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We study the morphology of lipid membranes subjected to intra-membrane viscous flows and interactions with elliptical cylinder substrates. From the non-linear theory of elastic surfaces, a linearized shape equation and admissible boundary conditions are formulated in elliptical coordinates via the Monge representation of a surface. In particular, the intra-membrane viscosity terms are linearized and mapped into elliptic coordinates in order to accommodate more general forms of viscous flow. The assimilated viscous flow is characterized by potential functions which satisfies the continuity condition. A complete solution in terms of Mathieu function is then obtained within the prescription of incremental deformations superposed on large. The results describe smooth morphological transitions over the domain of interest and, more importantly, predicts wrinkle formations in the presence of intra-membrane viscous flow in the surface. Lastly, the obtained solution accommodates the results from the circular cases in the limit of vanishing eccentricity and intra-membrane viscous flow.

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Deformation analysis of lipid membranes subjected to general forms of intra-membrane viscous flow and interactions with an elliptical-cross-section substrate