On the asymptotic behavior of the equations describing the motion of an incompressible binary fluid

We determine the asymptotic behavior of the system of Cahn-Hillard/Euler's equations that control the dynamics inside the thin boundary layer separating two inviscid, incompressible, and nearly immiscible fluids. This model was proposed recently in order to replace the classical moving boundary model of two immiscible fluids, separated by the interface with the surface tension. We formally verify that these two problems are related. Using the method of matched asymptotic expansions, we show that when the width of the interface and the miscibility of the fluids converge to zero, then the system of Cahn-Hillard/Euler's equations converges asymptotically to the classical moving boundary problem. In addition, we analyze for the different timescales the behavior of the mixture inside the boundary layer.