. Gondret and M. Rabaud, "Shear instability of two-fluid parallel flow in a Hele-Shaw cell", Physics of Fluids 9, 3267-3274 (1997).
Abstract : We study experimentally the parallel flow in a Hele-Shaw cell of two immiscible fluids, a gas and a viscous liquid, driven by a given pressure gradient. We observe that the interface is destabilized above a critical value of the gas flow and that waves grow and propagate along the cell. The experimental threshold corresponds to a velocity difference of the two fluids in good agreement with the inviscid Kelvin-Helmholtz instability, while the wave velocity corresponds to a pure viscous theory deriving from Darcy's law. We report our experimental results and analyze this instability by the study of a new equation where the viscous effects are added to the Euler equation through a unique drag term. The predictions made from the linear stability analysis of this equation agree with the experimental measurements.