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Helicity in dispersive fluid mechanics

S. L. Gavrilyuk, H. Gouin

Abstract

By dispersive models of fluid mechanics we are referring to the Euler-Lagrange equations for the constrained Hamilton action functional where the internal energy depends on high order derivatives of unknowns. The mass conservation law is considered as a constraint. The corresponding Euler-Lagrange equations include, in particular, the van der Waals--Korteweg model of capillary fluids, the model of fluids containing small gas bubbles and the model describing long free-surface gravity waves. We obtain new conservation laws generalizing the helicity conservation for classical barotropic fluids.

Helicity in dispersive fluid mechanics

Abstract

By dispersive models of fluid mechanics we are referring to the Euler-Lagrange equations for the constrained Hamilton action functional where the internal energy depends on high order derivatives of unknowns. The mass conservation law is considered as a constraint. The corresponding Euler-Lagrange equations include, in particular, the van der Waals--Korteweg model of capillary fluids, the model of fluids containing small gas bubbles and the model describing long free-surface gravity waves. We obtain new conservation laws generalizing the helicity conservation for classical barotropic fluids.
Paper Structure (7 sections, 6 theorems, 87 equations, 1 figure)

This paper contains 7 sections, 6 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fluid motion
  3. Basic lemmas
  4. Euler--van der Waals--Korteweg's fluids
  5. Fluids with internal inertia
  6. Conclusion and discussion
  7. Lie derivatives of one-form and two-form fields

Key Result

Lemma 1

The following identities are satisfied:

Figures (1)

  • Figure 1: The material curves $X^1(t, x^1,x^2)=X^{10}$ and $X^2(t, x^1,x^2)=X^{20}$ corresponding to coordinate lines $X^1=X^{10} =Cst$ and $X^2=X^{20}= Cst$ are drawn. At any point, these curves are tangent to the vectors $\boldsymbol E_2$ and $\boldsymbol E_1$. A priori, vectors $\boldsymbol u$ and $\boldsymbol K$ are not collinear.

Theorems & Definitions (8)

  • Lemma 1
  • Lemma 2
  • Remark 3
  • Lemma 4
  • Remark 5
  • Theorem 6
  • Theorem 7
  • Theorem 8