TY - JOUR
T1 - A liquid spheroidal drop in a viscous incompressible fluid under a steady electric field
AU - Zabarankin, Michael
PY - 2013
Y1 - 2013
N2 - An analytical approach to the problem of a liquid spheroidal drop in a viscous incompressible fluid under a steady electric field is presented. It is assumed that the drop and ambient fluid are slightly conducting (leaky dielectrics) with no net charge. The velocity field inside and outside the drop is governed by the Stokes equations and is represented in terms of four generalized analytic functions which are found from the velocity and stress boundary conditions. For the case of equal viscosities, the velocity field admits an integral-form solution, whereas for an arbitrary viscosity ratio λ, it is given by a series of spheroidal harmonics in the prolate spheroidal coordinates. Both solution forms hold for prolate and oblate spheroids. The axes ratio of a spheroid closest to a steady shape (or "steady" spheroid) is a minimizer of the kinematic condition error. However, the entire dependence of the "steady" spheroid's axes ratio on an electric capillary number CaE can be estimated analytically with the inverse problem: for each spheroid's axes ratio, find CaE that minimizes the kinematic condition error. The inverse problem, being quadratic with respect to 1/CaE, has a simple analytical solution, which identifies critical CaE for spheroidal drops and includes an "unstable" solution. Remarkably, as functions of CaE, the deformation parameters D for the "steady" spheroids and for "true" steady shapes (obtained based on boundary-integral equations) have the same qualitative behavior for various viscosity, conductivity, and permittivity ratios (λ, R, and Q, respectively), and are virtually identical within the range-0.4 ≤ D ≤ 0.4 for all investigated λ, R, and Q. This shows that the presented analytical approach for spheroidal drops is by far superior to Taylor's small-perturbation theory and Ajayi's second-order correction (O(CaE) and O(Ca2E) theories, respectively) and can replace numerical solutions from boundary-integral equations up to large deformations.
AB - An analytical approach to the problem of a liquid spheroidal drop in a viscous incompressible fluid under a steady electric field is presented. It is assumed that the drop and ambient fluid are slightly conducting (leaky dielectrics) with no net charge. The velocity field inside and outside the drop is governed by the Stokes equations and is represented in terms of four generalized analytic functions which are found from the velocity and stress boundary conditions. For the case of equal viscosities, the velocity field admits an integral-form solution, whereas for an arbitrary viscosity ratio λ, it is given by a series of spheroidal harmonics in the prolate spheroidal coordinates. Both solution forms hold for prolate and oblate spheroids. The axes ratio of a spheroid closest to a steady shape (or "steady" spheroid) is a minimizer of the kinematic condition error. However, the entire dependence of the "steady" spheroid's axes ratio on an electric capillary number CaE can be estimated analytically with the inverse problem: for each spheroid's axes ratio, find CaE that minimizes the kinematic condition error. The inverse problem, being quadratic with respect to 1/CaE, has a simple analytical solution, which identifies critical CaE for spheroidal drops and includes an "unstable" solution. Remarkably, as functions of CaE, the deformation parameters D for the "steady" spheroids and for "true" steady shapes (obtained based on boundary-integral equations) have the same qualitative behavior for various viscosity, conductivity, and permittivity ratios (λ, R, and Q, respectively), and are virtually identical within the range-0.4 ≤ D ≤ 0.4 for all investigated λ, R, and Q. This shows that the presented analytical approach for spheroidal drops is by far superior to Taylor's small-perturbation theory and Ajayi's second-order correction (O(CaE) and O(Ca2E) theories, respectively) and can replace numerical solutions from boundary-integral equations up to large deformations.
KW - Electric field
KW - Generalized analytic function
KW - Inverse problem
KW - Spheroidal drop
KW - Viscous fluid
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U2 - 10.1137/120888430
DO - 10.1137/120888430
M3 - Article
AN - SCOPUS:84878080534
SN - 0036-1399
VL - 73
SP - 677
EP - 699
JO - SIAM Journal on Applied Mathematics
JF - SIAM Journal on Applied Mathematics
IS - 2
ER -