Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yoshikazu Giga, Takahito Kashiwabara, Haruki Takemura

TL;DR

The paper analyzes periodic traveling waves in 1D isothermal Navier-Stokes-Korteweg and Euler-Korteweg diffuse-interface models for two-phase flow. It proves a no-go result for phase-transition traveling waves in NSK while establishing existence of phase-transition traveling waves in EK under a double-well energy and small Korteweg relaxation, with convergence to a monotone traveling wave as the period grows. The approach combines variational methods and Gamma-convergence (Modica–Mortola) to construct EK waves, and a rigorous energy-dissipation argument to rule out phase transitions in NSK; numerical simulations using a Strang-splitting CIP scheme corroborate the theory and illustrate phase-flux behavior at small viscosity. Together, these results clarify how viscosity and energy structure govern phase-transition patterns and connect diffuse-interface models to sharp-interface limits, supported by computational methods that capture traveling-wave profiles.

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.

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

TL;DR

The paper analyzes periodic traveling waves in 1D isothermal Navier-Stokes-Korteweg and Euler-Korteweg diffuse-interface models for two-phase flow. It proves a no-go result for phase-transition traveling waves in NSK while establishing existence of phase-transition traveling waves in EK under a double-well energy and small Korteweg relaxation, with convergence to a monotone traveling wave as the period grows. The approach combines variational methods and Gamma-convergence (Modica–Mortola) to construct EK waves, and a rigorous energy-dissipation argument to rule out phase transitions in NSK; numerical simulations using a Strang-splitting CIP scheme corroborate the theory and illustrate phase-flux behavior at small viscosity. Together, these results clarify how viscosity and energy structure govern phase-transition patterns and connect diffuse-interface models to sharp-interface limits, supported by computational methods that capture traveling-wave profiles.

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.

Paper Structure

This paper contains 6 sections, 10 theorems, 52 equations, 11 figures.

Table of Contents

  1. Introduction
  2. No phase transition for periodic traveling wave solutions for viscous case
  3. Existence of non-trivial periodic traveling wave solutions
  4. Monotone traveling wave
  5. Numerical experiments
  6. Appendix: numerical scheme

Key Result

Theorem 2.1

Assume that $\bar{\mu}(\rho)>0$ and $\varepsilon(\rho)>0$ for $\rho>0$. Let $(\rho,m)$ be a $C^2$ solution to EMS--EMMS with periodic boundary condition and $\rho\ge0$. Then $\rho$ must be a constant provided that $m\neq0$.

Figures (11)

  • Figure 1: graph of $\Psi$
  • Figure 2: phase plane for $\lambda=0$
  • Figure 3: Graph of the density $\rho(x, t)$, velocity $u(x, t)$, interface velocity $c(t)$, and mass flux $\rho(x, t)(u(x, t) - c(t))$ at time $t = 25$. The parameters are set to $\bar{\mu} = 10^{-1}$ and $\varepsilon = 10^{-4}$.
  • Figure 4: $\rho(x, t)$, $u(x, t)$, $c(t)$, and $\rho(x, t)(u(x, t) - c(t))$ at $t = 25$. The parameters are set to $\bar{\mu} = 10^{-3}$ and $\varepsilon = 10^{-4}$.
  • Figure 5: $\rho(x, t)$, $u(x, t)$, $c(t)$, and $\rho(x, t)(u(x, t) - c(t))$ at $t = 25$. The parameters are set to $\bar{\mu} = 10^{-1}$ and $\varepsilon = 10^{-5}$.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Theorem 3.1
  • Proposition 3.2
  • Lemma 3.3
  • proof : Sketch of the proof
  • proof : Proof of Theorem \ref{['TEx']}
  • Corollary 3.4
  • Remark 3.5
  • ...and 6 more