Modeling the effect of magnetic field on wave propagation in ferrofluids and elastic bodies with void fraction
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MODELING THE EFFECT OF MAGNETIC FIELD ON WAVE PROPAGATION IN FERROFLUIDS AND ELASTIC BODIES WITH VOID FRACTION I. T. Selezova and Yu. G. Krivonosb
UDC 537.84
Abstract. The paper presents two new generalized wave models. One considers the effect of magnetic field on the elastic solid with void fraction. The other is a new generalized ferrohydrodynamic model describing wave propagation with finite velocities. The existence of wave solutions is investigated. Keywords: magnetic field, elastic body, void fraction, ferrofluid, wave propagation, finite velocity. INTRODUCTION In the paper, we will consider two generalized wave models: a model of the effect of magnetic field on elastic body with void fraction and a new generalized ferrohydrodynamic model. The model of a deformable elastic body with voids [1, 2] is generalized to account for magnetoelastic effects. The medium is assumed to be electrically conducting and subject to an external magnetic field. The void fraction is small as compared with the elastic matrix. The medium inside the voids is non-conducting [3]. A strong effect of the magnetic Reynolds number and of the porosity of void fraction b on the propagation of plane waves in the body is shown. Some dynamic problems without magnetoelastic interactions were considered by now [4, 5]. The new generalized ferrohydrodynamic model accounts for the compressibility and thermal relaxation effects. The system of nonlinear equations is linearized with respect to the unperturbed field of pressure, density, temperature, velocity, magnetic field intensity, and magnetization. As a result, the original equations are reduced to the system of three resolving scalar partial differential hyperbolic–elliptic equations. This predicts wave propagation with finite velocity, which differentiates the system from the standard model. (The finiteness of the velocity of propagation of perturbations in liquid media was considered in [6–8].) We will investigate the resolvability of the corresponding problems on the propagation of plane waves and show that the problem of the propagation of stationary waves has no solution. The resolvability conditions will be established for the case of propagation of monochromatic waves. MODELS OF THE EFFECT OF MAGNETIC FIELD ON ELASTIC BODY WITH VOID FRACTION Equations of Motion of the Elastic Medium with Void Fraction in a Magnetic Field. The equations are written in the rationalized MKS system [9]: the equation of motion of elastic matrix r r r r r r ¶2u r r (1) GÑ 2 u + ( l + G )Ñ (Ñ × u ) = - r1 = J ´ B - bÑ h , ¶t 2 the equation of motion of void fraction r r ¶2h ¶h (2) aÑ 2 h - r1 k - w - xh = b Ñ × u , ¶t ¶t 2 a
Institute of Hydromechanics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 4, July–August, 2013, pp. 97–106. Original article submitted January 23, 2013.
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1060-0396/13/4904-0569
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