Eddies driven by eddy currents: Magnetokinetic flow in a conducting drop due to an oscillating magnetic field

\n\nAn oscillating magnetic field of amplitude H0 and angular frequency ω is applied across an electrically conducting non-magnetic drop of conductivity κ, viscosity η and radius R. The oscillating magnetic field generates an oscillating electric field due to Faraday's law, $\\boldsymbol {\\nabla } \\boldsymbol {\\times } \\boldsymbol {E} = -\\mu _{0} (\\partial \\boldsymbol {H}/\\partial t)$ , where E and H are the electric and magnetic field, and μ0 is the magnetic permeability. This generates a current density, $\\boldsymbol {J} = \\kappa \\boldsymbol {E}$ . The non-linear interaction between the oscillating current and magnetic field results in a time-independent body force density, $\\mu _{0} (\\boldsymbol {J} \\boldsymbol {\\times } \\boldsymbol {H})$ due to Ampere's circuital law. This drives a pair of axisymmetric circulation rolls in the two hemispheres of the drop in the viscous limit where inertia is neglected. The characteristic circulation velocity is $(\\mu _{0} H_{0}^{2} R/\\eta )$ times a function of the dimensionless parameter $\\beta R = \\sqrt {\\mu _{0}\\,\\kappa \\omega R^{2}}$ , the square root of the ratio of the magnetic field frequency and the eddy current relaxation rate. The velocity increases proportional to $(\\beta R)^{4}$ for $\\beta R \\ll1$ , has a maximum at $\\beta R \\simeq6$ , and decreases proportional to $(\\beta R)^{-1}$ for $\\beta \\gg1$ . Large strain rates in the range $1\\, \\textrm {s}^{-1}$ are generated in a metal drop of radius 0.1–1 mm for magnetic flux density as low as $3.5\\,\\times10^{-3}-3.5\\,\\times10^{-1} \\text{T}$ , provided relatively high frequencies of oscillation in the range $10\\, \\textrm {kHz}$ are used. Joule heating could increase the drop temperature by tens of degrees Kelvin in comparison to the ambient.\n