We develop a conservative, second-order accurate fully implicit discretization of the Navier-Stokes (NS) and Cahn-Hilliard (CH) system that has an associated discrete energy functional. This system provides a diffuse-interface description of binary fluid flows with compressible or incompressible flow components [R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998) 2617]. In this work, we focus on the case of flows containing two immiscible, incompressible and density-matched components. The scheme, however, has a straightforward extension to multi-component systems. To efficiently solve the discrete system at the implicit time-level, we develop a nonlinear multigrid method to solve the CH equation which is then coupled to a projection method that is used to solve the NS equation. We demonstrate convergence of our scheme numerically in both the presence and absence of flow and perform simulations of phase separation via spinodal decomposition. We examine the separate effects of surface tension and external flow on the decomposition. We find surface tension driven flow alone increases coalescence rates through the retraction of interfaces. When there is an applied external shear, the evolution of the flow is nontrivial and the flow morphology repeats itself in time as multiple pinchoff and reconnection events occur. Eventually, the periodic motion ceases and the system relaxes to a global equilibrium. The equilibria we observe appears has a similar structure in all cases although the dynamics of the evolution is quite different. We view the work presented in this paper as preparatory for a detailed investigation of liquid-liquid interfaces with surface tension where the interfaces separate two immiscible fluids [On the pinchoff of liquid-liquid jets with surface tension, in preparation]. To this end, we also include a simulation of the pinchoff of a liquid thread under the Rayleigh mistability at finite Reynolds number.
Bibliographical noteFunding Information:
The first (J.K.) and third (J.L.) authors acknowledge the support of the Department of Energy, Office of Basic Energy Sciences. J.L. also acknowledges the support of the National Science Foundation, Division of Mathematical Sciences. The second author (K.K.) was partially supported by NSF Grant No. DMS-9877055 and he thanks Professor B. Cockburn for helpful discussions. The authors are also grateful for the support of the Minnesota Supercomputer Institute and the Institute for Mathematics and its Applications.
- Cahn-Hilliard equation
- Fluid flow
- Interfacial tension
- Nonlinear multigrid method
ASJC Scopus subject areas
- Numerical Analysis
- Modelling and Simulation
- Physics and Astronomy (miscellaneous)
- Physics and Astronomy(all)
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics