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The Generalized Integral Transform Technique is employed in the hybrid numerical-analytical solution of the steady two-dimensional Navier±Stokes equations, de-®ned within arbitrarily shaped domains, for incompressible laminar channel¯ow. The formalism is illustrated for the classical test-case of laminar¯ow in a gradual expansion duct. Numerical results with automatic global accuracy control are obtained for suggested values of Reynolds numbers in the literature, and critically compared against previously reported benchmark solutions for the same problem. The relative merits of the proposed approach are then pointed out.The authors would like to acknowledge the ®nancial support provided by CNPq, FUJB, FAPERJ and CNEN, sponsoring agencies in Brazil.

The generalized integral transform technique is employed in the hybrid numerical-analytical solution of the Navier-Stokes equations in streamfunction-only formulation, which govern the incompressible laminar flow of a Newtonian fluid within a parallel plate channel. Owing to the analytic nature of this approach, the outflow boundary condition for an infinite duct is handled exactly, and the error involved in considering finite duct lengths is investigated. The present error-controlled solutions are used to inspect the relative accuracy of previously reported purely numerical schemes and to compare Navier-Stokes and boundary layer formulations for various combinations of inlet conditions and Reynolds number.

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The basic ideas in the generalized integral transform technique are further advanced to allow for the hybrid numerical-analytical solution of the two-dimensional steady Navier-Stokes equations in streamfunctiononly formulation. The classical lid-driven square cavity problem is selected for illustration of the approach. The corresponding biharmonic-type non-linear partial differential equation for the streamfunction is integral transformed in one of the co-ordinates and an infinite system of coupled non-linear ODES for the transformed potential results in the other independent variable. Upon truncation to an appropriate finite order, the ODE system is numerically solved by well-established algorithms with automatic error control devices. The convergence behaviour of the eigenfunction expansion is demonstrated and reference results are provided for typical values of Reynolds number.