Consider a dilute, insoluble surfactant monolayer on the free surface of a thin viscous film. A gradient in surfactant concentration generates a gradient in surface tension, driving a flow that redistributes the surfactant so that these gradients decay. The nonlinear evolution equations governing such flows, derived using lubrication theory, have previously been shown to admit a set of simple similarity solutions representing the spreading of a monolayer over an uncontaminated interface. Here, a much more general class of similarity solutions is considered, and a transformation is identified reducing the governing partial differential equations to a set of nonlinear ordinary differential equations, the solutions of which correspond to integral curves in a two‐dimensional phase plane. This allows the construction of solutions to a wide range of problems. Many new solutions are revealed, including one that cannot be determined by simpler techniques, namely the closing of an axisymmetric hole in a monolayer, the radius of which is shown to be proportional to (−t)δ as t→0−, where δ≊0.807 41; this solution corresponds to a heteroclinic orbit in the phase plane.
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March 1994
Research Article|
March 01 1994
Self‐similar, surfactant‐driven flows
O. E. Jensen
O. E. Jensen
Department of Mathematics and Statistics, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, England
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Physics of Fluids 6, 1084–1094 (1994)
Article history
Received:
May 11 1993
Accepted:
November 15 1993
Citation
O. E. Jensen; Self‐similar, surfactant‐driven flows. Physics of Fluids 1 March 1994; 6 (3): 1084–1094. https://doi.org/10.1063/1.868280
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