Transient heat conduction in a medium with multiple spherical cavities

• Published in: International journal for numerical methods in engineering 2009-02, Vol.77 (6), p.751-775
• Main Authors: Gordeliy, Elizaveta, Crouch, Steven L., Mogilevskaya, Sofia G.
• Format: Article
• Language: English
• Subjects: asymptotic series; Computational techniques; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Heat conduction; Heat transfer; Laplace transform; Mathematical methods in physics; parabolic partial differential equation; Physics; solids; surface spherical harmonics; thermal effects
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.2430

This paper considers a transient heat conduction problem for an infinite medium with multiple non‐overlapping spherical cavities. Suddenly applied, steady Dirichlet‐, Neumann‐ or Robin‐type boundary conditions are assumed. The approach is based on the use of the general solution to the problem of a single cavity and superposition. Application of the Laplace transform and the so‐called addition theorem results in a semi‐analytical transformed solution for the case of multiple cavities. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. A large‐time asymptotic series for the temperature is obtained. The limiting case of infinitely large time results in the solution for the corresponding steady‐state problem. Several numerical examples that demonstrate the accuracy and the efficiency of the method are presented. Copyright © 2008 John Wiley & Sons, Ltd.