The low surface Péclet number regime for surfactant-laden viscous droplets: Influence of surfactant concentration, interfacial slip effects and cross migration

We consider the motion of a viscous drop in an arbitrary unsteady Stokes flow such that the surface of the drop is fully covered with a stagnant surfactant layer. In particular the regime of low surface Péclet number is analyzed and we account for the effect interfacial slip on the overall behavior of the flow field. The hydrodynamic problem is solved by the solenoidal decomposition method and the drag force is computed in terms of Faxen's laws, using a perturbation ansatz in powers of the surface Péclet number. The surface equation of the deformed sphere has been determined by an iterative method up to the first order approximation. Analytical expressions for the migration velocity of the drop are likewise given. Based on this analysis we can completely characterize various flow situations including a given ambient flow as uniform flow, Couette flow and Poiseuille flow. Moreover, we find out that a surfactant-induced cross-stream migration of the drop occurs towards the center-line in both, Couette and Poiseuille flow cases. The variation of the drag force and the migration velocity is computed for different parameters such as the Péclet number and the Marangoni number. Finally, the theoretical findings are validated on with available experimental data.