Exact solutions for the formation of stagnant caps of insoluble surfactant on a planar free surface

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.

Fast reaction of soluble surfactant can remobilize a stagnant cap

“…For more recent work on superhydrophobic channel flows, Peaudecerf et al. (2017), Baier & Hardt (2021), Crowdy (2021 a , b ) and Mayer & Crowdy (2022) are useful references.…”

“…Importantly, for the first time, this paper constructs analytical solutions to a class of such problems. The mathematics involved is underpinned by recent work by Crowdy (2021 a , b ) that has lent new analytical insights into the dynamics of insoluble surfactant on a planar free surface at zero capillary and Reynolds numbers. In that work, a key role is played by complex partial differential equations (PDEs) of Burgers type that emerge from a complex variable reformulation of the physical problem.…”

“…In that work, a key role is played by complex partial differential equations (PDEs) of Burgers type that emerge from a complex variable reformulation of the physical problem. In particular, Crowdy (2021 a ) has given a class of explicit, time-evolving solutions for the evolution of the surfactant concentration on a planar free surface in the presence of a linear background strain akin to that at the rear of a rising bubble. At long times, the solutions tend to, but never quite reach, a stagnant cap equilibrium where, in the absence of diffusion or reaction effects, the surfactant piles up onto an immobilized portion of the interface.…”

“…Reaction kinetics have also been incorporated into the mathematical approach based on PDEs of complex Burgers type (Crowdy 2021 a ), and exact solutions are available in that case too. This simple model assumes that surfactant sublimates off the interface according to a first-order reaction model.…”

“…It is this deficiency that the work in the present paper eliminates. It is worth pointing out that Bickel (2019) studied the same first-order reaction model as Crowdy (2021 a ) using an approach based on Hilbert transforms, and without recognizing the connection to the complex Burgers equation, but has since made further advances (Bickel & Detcheverry 2022) in light of the connection to Burgers-type PDEs unveiled by Crowdy (2021 a , b ). The theoretical connection therefore appears to be a fruitful one, and this matter is surveyed again later in § 12.…”