Homogenization of nonlocal Navier-Stokes-Korteweg equations for compressible liquid-vapor flow in porous media. (English) Zbl 1454.76098
Summary: We consider a nonlocal version of the quasi-static Navier-Stokes-Korteweg equations with a nonmonotone pressure law. This system governs the low-Reynolds number dynamics of a compressible viscous fluid that may take either a liquid or a vapor state. For a porous domain that is perforated by cavities with diameter proportional to their mutual distance the homogenization limit is analyzed. We extend the results for compressible one-phase flow with polytropic pressure laws and prove that the effective motion is governed by a nonlocal version of the Cahn-Hilliard equation. Crucial for the analysis is the convolution-like structure of the nonlocal capillarity term that allows us to equip the system with a generalized convex free energy. Moreover, the capillarity term accounts not only for the energetic interaction within the fluid but also for the interaction with a solid wall boundary.
MSC:
| 76T10 | Liquid-gas two-phase flows, bubbly flows |
| 76S05 | Flows in porous media; filtration; seepage |
| 76M50 | Homogenization applied to problems in fluid mechanics |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
Keywords:
weak solution; polytropic pressure law; nonlocal Cahn-Hilliard equation; convex free energyReferences:
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