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1Author    D.-A Becker, E. W. RichterRequires cookie*
 Title    Partially Invariant Solutions for Two-dimensional Ideal MHD Equations  
 Abstract    A generalization of the usual method of similarity analysis of differential equations, the method of partially invariant solutions, was introduced by Ovsiannikov. The degree of non-invariance of these solutions is characterized by the defect of invariance d. We develop an algorithm leading to partially invariant solutions of quasilinear systems of first-order partial differential equations. We apply the algorithm to the non-linear equations of the two-dimensional non-stationary ideal MHD with a magnetic field perpendicular to the plane of motion. 
  Reference    Z. Naturforsch. 45a, 1219—1229 (1990); received July 4 1990 
  Published    1990 
  Keywords    Magnetohydrodynamics, Compressible flows, Group theory 
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 TEI-XML for    default:Reihe_A/45/ZNA-1990-45a-1219.pdf 
 Identifier    ZNA-1990-45a-1219 
 Volume    45 
2Author    H. KötzRequires cookie*
 Title    A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. I. Optimal Systems of Solvable Lie Subalgebras  
 Abstract    Lie group analysis is a powerful tool for obtaining exact similarity solutions of nonlinear (integro-) differential equations. In order to calculate the group-invariant solutions one first has to find the full Lie point symmetry group admitted by the given (integro-)differential equations and to determine all the subgroups of this Lie group. An effective, systematic means to classify the similarity solutions afterwards is an "optimal system", i.e. a list of group-invariant solutions from which every other such solution can be derived. The problem to find optimal systems of similarity solutions leads to that to "construct" the optimal systems of subalgebras for the Lie algebra of the known Lie point symmetry group. Our aim is to demonstrate a practicable technique for determining these optimal subalgebraic systems using the invariants relative to the group of the inner automorphisms of the Lie algebra in case of a finite-dimensional Lie point symmetry group. Here, we restrict our attention to optimal subsystems of solvable Lie subalgebras. This technique is applied to the nine-dimensional real Lie point symmetry group admitted by the two-dimensional non-stationary ideal magneto-hydrodynamic equations. 
  Reference    Z. Naturforsch. 47a, 1161—1173 (1992); received July 9 1992 
  Published    1992 
  Keywords    Group theory, Magnetohydrodynamics, Compressible flows 
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 TEI-XML for    default:Reihe_A/47/ZNA-1992-47a-1161.pdf 
 Identifier    ZNA-1992-47a-1161 
 Volume    47 
3Author    Mohamed Fahmy, El-SayedRequires cookie*
 Title    Hydromagnetic Instability Conditions for Viscoelastic Non-Newtonian Fluids  
 Abstract    The effect of a horizontal magnetic field and a non-Newtonian stress tensor, as described by the Wal-ters B' model, on the instability of two second order fluids of high kinematic viscosities and viscoelas-ticities is investigated. For the potentially stable configuration, it is found that the system is stable or unstable for a wavenumber range depending on the kinematic viscoelasticity. For the potentially un-stable configuration, it is found that the stability criterion is dependent on orientation and magnitude of the magnetic field which is found to stabilize a certain range of the unstable configuration related to the viscoelasticity values. The behaviour of growth rates with respect to Alfven velocities are examined analytically, and it is found that the magnetic field has a dual role on the instability problem. For the exponentially varying stratifications, the system is found to be stable or unstable for the stable and un-stable stratifications under certain physical conditions, and the growth rates are found to increase or de-crease with increasing the stratification parameter values, according to some restrictions satisfied by the chosen wavenumbers range. 
  Reference    Z. Naturforsch. 55a, 460—466 (2000); received August 31 1999 
  Published    2000 
  Keywords    Hydrodynamic Stability, Non-Newtonian Fluid Flows, Magnetohydrodynamics 
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 TEI-XML for    default:Reihe_A/55/ZNA-2000-55a-0460.pdf 
 Identifier    ZNA-2000-55a-0460 
 Volume    55 
4Author    Mohamed Fahmy, El-SayedRequires cookie*
 Title    Magnetohydrodynamic Stability of Two Streaming Superposed Viscoelastic Conducting Fluids  
 Abstract    The stability o f the plane interface separating two Oldroydian viscoelastic superposed moving fluids of uniform densities when immersed in a uniform horizontal magnetic field has been in­ vestigated. The stability analysis has been carried out, for mathematical simplicity, for two highly viscous fluids o f equal kinematic viscosities. It is found that the potentially stable configuration remains stable if the fluids are at rest, while it becomes unstable if the fluids move. The stability criterion is found to be independent o f the viscosity and viscoelasticity, and to be dependent on the orientation of the magnetic field and the magnitudes of the fluids and Alfven velocities. It is also found that the potentially unstable configuration remains unstable in the absence of average fluid velocities, or in the presence of fluid velocities and absence of a magnetic field. The magnetic field is found to stabilize a certain wavenumbers range o f the unstable configuration even in the presence o f the effects of viscoelasticity. The behaviour of growth rates with respect to the stress relaxation time, strain retardation time, fluid and Alfven velocity parameters is examined analytically, and the stability conditions are obtained and discussed. -Pacs: 47.20.-k; 47.50.+d; 47.65.+a. 
  Reference    Z. Naturforsch. 56a, 416 (2001); received December 13 2000 
  Published    2001 
  Keywords    Hydrodynamic Stability, Non-Newtonian Fluid Flows, Magnetohydrodynamics 
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 TEI-XML for    default:Reihe_A/56/ZNA-2001-56a-0416.pdf 
 Identifier    ZNA-2001-56a-0416 
 Volume    56 
5Author    Mohamed Fahmy, El-SayedRequires cookie*
 Title    The Effects of Collisions with Neutral Particles on the Instability of Two Superposed Composite Plasmas Streaming Through Porous Medium  
 Abstract    The effects of collisions with neutral atoms on the hydromagnetic stability of the plane interface separating two streaming superposed composite plasmas of uniform densities in a porous medium are investigated. In the absence of fluid velocities, it is found, for a potentially stable configuration, that the system remains stable, while for a potentially unstable configuraion, the unstable system becomes stable under a certain condition of the wavenumber depending on the values of the fluid densities, Alfven velocities, and the orientation of the magnetic field. The porosity of the porous medium does not have any significant effect on the stability criterion. In the presence of fluid velocities, it is found that, the instability criterion is independent of the permeability of the medium and the collision effects with neutral particles. The criterion determing the stability does not depend on the permeability of the medium but depends on the density of neutral particles. The porosity of the medium is found to have a significant effect on both the stability and instability criteria in this case. The role of the permeability of the medium, the collisional frequency, and the porosity of the porous medium on the growth rate of the unstable mode is examined analitically. Routh's test of stability is applied to confirm the above results. 
  Reference    Z. Naturforsch. 54a, 411—416 (1999); received April 19 1999 
  Published    1999 
  Keywords    Hydrodynamic Stability, Flows through Porous Media, Magnetohydrodynamics, Plasma flow, Instabilities in Plasma 
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 TEI-XML for    default:Reihe_A/54/ZNA-1999-54a-0411.pdf 
 Identifier    ZNA-1999-54a-0411 
 Volume    54