Laurel Ohm, Department of Mathematics, Princeton University

Location: CULM 611

Mathematical Foundations of Slender Body Theory

Slender body theory (SBT) facilitates computational simulations of thin filaments in a 3D viscous fluid by approximating the hydrodynamic effect of each fiber as the flow due to a line force density along a 1D curve. Despite the popularity of SBT in computational models, there had been no rigorous analysis of the error in using SBT to approximate the interaction of a thin fiber with fluid. In this talk, we develop a PDE framework for analyzing the error introduced by this approximation. In particular, given a 1D force along the fiber centerline, we define a notion of `true' solution to the full 3D slender body problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing. In addition, we perform a complete spectral analysis of the slender body PDE in a simple geometric setting, which sheds light on the use of SBT in approximating the `slender body inverse problem,' where we instead specify the fiber velocity and solve for the 1D force density. Finally, we make comparisons to slender body models based on the method of regularized Stokeslets and the Rotne-Prager-Yamakawa tensor.

October 4

Nick Moore, Mathematics Department, US Naval Academy

Location: Webex

Title: TBA

Abstract: TBA

October 18

Shima Parsa, School of Physics and Astronomy, Rochester Institute of Technology

Tunable Collective Dynamics of Inclusions in Viscous Membranes

The typical cell membrane is a crowded assembly of molecular motors and biomolecules embedded in a 2D fluid mosaic. Active molecular motors perform complex cellular tasks by binding, inserting, polymerizing, and changing conformations, inducing disturbance flows in the membrane and the surrounding fluid. These long-ranged hydrodynamic fields perturb neighboring inclusions, potentially leading to coordinated motion. I will build on classic theories of Newtonian fluid dynamics of viscous membranes to illustrate unique oscillations and aggregation dynamics in pairs of active membrane inclusions. The phase behavior of the pair problem reveals the underlying mechanisms and suggests strategies to tune large-scale aggregation. I will also show numerical simulations of large numbers of interacting inclusions whose collective dynamics can be tuned based on these basic insights. If time permits, I will then describe the first steps in the analysis of inclusions in membranes with a nontrivial rheology. Real membranes are often strongly non-Newtonian. I will illustrate a formulation based on the Lorentz reciprocal theorem to asymptotically capture effects of non-constant surface viscosity of phospholipids that comprise most biological membranes. I will highlight the qualitative differences that ensue, and potential implications in crowded membranes.