Publication 17-CNA-002
Dynamics And Stability Of Surface Waves With Bulk-Soluble
Surfactants
Ian Tice
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
ian.tice@andrew.cmu.edu
Lei Wu
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA
lwu2@andrew.cmu.edu
Abstract: In this paper we study the dynamics of a layer of incompressible viscous
fluid bounded below
by a rigid boundary and above by a free boundary, in the presence of a uniform gravitational field. We
assume that a mass of surfactant is present both at the free surface and in the bulk of
fluid, and that
conversion from one species to the other is possible. The surfactants couple to the
fluid dynamics through
the coefficient of surface tension, which depends on the the surface density of surfactants. Gradients in this
concentration give rise to Marangoni stress on the free surface. In turn, the
fluids advect the surfactants and
distort their concentration through geometric distortions of the free surface. We model the surfactants in a
way that allows absorption and desorption of surfactant between the surface and bulk. We prove that small
perturbations of the equilibrium solutions give rise to global-in-time solutions that decay to equilibrium at
an exponential rate. This establishes the asymptotic stability of the equilibrium solutions.
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