A diffuse domain method for two-phase flows with large density ratio in complex geometries

We present a quasi-incompressible Navier-Stokes-Cahn-Hilliard (q-NSCH) diffuse interface model for two-phase fluid flows with variable physical properties that maintains thermodynamic consistency. Then, we couple the diffuse domain method with this two-phase fluid model - yielding a new q-NSCH-DD model - to simulate the two-phase flows with moving contact lines in complex geometries. The original complex domain is extended to a larger regular domain, usually a cuboid, and the complex domain boundary is replaced by an interfacial region with finite thickness. A phase-field function is introduced to approximate the characteristic function of the original domain of interest. The original fluid model, q-NSCH, is reformulated on the larger domain with additional source terms that approximate the boundary conditions on the solid surface. We show that the q-NSCH-DD system converges to the q-NSCH system asymptotically as the thickness of the diffuse domain interface introduced by the phase-field function shrinks to zero (epsilon -> 0) with O(epsilon). Our analytic results are confirmed numerically by measuring the errors in both L-2 and L-infinity norms. In addition, we show that the q-NSCH-DD system not only allows the contact line to move on curved boundaries, but also makes the fluid-fluid interface intersect the solid object at an angle that is consistent with the prescribed contact angle.