Transient Stokes flow past a spherical droplet with a stagnant cap due to contaminated surfactant layer

Abstract

In this paper, we consider a viscous droplet migrating in a viscous fluid of a different viscosity. Further, we assume that the surface of the droplet is partially contaminated with a stagnant layer of surfactant (surface active agent which reduces the interfacial tension). We analyze the effects of the following phenomena associated with the thermocapillary migration of a droplet in a transient Stokes flow. The first is the influence of surfactant cap for an arbitrary cap angle which is partially coated on the droplet surface for both high and low surface Péclet number cases. The second is the influence of the energy changes associated with stretching and shrinkage of the interfacial area elements, when the droplet is in motion. It can be noted that for the vanishing cap angle, both high and low surface Péclet number limits reduce to the case of a pure thermocapillary migration of a droplet in a transient Stokes flow. For a given ambient flow, the migration of the droplet is controlled by the magnitude of the ambient velocity and the surface tension variations due to temperature and surfactant concentration. In particular, these surface tension variations balance the tangential stress balance. Considering axisymmetric transient Stokes flow, we obtain analytical solutions in two limiting cases, namely low and high surface Péclet number. This work considers linear variation of interfacial tension on both thermal and surfactant gradients. The main contribution is pertaining to the capillary drift and the corresponding surfactant transport on the droplet for an axisymmetric hydrodynamic as well as thermal and surfactant fields. We have analyzed the level curves corresponding to stream function and temperature fields, i.e., streamlines and isotherms for various parameters in order to develop a realistic picture of the migration pattern and the influence of thermal fields. We observe that the streamlines in the vicinity of rear end of the droplet show asymmetry due to the surfactant accumulation at that region. Increasing cap angle breaks the symmetry of the induced stream. It is seen that increasing values of nondimensional parameter that accounts for the stretching and shrinkage of the droplet surface immobilizes the surface and offers retardation to the migrating droplet. The variation of migration velocity with time suggests a control mechanism for the migration of the drop under external/surface gradients and hence may serve as a useful tool in applications like targeted drug delivery systems.