||A numerical study of the streamfunction-vorticity formulation of the two-dimensional Navier-Stokes equations is developed using a boundary integral condition for the vorticity. The method is appropriate for steady viscous channel flows and flows past rigid bodies where vorticity is generated by the viscous shear, and concentrated near the no-slip boundaries. The boundary integral method is presented along with the discussion of the kinetic and kinematic aspects of the problem. An iterative method is required to reduce the integral boundary condition to a sequence of Dirichlet boundary conditions. The iterative process is incorporated into time marching and nonlinear iterations algorithm used for the vorticity transport equation. The convergence of the iterative process is very sensitive to relaxation parameters. A convergence analysis of the method for the steady Stokes problem leads to the selection of these parameters; the technique is applied to the Navier-Stokes equations. The vorticity transport equation is discretized in space using a Galerkin/Least-squares finite element formulation; this leads to a system of ordinary differential equations, which is solved using a predictor-corrector algorithm.