Fall 2016
Colloquia are held on Fridays at 11:30 a.m. in Cullimore Lecture Hall II, unless noted otherwise. Refreshments are served at 11:30 am. For questions about the seminar schedule, please contact Yassine Boubendir.
Date: September 9, 2016
Speaker: Rolf Ryham
Department of Mathematics,
Fordham University
Title: "Explicit Stokes Flows for a Moving Internal Boundary with Applications to Pore Dynamics in Liposomes"
Abstract:
Calculations for the streaming flow around deformable bodies are of interest in a number of areas including physiology and chemical engineering. Electroporation is one experimental technique in which the application of a large electric pulse results in the dielectric breakdown of a cell membrane; the formation of a pore allows the introduction of once-impermeant molecules into the cell. In order to make progress in modeling this and related phenomena, and eventually use the techniques for medical applications, it is useful to know the details of the flow patterns and quantify the rate at which electric or mechanical energy is dissipated by viscous losses. A fairly complete analytical description is possible in this situation because even though the initial deformations can be large, pore widening and closure occur over a long time scale while the cell is nearly spherical.
The present talk describes the streaming flow for motions of the spherical cap--a hallow sphere with a single hole. The method employed uses the complementary integral representation developed by the late Keith Ranger. With this method, it is possible to determine the stream function in a closed form, so that the drag coefficient for pore closure can be calculated exactly. The formulas are valid for all angles defining the pore, and allow one to interpolate between classical results for a hole in a plane wall and for a circular disk, in the limit of vanishing curvature of the sphere. Further calculations have now revealed the flow pattern for leak-out. A formula generalizing the pressure drop across a hole in a plane wall to a hole in a sphere is presented. Apart from possible applications in physiology and chemical engineering, the present determinations have mathematical interest because of their role in the numerical analysis of mixed boundary-value problems where the internal boundary possesses an edge. In particular, the analytical results show that any numerical scheme modeling a porated membrane by a zero thickness surface must carefully evaluate fluid velocity through mesh refinement or the construction of suitable local solutions in order to diminish the effect of singularities in the pressure and stress fields.