External and internal streamlines and deformation of drops in linear two-dimensional flows

Abstract

The creeping flow of a Newtonian fluid around a neutrally buoyant, immiscible spherical liquid drop has been studied theoretically. The streamlines inside and outside the drop and its deformation have been calculated for the family of linear two-dimensional flows, members of which are determined by a dimensionless parameter −1 ⩽ α ⩽ 1. These flows include pure shear flow (α = 1) as one limit and pure rotation (α = −1) as the other, with simple shear (α = 0) as an intermediate case. In the bulk medium, it is found that both an open and a closed streamline region exists for 0 ⩽ α < 1, with the distance from the drop to the limiting streamline dividing the two regions being determined by the ratio of the drop viscosity to that of the medium, λ. All streamlines are open in pure shear flows (α = 1) regardless of λ and for all flows when 0 ⩽ α ⩽ 1 and λ = 0. For flows having −1 ⩽ α < 0, streamlines are always closed. The circulation established in the drop due to the external flow varies according to α and λ. In particular, as many as four pockets of circulation are found for flows with 0 < α ⩽ 1 while for −1 < α ⩽ 0 only two such “pockets” exist. In the trivial case α = −1 there is a single pocket in the drop. Equations giving the axis ratio and orientation of a deformed drop are also derived.