Modeling and Analysis of Thin Viscous Liquid Films in Spherical Geometry

Start: 03/26/2018 - 4:15pm
End  : 03/26/2018 - 5:15pm

Applied Math Seminar

Di Kang (CGU)


This talk studies the dynamics of a thin viscous liquid film coating the inner or outer surface of a sphere in the presence of gravity, surface tension and Marangoni effects. We also allow the sphere to rotate around its vertical axis. The surface tension coefficient can be considered as a constant, or a function of temperature or surfactant concentration. An asymptotic model describing the evolution of the film thickness is derived based on the lubrication approximation.

When the surface tension coefficient is a constant, the model includes the centrifugal and gravity forces and the stabilizing effect of surface tension. This thesis shows that the steady states are of three different types: uniformly positive film thickness, or the states with one or two dry zones on the sphere, depending on the relative strength of the centrifugal force to that of gravity. The transient dynamics in approaching those states are also described. This thesis also provides a constructive proof for the existence of non-negative weak solutions in a weighted Sobolev space.

When the surface tension coefficient is a non-constant function, an additional term representing the Marangoni effect is added to the equation. This thesis studies the cases when the surface tension gradient is due to an externally imposed temperature field or the presence of surfactant molecules. In the former case, we consider two different heating regimes with axial or radial thermal gradients and discuss the resulting dynamics. In the latter case, this thesis derives and studies a model for the coating flow inside the alveolar compartment of the lungs, taking into account the effect of pulmonary surfactant and its production and degradation. We derive a degenerate system of two coupled parabolic partial differential equations that describe the time evolution of the thickness of the coating film together with that of the surfactant concentration at the liquid-air interface. This thesis presents numerical simulations of the dynamics using parameter values consistent with experimental measurements.

Emmy Noether Rm Millikan 1021 Pomona College

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