Eigenfunction Expansion Solution for Boundary-Layer Equations in Cylindrical Coordinates: Simultaneously Developing Flow in Circular Tubes

Abstract

The Generalized Integral Transform Technique (GITT) is employed, via a novel eigenfunction expansion, in the solution of the steady-state continuity, momentum, and energy equations under the boundary-layer formulation and cylindrical coordinates, and applied to the solution of simultaneously developing laminar flow inside circular ducts. The streamfunction formulation is adopted to automatically satisfy the continuity equation and to eliminate the pressure field. A fourth-order eigenvalue problem is thus considered for the velocity field, eliminating the difficulties associated with the singularity at the channel centerline through this recently introduced expansion basis. A thorough analysis of convergence behavior is undertaken for both the velocity and temperature proposed eigenfunction representations, and here illustrated for representative values of governing parameters and positions along the channel. Results for quantities associated with applications, such as the product of the friction factor–Reynolds number and Nusselt numbers, are also computed along the entrance region for different values of the governing parameters, and tabulated for reference purposes. Critical comparisons with previous results in the literature are also performed, in order to validate the numerical code developed and to inspect the adequacy of previously proposed approximate solutions.

The authors would like to acknowledge the financial support provided by CNPq and FAPERJ.

Notes

^a Hornbeck [35].

^b Liu [36].

N = NV = NT.

^a Pr = 0.7.

^b Pr = 2.

^c Pr = 5.

N  = NV = NT.

^a Graphical results of Hornbeck [37].

^b Manohar [32].

^c Hwang [32].

^a Graphical results of Hornbeck [37].

^a Graphical results of Hornbeck [37].