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Inverse Problem for Drop Deformation in Nonlinear Electrohydrodynamics
The problem of deformation of a drop freely suspended in another fluid and subjected to an electric field has long been of theoretical and practical interest. In this problem, the velocity and electric fields are coupled through the boundary conditions, one of which corresponds to the nonlinear effect of surface charge convection. The balance between the electric stress and surface tension is characterized by electric capillary number CaE, whereas the effect of surface charge convection is characterized by electric Reynolds number ReE (the ratio of timescales for charge relaxation and convective flow). The drop experiences no deformation when CaE=0, and there is no surface charge convection when ReE=0. Finding drop steady shape even when ReE=0 is a computationally expensive iterative procedure with generally slow convergence. However, several experimental and theoretical studies show that when drop deformation is not large, the drop steady shape is close to spheroidal. This observation and the fact that Ca−1E enters the boundary conditions linearly suggest to solve the inverse problem: Find CaE for which a spheroidal shape with given axes ratio d is as close to being steady as possible. The velocity and electric fields and CaE are expanded into series with respect to a parameter proportional to ReE, and the inverse problem is reduced to a system of successive linear problems which can be efficiently solved. Values of (CaE(d),d), obtained based on that system for various values of ReE and for various ratios of phases' electric conductivities, dielectric constants, and viscosities, are in good agreement with existing results for nonsmall deformations and nonsmall ReE.
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