On periodic traveling wave solutions with or without phase transition to the Navier-Stokes-Korteweg and the Euler-Korteweg equations

TL;DR

This paper investigates the existence and properties of periodic traveling wave solutions with or without phase transition in the Navier-Stokes-Korteweg and Euler-Korteweg equations, revealing conditions for their existence and stability.

Contribution

It provides new theoretical results on the existence of phase transition solutions and their asymptotic behavior, along with numerical insights into their stability.

Findings

No phase transition solutions for Navier-Stokes-Korteweg equations.

Existence of phase transition solutions for Euler-Korteweg equations with small relaxation parameter.

Stability of certain periodic solutions under small viscosity and perturbations.

Abstract

The Navier-Stokes-Korteweg and the Euler-Korteweg equations are considered in isothermal setting. These are diffuse interface models of two-phase flow. For the Navier-Stokes-Korteweg equations, we show that there is no periodic traveling wave solution with phase transition although there exists a non-constant periodic traveling wave solution with no phase transition. For the Euler-Korteweg equations, we show that there always exists a periodic traveling wave solution with phase transition for any period if the Korteweg relaxation parameter is small compared with the period provided that the available energy is double-well type. We also show that such a periodic traveling wave solution tends to a monotone traveling wave solution as the period tends to infinity under suitable spatial translation. Our numerical experiment suggests that there is periodic traveling wave with phase transition which is stable under periodic perturbation for small viscosity but it seems that this is a transition pattern.