Long waves at the interface between two viscous fluids

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Abstract
Using a perturbation method up to the second order, the equation for long waves at the interface between two viscous fluids is derived for plane Couette–Poiseuille flow, and for moderate surface tension. The leading order equation is the Burgers equation. Higher order terms take into account linear dispersive effects and stabilizing effect of surface tension, and involve two more nonlinear terms. The exactness of the coefficients of this equation has been checked by using symmetry properties. For zero gravity, Poiseuille flow is shown to be stable if and only if the velocity profile is convex. Stable Couette flow can become unstable when the Reynolds number of one fluid is decreased, while keeping the other dimensionless parameters unchanged. Introducing characteristic length scales, the interface equation is put into ‘‘canonical’’ form.