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Hydrodynamics on Smooth 2-Manifolds with Spherical Topology


The investigation of soft materials poses many important challenges having implications for application areas that range from the design of better materials for use in engineering practice to gaining further insights into the functioning of biological organisms. In soft materials there is often an interplay between direct microstructure-level interactions and fluctuations in yielding observed bulk macroscopic material properties. As part of these interactions geometry and hydrodynamic interactions often play a central role. We shall investigate interfacial phenomena associated with soft materials, particularly relevant to the study of lipid bilayer membranes. We shall address the problem of how to formulate and numerically approximate continuum mechanics on 2-manifolds in the case of non-trivial geometries

having spherical topology. We shall be particularly interested in developing methods for investigating the case of hydrodynamic flow responses on curved surfaces. We shall present results for an initial model assuming that we have smooth, star-shaped (radial) membrane geometry. We show how spectral methods of approximation can be developed based on use of spherical harmonics expansions, Lebedev quadrature. We use these approaches to investigate how hydrodynamic flow responses depend on the surface geometry. We find that the surface curvature can significantly effect dissipation rates and augment flow responses. We then develop more general methods for the case of any smooth geometry having spherical topology using numerical approaches based on

Generalized Moving Least Squares (GMLS). We use these to further investigate hydrodynamic flows in this setting. We conclude by briefly discussing our current work to extend these numerical approaches to even more general smooth compact manifolds without the need for spherical topology.

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