Approximation of Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier–Stokes/Allen–Cahn System

Abstract We show convergence of the Navier–Stokes/Allen–Cahn system to a classical sharp interface model for the two-phase flow of two viscous incompressible fluids with same viscosities in a smooth bounded domain in two and three space dimensions as long as a smooth solution of the limit system exists. Moreover, we obtain error estimates with the aid of a relative entropy method. Our results hold provided that the mobility m_ >0 m ε > 0 in the Allen–Cahn equation tends to zero in a subcritical way, i. e. , m_ = m₀ ^ m ε = m 0 ε β for some (0, 2) β ∈ (0, 2) and m₀>0 m 0 > 0. The proof proceeds by showing via a relative entropy argument that the solution to the Navier–Stokes/Allen–Cahn system remains close to the solution of a perturbed version of the two-phase flow problem, augmented by an extra mean curvature flow term m_ H 䂻 m ε H Γ t in the interface motion. In a second step, it is easy to see that the solution to the perturbed problem is close to the original two-phase flow.