Deformation of a liquid drop adhering to a plane wall: Significance of the drop viscosity and the effect of an insoluble surfactant

Abstract

The transient deformation of a viscous drop attached to a plane wall and subjected to an overpassing simple shear flow is studied in the limit of a vanishing Reynolds number. The buoyancy force is negligible, the contact line has and retains a prescribed circular shape, and the interface is either devoid of surfactants, in which case it exhibits constant surface tension, or it is occupied by an insoluble surfactant, in which case the surface tension is related to the surfactant concentration by means of a linear constitutive equation. In the numerical procedure, the interfacial velocity is computed by solving the equations of Stokes flow using a boundary-element method in which the interface is discretized into six-node curved triangles. The convection–diffusion equation for the concentration of the surfactant is integrated in time over the evolving interface using a semi-implicit finite-volume method. In the numerical investigations, the deformation of a drop is simulated from the initial state, where the interface has the prescribed shape of a section of a sphere and a uniform surfactant distribution, to either a steady deformed shape, or up to the point where evidence for continued deformation is obtained. The results show that the presence of surfactant promotes the deformation of drops whose viscosity is low compared to that of the ambient fluid, but has a small influence on the deformation of drops whose viscosity is comparable to, or higher than, that of the ambient fluid. At a fixed capillary number, the deformation of the drop equilibrium shape for constant or varying surface tension increases monotonically with raising the drop viscosity because of the increasing importance of the image flow due to the wall. Thus, whereas a viscous drop suspended in an infinite shear flow extends indefinitely into a slender ligament only when its viscosity is less than about four times the viscosity of the ambient fluid and the shear rate is sufficiently high, an analogous threshold for a drop attached to a wall does not seem to arise.