Exact solutions and conservation laws in dissipative fluid dynamics (original) (raw)
A Study of Solutions to Euler Equations for a One Dimensional Unsteady Flow
RESEARCH PAPER, 2013
In this paper we deal with the Euler equations for Isothermal gas. In analyzing the equations we obtain two real and distinct eigenvalues which enables us to determine the wave structure of the possible solutions to the Riemann problem set up. By considering the Rankine-Hugoniot condition we obtain the shock wave solution analytically. The rarefaction wave solution is determined analytically by considering the fact that rarefaction wave lies along integral curves. To obtain the numerical solution to the Riemann problem that we set up, we use a relaxation scheme to discretize the Euler equations for isothermal gas. Finally we present the simulation results of the numerical solutions, that is, the approximate shock and rarefaction wave solutions are shown, graphically, and explained.
Euler Equations and Related Hyperbolic Conservation Laws
Handbook of Differential Equations Evolutionary Equations, 2005
Some aspects of recent developments in the study of the Euler equations for compressible fluids and related hyperbolic conservation laws are analyzed and surveyed. Basic features and phenomena including convex entropy, symmetrization, hyperbolicity, genuine nonlinearity, singularities, BV bound, concentration and cavitation are exhibited. Global well-posedness for discontinuous solutions, including the BV theory and the L ~ theory, for the one-dimensional Euler equations and related hyperbolic systems of conservation laws is described. Some analytical approaches including techniques, methods and ideas, developed recently, for solving multidimensional steady problems are presented. Some multidimensional unsteady problems are analyzed. Connections between entropy solutions of hyperbolic conservation laws and divergence-measure fields, as well as the theory of divergence-measure fields, are discussed. Some further trends and open problems on the Euler equations and related multidimensional conservation laws are also addressed.
Dissipative continuous Euler flows
Inventiones mathematicae, 2013
We show the existence of continuous periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy.
Euler Equations Solutions for Incompressible Fluid Flow
viXra, 2015
This paper covers the solutions of the Euler equations in 3-D and 4-D for incompressible fluid flow. The solutions are the spin-offs of the author's previous analytic solutions of the Navier-Stokes equations (vixra:1405.0251 of 2014). However, some of the solutions contained implicit terms. In this paper, the implicit terms have been expressed explicitly in terms of x, y, z and t. The author applied a new law, the law of definite ratio for fluid flow. This law states that in incompressible fluid flow, the other terms of the fluid flow equation divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. This law evolved from the author's earlier solutions of the Navier-Stokes equations. In addition to the usual approach of solving these equations, the Euler equations have also been solved by a second method in which the three equations in the system are added to produce a single equation which is ...
Inverse problem on conservation laws
Physica D: Nonlinear Phenomena
The explicit formulation of the general inverse problem on conservation laws is presented for the first time. In this problem one aims to derive the general form of systems of differential equations that admit a prescribed set of conservation laws. The particular cases of the inverse problem on first integrals of ordinary differential equations and on conservation laws for evolution equations are studied. We also solve the inverse problem on conservation laws for differential equations admitting an infinite dimensional space of zeroth-order conservation-law characteristics. This particular case is further studied in the context of conservative first-order parameterization schemes for the two-dimensional incompressible Euler equations. We exhaustively classify conservative first-order parameterization schemes for the eddy-vorticity flux that lead to a class of closed, averaged Euler equations possessing generalized circulation, generalized momentum and energy conservation.
Physical Review E, 2015
We consider incompressible Euler flows in terms of the stream function in two dimensions and the vector potential in three dimensions. We pay special attention to the case with singular distributions of the vorticity, e.g., point vortices in two dimensions. An explicit equation governing the velocity potentials is derived in two steps. (i) Starting from the equation for the stream function [Ohkitani, Nonlinearity 21, T255 (2009)], which is valid for smooth flows as well, we derive an equation for the complex velocity potential. (ii) Taking a real part of this equation, we find a dynamical equation for the velocity potential, which may be regarded as a refinement of Bernoulli theorem. In three-dimensional incompressible flows, we first derive dynamical equations for the vector potentials which are valid for smooth fields and then recast them in hypercomplex form. The equation for the velocity potential is identified as its real part and is valid, for example, flows with vortex layers. As an application, the Kelvin-Helmholtz problem has been worked out on the basis the current formalism. A connection to the Navier-Stokes regularity problem is addressed as a physical application of the equations for the vector potentials for smooth fields.
Applied Mathematical Modelling
Flows of one-dimensional continuum in Lagrangian coordinates are studied in the paper. Equations describing these flows are reduced to a single Euler-Lagrange equation which contains two undefined functions. Particular choices of the undefined functions correspond to isentropic flows of an ideal gas, different forms of the hyperbolic shallow water equations. Complete group classification of the equation with respect to these functions is performed. Using Noether's theorem, all conservation laws are obtained. Their analogs in Eulerian coordinates are given.
Conservation Laws of a Nonlinear Incompressible Two-Fluid Model
2016
We study the conservation laws of the Choi-Camassa two-fluid model (1999) which is developed by approximating the two-dimensional (2D) Euler equations for incompressible motion of two non-mixing fluids in a channel. As preliminary work of this thesis, we compute the basic local conservation laws and the point symmetries of the 2D Euler equations for the incompressible fluid, and those of the vorticity system of the 2D Euler equations. To serve the main purpose of this thesis, we derive local conservation laws of the Choi-Camassa equations with an explicit expression for each locally conserved density and corresponding spatial flux. Using the direct conservation law construction method, we have constructed seven conservation laws including the conservation of mass, total horizontal momentum, energy, and irrotationality. The conserved quantities of the Choi-Camassa equations are compared with those of the full 2D Euler equations of incompressible fluid. We review periodic solutions, s...
THE RIEMANN PROBLEM FOR THE MULTI-PRESSURE EULER SYSTEM
Journal of Hyperbolic Differential Equations, 2005
We prove the existence and uniqueness of the Riemann solutions to the Euler equations closed by N independent constitutive pressure laws. This model stands as a natural asymptotic system for the multi-pressure Navier-Stokes equations in the regime of infinite Reynolds number. Due to the inherent lack of conservation form in the viscous regularization, the limit system exhibits measure-valued source terms concentrated on shock discontinuities. These non-positive bounded measures, called kinetic relations, are known to provide a suitable tool to encode the small-scale sensitivity in the singular limit. Considering N independent polytropic pressure laws, we show that these kinetic relations can be derived by solving a simple algebraic problem which governs the endpoints of the underlying viscous shock profiles, for any given but prescribed ratio of viscosity coefficient in the viscous perturbation. The analysis based on traveling wave solutions allows us to introduce the asymptotic Euler system in the setting of piecewise Lipschitz continuous functions and to study the Riemann problem.
Nonlinear Conservation Laws and Related Problems
2010
Typical examples of ``nonlinear conservation laws" are the Euler equations, MHD equations, Navier-Stokes equations, Boltzmann equation, and other important models arising in Elasticity, Fluid Dynamics, Combustion, and Kinetic theory. The Euler equations for inviscid compressible fluid flow are fundamental and important to many applications, yet the multidimensional theory is difficult and challenging. Presentation Highlights In this section we present a description of the keynote lectures of the meeting as well as the topics discussed during the panel discussions. The main themes of the keynote lectures are: (a) Singular Limits in Hydrodynamics (b) Recent Developments in Numerical Methods for Nonlinear PDEs (c) Local Isometric Embedding of Surfaces with Nonpositive Gaussian Curvature (d) Kaehler Geometry from PDE Perspective.