Surfactant-induced stagnant zones in the Jeong-Moffatt free surface Stokes flow problem

Abstract

We investigate the effect of surfactants on the problem introduced by Jeong and Moffatt [J. Fluid Mech. 241, 1–22 (1992)] to model the free surface deformation induced by two rollers beneath an infinite interface in a bath of viscous fluid. We demonstrate that the presence of surfactant dramatically affects the shape of the response curve for steady state equilibria. It is also shown that there is a well defined, albeit non-uniform, passage to the clean flow limit as the influence of the surfactant vanishes. Our analysis proceeds via a new mathematical approach to mixed boundary value problems arising in free surface Marangoni flows at infinite Péclet number in two-dimensional Stokes flows. Such problems often involve interfaces with a mixture of no-slip zones and regions where a capillary stress balance holds. By a conformal mapping technique accounting for the square root singularities inherent in such mixed boundary value problems it is shown that the latter can be transformed to the classical modified Schwarz problem of complex analysis solvable by standard methods. As a second application of the method we give a new derivation, and representation, of the solutions for steady surfactant-laden stagnant-cap bubbles in a linear strain presented by Siegel [SIAM J. Appl. Math. 59, 1998–2027 (1999)].