Analytical aspects of heat flow in random two-component laminates
Authors:
J. M. Hill and J. R. Willis
Journal:
Quart. Appl. Math. 46 (1988), 353-364
MSC:
Primary 73B30; Secondary 35K57, 80A20
DOI:
https://doi.org/10.1090/qam/950607
MathSciNet review:
950607
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Abstract: Analytical aspects are examined for the coupled reaction-diffusion equations arising for the expected conditional temperatures for heat flow in a random two-component laminate with heat flowing in a direction perpendicular to the laminae. The basic equations derived by Clarke [1] involve new terms not previously encountered within the contexts of either diffusion or heat flows. Various results and solution procedures are developed for both the coupled system and the underlying fourth-order equation. Basic source solutions for the coupled system are derived which are extended to general solutions of the coupled system with the new terms. These general solutions involve two arbitrary heat functions and display explicitly the dependence of solutions on the various parameters of the model. However, these general solutions appear not to be as useful in the context of the solution of boundary value problems as the corresponding results for the coupled system without the new terms. A simple illustrative example is provided for the utilization of such general solutions for a specific problem.
N. S. Clarke, Heat diffusion in random laminates, Quart. J. Mech. Appl. Math. 37, 195–230 (1984)
N. S. Clarke, On the effective diffusivity model for heat diffusion in inclined random laminates, Quart. J. Mech. Appl. Math. 39, 139–152 (1986)
- J. M. Hill, On the solution of reaction-diffusion equations, IMA J. Appl. Math. 27 (1981), no. 2, 177–194. MR 626272, DOI https://doi.org/10.1093/imamat/27.2.177
- A. I. Lee and J. M. Hill, On the solution of boundary value problems for fourth order diffusion, Acta Mech. 46 (1983), no. 1-4, 23–35. MR 696459, DOI https://doi.org/10.1007/BF01176762
- Alexander I. Lee and James M. Hill, On the general linear coupled system for diffusion in media with two diffusivities, J. Math. Anal. Appl. 89 (1982), no. 2, 530–557. MR 677744, DOI https://doi.org/10.1016/0022-247X%2882%2990116-0
J. R. Willis and J. M. Hill, On heat diffusion in random laminates, Quart. J. Mech. Appl. Math. 41 (1988)
N. S. Clarke, Heat diffusion in random laminates, Quart. J. Mech. Appl. Math. 37, 195–230 (1984)
N. S. Clarke, On the effective diffusivity model for heat diffusion in inclined random laminates, Quart. J. Mech. Appl. Math. 39, 139–152 (1986)
J. M. Hill, On the solution of reaction-diffusion equations, IMA J. Appl. Math. 27, 177–194 (1981)
A. I. Lee and J. M. Hill, On the solution of boundary value problems for fourth order diffusion, Acta Mech. 46, 23–35 (1983)
A. I. Lee and J. M. Hill, On the general linear coupled system for diffusion in media with two diffusivities, J. Math. Anal. Appl. 89, 530–557 (1982)
J. R. Willis and J. M. Hill, On heat diffusion in random laminates, Quart. J. Mech. Appl. Math. 41 (1988)
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Article copyright:
© Copyright 1988
American Mathematical Society
