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Download magnetic permeability and electric conductivity of magnetic emulsions
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MAGNETIC PERMEABILITY AND ELECTRIC CONDUCTIVITY OF MAGNETIC EMULSIONS Alexander N. Tyatyushkin Institute of Mechanics, Moscow State University, Michurinskiy Pr., 1, Moscow 117192, Russia Simultaneous effect of electric and magnetic fields on magnetic emulsions has peculiarities which are not characteristic for both magnetic and electric fields acting separately. In order to obtain a magnetic emulsion, a kerosene-based magnetic liquid is heated up, then some amount of an oil is added to it, and the obtained mixture is then cooled down back to the room temperature [1]. The dispersed phase of such an emulsion consists of small drops of more concentrated magnetic liquid, and its dispersive phase is a less concentrated magnetic liquid, in which the drops are suspended. In the absence of electric and magnetic fields the drops of the dispersed phase of a magnetic emulsion are spherical. An electric or magnetic field distorts the shape of the drops and thus makes the emulsion anisotropic. The goal of the present work is to theoretically investigate the influence of the anisotropy induced by simultaneously acting electric and magnetic fields on the electric conductivity and magnetic permeability. Kerosene-based magnetic emulsions are weakly electrically conducting, and the effect of an electric field on them can be described within the electrohydrodynamic approximation [2]. Due to weak conductivity, the magnetic field caused by the electric current in an electric field can be neglected. Single drop of the dispersed phase. Consider a single drop of the dispersed phase, whose radius in the absence of fields is R, in simultaneously acting uniform constant electric and magnetic fields. Let the viscosity, electric conductivity, dielecric and magnetic permeability of the dispersed and dispersive phases be, respectively, i, i, i, i, e, e, e, e, the surface tension of the interface between the phases be and the magnetic and electric intensity vectors be, respectively, H and E. If i and e are sufficiently small, then, to the first order approximation with respect to the dimensionless small parameters H2R/ and E2R/, the surface of the deformed drop is an ellipsoid whose semi-axes, a1, a2 and a3, are determined by the following expressions [3] a 2 1 a 2 2 R2 R2 E 2 H 2 3D R 12 , (1) R2 R2 E 2 H 2 3D R 12 , (2) a (3) 2 3 R 2 R 2 E 2 H 2 R 6 . The orientation of the ellipsoid is given by the unit vectors, l1, l2 and l3, directed along its main axes l1 sin HE sin EH E H , (4) l1 cosHE cosEH E H , (5) l1 E H E H , (6) Here, , , D, and are determined as follows 2 9e 16e 19i i e e i 9 e i e , 2 8 5e i 2e i 8 2e i 9e i e 8 2e i D E 2 (7) 2 , (8) H 2 4 E H , 2 2 (9) sin 2 2 E H E H H 2 D, (10) cos2 2 E H E H H 2 D . (11) Here, and denote the scalar and vector products, respectively, and |b| denotes the module of the vector b. The conditions for the parameters of the problem, R, i, i, i, i, e, e, e, e, , Ha and Ea, which should be satisfied in order (1)–(11) to be valid, are written down in Ref. [3]. Electric conductivity and magnetic permeability. Consider a sufficiently dilute monodispersed magnetic emulsion. Applying the method used in [4] for obtaining the electric permeability of dilute suspension of spherical particles, one obtains the following expressions for the tensors of electric conductivity and magnetic permeability, and , of such an emulsion eI I i e N e c , 1 (12) μ e I I i e N e c , 1 (13) where c is the volume concentration of the dispersed phase, I is the unit tensor, S-1 denotes the inverse tensor with respect to the tensor S, and N is the demagnetization tensor (see [4]) of a single drop. To a first order approximation with respect to the small parameters (ai2-R2)/R2 (i=1,2,3), I 3 T 3R 2I 5R 2 , (14) where T = a12l1l1 + a22l2l2 + a32l3l3 (bd denotes the dyadic product of the vectors b and d). Substituting (14) into (12) and (13) and using the smallness of the parameters (ai2-R2)/R2, one obtains 0 I 2 52e i 2 R 2 c, 9e i e T 3R 2 I (15) 2 μ 0 I where 9e i e T 3R 2 I 52e i 2 R 2 c, (16) 0 e 3e i e c 2e i , (17) 0 e 3e i e c 2e i , (18) Transverse current and magnetization. Due to the anisotropy, the electric current density, j=Ea, and the magnetization, M=(-I)Ha/(4), in a magnetic emulsion have, in general, both longitudinal and transverse components relative to Ea and Ha, respectively (Sb denotes the contraction of the tensor S with the vector b. Using (1)–(11) and (14)–(16), one obtains the following expression for the longitudinal and transverse components jl 0 E 9e i e 2E 4 3 E H E 2 H 2 R 2 52e i 2 2 6E 2 cE , (19) 9e i e 8RE H E H E 2 jt Ml 52e i E 2 0 1 2 c , (20) H 4 2 2 9 e i e 2 H 4 3 E H E 2 H 2 R 52 e i 24H 2 2 9e i e 2RE H H E H cH , (21) 2 Mt 52e i 2 H 2 c, (22) It follows from (20) and (22) that the transverse components are equal to zero when Ea and Ha are parallel or perpendicular and are maximal for given values of Ea and Ha when the angle between Ea and Ha is equal to 45 or 135. Acknowledgements. This work is supported by Russian Foundation for Basic Research grants No. 01-01-00010 and No. 01-01-00423. References 1. Yu. I. Dikanskii, N. G. Polikhronidi and K. A. Balabanov, Magnetohydrodynamics 24 (1988) 211. 2. J. R. Melcher and G. I. Taylor, Ann. Rev. Fluid Mech. 1 (1969), 111. 3. A. N. Tyatyushkin and M. G. Velarde, J. Colloid Interface Sci. 235 (2001) 46. 4. L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media, (Pergamon Press, 1984).