The sharp-interface limit of the Cahn–Hilliard/Navier–Stokes model for binary fluids

Abstract

The Cahn–Hilliard model is increasingly often being used in combination with the incompressible Navier–Stokes equation to describe unsteady binary fluids in a variety of applications ranging from turbulent two-phase flows to microfluidics. The thickness of the interface between the two bulk fluids and the mobility are the main parameters of the model. For real fluids they are usually too small to be directly used in numerical simulations. Several authors proposed criteria for the proper choice of interface thickness and mobility in order to reach the so-called ‘sharp-interface limit’. In this paper the problem is approached by a formal asymptotic expansion of the governing equations. It is shown that the mobility is an effective parameter to be chosen proportional to the square of the interface thickness. The theoretical results are confirmed by numerical simulations for two prototypal flows, namely capillary waves riding the interface and droplets coalescence. The numerical analysis of two different physical problems confirms the theoretical findings and establishes an optimal relationship between the effective parameters of the model.

Motion and coalescence of sessile drops driven by substrate wetting gradient and external flow

“…Alternative choices of the mobility m have been explored, for example by Magaletti et al (2013) and Sibley, Nold & Kalliadasis (2013a). For moving contact lines in a one-component liquid-vapour system, Sibley et al (2013b) have analysed the asymptotic limit of the diffuse-interface model at the contact line.…”

Phase-field modeling of interfacial dynamics in emulsion flows: Nonequilibrium surface tension

“…On the other hand, Jacqmin (20 0 0) performed a careful matched asymptotic analysis showing that, in the limit of vanishing interfacial width, the diffuse-interface model is consistent with the usual Marangoni-type boundary conditions that arise in the classical formulation of two-phase flow. Note that the asymptotic analysis of diffuse-interface models has been furthered considerably ( Magaletti et al, 2013;Sibley et al, 2013a ) since the Jacqmin (20 0 0) paper. In particular, recent work by Sibley et al (2013a ) has shown that when a binary fluid diffuse-interface model is employed in conjunction with a tensorial mobility, the model allows the classical two-phase flow equations to be recovered to all orders in the Cahn number, in the limit as it tends to zero.…”

A diffuse-interface immersed-boundary method for two-dimensional simulation of flows with moving contact lines on curved substrates

“…A constant mobility is assumed for all the components here, and the value of P e represents the relative importance of the convective fluxes uC to the diffusive fluxes ∇Ψ. As suggested in [27], we select P e = α/Cn, so that the DI model approaches the sharp interface limit with the vanishing of Cn, where α ∼ O(1) is a constant. Based on our numerical experiences, P e = 0.9/Cn is used in the present study.…”