The lift on a small sphere in a linear shear flow near the interface of two immiscible fluids

Abstract

AbstractAnalytical expressions have been derived which predict, to lowest order, the inertial lift and the lateral migration velocity of a rigid sphere translating and rotating in a linear shear flow field near the flat interface of two immiscible fluids. This asymptotic analysis is primarily based on the assumption that the two Reynolds numbers defined by the gap width between the interface and the sphere, the shear rate and the translational slip velocity with which the spherical particle moves parallel to the interface are small. Furthermore, the radius of the sphere is assumed to be small compared to the gap width. To leading order in this creeping flow regime, the linear Stokes equations are obtained and a symmetry argument can be used to show that the Stokes solution does not predict any lift force. The transverse force experienced by the sphere and its migration velocity are due to the small but finite inertial terms in the Navier‐Stokes equations, which can be studied by perturbation techniques. By applying a Green's function approach and matched asymptotic methods, which also incorporate the effects of the outer Oseen‐like flow regime, the three components comprising the lift velocity have been calculated in closed form: the one induced by the shear rate only, the purely slip induced one and the one due to the interaction of the slip velocity with the shear flow field. The thus obtained expressions for the case of two immiscible fluids with arbitrary density and viscosity ratios extend the results that already exist in the literature for other flow configurations, such as an unbounded shear flow field [1] or a wall‐bounded one, where the wall lies either within the leading order Stokes region [2] or in the outer Oseen region [3]. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)