  The chemotaxis--Navier--Stokes system \begin{equation*}\label{0.1}
  \left\{\begin{array}{ll}
  n_t+u\cdot \nabla n=\triangle n-\chi\nabla\cdotp \left(\displaystyle\frac n
{c}\nabla c\right)+n(r-\mu n),
  c_t+u\cdot \nabla c=\triangle c-nc,
  u_t+ (u\cdot \nabla) u=\Delta u+\nabla P+n\nabla\phi,
  \nabla\cdot u=0,
  \end{array}\right. \end{equation*} is considered in a bounded smooth domain
$\Omega \subset \mathbb{R}^2$, where $\phi\in W^{1,\infty}(\Omega)$, $\chi>0$,
$r\in \mathbb{R}$ and $\mu> 0$ are given parameters. It is shown that there
exists a value $\mu_*(\Omega,\chi, r)\geq 0$ such that whenever $
\mu>\mu_*(\Omega,\chi, r)$, the global-in-time classical solution to the system
is uniformly bounded with respect to $x\in \Omega$. Moreover, for the case
$r>0$, $(n,c,\frac {|\nabla c|}c,u)$ converges to $(\frac r \mu,0,0,0)$ in
$L^\infty(\Omega)\times L^\infty(\Omega)\times L^p(\Omega)\times
L^\infty(\Omega)$ for any $p>1$ exponentially as $t\rightarrow \infty$, while
in the case $r=0$, $(n,c,\frac {|\nabla c|}c,u)$ converges to $(0,0,0,0)$ in
$(L^\infty(\Omega))^4$ algebraically. To the best of our knowledge, these
results provide the first precise information on the asymptotic profile of
solutions in two dimensions.
