  We study the nonhomogeneous boundary value problem for Navier--Stokes
equations of steady motion of a viscous incompressible fluid in a
two--dimensional bounded multiply connected domain
$\Omega=\Omega_1\setminus\bar{\Omega}_2, \;\bar\Omega_2\subset \Omega_1$. We
prove that this problem has a solution if the flux $\F$ of the boundary value
through $\partial\Omega_2$ is nonnegative. The proof of the main result uses
the Bernoulli law for a weak solution to the Euler equations and the one-side
maximum principle for the total head pressure corresponding to this solution.
