Zero-viscosity limit of the chemotaxis-Navier-Stokes equations with the Navier-slip boundary condition

TL;DR

This paper rigorously analyzes the zero-viscosity limit of the chemotaxis-Navier-Stokes equations with Navier-slip boundary conditions, revealing boundary layer behaviors in a 2D setting.

Contribution

It derives boundary layer equations and proves the vanishing viscosity limit for the coupled chemotaxis-fluid system in a half-space.

Findings

Established boundary layer equations for chemotaxis-Navier-Stokes system.

Proved the vanishing viscosity limit in anisotropic conormal Sobolev spaces.

Analyzed the effects of chemotactic and velocity viscosities on flow behavior.

Abstract

The interplay of chemotaxis and diffusion of nutrients or signaling chemicals in bacterial suspensions can produce a variety of structures with locally high concentrations of cells, including phyllotactic patterns, filaments, and concentrations in fabricated microstructures, which is described by the chemotaxis-Navier-Stokes flow by Tuval et al. in 2005. Dombrowski et al. also observed that Bacterial flow in a sessile drop related to those in the Boycott effect of sedimentation can carry bioconvective plumes, viewed from below through the bottom of a petri dish, and the horizontal "turbulence" white line near the top is the air-water-plastic contact line. It's interesting to verify these turbulent phenomena mathematically. For varying chemotactic and velocity viscosities, we derive the boundary layer equations of the chemotaxis-Navier-Stokes system rigorously in a two-dimensional half-space under the Navier-slip boundary condition and obtain the vanishing viscosity limit of the 2D chemotaxis-fluid coupled system in the anisotropic conormal Sobolev spaces.