Two-phase incompressible flows with variable density: an energetic variational approach. (English) Zbl 1361.35129
Summary: In this paper, we study a diffuse-interface model for two-phase incompressible flows with different densities. First, we present a derivation of the model using an energetic variational approach. Our model allows large density ratio between the two phases and moreover, it is thermodynamically consistent and admits a dissipative energy law. Under suitable assumptions on the average density function, we establish the global existence of a weak solution in the 3D case as well as the global well-posedness of strong solutions in the 2D case to an initial-boundary problem for the resulting Allen-Cahn-Navier-Stokes system. Furthermore, we investigate the longtime behavior of the 2D strong solutions. In particular, we obtain existence of a maximal compact attractor and prove that the solution will converge to an equilibrium as time goes to infinity.
MSC:
| 35Q30 | Navier-Stokes equations |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76T99 | Multiphase and multicomponent flows |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B32 | Bifurcations in context of PDEs |
| 35A15 | Variational methods applied to PDEs |
| 35D30 | Weak solutions to PDEs |
| 35D35 | Strong solutions to PDEs |
Keywords:
two-phase flow; incompressible Navier-Stokes; variable density; global existence; longtime behaviorReferences:
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