Physical formulation and numerical algorithm for simulating N immiscible incompressible fluids involving general order parameters
Abstract
We present a family of physical formulations, and a numerical algorithm, based on a class of general order parameters for simulating the motion of a mixture of N (N ⩾ 2) immiscible incompressible fluids with given densities, dynamic viscosities, and pairwise surface tensions. The Nphase formulations stem from a phase field model we developed in a recent work based on the conservations of mass/momentum, and the second law of thermodynamics. The introduction of general order parameters leads to an extremely stronglycoupled system of (N  1) phase field equations. On the other hand, the general form enables one to compute the Nphase mixing energy density coefficients in an explicit fashion in terms of the pairwise surface tensions. We show that the increased complexity in the form of the phase field equations associated with general order parameters in actuality does not cause essential computational difficulties. Our numerical algorithm reformulates the (N  1) stronglycoupled phase field equations for general order parameters into 2 (N  1) Helmholtztype equations that are completely decoupled from one another. This leads to a computational complexity comparable to that for the simplified phase field equations associated with certain special choice of the order parameters. We demonstrate the capabilities of the method developed herein using several test problems involving multiple fluid phases and large contrasts in densities and viscosities among the multitude of fluids. In particular, by comparing simulation results with the Langmuirde Gennes theory of floating liquid lenses we show that the method using general order parameters produces physically accurate results for multiple fluid phases.
 Publication:

Journal of Computational Physics
 Pub Date:
 February 2015
 DOI:
 10.1016/j.jcp.2014.11.039
 arXiv:
 arXiv:1408.3660
 Bibcode:
 2015JCoPh.283...98D
 Keywords:

 Nphase flow;
 General order parameters;
 Pairwise surface tensions;
 Large density contrast;
 Phase field;
 Multiphase flow;
 Physics  Fluid Dynamics;
 Mathematics  Numerical Analysis;
 Physics  Computational Physics
 EPrint:
 38 pages, 9 figures, 6 tables