Transient electrohydrodynamics of compound drops

The transient electrohydrodynamics of a compound drop under a uniform electric field of small strength is investigated. A closed form analytical solution is developed for creeping flow regimes and fluid systems and drop sizes with moderate and/or large Ohnesorge numbers, in the framework of leaky dielectric theory. For small distortion from a spherical shape, the inner and the outer drops deform to ellipsoids, and their deformation–time histories can be represented by Dij=Aijexp(-t/τ1)+Bijexp(-t/τ2)+Dij∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}_{ij}=A_{ij} {\rm exp}(-t/\tau_1) + B_{ij} {\rm exp}(-t/\tau_2) + \mathcal{D}^\infty_{ij}}$$\end{document} , where ij = 12,23 refers to the surfaces of the inner and the outer drops, τ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau_1}$$\end{document} and τ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau_2}$$\end{document} are the characteristic times, Aij and Bij are the coefficients that depend on the input parameters of the system, and Dij∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}^\infty_{ij}}$$\end{document} are the steady-state deformation parameters. The evolution of the flow field for several fluid systems was explored, and it was shown that the ratios of electric conductivities and permittivities of the participating fluids play a key role in determining the evolution of the flow field toward the steady state and that the steady-state flow is established by the motion of toroidal vortices that are formed in the drops and move outward, or formed in the ambient fluid and move inward.