Sir Geoffrey Ingram Taylor
Proceedings of the Royal Society, A,
vol. ccxci (1966), pp. 159-166
In which the author determines the conditions for equilibrium of a spherical interface between two conducting liquids in an imposed field and determines the conditions for elongation into prolate and oblate spheroids.
In this paper, Taylor (who was in his late seventies) determines the equilibrium conditions on the spherical interface between two conducting liquids. This calculation involves solving for the electric field in such a system, performing a force balance on the interface and using the resulting boundary conditions for the stress arising from fluid motion to solve for the flow field in each conductor. The analysis involves the use of the Maxwell Stress Tensor to determine the electrical forces on the interface between fluids, therefore we introduce this concept first. Afterwards, a force balance is performed on a differential surface element. This yields relations for normal and tangential stresses due to fluid motion. The flow inside and outside is assumed axisymmetric with respect to the direction of the imposed electric field, therefore a stream function solution is attempted. The form of the desired normal and tangential stresses suggests a form for the stream function. Once the fluid mechanical problem is solved, an equilibrium relation between the physical parameters is found. Next, deformations from the spherical shape are analyzed by assuming that the sphere deforms into either a prolate or oblate spheroid. A discriminating function of the physical parameters determines which deformation regime prevails.