Incompressible Multiphase Flows: Physical Formulation and Numerical Algorithm

<p>We present a family of physical formulations,&nbsp;and a&nbsp; numerical algorithm,&nbsp;based on a class of general order parameters for simulating&nbsp;the motion of a mixture of N&nbsp;(N &gt;=&nbsp;2)&nbsp;immiscible incompressible fluids&nbsp;with given densities, dynamic viscosities, and pairwise&nbsp;surface tensions.&nbsp;The N-phase formulations stem from a phase field model&nbsp;we developed in a recent work based on&nbsp;the conservations of mass/momentum, and the&nbsp;second law of thermodynamics.&nbsp;The introduction of general order&nbsp;parameters leads to an extremely strongly-coupled system&nbsp;of (N-1)&nbsp;phase field equations.&nbsp;On the other hand, the general form enables one to compute the N-phase&nbsp;mixing energy density coefficients in&nbsp;an explicit&nbsp;fashion in terms of&nbsp;the pairwise surface tensions.&nbsp;We show that the increased complexity in the form of&nbsp;the&nbsp;phase field equations&nbsp;associated with general order parameters&nbsp;in actuality&nbsp;does not cause&nbsp;essential computational difficulties.&nbsp;Our numerical algorithm reformulates the (N-1) strongly-coupled&nbsp;phase field equations for general order parameters&nbsp;into 2(N-1)&nbsp;Helmholtz-type equations that are completely de-coupled&nbsp;from one another.

This leads to a computational complexity&nbsp;comparable to&nbsp;that for the simplified phase field equations&nbsp;associated with certain special choice of the order parameters.&nbsp;We demonstrate the capabilities of the method&nbsp;developed herein using several test problems involving multiple fluid phases&nbsp;and large&nbsp;contrasts in densities and viscosities&nbsp;among&nbsp;the multitude of fluids. In particular, by comparing&nbsp;simulation results with the Langmuir-de Gennes&nbsp;theory of floating liquid&nbsp;lenses we show that the method using general&nbsp;order parameters&nbsp;produces physically accurate results for multiple fluid phases.&nbsp;</p>