Axisymmetric magnetohydrodynamic equations: Exact solutions for stationary incompressible flows

1992 ◽  
Vol 4 (1) ◽  
pp. 35-43 ◽  
Author(s):  
Francesca Bacciotti ◽  
Claudio Chiuderi
Keyword(s):  
Exact Solutions ◽  
Incompressible Flows ◽  
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.

A Geometrically Consistent Linearization Method

1985 ◽  
Vol 9 (2) ◽  
pp. 84-89
Author(s):  
S.M. Calisal
Keyword(s):  
Exact Solutions ◽  
Incompressible Flows ◽  
Three Dimensional ◽  
Leading Edge ◽  
Iterative Solution ◽  
Linearization Method ◽  
The Body ◽  
First Order ◽  
Flow Problems ◽  
Consistent Linearization

The study of irrotational incompressible flows about thin geometries can be carried out using the well known perturbation procedures. In two-dimensional flows exact solutions based on mappings can be used to compare the accuracy of first order solutions. For most airfoil sections a first order perturbation solution is not sufficiently accurate in representing the pressure and velocity distribution, especially about the leading edge. For three-dimensional flows exact solutions are rare and for more complex problems such as ship wave resistance formulations an exact solution does not exist for comparison of results. In this last case second-order solutions exist but are very difficult to calculate. Therefore, it would appear advantageous to improve first-order calculations. To this end a perturbation method that incorporates the geometric properties of the body is studied. This method is applied to a symmetric Joukowski airfoil and to an elipse. This method, here called the “geometrically-consistent linearization method” predicts the leading edge pressure variations correctly in the two cases studied and appears to be superior to the classical first order solutions. An iterative solution following this procedure further improves the calculation especially for thicker foils. The method discussed and the following iteration procedure seem to form an efficient numerical solution to airfoil flow problems.

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