Drops with conical ends in electric and magnetic fields

摘要

Slender-body theory is used to determine the approximate static shape of a conically ended dielectric drop in an electric field. The shape and the electric-field distribution follow from solution of a second-order nonlinear ordinary differential equation that can be integrated numerically or analytically. An analytic formula is given for the dependence of the equilibrium cone angle on the ratio, (implied by)/(implied by)-bar, of the dielectric constants of the drop and the surrounding fluid. A rescaling of the equations shows that the dimensionless shape depends only on a single combination of (implied by)/(implied by)-bar and the ratio of electric stresses and interfacial tension. In combination with numerical solution of the equations, the rescaling also establishes that, to within logarithmic factors, there is a critical field E_(min) for cone formation proportional to ((implied by)/(implied by)-bar - 1)~(-5/12), at which the aspect ratio of the drop is proportional to ((implied by)/(implied by)-bar - 1)~(1/2). Drop shapes are computed for E_(infinity) > E_(min). For E_(infinity) E_(min) the aspect ratio of the drop is proportional to E_(infinity)~(6/7). Analogous results apply to a ferrofluid in a magnetic field.