Keywords: electro-magneto-phoresis, magneto-hydrodynamics, droplets, Stokes flow, reflection method, interactions.

Solid and either insulating or conducting particles immersed in a conducting liquid are known to move [1] under ambient electric and magnetic fields. Since such a phenomenon, also termed electro-magneto-phoresis, is likely to receive basic applications in impurities removal (for instance in steel casting or glass processing) it is of prime interest to determine the motion of one or several particles. Recently Reference [2] provided the general behaviour of a solid particle depending upon its geometrical symmetries whereas a boundary formulation and the associated numerical implementation was given in [3] to acurately compute at a reasonable CPU cost the rigid-body motion of a solid and arbitrary-shaped particle. Since particle-particle interactions are likely to affect the migration predicted the proposed approach was also further extended in [4,5] to the challenging case of arbitrary clusters made of solid particles.

In practice one also encounters non-solid particles such as spherical bubbles or droplets. The migration of a single bubble has been analytically investigated in [6] together with bubble-bubble interactions for a two-bubble cluster made of distant bubbles. Depending upon the orientations of the ambient electric and magnetic fields either small long-range or very weak short-range bubble-bubble interactions were obtained for each bubble velocity. The case of droplets appears to be more tricky since a flow also takes place inside each droplet.

This paper examines the case of two interacting and distant droplets using a method of reflection. Net forces exerted on the droplets and each droplet velocity are asymptotically expanded in terms of a small parameter which compares the droplets typical radius with the droplets center-to-center distance. As for two bubbles, it is found that droplet-droplet interactions are very sensitive to the electric and magnetic field orientations and the addressed quantity and that one may either obtain small long-range or very weak short-range interactions.

1

D. Leenov, A. Kolin,"Theory of electromagnetophoresis. I. Magnetohydrodynamic forces experienced by spherical and symmetrically oriented cylindrical particles", J. Chem. Phys, 22(4), 683-688, 1954. doi:10.1063/1.1740149

2

H.K. Moffatt, A. Sellier, "Migration of an insulating particle under the action of uniform ambient electric and magnetic fields. Part 1. General theory", J. Fluid Mech., 464, 279-286, 2002. doi:10.1017/S0022112002001131

3

A. Sellier, "Slow viscous migration of a conducting solid particle under the action of uniform ambient electric and magnetic fields", CMES, 21(2), 105-132, 2007. doi:10.3970/cmes.2007.021.105

4

A. Sellier, "On particle-particle interactions in solid and non-conducting impurities removal", Comptes Rendus Mécanique, 331, 753-758, 2003. doi:10.1016/j.crme.2003.07.002

5

A. Sellier, "Electromagneto-phoresis of interacting, solid and conducting particles", Submitted to J. Fluid Mech.

6

A. Sellier, "Migrating and interacting micro-bubbles immersed in a conducting liquid under the action of uniform electric and magnetic fields", Magneto-Hydro-Dynamics, 45(3), 377-384, 2009.

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