  This paper is concerned with the following quasilinear
chemotaxis--Navier--Stokes system with nonlinear diffusion and rotation $$
\left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta
n^m-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in \Omega, t>0,
c_t+u\cdot\nabla c=\Delta c-nc,\quad x\in \Omega, t>0,\\ u_t+\kappa(u \cdot
\nabla)u+\nabla P=\Delta u+n\nabla \phi ,\quad x\in \Omega, t>0,\\ \nabla\cdot
u=0,\quad x\in \Omega, t>0 \end{array}\right.\eqno(CNF) $$ is considered under
the no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition
for $u$ in a three-dimensional convex domain $\Omega\subseteq \mathbb{R}^3$
with smooth boundary, which describes the motion of oxygen-driven bacteria in a
fluid. Here % $\Omega\subseteq \mathbb{R}^3$ is a , $\kappa\in \mathbb{R}$ and
$S$ denotes the strength of nonlinear fluid convection and a given
tensor-valued function, respectively. Assume $m>\frac{10}{9}$ and $S$ fulfills
$|S(x,n,c)| \leq S_0(c)$ for all $(x,n,c)\in \bar{\Omega} \times [0,
\infty)\times[0, \infty)$ with $S_0(c)$ nondecreasing on $[0,\infty)$, then for
any reasonably regular initial data, the corresponding initial-boundary problem
$(CNF)$ admits at least one global weak solution.
