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

## TL;DR

This paper analyzes the homogenization of a nonlocal Navier-Stokes-Korteweg system modeling compressible liquid-vapour flow in porous media, deriving an effective nonlocal Cahn-Hilliard equation that captures complex capillarity effects.

## Contribution

It extends homogenization results to nonlocal compressible flow equations with non-monotone pressure laws, revealing the effective dynamics as a nonlocal Cahn-Hilliard type equation.

## Key findings

- Effective motion governed by a nonlocal Cahn-Hilliard equation
- Incorporates capillarity interactions with solid boundaries
- Extends homogenization to nonlocal, compressible flow systems

## Abstract

We consider a nonlocal version of the quasi-static Navier-Stokes-Korteweg equations with a non-monotone pressure law. This system governs the low-Reynolds number dynamics of a compressible viscous fluid that may take either a liquid or a vapour 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 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.

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Source: https://tomesphere.com/paper/1902.07100