## Green's Functions For Heat Conduction in Adjacent Materials

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#### Abstract

This paper considers classical linear, transient heat conduction problems set in Regions 1 and 2 defined by the half planes x>0 and x<0. These regions are initially at zero temperature and have distinct, constant physical properties. The two regions are considered to be in contact with each other along the boundary x=0. The measure of contact can range from perfect contact with maximum heat transfer to highly resistive contact with zero heat transfer. The energy source is an instantaneous point source of heat at time zero of strength Q at the point (x',y'). In a three-dimensional setting we have semi-infinite slabs with a heat resistive boundary on the plane x=0. The instantaneous source emits a uniform pulse of heat along a straight line in the z direction piercing the (x,y) plane at the point (x',y').

The case of perfect contact along the common boundary is considered first and then generalized to the thermally resistive case with a heat transfer coefficient h. The solution for temperatures in each half plane for both the instantaneous and continuous point sources is obtained by a classical application of the Laplace transform. The results are integral representations which, when taken to be real, only converge for a subset of the (x,y) plane. However, by considering paths of integration in the complex plane, complete solutions are generated as line integrals. Quadratures are required for numerical evaluation and references to the computation of the non-elementary complex functions are given. Sources placed along the boundary are of special interest and known results for a uniform source emitting heat continuously along the negative y-axis are duplicated using the Green's function approach.

By using the method of images, Green's functions for the two-region upper half plane problem in quadrants 1 and 2 having zero temperature or zero flux on the boundary y=0 can be constructed.

Keywords Two Regions Heat Conduction Unsteady State Laplace Transform

#### References

References

[1] Abramowitz S, Stegun IA (1965) Handbook of Mathematical Functions, AMS 55, Dover Publications Inc., New York, 1046pp

[2] Amos DE (2012) Handbook of Integrals Related to Heat Conduction and Diffusion, http://nanohub.org/resources/13874

[3] Amos DE, Beck JV, de Monte F (2011) Transient Heat Conduction in Adjacent Quadrants Separated by a Thermal Resistance, http://nanohub.org/resources/12465

[4] Amos DE (2011) Transient Heat Conduction in Adjacent Materials Heated on Part of the Common Boundary, https://nanohub.org/resources/12390

[5] Cole KD, Beck JV, Haji-Sheikh A, Litkouhi B (2010) Heat Conduction Using Green's Functions, 2nd Edition, CRC Press, Boca Raton, 643pp

[6] Carslaw HS, Jaeger JC (1948) Conduction of Heat in Solids, Oxford Univ Press, London, 386pp

[7] Erdelyi A. et. al. (1954) Tables of Integral Transforms, Vol. 1, Mc Graw-Hill, New York, 391pp

[8] Morse PM, Feshbach H (1953) Methods of Theoretical Physics, Part I, McGraw-Hill, New York, 997pp

[9] Amos DE (1980) Computation of Exponential Integrals, ACM Transactions on Mathematical Software, Vol 6, No. 3, pp365-377

[10] Amos DE (1980) Algorithm 556, Exponential Integrals [S13], ACM Transactions on Mathematical Software, Vol 6, No. 3, pp420-428

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