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Conservative multigrid methods for Cahn–Hilliard fluids. (English) Zbl 1109.76348
Summary: 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 [J. S. Lowengrub and L. Truskinovsky, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454, No. 1978, 2617–2654 (1998; Zbl 0927.76007)]. 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 [Junseok Kim et al., On the pinchoff of liquid–liquid jets with surface tension, In preparation for submission to Phys. Fluids]. To this end, we also include a simulation of the pinchoff of a liquid thread under the Rayleigh instability at finite Reynolds number.

##### MSC:
 76M25 Other numerical methods (fluid mechanics) (MSC2010) 65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs 76A05 Non-Newtonian fluids 76D45 Capillarity (surface tension) for incompressible viscous fluids 76E17 Interfacial stability and instability in hydrodynamic stability
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