Dynamics of Swimmers in Fluids with Resistance

Abstract

Micro-swimmers such as spermatozoa are able to efficiently navigate through viscous fluids that contain a sparse network of fibers or other macromolecules. We utilize the Brinkman equation to capture the fluid dynamics of sparse and stationary obstacles that are represented via a single resistance parameter. The method of regularized Brinkmanlets is utilized to solve for the fluid flow and motion of the swimmer in 2-dimensions when assuming the flagellum (tail) propagates a curvature wave. Extending previous studies, we investigate the dynamics of swimming when varying the resistance parameter, head or cell body radius, and preferred beat form parameters. For a single swimmer, we determine that increased swimming speed occurs for a smaller cell body radius and smaller fluid resistance. Progression of swimmers exhibits complex dynamics when considering hydrodynamic interactions; attraction of two swimmers is a robust phenomenon for smaller beat amplitude of the tail and smaller fluid resistance. Wall attraction is also observed, with a longer time scale of wall attraction with a larger resistance parameter.

Flagellar Cooperativity and Collective Motion in Sperm

“…Mathematical models for flagellar swimming in groups are an active area of research. Some studies have considered 2D fluids, which demonstrated a tendency of flagella to attract and that this attraction is energetically favorable [23,[28][29][30]. In 3D fluids, models have shown both attraction and repulsion of neighboring flagellar pairs, but the results are dependent on configurations of the flagella, the fluid domain, timescales of simulation, and the flagellar model.…”

“…Recent work has simulated flagella in pairs for up to 10-20 beats [35] but we will go beyond this here to address any lingering questions about long-range stability of flagellar interactions like those posed in [30]. Simulations involving all of these components are computational expensive due to the finer scale details we aim to resolve but they are plausible, as long as we relax some of other ideal features of a model.…”

“…Simulations involving all of these components are computational expensive due to the finer scale details we aim to resolve but they are plausible, as long as we relax some of other ideal features of a model. In our case, we will focus on purely viscous, Newtonian fluid flow instead of non-Newtonian fluids like those modeled in [30,40]. We will also not consider sperm heads as done in [26], external fluid flow such as [41], or more detailed biochemical interactions like those described in [37,42].…”

“…A comparison of pairs of coplanar flagella in 2D and 3D fluids was done in [29], in which attraction and synchronization was observed in both fluids, and efficiency as well as velocity were shown to increase in some parameter regimes. In [30], pairwise attraction of flagella was demonstrated in a 2D fluid with a resistance term meant to mimic swimming through a complex fluid containing other cells and polymers, though the question of long-term stability of this attraction was posed due to final configurations of the simulations. The work in [33] also included a 2D framework with large populations of flagella (10-1000) and clustering phenomena were demonstrated, though the clustering was sensitive stochasticity in beat frequencies.…”

Eulerian simulation of complex suspensions and biolocomotion in three dimensions

“…We also revisit the classic problem of lid-driven cavity flows ( 33 – 35 ), now containing an immersed deformable sphere, and provide detailed data to serve as benchmarks. There has been sustained interest in swimming and biolocomotion in recent decades, yet reduced-dimension models of the swimmer ( 36 – 38 ) are often the state of the art. We present a model of a swimming organism using full volumetric actuation, demonstrating the RMT as a simulation tool for exploring a broader range of swimming modalities.…”