A drop in an axisymmetric extensional ow is studied using boundary integral
methods to understand the effects of a monolayer-forming surfactant on a strongly
deforming interface. Surfactants occupy area, so there is an upper bound to the
surface concentration that can be adsorbed in a monolayer, Γ∞.
The surface tension is a highly nonlinear function of the surface concentration
Γ because of this upper
bound. As a result, the mechanical response of the system varies strongly with
Γ for realistic material parameters. In this work, an insoluble surfactant is
considered in the limit where the drop and external fluid viscosities are equal.
For Γ<Γ∞, surface convection sweeps
surfactant toward the drop poles. When surface diffusion is negligible, once the stable drop
shapes are attained, the interface
can be divided into stagnant caps near the drop poles, where
Γ is non-zero, and
tangentially mobile regions near the drop equator, where the surface concentration is
zero. This result is general for any axisymmetric fluid particle. For
Γ near Γ∞, the
stresses resisting accumulation are large in order to prevent the local concentration
from reaching the upper bound. As a result, the surface is highly stressed tangentially
while Γ departs only slightly from a uniform distribution. For this case,
Γ is never zero, so the tangential surface velocity is zero for
the steady drop shape.
This observation that Γ dilutes nearly uniformly for high surface
concentrations is used to derive a simplified form for the surface mass balance that applies in
the limit of high surface concentration. The balance requires that the tangential flux
should balance the local dilatation in order that the surface concentration profile will
remain spatially uniform. Throughout the drop evolution, this equation yields results
in agreement with the full solution for moderate deformations, and underscores the
dominant mechanism at high deformation. The simplified balance reduces to the
stagnant interface condition at steady state.
Drop deformations vary non-monotonically with concentration; for
reduction of the surface tension near the poles leads to higher deformations than the
clean interface case. For Γ near Γ∞,
however, Γ dilutes nearly uniformly, resulting in
higher mean surface tensions and smaller deformations. The drop contribution to the
volume averaged stress tensor is also calculated and shown to vary non-monotonically
with surface concentration.