# Support

## Support Options

• ### Knowledge Base

Find information on common issues.

• ### Ask the Community

Ask questions and find answers from other users.

• ### Wish List

Suggest a new site feature or improvement.

• ### Support Tickets

Check on status of your tickets.

## Green's Function For Radial Heat Conduction in Two-Region Composite Cylinders With Perfect Boundary Contact

Sandia National Laboratories, Retired

#### Abstract

This paper presents the derivation of the Green's function for composite cylinders 0<r<a and r>a in perfect contact on the surface r=a. Because the source function can be in either region, there are two pairs of functions which define the Green's function. Each pair is the solution to a two-region conduction problem with zero initial temperatures and continuity of temperature and flux on the cylinder r=a. These pairs are used in conjunction with a general formula to get the solution to other problems where the cylinders are in perfect contact, but may have non-zero initial conditions and/or possibly a distribution of internal heat sources. The Green's function approach and a direct approach agree when applied to three problems with known solutions. A fourth problem illustrates a complication where the Laplace transform of the general solution is more useful.

Keywords Two Regions Heat Conduction Unsteady State Laplace Transform

#### References

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

[2] Amos DE (2006) 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, http://nanohub.org/resources/12390

[5] Amos, DE (2012), Green's Functions For Heat Conduction in Adjacent Materials, http://nanohub.org/resources/12856

[6] Amos, DE (2012), Theory of Heat Conduction for Two-region Problems Using Green's Functions, http://nanohub.org/resources/13671

[7] Amos, DE (2012), 1-D Green's Functions For Heat Conduction Between Semi-infinite Slabs With Perfect and Imperfect Boundary Contact, http://nanohub.org/resources/15237.

[8] Cole DC, Beck JV, Haji-SheikhA, Litkouhi B (2010) Heat Conduction Using Green's Functions, 2nd Ed., CRC Press, 643p.

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

#### Cite this work

Researchers should cite this work as follows:

• Donald E. Amos (2013), "Green's Function For Radial Heat Conduction in Two-Region Composite Cylinders With Perfect Boundary Contact," https://nanohub.org/resources/16482.

### Legal

nanoHUB.org, a resource for nanoscience and nanotechnology, is supported by the National Science Foundation and other funding agencies. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.