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We consider a nonlocal version of the quasi-static Navier-Stokes-Kortewegequations with a non-monotone pressure law. This system governs thelow-Reynolds number dynamics of a compressible viscous fluid that may takeeither a liquid or a vapour state. For a porous domain that is perforated bycavities with diameter proportional to their mutual distance the homogenizationlimit is analyzed. We extend the results for compressible one-phase flow withpolytropic pressure laws and prove that the effective motion is governed by anonlocal version of the Cahn-Hilliard equation. Crucial for the analysis is theconvolution-like structure of the nonlocal capillarity term that allows toequip the system with a generalized convex free energy. Moreover, thecapillarity term accounts not only for the energetic interaction within thefluid but also for the interaction with a solid wall boundary.

Homogenization of Non-Local Navier-Stokes-Korteweg Equations for Compressible Liquid-Vapour Flow in Porous Media