Annals of Mathematical Sciences and Applications

Volume 9 (2024)

Number 1

A second-order partitioned method for bioconvective flows with concentration dependent viscosity

Pages: 141 – 184

DOI: https://dx.doi.org/10.4310/AMSA.2024.v9.n1.a5

Authors

Madeline Edwards (Department of Neuroscience, University of Pittsburgh, Pittsburgh, Pennsylvania, U.S.A.)

Martina Bukač (Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Indiana, U.S.A.)

Catalin Trenchea (Department of Mathematics, University of Pittsburgh, Pennsylvania, U.S.A.)

Abstract

This work is focused on the mathematical and computational modeling of bioconvection, which describes the mixing of fluid and micro-organisms exhibiting negative geotaxis movement under the force of gravity. The collective population moves towards the surface of the fluid, generating a Rayleigh–Taylor instability, where initial fingers of organisms plummet to the bottom. The inherent drive to swim vertically generates large collective flow patterns that persist in time. We model the flow using the Navier–Stokes equations for an incompressible, viscous fluid, coupled with the transport equation describing the concentration of the micro-organisms. We use a nonlinear semigroup approach to prove the existence of solutions. We propose a partitioned, second-order, time adaptive numerical method based on the Cauchy’s one-legged ‘$\theta$-like’ scheme. We prove that the method is energy-stable, and for small time steps, the iterative procedure in the partitioned algorithm is linearly convergent. The numerical results confirm the expected second-order of accuracy. We also present a computational study of a chaotic system describing bioconvection of motile flagellates.

Keywords

bioconvection, partitioned method, adaptive time-stepping, chaotic flow

2010 Mathematics Subject Classification

35B30, 65M12, 76Z99

Received 19 June 2023

Accepted 5 February 2024

Published 5 April 2024