Viscous Current Induced by Kelvin Force in Ordinary Fluids with Magnetic Susceptibility Contrasts

The magnetic susceptibilities of various electrically insulating ordinary fluids depend on their local states, such as their density and temperature. When such fluids, which can be characterized as either paramagnetic or diamagnetic and occur commonly in nature, are subjected to magnetic field gradients, it induces an effective body force—the Kelvin force. This force, which depends on the susceptibility and the gradient of the square of the magnetic field strength, can become one of the effective mechanisms for modulating the flow and transport, particularly where terrestrial gravity becomes negligible, such as in free space or under microgravity conditions. For the first time, we developed a theoretical model demonstrating that a viscous current can be generated due to the contrasts between the magnetic susceptibilities of the intruding and ambient fluids in the presence of gradients in magnetic fields, analogous to the viscous gravity current in terrestrial situations. We derived similarity solutions for the two-dimensional and axisymmetric currents arising from a balance between the Kelvin buoyancy and viscous forces with a prescribed power law for the magnetic field strength. These determine the shape and various spreading relationships of the viscous current. For a prescribed time variation in the source flux, it is shown that a family of scaling laws exists for the spreading rate and the thickness of the current, which depend on the steepness of the magnetic field gradient. Unlike gravity, since the driving horizontal buoyancy arising from the Kelvin force is externally specified, it potentially offers a mechanism to control the characteristic shape and the rate of motion of the viscous current.