This paper presents a direct domain decomposition method, coupled with a Chebyshev collocation approximation, for solving the incompressible Navier-Stokes equations in the vorticity-streamfunction formulation. The method is based on the influence matrix technique used to treat the lack of vorticity boundary conditions on no-slip walls as well as to enforce the continuity conditions at the interfaces between adjacent subdomains. The multi-domain approach is proposed in order to extend the use of spectral approximations to non-rectangular geometries and singular solutions. It is applied to the computation of a four domain configuration, corresponding to a forced throughflow in a rotating channel-cavity system which is important in air cooling devices and cannot be modeled by single-domain spectral approximations.