Numerical Analysis and Methods for Fluid Deformable Surfaces and Their Interaction with the Bulk

University Of Houston, Houston TX

Investigators

Abstract

Fluid surfaces that deform are ubiquitous in cell and tissue biology as well as in modeling of emulsions and foams. Computer modeling plays an increasingly important role in better understanding of processes involving such surfaces as well as interfacial phenomena. The present project aims to develop accurate and reliable numerical methods for the simulation of fluidic deformable surfaces and their interaction with the bulk. Development of such methods will facilitate understanding the functionality of lipid bilayers, the actin cortex, epithelial cell sheets, and the properties of other thin structures exhibiting in-plane viscosity and lateral mobility. The project provides research training for a graduate student. The project will consider models of fluids on surfaces. For these, continuum-based modeling leads to systems of partial differential equations posed on time-dependent manifolds. For example, lipid membranes are fluidic thin layers that can be modeled as two-dimensional viscous surface fluids with bending elasticity. The project will develop and analyze a geometrically unfitted finite element method for fluid systems posed on deformable surfaces such as surface Stokes and surface Navier-Stokes equations, tangential fluid equations coupled with in-plane elasticity and equations governing out-of-plane (geometrical) motions, as well as interface-bulk coupled fluid systems. The project focus is on splitting schemes for the resulting coupled systems. The methods build on formulation of governing equations in terms of tangential differential calculus and employ time-independent unfitted background meshes. The approach will allow for implicitly given complex shapes that may undergo topological transitions. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.