Boundary layer problem on chemotaxis-Navier-Stokes system with Robin boundary conditions

TL;DR

This paper investigates the boundary layer formation in a chemotaxis-Navier-Stokes system with Robin boundary conditions, analyzing how solutions behave as oxygen diffusion rate approaches zero, revealing boundary layer effects.

Contribution

It provides a rigorous analysis of boundary layer effects in a chemotaxis-Navier-Stokes system with Robin boundary conditions, including the asymptotic behavior of solutions as diffusion vanishes.

Findings

Boundary layer effects occur as oxygen diffusion rate approaches zero.

Radial solutions exhibit boundary layer thickness of order ( ext{ extbackslash}varepsilon^ ext{ extbackslash}alpha) with 0< ext{ extbackslash}alpha<rac{1}{2}.

Solutions' gradients become concentrated near the boundary in the zero-diffusion limit.

Abstract

This paper is concerned with the boundary layer problem on a chemotaxis-Navier-Stokes system modelling boundary layer formation of aerobic bacteria in fluid. Completing the system with physical Robin-type boundary conditions for oxygen, no-flux and Dirichlet boundary conditions for bacteria and fluid velocity, we show that the gradients of its radial solutions in a region between two concentric spheres possessing boundary layer effects as the oxygen diffusion rate $\varepsilon$ goes to zero and the boundary-layer thickness is of order $\mathcal{O}(\varepsilon^\alpha)$ with $0<\alpha<\frac{1}{2}$.