Pergamon Int. Comm. Heat Mass Transfer, Vol. 26, No. 1, pp. 125-134, 1999 Copyright © 1999 Elsevier Science Ltd Printed in the USA. All rights reserved 0735-1933/99/S-see front matter PII S0735-1933(98)00128-6 SOLUTION OF THE COMPLEX EIGENVALUE PROBLEM FOR TRANSIENT LAMINAR FORCED CONVECTION INSIDE TUBES Mazhar Onsal Department of Mechanical Engineering University of Gaziantep, 27310 Gaziantep, Turkey (Communicated by J.W. Rose) ABSTRACT Two solutions are presented for the complex eigenvalue problem arising in laminar transient forced convection inside tubes. The "method of matched asymptotic expansions" is applied to the complex eigenvalue problem yielding a closed form algebraic expression for the complex eigenvalues and two asymptotic expansions, one valid near the tube centerline and another valid near the tube wall. These expansions are combined yielding a single uniformly valid composite asymptotic expansion for the complex eigenfunctions. The present asymptotic solution is accurate for all normalized complex eigenfunctions except for those corresponding to the smaller eigenvalues. Another solution of the complex eigenvalue problem, which is accurate for the smaller eigenvalues, is presented which is based on the finite Bessel transform method. This latter solution is utilized for checking the accuracy of the closed form asymptotic solutions for the eigenvalues and the eigenfunctions of the complex eigenvalue problem. © 1999 Elsevier Science Ltd Introduction Difficulties associated with analytical solution of the complex eigenvalue problems arising in internal transient forced convection were previously recognized by Mikhailov[1] and by Kakaq and Yener[2]. In order to by-pass the analysis of complex eigenvalue problems, Cotta and OZl~tk[3] applied a variation of the "generalized integral transform technique" and obtained solutions to the transient laminar forced convection problem inside parallel plate channels and tubes considering fully developed velocity distributions and sinusoidal variations of inlet temperature. Kim, Cotta and (~)Zl§lk[4] presented solutions to the same problem for arbitrarily shaped temporal inlet temperature variations. Solutions of the problems reported in References [3] and [41 are based on applications of a variation of the "generalized integral transform technique" derived from classical eigenvalue problems and require numerical solutions of systems of algebraic or ordinary differential equations. 125