scholarly journals Self-intersecting Interfaces for Stationary Solutions of the Two-Fluid Euler Equations

Annals of PDE ◽  
2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Diego Córdoba ◽  
Alberto Enciso ◽  
Nastasia Grubic
Keyword(s):  
Euler Equations ◽  
Stationary Solutions ◽  
Two Fluid

Fully nonlinear internal waves in a two-fluid system

1999 ◽  
Vol 396 ◽  
pp. 1-36 ◽  
Author(s):  
WOOYOUNG CHOI ◽  
ROBERTO CAMASSA
Keyword(s):  
Solitary Wave ◽  
Euler Equations ◽  
Deep Water ◽  
Large Amplitude ◽  
Nonlinear Models ◽  
Weak Nonlinearity ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
The Waves ◽  
Two Fluid

Model equations that govern the evolution of internal gravity waves at the interface of two immiscible inviscid fluids are derived. These models follow from the original Euler equations under the sole assumption that the waves are long compared to the undisturbed thickness of one of the fluid layers. No smallness assumption on the wave amplitude is made. Both shallow and deep water configurations are considered, depending on whether the waves are assumed to be long with respect to the total undisturbed thickness of the fluids or long with respect to just one of the two layers, respectively. The removal of the traditional weak nonlinearity assumption is aimed at improving the agreement with the dynamics of Euler equations for large-amplitude waves. This is obtained without compromising much of the simplicity of the previously known weakly nonlinear models. Compared to these, the fully nonlinear models' most prominent feature is the presence of additional nonlinear dispersive terms, which coexist with the typical linear dispersive terms of the weakly nonlinear models. The fully nonlinear models contain the Korteweg–de Vries (KdV) equation and the Intermediate Long Wave (ILW) equation, for shallow and deep water configurations respectively, as special cases in the limit of weak nonlinearity and unidirectional wave propagation. In particular, for a solitary wave of given amplitude, the new models show that the characteristic wavelength is larger and the wave speed is smaller than their counterparts for solitary wave solutions of the weakly nonlinear equations. These features are compared and found in overall good agreement with available experimental data for solitary waves of large amplitude in two-fluid systems.

scholarly journals Blow-up criteria for a two-fluid model of the truncated Euler equations

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
Yonglin Xu ◽  
Haizhou Mao ◽  
Xiaohong Fan ◽  
Liqiang Ma
Keyword(s):  
Euler Equations ◽  
Blow Up ◽  
Fluid Model ◽  
Two Fluid Model ◽  
Two Fluid

Stationary solutions to Euler equations with spherical symmetry

2005 ◽  
Vol 61 (1-2) ◽  
pp. 261-267
Author(s):  
Yanhong Liu ◽  
Changjiang Zhu
Keyword(s):  
Euler Equations ◽  
Spherical Symmetry ◽  
Stationary Solutions

Selected topics on the topology of ideal fluid flows

2016 ◽  
Vol 13 (Supp. 1) ◽  
pp. 1630012 ◽  
Author(s):  
Daniel Peralta-Salas
Keyword(s):  
Euler Equations ◽  
Topological Structure ◽  
Fluid Flows ◽  
Stationary Solutions ◽  
Knot Invariants ◽  
Vortex Lines ◽  
The Subject ◽  
Complicated Topology ◽  
Incompressible Fluid Flows ◽  
Main Ideas

This is a survey of certain geometric aspects of inviscid and incompressible fluid flows, which are described by the solutions to the Euler equations. We will review Arnold’s theorem on the topological structure of stationary fluids in compact manifolds, and Moffatt’s theorem on the topological interpretation of helicity in terms of knot invariants. The recent realization theorem by Enciso and Peralta-Salas of vortex lines of arbitrarily complicated topology for stationary solutions to the Euler equations will also be introduced. The aim of this paper is not to provide detailed proofs of all the stated results but to introduce the main ideas and methods behind certain selected topics of the subject known as Topological Fluid Mechanics. This is the set of lecture notes, the author gave at the XXIV International Fall Workshop on Geometry and Physics held in Zaragoza (Spain) during September 2015.

scholarly journals On the Semi-Analytical Solutions in Hydrodynamics of Ideal Fluid Flows Governed by Large-Scale Coherent Structures of Spiral-Type

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2307
Author(s):  
Sergey V. Ershkov ◽  
Alla Rachinskaya ◽  
Evgenii Yu. Prosviryakov ◽  
Roman V. Shamin
Keyword(s):  
Euler Equations ◽  
Coherent Structures ◽  
Large Scale ◽  
Stationary Solutions ◽  
Incompressible Fluids ◽  
Bottom Surface ◽  
Stationary Flows ◽  
Long Time ◽  
Partial Type ◽  
Spiral Type

We have presented here a clearly formulated algorithm or semi-analytical solving procedure for obtaining or tracing approximate hydrodynamical fields of flows (and thus, videlicet, their trajectories) for ideal incompressible fluids governed by external large-scale coherent structures of spiral-type, which can be recognized as special invariant at symmetry reduction. Examples of such structures are widely presented in nature in “wind-water-coastline” interactions during a long-time period. Our suggested mathematical approach has obvious practical meaning as tracing process of formation of the paths or trajectories for material flows of fallout descending near ocean coastlines which are forming its geometry or bottom surface of the ocean. In our presentation, we explore (as first approximation) the case of non-stationary flows of Euler equations for incompressible fluids, which should conserve the Bernoulli-function as being invariant for the aforementioned system. The current research assumes approximated solution (with numerical findings), which stems from presenting the Euler equations in a special form with a partial type of approximated components of vortex field in a fluid. Conditions and restrictions for the existence of the 2D and 3D non-stationary solutions of the aforementioned type have been formulated as well.

scholarly journals A note on the stationary Euler equations of hydrodynamics

2015 ◽  
Vol 37 (2) ◽  
pp. 454-480 ◽  
Author(s):  
K. CIELIEBAK ◽  
E. VOLKOV
Keyword(s):  
Vector Field ◽  
Euler Equations ◽  
Hamiltonian Structure ◽  
Stationary Solutions ◽  
Torus Bundle ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Real Analyticity ◽  
Bernoulli Function ◽  
Real Analytic

This note concerns stationary solutions of the Euler equations for an ideal fluid on a closed 3-manifold. We prove that if the velocity field of such a solution has no zeroes and real analytic Bernoulli function, then it can be rescaled to the Reeb vector field of a stable Hamiltonian structure. In particular, such a vector field has a periodic orbit unless the 3-manifold is a torus bundle over the circle. We provide a counterexample showing that the correspondence breaks down without the real analyticity hypothesis.

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