A Diffuse-Domain Based Numerical Method for a Chemotaxis-Fluid Model

TL;DR

This paper introduces a novel diffuse-domain numerical method for simulating chemotaxis-fluid interactions in complex geometries, capturing bacterial behavior in sessile drops with high accuracy and positivity preservation.

Contribution

The paper develops a new diffuse-domain approach and a high-resolution numerical scheme for coupled chemotaxis-fluid systems in complex geometries, ensuring convergence and positivity.

Findings

The method accurately captures bacterial patterns in various droplet shapes.

Numerical experiments demonstrate the model's ability to simulate chemotactic phenomena.

The diffuse-domain approach simplifies boundary handling in complex geometries.

Abstract

In this paper, we consider a coupled chemotaxis-fluid system that models self-organized collective behavior of oxytactic bacteria in a sessile drop. This model describes the biological chemotaxis phenomenon in the fluid environment and couples a convective chemotaxis system for the oxygen-consuming and oxytactic bacteria with the incompressible Navier-Stokes equations subject to a gravitational force, which is proportional to the relative surplus of the cell density compared to the water density. We develop a new positivity preserving and high-resolution method for the studied chemotaxis-fluid system. Our method is based on the diffuse-domain approach, which we use to derive a new chemotaxis-fluid diffuse-domain (cf-DD) model for simulating bioconvection in complex geometries. The drop domain is imbedded into a larger rectangular domain, and the original boundary is replaced by a diffuse interface with finite thickness. The original chemotaxis-fluid system is reformulated on the larger domain with additional source terms that approximate the boundary conditions on the physical interface. We show that the cf-DD model converges to the chemotaxis-fluid model asymptotically as the width of the diffuse interface shrinks to zero. We numerically solve the resulting cf-DD system by a second-order hybrid finite-volume finite-difference method and demonstrate the performance of the proposed approach on a number of numerical experiments that showcase several interesting chemotactic phenomena in sessile drops of different shapes, where the bacterial patterns depend on the droplet geometries.