Drop motion with surfactant transfer in a homogeneous surrounding

Abstract

The influence of a soluble surfactant on the stationary motion of a drop in an infinite motionless homogeneous surfactant solution is considered when the surfactant undergoes a first-order isothermal chemical reaction on the surface of the drop. The study is carried out for the asymptotic case of small Reynolds and Péclet numbers. First, the linear approximation is considered. It appears, in particular, that the hydrodynamical force acting on the drop provides either thrust or drag according to the parameter values. It also appears that the motion can be unstable and critical Marangoni numbers corresponding to instability thresholds of the motionless state of the fluids in the absence of buoyancy are provided. Emphasis is given to instability of the drop to its translations. A weakly nonlinear analysis past translational instability shows the possibility of multiple stationary states. On the one hand the hydrodynamical force dependence on the drop velocity can be nonmonotonous. Then, in particular, one may have either the rest state or self-sustained (autonomous), Marangoni-driven motion in the absence of buoyancy or any other external forcing factors. On the other hand, the velocity dependence on the hydrodynamical force can also be nonmonotonous. Then, in particular, in the absence of any external compensating forces one may have the drop levitating even under nonzero buoyancy. Finally, the relative stability of these nonlinear regimes and possible experimental observations are discussed.