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.

Existence Theory for the Isentropic Euler Equations

Archive for Rational Mechanics and Analysis, 2003

We establish an existence theorem for entropy solutions to the Euler equations modeling isentropic compressible fluids. We develop a new approach for constructing mathematical entropies for the Euler equations, which are singular near the vacuum. In particular, we identify the optimal assumption required on the singular behavior on the pressure law at the vacuum in order to validate the two-term asymptotic expansion of the entropy kernel we proposed earlier. For more general pressure laws, we introduce a new multiple-term expansion based on the Bessel functions with suitable exponents, and we also identify the optimal assumption needed to validate the multiple-term expansion and to establish the existence theory. Our results cover, as a special example, the density-pressure law p(ρ) = κ 1 ρ γ 1 + κ 2 ρ γ 2 where γ 1 , γ 2 ∈ (1, 3) and κ 1 , κ 2 > 0 are arbitrary constants.

Conservation laws of the one-dimensional isentropic gas dynamics equations in Lagrangian coordinates

MODERN TREATMENT OF SYMMETRIES, DIFFERENTIAL EQUATIONS AND APPLICATIONS (Symmetry 2019)

Isentropic flows of a one-dimensional gas dynamics equations in Lagrangian coordinates are studied in the paper. Equations describing these flows are reduced to the Euler-Lagrange equation. Using group classification and Noether's theorem, conservation laws are obtained. Their analogs in Eulerian coordinates are given.

Modeling of linear isentropic flow systems

IEEE International Conference on Systems Engineering, 1991

A modeling approach for linear isentropic flow systems based on the quasi-one dimensional Euler equations of non-viscous, compressible flow is presented. Such systems are representative of certain high speed propulsion systems. Accurate models useful in control system studies are developed. A supersonic inlet is considered and the resulting set of partial differential equations with boundary conditions is solved for a linear transfer matrix using Laplace transforms.

Euler solutions as limit of infinite Reynolds number for separation flows and flows with vortices

Lecture Notes in Physics, 2000

A combination of a finite volume discretisation in conjunction with carefully designed dissipative terms of third order, and a fourth order Runge Kutta time stepping scheme, is shown to yield an efficient and accurate method for solving the time-dependent Euler equations in arbitrary geometric domains. Convergence to the steady state has been accelerated by the use of different techniques described briefly. The main attempt of the present paper however is the demonstration of inviscid compressible flow computations as solutions to the full time dependent Euler equations over two-and three-dimensional configurations with separation. It is clearly shown that in inviscid flow separation can occur on sharp corners as well as on smooth surfaces as a consequence of compressibility effects. Results for nonlifting and lifting two-and three-dimensional flows with separation from round and sharp corners are presented.

A Variational Principle for Dissipative Fluid Dynamics

2011

In the variational principle leading to the Euler equation for a perfect fluid, we can use the method of undetermined multiplier for holonomic constraints representing mass conservation and adiabatic condition. For a dissipative fluid, the latter condition is replaced by the constraint specifying how to dissipate. Noting that this constraint is nonholonomic, we can derive the balance equation of momentum for viscous and viscoelastic fluids by using a single variational principle. We can also derive the associated Hamiltonian formulation by regarding the velocity field as the input in the framework of control theory.