Translation of a spherical viscous drop driven by localized forcing in Stokes flow

Abstract

Localized forcing in the fluid inside or outside a viscous drop can drive drop translation. Using the Lorentz reciprocal theorem, we derive an integral expression for the translational velocity of a spherical Newtonian drop subject to localized force and source distributions in either fluid and to interfacial traction. For a clean drop, we obtain explicit responses to Stokeslets, force dipoles, rotlets, general second force moments, and source dipoles as functions of position, orientation, and viscosity ratio. Interior forcing obeys a finite selection rule: only force moments through second order and the first source moment contribute directly to translation. Exterior forcing can couple to multipoles of all orders and produces distance-dependent responses. Although different enclosed singularities can produce the same drop velocity, resolving their exterior flows in drop-centered spherical Stokes modes provides additional constraints on the underlying forcing. We also distinguish regularized force distributions, governed by prescribed kernel moments, from resolved rigid particles, governed by low-order surface-traction moments and prescribed slip. The framework unifies these representations and shows how exterior-flow measurements provide information beyond drop translation, laying the foundation for constructing squirmer-like viscous drop solutions with controllable far-field behaviors.