Abstract
A class of exact solutions is presented describing the time evolution of insoluble surfactant to a stagnant cap equilibrium on the surface of deep water in the Stokes flow regime at zero capillary number and infinite surface Péclet number. This is done by demonstrating, in a two-dimensional model setting, the relevance of the forced complex Burgers equation to this problem when a linear equation of state relates the surface tension to the surfactant concentration. A complex-variable version of the method of characteristics can then be deployed to find an implicit representation of the general solution. A special class of initial conditions is considered for which the associated solutions can be given explicitly. The new exact solutions, which include both spreading and compactifying scenarios, provide analytical insight into the unsteady formation of stagnant caps of insoluble surfactant. It is also shown that first-order reaction kinetics modelling sublimation or evaporation of the insoluble surfactant to the upper gas phase can be incorporated into the framework; this leads to a forced complex Burgers equation with linear damping. Generalized exact solutions to the latter equation at infinite surface Péclet number are also found and used to study how reaction effects destroy the surfactant cap equilibrium.
Highlights
The aim of this paper is to contribute some new theoretical results to the rapidly growing literature on the dynamics of insoluble surfactants on free surfaces in the Stokes regime
An important paper by Pozrikidis and Li [1] laid the foundation for much of that literature: in it the authors carried out a numerical investigation of the effects of insoluble surfactants on time-dependent drop deformations in background flows, such as simple shear and extensional flows, at low Reynolds numbers
We identify analytical solutions only for infinite surface Péclet number, but other theoretical ramifications of this new connection with the complex Burgers equation, including other exact solutions, for finite surface Péclet number have been surveyed by the author elsewhere [24]
Summary
The aim of this paper is to contribute some new theoretical results to the rapidly growing literature on the dynamics of insoluble surfactants on free surfaces in the Stokes regime. An important paper by Pozrikidis and Li [1] laid the foundation for much of that literature: in it the authors carried out a numerical investigation of the effects of insoluble surfactants on time-dependent drop deformations in background flows, such as simple shear and extensional flows, at low Reynolds numbers. The influence of surfactants on steadily-translating bubbles or drops is an especially well-studied problem [5,10,11,12]. One interesting feature of travelling bubbles in an otherwise quiescent fluid is that surfactants can accumulate at their rear and retard motion [10,11,13,14]. A common phenomenon is the formation of what have become known as
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