  We study the chemotaxis-fluid system \begin{align*}
\left\{\begin{array}{r@{\,}l@{\quad}l@{\,}c} n_{t}&=\Delta
n-\nabla\!\cdot(n\nabla c)-u\cdot\!\nabla n,\ &x\in\Omega,& t>0,\\
c_{t}&=\Delta c-c+f(n)-u\cdot\!\nabla c,\ &x\in\Omega,& t>0,\\ u_{t}&=\Delta
u+\nabla P+n\cdot\!\nabla\phi,\ &x\in\Omega,& t>0,\\ \nabla\cdot u&=0,\
&x\in\Omega,& t>0, \end{array}\right. \end{align*} where
$\Omega\subset\mathbb{R}^2$ is a bounded and convex domain with smooth
boundary, $\phi\in W^{1,\infty}\left(\Omega\right)$ and $f\in C^1([0,\infty))$
satisfies $0\leq f(s)\leq K_0 s^\alpha$ for all $s\in[0,\infty)$, with $K_0>0$
and $\alpha\in(0,1]$. This system models the chemotactic movement of actively
communicating cells in slow moving liquid.
  We will show that in the two-dimensional setting for any $\alpha\in(0,1)$ the
classical solution to this Keller-Segel-Stokes-system is global and remains
bounded for all times.
