Pseudo-similarity solutions of the one-dimensional diffusion equation with applications to the phase change problem
Author:
David Langford
Journal:
Quart. Appl. Math. 25 (1967), 45-52
MSC:
Primary 35.62; Secondary 80.00
DOI:
https://doi.org/10.1090/qam/209686
MathSciNet review:
209686
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Abstract: New solutions of the diffusion equation may be used to prescribe both the diffusion potential and the diffusional flow rate, along the moving curve $X = \alpha {\left ( {1 + \beta \cdot T} \right )^{1/2}}$, as arbitrary power series in the variable $\left ( {\alpha {{\left ( {1 + \beta \cdot T} \right )}^{1/2}}} \right )$, where $\alpha$ and $\beta$ are arbitrary constants and $T$ is time.
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
- H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford, at the Clarendon Press, 1947. MR 0022294
- Ruel V. Churchill, Operational mathematics, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1958. 2nd ed. MR 0108696
- F. C. Frank, Radially symmetric phase growth controlled by diffusion, Proc. Roy. Soc. London Ser. A 201 (1950), 586–599. MR 36925, DOI https://doi.org/10.1098/rspa.1950.0080
A. Huber, On the propagation of the melting boundary in a linear conductor (Über das Fortschreiten der Schmelzgrenze in einen linearen Leiter), Z. Angew. Math. Mech. 19, 1–21 (1939). (In German). (Cited by Carslaw and Jaeger [2] and Frank [4])
Lamé and Clapeyron, Memorandum on the solidification by freezing of a liquid sphere, (Memoire sur la solidification par refroidissement d’un globe liquide). Ann. Chimie et Physique, 47, 250–256 (1831). (In French)
David Langford, A closed form solution for the constant velocity solidification of spheres initially at the fusion temperature, British J. Applied Physics 17(2), 286 (Feb. 1966)
David Langford, The freezing of spheres, Int. J. Heat and Mass Transfer, 9(8), 827–828 (August 1966)
David Langford, Stefan’s melting problem, Doctoral Dissertation, Rensselaer Polytechnic Institute, Troy, N. Y., January 1965
- G. A. Martynov, Solution of the inverse Stefan problem in the case of spherical symmetry, Soviet Physics. Tech. Phys. 5 (1960), 215–218 (Russian). MR 0113592
D. V. Redozubov, The solution of certain types of linear thermal problems in limited and semiinfinite regions with motion of the boundary according to $\alpha \beta {\left ( t \right )^{1/2}}$ law, Soviet Physics—Technical Physics, 7(5), 459–461 (Nov. 1962). Translated from Zhurnal Tekhnicheskoi Fiziki, 32(5), 632–637 (May 1962)
J. Stefan, On the theory of ice formation, especially on ice formation in polar seas, (Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere). Sitzungsberichte der Ka[ill]rlichen Akademie Wiss. Wien., Math.-naturwiss. Kl, 98(2a), 965–983 (1890). (In German)
Milton Abramowitz and Irene A. Stegun, Ed., Handbook of mathematical functions, U. S. National Bureau of Standards, Applied Mathematics Series #55, Washington, D. C., June 1964. Second printing, with corrections, Nov. 1964
H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids, 2nd Ed. Oxford Univ. Press, London, 1959
Ruel V. Churchill, Operational mathematics, 2nd Ed. McGraw-Hill, New York, 1958; p. 259 and chapter 9
F. C. Frank, Radially symmetric phase growth controlled by diffusion, Proc. Roy. Soc. Ser. A 201, 586–599 (1950)
A. Huber, On the propagation of the melting boundary in a linear conductor (Über das Fortschreiten der Schmelzgrenze in einen linearen Leiter), Z. Angew. Math. Mech. 19, 1–21 (1939). (In German). (Cited by Carslaw and Jaeger [2] and Frank [4])
Lamé and Clapeyron, Memorandum on the solidification by freezing of a liquid sphere, (Memoire sur la solidification par refroidissement d’un globe liquide). Ann. Chimie et Physique, 47, 250–256 (1831). (In French)
David Langford, A closed form solution for the constant velocity solidification of spheres initially at the fusion temperature, British J. Applied Physics 17(2), 286 (Feb. 1966)
David Langford, The freezing of spheres, Int. J. Heat and Mass Transfer, 9(8), 827–828 (August 1966)
David Langford, Stefan’s melting problem, Doctoral Dissertation, Rensselaer Polytechnic Institute, Troy, N. Y., January 1965
G. A. Martynov, Solution of the inverse Stefan problem in the case of spherical symmetry, Soviet Physics—Technical Physics, 5(2), 215–218 (Aug. 1960). Translated from Zhurnal Tekhnicheskoi Fiziki, 30(2), 239–241 (Feb. 1960)
D. V. Redozubov, The solution of certain types of linear thermal problems in limited and semiinfinite regions with motion of the boundary according to $\alpha \beta {\left ( t \right )^{1/2}}$ law, Soviet Physics—Technical Physics, 7(5), 459–461 (Nov. 1962). Translated from Zhurnal Tekhnicheskoi Fiziki, 32(5), 632–637 (May 1962)
J. Stefan, On the theory of ice formation, especially on ice formation in polar seas, (Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere). Sitzungsberichte der Ka[ill]rlichen Akademie Wiss. Wien., Math.-naturwiss. Kl, 98(2a), 965–983 (1890). (In German)
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Article copyright:
© Copyright 1967
American Mathematical Society
