Surfactant-induced migration of a spherical droplet in non-isothermal Stokes flow

Abstract

The motion of viscous droplet in an unbounded arbitrary (non-axisymmetric) Stokes flow under the combined influence of bulk-insoluble surfactant Marangoni stresses and thermal Marangoni stresses is studied analytically in two limiting cases, namely, low and high surface Péclet numbers. This work considers a more general model where the nonlinear variation of the interfacial tension is due to both thermal and surfactant gradients. It is well known that linear thermocapillary stresses assist migration of the droplet, while surfactants resist when the ambient thermal field is along the direction of the ambient hydrodynamic field. We have observed that this behavior prevails even when a nonlinear combination of thermal and surfactant stresses is considered. However, since the retardation due to surfactants is marginal, when combined linear thermal and linear surfactant stresses are superimposed, the thermal forces dominate the overall migration. The present work derives closed form expressions for the drift and the migration velocity where the capillary stresses can be non-axisymmetric and along the axial or transverse direction. This would enable one to design parameter combinations to control the droplet migration for possible use in various applications. Since the results are for any arbitrary ambient flow, we have provided the corresponding analysis when the ambient hydrodynamic flow is due to Poiseuille flow. The corresponding results when the thermal gradients are axial or transverse to the flow direction are discussed. In this regard, we contribute some important findings on the cross migration of the droplet. We observe that the droplet can move towards or away from the centerline depending on the critical thermal Marangoni number. We have seen the variation of velocity fields in different planes when the ambient flow field is a Poiseuille flow. Furthermore, we have computed the power (rate of work) on the surface of the droplet. The obtained analytical results are compared with the existing literature in all possible limiting cases. Finally, we draw a striking analogy with flow through porous media that the centroid of the droplet migrates with a velocity that is thought of as the volume averaged velocity inside a resistive medium.