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
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
Copyright © 1996 Xanming Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.