The effect of surface charge convection and shape deformation on the settling velocity of drops in nonuniform electric field

The electrohydrodynamic settling of a leaky dielectric drop suspended in another leaky dielectric medium of unbounded extent in the combined presence of gravity and a nonuniform (combination of uniform and quadrupole) electric field is investigated theoretically in the Stokes flow limit. The present study incorporates both the effects of surface charge convection and shape deformation on the drop settling speed. The drop settling speed in the presence of an electric field is governed by three dimensionless groups: (i) capillary number Ca (the ratio of viscous to capillary stresses), (ii) electric Reynolds number Re-E (the ratio of charge relaxation to convection time scales), and (iii) Masson number M (the ratio of electric to viscous stresses). Depending on the material properties of the drop and suspending medium, the strength of the applied electric field, and the drop radius, the following two different kinds of physical systems are identified for which asymptotic solutions for the settling velocity is obtained: (i) weak charge convection and small shape deformation (i.e., Re-E << 1, Ca << 1) but with appreciable electric field strength (M similar to 1) and (ii) weak electric field (i.e., M << 1) but with appreciable charge convection Re-E similar to 1. For a perfectly conducting drop suspended in a leaky (or perfectly) dielectric medium, the application of a uniform and/or quadrupole electric field always deforms the drop into a prolate shape which leads to an increase in drop settling speed. For a perfectly dielectric drop suspended in a perfectly dielectric medium, the application of a quadrupole electric field can deform the drop into either a prolate or an oblate shape which leads to an increase or decrease in the drop speed depending on the drop to medium permittivity ratio. Combined presence of uniform and quadrupole fields induces a dielectrophoretic force on the drop which not only alters the drop speed but also can change the direction of drop motion depending on the values of physical parameters. The present study establishes that surface charge convection plays an important role in determining the drop settling speed for a leaky dielectric drop suspended in a leaky dielectric medium. It is shown that the charge-convection-induced correction in the drop settling speed is larger in magnitude in the presence of a quadrupole electric field as compared to a uniform electric field; this is due to a more intense circulation inside the drop in the presence of a quadrupole electric field. Published by AIP Publishing.