A novel pseudospectral solution procedure for the Navier-Stokes equations in the velocity-vorticity formulation, suitable to simulate flows with two nonperiodic directions, is proposed. An influence matrix method, including a tau correction procedure, has been employed to elicit an algorithmic substitute for the a priori lacking boundary conditions for the vorticity. Following O. Daube ( J. Comput. Phys. 103, 402 (1992)) the influence matrix is built by enforcing either the definition of the vorticity or the continuity equation at the boundary of the domain. The influence matrix is nonsingular in both cases. The order of the influence matrix is twice larger than one would expect from analogous solution methods based on finite differences or finite elements. The spatial discretization is based on a 2D Chebyshev expansion on a nonstaggered grid of collocation points, and the boundary conditions are imposed via the Lanczos tau procedure. The time marching scheme is Adams-Bashforth for the advection term and Crank-Nicolson for the viscous term. The proposed scheme yields machine accurate divergence-free flow fields, and the definition of the vorticity is satisfied within machine accuracy.