  The local existence of solutions to nonhomogeneous Navier-Stokes equations in
cylindrical domains with arbitrary large flux is demonstrated. The existence is
proved by the method of successive approximations. To show the existence with
the lowest possible regularity the special Besov spaces called the
Sobolev-Slobodetskii spaces are used. The inflow and outflow are prescribed on
the parts of the boundary which are perpendicular to the $x_3$-axis. Since the
inflow and outflow are positive the crucial point of this paper is to verify
that $x_3$-coordinate of velocity is also positive. Finally, we conclude the
local existence such that the velocity belongs to $W_{\sigma}
^{2+{s},1+{s}/2}(\Omega^t)$, the gradient of pressure to $W_{\sigma}
^{{s},{s}/2}(\Omega^t)$ and the density to $W_{r,\infty}^{1,1}(\Omega^t)$,
where ${s}\in(0,1)$, ${\sigma} >3/{s}$, $r>5/{s}$, $r>{\sigma} $.
