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TOPOLOGY OF STEADY HEAT CONDUCTION IN A SOLID SLAB SUBJECT TO A NONUNIFORM BOUNDARY CONDITION: THE CARSLAW–JAEGER SOLUTION REVISITED

Published online by Cambridge University Press:  18 February 2013

R. G. KASIMOVA  and
YU. V. OBNOSOV
R. G. KASIMOVA*
Affiliation:
German University of Technology in Oman, Muscat, Sultanate of Oman
YU. V. OBNOSOV*
Affiliation:
Institute of Mathematics and Mechanics, Kazan Federal University, Kazan, Russia
*
✉rouzalia.kasimova@gutech.edu.om
✉yurii.obnosov@gmail.com
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Abstract

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Temperature distributions recorded by thermocouples in a solid body (slab) subject to surface heating are used in a mathematical model of two-dimensional heat conduction. The corresponding Dirichlet problem for a holomorphic function (complex potential), involving temperature and a heat stream function, is solved in a strip. The Zhukovskii function is reconstructed through singular integrals, involving an auxiliary complex variable. The complex potential is mapped onto an auxiliary half-plane. The flow net (orthogonal isotherms and heat lines) of heat conduction is compared with the known Carslaw–Jaeger solution and shows a puzzling topology of three regimes of energy fluxes for temperature boundary conditions common in passive thermal insulation. The simplest regime is realized if cooling of a shaded zone is mild and heat flows in a slightly distorted “resistor model” flow tube. The second regime emerges when cooling is stronger and two disconnected separatrices demarcate the back-flow of heat from a relatively hot segment of the slab surface to the atmosphere through relatively cold parts of this surface. The third topological regime is characterized by a single separatrix with a critical point inside the slab, where the thermal gradient is nil. In this regime the back-suction of heat into the atmosphere is most intensive. The closed-form solutions obtained can be used in assessment of efficiency of thermal protection of buildings.

Keywords

two-dimensional heat conductionLaplace’s equationheat linesisothermscomplex potentialconformal mappings
Type
Research Article
Information
The ANZIAM Journal , Volume 53 , Issue 4 , April 2012 , pp. 308 - 320
Copyright
Copyright ©2013 Australian Mathematical Society

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