Flows of two immiscible fluids over an obstacle

In this paper, we consider superposed flows of two fluids over an obstacle lying on the bottom of a plane channel. Asymptotically upstream and downstream, the free boundary (fluid-air) as well as the interface (separating the two fluids) are flat but near the obstacle a perturbation appears. Each flow is irrotational, stationary and the fluids are ideal and incompressible. We take gravity into account and we neglect the effects of superficial tension. The problem is formulated using nonlinear equations defined on the free surface and on the interface (derived from Bernoulli’s equation). The unknowns are the velocities fields, the free boundary and the interface. A result of existence and uniqueness is given in Banach algebras by using the implicit functions theorem. The originality of our model consists of considering two fluids. So we are challenged by three difficulties: a nonlinear coupled equation on the interface, a nonlinear equation on the free boundary and a free boundary which is itself an unknown of the problem.