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A free boundary stationary magnetohydrodynamic problem in connection with the electromagnetic casting process

Tomasz Roliński (1995)

Annales Polonici Mathematici

We investigate the behaviour of the meniscus of a drop of liquid aluminium in the neighbourhood of a state of equilibrium under the influence of weak electromagnetic forces. The mathematical model comprises both Maxwell and Navier-Stokes equations in 2D. The meniscus is governed by the Young-Laplace equation, the data being the jump of the normal stress. To show the existence and uniqueness of the solution we use the classical implicit function theorem. Moreover, the differentiability of the operator...

A plane problem of incompressible magnetohydro-dynamics with viscosity and resistivity depending on the temperature

Giovanni Cimatti (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The plane flow of a fluid obeying the equations of magnetohydrodynamics is studied under the assumption that both the viscosity and the resistivity depend on the temperature. Some results of existence, non-existence, and uniqueness of solution are proved.

A regularity criterion for the 2D MHD and viscoelastic fluid equations

Zhuan Ye (2015)

Annales Polonici Mathematici

This paper is dedicated to a regularity criterion for the 2D MHD equations and viscoelastic equations. We prove that if the magnetic field B, respectively the local deformation gradient F, satisfies B , F L q ( 0 , T ; L p ( ² ) ) for 1/p + 1/q = 1 and 2 < p ≤ ∞, then the corresponding local solution can be extended beyond time T.

A singular radially symmetric problem in electrolytes theory

Tadeusz Nadzieja, Andrzej Raczyński (1998)

Applicationes Mathematicae

Existence of radially symmetric solutions (both stationary and time dependent) for a parabolic-elliptic system describing the evolution of the spatial density of ions in an electrolyte is studied.

A WKB analysis of the Alfvén spectrum of the linearized magnetohydrodynamics equations

Manuel Núñez, Jesús Rojo (1993)

Applications of Mathematics

Small perturbations of an equilibrium plasma satisfy the linearized magnetohydrodynamics equations. These form a mixed elliptic-hyperbolic system that in a straight-field geometry and for a fixed time frequency may be reduced to a single scalar equation div A 1 Δ u + A 2 u = 0 , where A 1 may have singularities in the domaind U of definition. We study the case when U is a half-plane and u possesses high Fourier components, analyzing the changes brought about by the singularity A 1 = . We show that absorptions of energy takes...

A zoology of boundary layers.

David Gérard-Varet, Emmanuel Grenier (2002)


In meteorology and magnetohydrodynamics many different boundary layers appear. Some of them are already mathematically well known, like Ekman or Hartmann layers. Others remain unstudied, and can be much more complex. The aim of this paper is to give a simple and unified presentation of the main boundary layers, and to propose a simple method to derive their sizes and equations.

Almost global solutions of the free boundary problem for the equations of a magnetohydrodynamic incompressible fluid

Piotr Kacprzyk (2004)

Applicationes Mathematicae

Almost global in time existence of solutions for equations describing the motion of a magnetohydrodynamic incompressible fluid in a domain bounded by a free surfaced is proved. In the exterior domain we have an electromagnetic field which is generated by some currents which are located on a fixed boundary. We prove that a solution exists for t ∈ (0,T), where T > 0 is large if the data are small.

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