Steady state solutions of ferrofluid flow models

Youcef Amirat; Kamel  Hamdache

doi:10.3934/cpaa.2016039

Article Contents

2016, Volume 15, Issue 6: 2329-2355. Doi: 10.3934/cpaa.2016039

This issue Previous Article Finite dimensional smooth attractor for the Berger plate with dissipation acting on a portion of the boundary Next Article Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces

Steady state solutions of ferrofluid flow models

Youcef Amirat^1, and
Kamel Hamdache^2,

1.
Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière cedex
2.
Léonard de Vinci, Pôle Universitaire. Research Center, 92916 Paris la Défense Cedex

Received: February 2016

Revised: May 2016

Published: November 2016

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Abstract

We study two models of differential equations for the stationary flow of an incompressible viscous magnetic fluid subjected to an external magnetic field. The first model, called Rosensweig's model, consists of the incompressible Navier-Stokes equations, the angular momentum equation, the magnetization equation of Bloch-Torrey type, and the magnetostatic equations. The second one, called Shliomis model, is obtained by assuming that the angular momentum is given in terms of the magnetic field, the magnetization field and the vorticity. It consists of the incompressible Navier-Stokes equation, the magnetization equation and the magnetostatic equations. We prove, for each of the differential systems posed in a bounded domain of $\mathbb{R}^3$ and equipped with boundary conditions, existence of weak solutions by using regularization techniques, linearization and the Schauder fixed point theorem.

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 76D05.

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