Analytic Solutions of a Two-Fluid Hydrodynamic Model (original) (raw)

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...

Self-Similar Analytic Solution of the Two-Dimensional Navier-Stokes Equation with a Non-Newtonian Type of Viscosity

Mathematical Modelling and Analysis, 2016

We investigate Navier-Stokes (NS) and the continuity equations in Cartesian coordinates and Eulerian description for the two dimensional incompressible nonNewtonian fluids. Due to the non-Newtonian viscosity we consider the Ladyzenskaya model with a non-linear velocity dependent stress tensor. The key idea is the multidimensional generalization of the well-known self-similar Ansatz, which has already been used for non-compressible and compressible viscous flow studies. Geometrical interpretations of the trial function are also discussed. Our recent results are compared to the former Newtonian ones.

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 ...

A fully conservative model for compressible two‐fluid flow

… journal for numerical methods in fluids, 2005

A ÿve-equation model for compressible two-uid ow is proposed, that is based on physical ow equations only. The model is conservative and pressure-oscillation free. Equations for continuous ow and jump conditions for discontinuities are given, as well as a discretization of the equations and an adaptation of the HLL Riemann solver to two-uid ow. Numerical tests in 1D and 2D show the accuracy of the method.

Analytic solutions for the three-dimensional compressible Navier–Stokes equation

Fluid Dynamics Research, 2014

We investigate the three dimensional compressible Navier-Stokes and the continuity equations in Cartesian coordinates for Newtonian fluids. The problem has an importance in different fields of science and engineering like fluid, aerospace dynamics or transfer processes. Finding an analytic solution may bring a considerable progress in understanding the transport phenomena and in the design of different equipments where the Navier Sotkes equation is applicable. For solving the equation the polytropic equation of state is used as closing condition. The key idea is the three-dimensional generalization of the well-known self-similar Ansatz which was already used for non-compressible viscous flow in our former study. The geometrical interpretations of the trial function is also discussed. We compared our recent results to the former non-compressible ones.

Analytic solutions of hydrodynamics equations

Physics of Fluids A: Fluid Dynamics, 1991

Many similarity solutions have been found for the equations of one-dimensional (1-D) hydrodynamics. These special combinations of variables allow the partial differential equations to be reduced to ordinary differential equations, which must then be solved to determine the physical solutions. Usually, these reduced ordinary differential equations are solved numerically. In some cases it is possible to solve these reduced equations analytically to obtain explicit solutions. In this work a collection of analytic solutions of the 1-D hydrodynamics equations is presented. These can be used for a variety of purposes, including (i) numerical benchmark problems, (ii) as a basis for analytic models, and (iii) to provide insight into more complicated solutions.

A physical five-equation model for compressible two-fluid flow, and its numerical treatment

2009

A novel five-equation model for inviscid, non-heat-conducting, compressible two-fluid flow is derived, together with an appropriate numerical method. The model uses flow equations based on conservation laws and exchange laws only. The two fluids exchange momentum and energy, for which source terms are derived from fundamental physical laws. The Riemann invariants of the governing equations are derived, and used in the construction of an Osher-type approximate Riemann solver. A consistent finite-volume discretization of the source terms is proposed. The source terms have distinct contributions in the cell domain and at the cell faces. For the source-term evaluation at the cell faces, the Riemann solver is elegantly exploited. Numerical results are presented for shock-tube and shock-bubble-interaction problems. The resemblance with experimental results is very good. Free-surface pressure oscillations do not occur, without any precaution. The paper contributes to state of the art in computing two-fluid flows.

Vanishing viscosity limit of the Navier-Stokes equations to the euler equations for compressible fluid flow

Communications on Pure and Applied Mathematics, 2010

We establish the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup-norm of solutions with respect to the physical viscosity coefficient may not be directly controllable and, furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy-type estimates with respect to the viscosity coefficient for the solutions of the Navier-Stokes equations and establish the existence of measure-valued solutions of the isentropic Euler equations generated by the Navier-Stokes equations. Based on the uniform energy-type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier-Stokes equations for weak entropy-entropy flux pairs, generated by compactly supported C 2 test functions, are confined in a compact set in H −1 , which lead to the existence of measure-valued solutions that are confined by the Tartar-Murat commutator relation. A careful characterization of the unbounded support of the measure-valued solution confined by the commutator relation yields the reduction of the measure-valued solution to a Delta mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations.

Vortical and self-similar flows of 2D compressible Euler equations

Communications in Nonlinear Science and Numerical Simulation, 2014

This paper presents the vortical and self-similar solutions for 2D compressible Euler equations using the separation method. These solutions complement Makino's solutions in radial symmetry without rotation. The rotational solutions provide new information that furthers our understanding of ocean vortices and reference examples for numerical methods. In addition, the corresponding blowup, time-periodic or global existence conditions are classified through an analysis of the new Emden equation. A conjecture regarding rotational solutions in 3D is also made. MSC: 76U05, 35C05, 35C06, 35B10, 35R35