The initial boundary value problem for the non-stationary Navier-Stokes equations is studied in 2D bounded domain with a power-cusp singular point $O$ on the boundary. The case of the boundary value with a nonzero flow rate is considered. In this case there is a source/sink in O and the solution necessary has infinite energy integral.The formal asymptotic expansion of the solution near the singular point is constructed. The justification of the asymptotic decomposition and the existence of a solution are proved.
