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