Transient heat conduction in a medium with multiple spherical cavities. (English) Zbl 1156.80324
Summary: 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.
MSC:
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
| 80M35 | Asymptotic analysis for problems in thermodynamics and heat transfer |
Keywords:
solids; thermal effects; parabolic partial differential equation; Laplace transform; surface spherical harmonics; asymptotic seriesReferences:
| [1] | Traytak, The diffusive interaction in diffusion-limited reactions: the steady-state case, Chemical Physics Letters 197 (3) pp 247– (1992) |
| [2] | Traytak, The diffusive interaction in diffusion-limited reactions: the time-dependent case, Chemical Physics 193 pp 351– (1995) |
| [3] | Lamb, Hydrodynamics pp 130– (1932) |
| [4] | Jeffery, On a form of the solution of Laplace’s equation suitable for problems relating to two spheres, Proceedings of the Royal Society of London, Series A 87 (593) pp 109– (1912) · JFM 43.0855.01 |
| [5] | Davis, Two charged spherical conductors in a uniform electric field: forces and field strength, Quarterly Journal of Mechanics and Applied Mathematics 17 (4) pp 499– (1964) · Zbl 0123.23607 |
| [6] | Ross, The Dirichlet boundary value problem for two non-overlapping spheres, Bulletin of the Australian Mathematical Society 2 pp 237– (1970) · Zbl 0186.43201 |
| [7] | Jeffrey, Conduction through a random suspension of spheres, Proceedings of the Royal Society of London, Series A 335 pp 355– (1973) |
| [8] | Cheng, Effective conductivity of periodic arrays of spheres with interfacial resistance, Proceedings: Mathematical, Physical and Engineering Sciences 453 (1956) pp 145– (1997) |
| [9] | Cheng, On the method of images for systems of closely spaced conducting spheres, SIAM Journal on Applied Mathematics 61 (4) pp 1324– (2001) |
| [10] | Janković, Three-dimensional flow through large numbers of spheroidal inhomogeneities, Journal of Hydrology 226 pp 224– (1999) |
| [11] | Felderhof, Concentration dependence of the rate of diffusion-controlled reactions, Journal of Chemical Physics 64 (11) pp 4551– (1976) |
| [12] | Chang, Rate of heat conduction from a heated sphere to a matrix containing passive spheres of a different conductivity, Journal of Applied Physics 59 (10) pp 3375– (1986) |
| [13] | Traytak, Convergence of a reflection method for diffusion-controlled reactions on static sinks, Physica A 362 pp 240– (2006) |
| [14] | Traytak, Boundary-value problems for the diffusion equation in domains with disconnected boundary, Diffusion Fundamentals 2 pp 38– (2005) |
| [15] | Carslaw, Conduction of Heat in Solids (1959) |
| [16] | Greengard, The Rapid Evaluation of Potential Fields in Particle Systems (1988) · Zbl 1001.31500 |
| [17] | Shen, An adaptive fast multipole boundary element method for three-dimensional potential problems, Computational Mechanics 39 pp 681– (2007) · Zbl 1198.74113 |
| [18] | Erofeenko, Teoremy Slozheniia: Spravochnik (1989) |
| [19] | Lopez-Fernandez, On the numerical inversion of the Laplace transform of certain holomorphic mappings, Applied Numerical Mathematics 51 pp 289– (2004) |
| [20] | Uhm, Generalization of Wilemski-Fixman-Weiss decoupling approximation to the case involving multiple sinks of different sizes, shapes, and reactivities, Journal of Chemical Physics 125 pp 054911-1-8– (2006) |
| [21] | Tsao, Rate of diffusion-limited reactions in a cluster of spherical sinks, Journal of Chemical Physics 115 (8) pp 3827– (2001) |
| [22] | Feng, Electrostatic interaction between two charged dielectric spheres in contact, Physical Review E 62 (2) pp 2891– (2000) |
| [23] | Gordeliy, Transient heat conduction in a medium with two circular cavities: Semi-analytical solution, International Journal of Heat and Mass Transfer 51 pp 3556– (2008) · Zbl 1148.80315 |
| [24] | Arfken, Mathematical Methods for Physicists (1970) |
| [25] | Abramowitz, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1972) · Zbl 0543.33001 |
| [26] | Carslaw, Operational Methods in Applied Mathematics pp 279– (1963) |
| [27] | Koshlyakov, Differential Equations of Mathematical Physics (1964) |
| [28] | Golub, Matrix Computations (1996) |
| [29] | Hetnarski, An algorithm for generating some inverse Laplace transforms of exponential form, Journal of Applied Mathematics and Physics (ZAMP) 26 pp 249– (1975) · Zbl 0313.44001 |
| [30] | Greengard, A fast algorithm for particle simulations, Journal of Computational Physics 73 pp 325– (1987) · Zbl 0629.65005 |
| [31] | Anderson, An implementation of the fast multipole method without multipoles, SIAM Journal on Scientific and Statistical Computing 13 (4) pp 923– (1992) · Zbl 0754.65101 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
