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# Rheology of a suspension of conducting particles in a magnetic field

Kumaran, V (2019) Rheology of a suspension of conducting particles in a magnetic field. In: JOURNAL OF FLUID MECHANICS, 871 . pp. 139-185.

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Official URL: https://dx.doi.org/ 10.1017/jfm.2019.295

## Abstract

When a suspension of conducting particles is sheared in a magnetic field, the fluid vorticity causes particle rotation. Eddy currents are induced in a conductor rotating in a magnetic field, resulting in magnetic moment, and a magnetic torque due to the external field. In the absence of inertia, the angular velocity of a particle is determined from the condition that the sum of the hydrodynamic and magnetic torques is zero. When the particle angular velocity is different from the fluid rotation rate, the torque exerted by the particles on the fluid results in an antisymmetric particle stress. The stress is of the form . p /D j!j.. 1 / c.O V !O /C . 2 / c O V. H O !O.!O H O //=. q 1 H O /2 /C . 3 / c.!O H O H O !O /= q 1 H O /2 /, where ! is the fluid vorticity at the centre of the particle, !O and H O are the unit vectors in the direction of the fluid vorticity and the magnetic field, O is the third order Levi-Civita antisymmetric tensor and . 1 / c, . 2 / c and . 3 / c are called the first, second and third couple stress coefficients. The stress proportional to . 1 / c is in the plane perpendicular to !O, that proportional to . 2 / c is in the plane perpendicular to the unit normal to !O in the !O -H O plane, and that proportional to . 3 / c is in the !O -H O plane. A relation . 2 / c D H O . 1 / c = q 1 H O /2 /results from the condition that the component of the eddy current torque along the magnetic field is zero. The couple stress coefficients are obtained for two geometries, a uniform spherical particle of radius R and a thin spherical shell of radius R and thickness ffi R with ffi 1, in the dilute (non-interacting) limit in the absence of fluid inertia. These couple stress coefficients are functions of two dimensionless parameters, D. 0H2 0 = 4 pj!j/, the ratio of the characteristic magnetic and hydrodynamic torques, and fi, the product of the vorticity and current relaxation time. Here 0 is the magnetic permeability, H0 is the magnetic field and is the fluid viscosity. The parameter fi has the form fi p D.j!j 0R2 = 2 %/for a uniform particle and fi s D.j!j 0R2 ffi= 2 %/for a thin shell, where % is the electrical resistivity. Scaled couple stress coefficients are defined, 1 D.. 1 / c =.. 3 = 2 /. 1.!O H O /2 /// and 3 D.. 3 / c =.. 3 = 2 /!O H O q 1 H O /2 //, which are independent of the fluid viscosity and the particle volume fraction, and which do not depend on !O and H O in the limits 1 and 1. Here, is the volume fraction of the particles. Asymptotic analysis is used to determine the couple stress coefficients in the limits 1 and 1, and a numerical solution procedure is formulated for 1. For 1, the particle angular velocity is aligned close to the fluid vorticity, and the scaled couple stress coefficients are times a function of fi. For 1, the particle angular velocity is aligned close to the magnetic field, 1 ! 1 and 3 / 1. When the magnetic field is perpendicular to the fluid vorticity, !O H O D 0, the particle angular velocity is aligned along the vorticity, and only the first couple stress coefficient is non-zero. For high fi, there are multiple solutions for the couple stress coefficient. Multiple steady states are also observed for a near perpendicular magnetic field, !O H O <. 1 = 3 /, for a reason different from that for a perpendicular magnetic field. Asymptotic analysis is used to explain the existence of multiple steady states in both cases.

Item Type: Journal Article JOURNAL OF FLUID MECHANICS CAMBRIDGE UNIV PRESS The copyright for this article belongs to Cambridge University Press magnetic fluids; rheology; suspensions Division of Mechanical Sciences > Chemical Engineering 24 Jun 2019 18:33 24 Jun 2019 18:33 http://eprints.iisc.ac.in/id/eprint/62762 View Item