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

• Published in: The ANZIAM journal Vol. 53; no. 4; pp. 308 - 320
• Main Authors: KASIMOVA, R. G., OBNOSOV, YU. V.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.04.2012
• Subjects: Analytic functions; Boundary conditions; Complex variables; Conduction; Conduction heating; Conductive heat transfer; Cooling; Critical point; Dirichlet problem; Flow nets; Heat; Heat conduction; Heat transfer; Heat transmission; Mathematical analysis; Mathematical models; Slabs; Suction; Temperature gradients; Thermal insulation; Thermal protection; Thermocouples; Topology; Two dimensional models; two-dimensional heat conduction; complex potential; isotherms; Laplace’s equation; conformal mappings; heat lines
• ISSN: 1446-1811; 1446-8735
• DOI: 10.1017/S1446181112000260

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.