Summary: Our aim is to derive an upper bound on the dimension of the attractor for Navier-Stokes equations with nonhomogeneous boundary conditions. In space dimension two, for flows in general domains with prescribed tangential velocity at the boundary, we obtain a bound on the dimension of the attactor of the form \(c\text{Re}^{3/2}\), where Re is the Reynolds number. This improves significantly on previous bounds which were exponential in Re.