الفهرس 
IntroductionOnly 14 pages are availabe for public view
 |  | from | 96 |  |  |
| | | from | 96 | | |
Abstract
The aim of this thesis is to study some variational formulations and to obtain analytical solutions for the equilibrium equations for some magnetohydrodynamics flows. The study of equations governing fluid motion is of a great importance in plasma physics. These equations are known in physics and the fluid mechanics, where these equations were reduced to reach an easier picture that is dealt with physically and mathematically. After obtaining this image, a set of solutions was reached and the physical interpretation of it was developed, thus facilitating the understanding of many astronomical and solar phenomena. This thesis consists of an introduction, four chapters, two summaries, one in English and one in Arabic, thirty-four diagrams, five tables, a bibliography and a summary as follows: Chapter 1
. Which is considered as a background for the material used in this thesis. It covers the fundamental concepts of known results concerning our objects to make this thesis somewhat self contained.
Chapter 2
. We have proposed a JEFs method to obtain classes of exact solutions to MHD equations describing incompressible flows. We have presented a modification on the balancing methods which imply new solutions can be obtained for translational symmetric equilibria with incompressible flow. The presented exact solutions may describe various new features of waves and then may be useful in the theoretical and numerical studies of the considered equations. One of the most important advantages of the solution method presented here is that it deals with the problems which contain nonlinear terms raised to fractional powers without making transformations of the original problem to another one. JEFs cover most of functions like trigonometric functions, hyperbolic functions and exponential functions. We have examined the applicability of some of the obtained solutions for describing magnetic structures of a bipolar sunspots groups.
Chapter 3.
We have investigated the symmetric equilibrium of ideal MHD incompressible flows in a Cartesian geometry. The MHD equilibrium is governed by an elliptic second-order NPDE for the poloidal magnetic flux function. Lie point symmetries and conservation laws for an incompressible MHD flow governed by a two-dimensional GGSE are formulated. Several exact solutions to the latter equation are obtained. The selfadjointness of the GGSE is discussed where we noted that it can be self-adjoint or quasi-self-adjoint or nonlinear self-adjoint according to the form of the function f(u) appears in Eq. (3.8). First and second order Lagrangians for the GGSE are proved. A construction for obtaining the solutions of the whole MHD system of incompressible flows is explained via the obtained solutions of the GGSE. The obtained solutions cover previously configurations and include new considerations on the nonlinearity of magnetic flux stream variables. In Petrie et al (2005) the velocity was taken to be parallel to the magnetic flux function and the Alfv´enic 66 mach number was taken to be a function of a dimensionless horizontal distance M2 = M2(x), beside the velocity and the magnetic fields have an exponential dependence on z. A solution was presented in Cheung (2014) in terms of tanh ξ, tan ξ and cot ξ where ξ = x + αy + βt can be obtained as a special case comparing with our results. In Nabert et al (2013) the gas pressure was taken to be an isotropic while in the present paper we did not provide this assumption on the pressure. In Wu et al (2016) the travelling wave method was used to get solutions of incompressible ideal Hall MHD, also the velocity and magnetic field were taken parallel to the wave vector. Numerical solutions were presented in Aslan (1996), and Nath and Kalra (2014). The present thesis is devoted to the investigation of generalized forms of the GGSE that describes symmetric plasma equilibria in the presence of poloidal and axial incompressible flows via Lie-point symmetries and conservation laws.