Some Exact Solutions to the Magnetohydrodynamic Equations for Incompressible Flow

1959 ◽  
Vol 26 (4) ◽  
pp. 246-247
Author(s):  
F. D. Hains
Keyword(s):  
Exact Solutions ◽  
Incompressible Flow ◽  
Magnetohydrodynamic Equations

scholarly journals Nonlinear Galerkin Methods for 3D Magnetohydrodynamic Equations

1997 ◽  
Vol 07 (07) ◽  
pp. 1497-1507 ◽  
Author(s):  
Olaf Schmidtmann ◽  
Fred Feudel ◽  
Norbert Seehafer
Keyword(s):  
Partial Differential Equations ◽  
Exact Solutions ◽  
Galerkin Method ◽  
Numerical Solution ◽  
Galerkin Approximation ◽  
Inertial Manifold ◽  
Galerkin Methods ◽  
Magnetohydrodynamic Equations ◽  
Nonlinear Galerkin Method ◽  
Nonlinear Galerkin Methods

The usage of nonlinear Galerkin methods for the numerical solution of partial differential equations is demonstrated by treating an example. We describe the implementation of a nonlinear Galerkin method based on an approximate inertial manifold for the 3D magnetohydrodynamic equations and compare its efficiency with the linear Galerkin approximation. Special bifurcation points, time-averaged values of energy and enstrophy as well as Kaplan–Yorke dimensions are calculated for both schemes in order to estimate the number of modes necessary to correctly describe the behavior of the exact solutions.

On some exact solutions in magnetohydrodynamics with astrophysical applications

1972 ◽  
Vol 51 (1) ◽  
pp. 33-38 ◽  
Author(s):  
C. Sozou
Keyword(s):  
Free Surface ◽  
Angular Velocity ◽  
Incompressible Fluid ◽  
Exact Solutions ◽  
Equilibrium Configuration ◽  
Constant Angular Velocity ◽  
Magnetohydrodynamic Equations ◽  
The Stability

Some exact solutions of the steady magnetohydrodynamic equations for a perfectly conducting inviscid self-gravitating incompressible fluid are discussed. It is shown that there exist solutions for which the free surface of the liquid is that of a planetary ellipsoid and rotates with constant angular velocity about its axis. The stability of the equilibrium configuration is not investigated.

A New Derivation of Exact Solutions for Incompressible Magnetohydrodynamic Plasma Turbulence

2021 ◽  
Vol 10 (1) ◽  
pp. 98-105
Author(s):  
Adel M. Morad ◽  
S. M. A. Maize ◽  
A. A. Nowaya ◽  
Y. S. Rammah
Keyword(s):  
Magnetic Field ◽  
Schrödinger Equation ◽  
Exact Solutions ◽  
Schrodinger Equation ◽  
Nonlinear Partial Differential Equation ◽  
Reductive Perturbation Method ◽  
Constant Density ◽  
Magnetohydrodynamic Equations ◽  
Nls Equation ◽  
Ansatz Method

The objective of this paper is to study the propagation of nonlinear, quasi-parallel, magnetohydrodynamic waves of small-amplitude in a cold Hall plasma of constant density. Magnetohydrodynamic equations, along with the cold plasma were expanded using the reductive perturbation method, which leads to a nonlinear partial differential equation that complies with a modified form of the derivative nonlinear evolution Schrödinger equation. Exact solutions of the turbulent magnetohydrodynamic model in plasma were formulated for the physical quantities that describe the problem completely by using the complex ansatz method. In addition, the complete set of equations was used taking into account the magnetic field, which can be considered to be constant in the x-direction to create stable vortex waves. Vortex solutions of the modified nonlinear Schrödinger equation (MNLS) were compared with the solutions of incompressible magnetohydrodynamic equations. The obtained equations differ from the standard NLS equation by one additional term that describes the interaction of the nonlinear waves with the constant density. The behavior of both the velocity profile and the waveform of pressure were studied. The results showed that there are clear disturbances in the identity of the velocity and thus pressure. The identity of both velocity and pressure results from that a magnetic field is formed.

scholarly journals II. Flow with circulation

1949 ◽  
Vol 196 (1044) ◽  
pp. 1-31 ◽  
Keyword(s):  
Circular Cylinder ◽  
Exact Solutions ◽  
Compressible Flow ◽  
Incompressible Flow ◽  
Explicit Solution ◽  
Linear Equations ◽  
Stagnation Points ◽  
Direct Generalization ◽  
Flow Past A Cylinder ◽  
Infinite Set

This is a sequel to an earlier paper (Cherry 1947) in which was found a family of exact solutions for compressible flow past a cylinder. In the present paper the solution is extended to the case where the circulation round the cylinder is not zero. The formulae are developed for the case where the circulation is sufficiently small for the existence of a pair of stagnation points on the surface of the cylinder, under the condition that the speed at infinity is subsonic. One substantial point which arises in the present investigation is that the most direct generalization of the formulae for incompressible flow yield multiple-valued formulae for compressible flow. To get a single-valued solution it is necessary to add another multiple-valued solution, involving a set of constants which are to be determined from an infinite set of linear equations. The explicit solution of these equations is found, and hence the flow around a profile which is a slightly distorted circular cylinder.

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