Mathematical Problems in Engineering 
Volume 3 (1996), Issue 1, Pages 75-94
doi:10.1155/S1024123X97000495

On the solution of the differential equation occurring in the problem of heat convection in laminar flow through a tube with slip—flow

Xanming Wang,1 Huaijin Gu,2 and Duli Yu1

 1Mechanical & Industrial Engineering Department, Louisiana Tech University, Ruston 71272, Louisiana, USA
2Department of Mathematics & Statistics, Louisiana Tech University, Ruston 71272, Louisiana, USA
Received 7 March 1996

Abstract

A technique is developed for evaluation of eigenvalues in solution of the differential equation d2y/dr2+(1/r)dy/dr+λ2(β−r2)y=0 which occurs in the problem of heat convection in laminar flow through a circular tube with silp-flow (β>1). A series solution requires the expansions of coeffecients involving extremely large numbers. No work has been reported in the case of β>1, because of its computational complexity in the evaluation of the eigenvalues. In this paper, a matrix was constructed and a computational algorithm was obtained to calculate the first four eigenvalues. Also, an asymptotic formula was developed to generate the full spectrum of eigenvalues. The computational results for various values of β were obtained.