A drop in a nonlinear shear flow

Abstract

The fluid mechanics problem of an initially spherical drop subjected to a two-directional nonlinear cubic shear flow, under creeping flow conditions, is theoretically studied. This physical situation is described by four dimensionless numbers: the capillary number (Ca), the viscosity ratio (λ), the linear velocity ratio (G) and the nonlinear intensity of the flow (E). Analytical expressions for the external and internal flows of a spherical drop as well as for the shape of the deformed drop were obtained by applying Lamb's general solution to the Stokes equations. The internal and external streamlines of a spherical drop (Ca = 0) suggest interesting and unusual shapes which are sometimes also accompanied by many stagnation points. Whenλ = ∞, the drop becomes a solid sphere with an angular velocity$\omega ={-} (1 - G)/2$independent ofE. For the familiar simple shear flow in thex-direction (G = 0,E = 0), Taylor's theory, valid at$Ca \ll 1$andλ = O(1), predicts an ellipsoidal drop oriented at an azimuthal angleϕ = 45°; asCaincreases, the deformation increases and the drop attains a peanut shape. For other linear or nonlinear cases, the drop is aligned atϕ = ±45°, and a spherical drop with no deformation is possible atCa ≠ 0. In a highly viscous drop$(\lambda \gg 1)$, the drop remains almost spherical even atCa = O(1), oriented atϕ = 0° or 90° (G ≠ 1), and its deformation is independent ofCa.