  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
$0<\alpha<\frac{1}{2}$.

Analysis of PDEs