Solutions of Heat-Conduction Problems With Nonseparable Domains

[+] Author and Article Information
Daniel Dicker

Department of Engineering Analysis, State University of New York, Stony Brook, N. Y.

M. B. Friedman

Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, N. Y.

J. Appl. Mech 30(4), 493-499 (Dec 01, 1963) (7 pages) doi:10.1115/1.3636608 History: Received September 29, 1961; Revised February 23, 1963; Online September 16, 2011


A method is presented for obtaining eigenfunctions of and solutions to the transient heat-conduction equation for a wide class of three-dimensional convex hexahedral domains and two-dimensional convex quadrilateral domains having straight or curved boundaries for which separation of variables cannot be applied. The method is employed to solve for the temperature distribution in a trapezoidal domain, initially at zero temperature, the boundaries of which are subjected to suddenly applied values at the initial instant. The solution is obtained in the form of a series and an examination of successive terms indicates fairly rapid convergence; it is found that the one-term solution yields almost as good values as a four-term solution, which is significant since the former is obtained with little effort. An independent method is utilized for obtaining the steady-state solution, i.e., t → ∞, and it is found that all approximations by the former method are substantially equal to the correct value for this case.

Copyright © 1963 by ASME
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