In this paper, the chemotaxis-fluid system is considered \(\begin{aligned} \left\{ \begin{array}{lllc} & n_t & = \Delta n-\nabla \cdot (n F(|\nabla c|^2)\nabla c)-u\cdot \nabla n, & {(x,t)\in } ~\Omega \times (0,T),\\ \displaystyle & c_t & =\Delta c-c+n-u\cdot \nabla c, & {(x,t)\in } ~\Omega \times (0,T),\\ \displaystyle & u_t & =\Delta u+\nabla P+n\nabla \phi , \quad \nabla \cdot u=0,& {(x,t)\in } ~\Omega \times (0,T) \end{array} \right. \end{aligned}\) in a bounded smooth domain \(\Omega \subset \mathbb {R}^3\) associated with homogenous Neumann boundary conditions for n and c, and Dirichlet boundary condition for u. Here, \(0\le F(s)\le K(1+s)^{-\frac{\alpha }{2}}\) for \(s\ge 0\) . It was obtained in (M. Winkler, Conditional estimates in three-dimensional chemotaxis-Stokes systems and application to a Keller-Segel-fluid model accounting for gradient-dependent flux limitation. J. Differential Equations 281: 33–57, 2021) that if \(\alpha >\frac{1}{2}\) , the system possesses a global bounded solution. However, no information is known when \(\alpha =\frac{1}{2}\) . In the present work, we consider the critical case and show existence of global bounded solution if the total mass \(\int _{\Omega }n_0\) is small.
