A compound drop in a nonlinear extensional flow

Abstract

The fluid mechanics problem of an initially spherical compound drop suspended in another fluid undergoing a nonlinear extensional creeping flow, is the subject of this theoretical report. The compound drop is originally composed from an inner spherical drop positioned at the center of a spherical fluid shell. The problem is governed by six dimensionless parameters: the external capillary number (Ca), two viscosity ratios: shell over external (λ21) and internal over shell (λ32), the radii ratio: inner over outer (κ), the surface tensions ratio: inner-shell over outer-shell (Ω), and the nonlinear intensity of the flow (E). When the extensional flow is linear (E=0), a case already treated in the literature, a uniaxial flow (Ca > 0) deforms the outer surface into a prolate spheroid and the inner drop into an oblate spheroid, while the reverse occurs for the biaxial flow (Ca < 0). If the extensional flow is nonlinear (E≠ 0), the external fluid behaves different than the linear case suggesting sometimes closed circulations (E > 0) and always separating surfaces (E < 0). As a result, each surface of the compound drop may experience both uniaxial and biaxial flow simultaneously, and the number of shell and internal drop circulations may be doubled. This weird situation, suggests new and exciting deformation and breakup patterns, and it is the result of the inclusions of nonlinear terms to the flow field.