Equilibrium deformation of a transversely electrified, poorly conducting liquid column

The large equilibrium deformation of a liquid column in a uniform transverse electric field is explored in the limit of small Reynolds number and in the framework of the Taylor–Melcher leaky dielectric theory. The deformation of the liquid column, measured by the aspect ratio, is solved approximately via the two-point method and the least squares method. The deformation is determined by four dimensionless numbers, i.e., the electric capillary number Ca, the electrical conductivity ratio kr, the dielectric constant ratio εr, and the viscosity ratio μr. The deformation types (prolate or oblate), the fluid circulation patterns (from the equator to the poles or from the poles to the equator), as well as the distribution of interfacial charge in different regions in the kr-εr plane are elucidated, which turn out to be analogous to the case of small deformation. The electrical conductivity ratio kr and the dielectric constant ratio εr are found to affect prolate deformation and oblate deformation in different ways. Viscosity tends to resist the flow. On the other hand, viscosity is not always against deformation; it may help the liquid column to deform when the liquid column has an electrical relaxation time smaller than the surrounding medium or the deformation is oblate-type.