Integral transform solution of the Navier-Stokes equations in full cylindrical regions with streamfunction formulation

Abstract

A hybrid numerical-analytical solution based on the generalized integral transform technique is proposed to handle the two-dimensional Navier-Stokes equations in cylindrical coordinates, expressed in terms of the streamfunction-only formulation. The proposed methodology is illustrated in solving steady-state incompressible laminar flow of Newtonian fluids in the developing region of a circular tube. The flow modeling also considers two limiting inlet conditions, namely, uniform velocity profile representing a parallel flow, and zero vorticity that characterizes irrotational inlet flow. The integral transform analysis for such a full cylindrical region brings up singularities at the channel centerline, and, as previously described in a work dealing with the boundary-layer formulation, a way to alleviate this difficulty is to adopt a recently introduced fourth-order eigenvalue problem as the basis for the eigenfunction expansion. A thorough convergence analysis of the proposed expansion is then undertaken, for different values of Reynolds number, and a set of reference results for the velocity distributions and friction factors are then presented in tabular and graphical forms. © 2009 John Wiley & Sons, Ltd.