Topology of Steady Heat Conduction in a Solid Slab Subject to a Nonuniform Boundary Condition: The Carslaw–jaeger Solution revisited

Abstract

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.2010 Mathematics subject classification: primary 35J25; secondary 65E04, 80A05.

Existence theory for a system of coupled multi‐term fractional differential equations with integral multi‐strip coupled boundary conditions

“…Here we emphasize that the first integral multi‐strip boundary condition (likewise the second integral multi‐strip condition) states that distribution of x ( s ) on the interval [0,1] equals the multi‐strip contributions for the function y ( s ). The nonlocal strip conditions have interesting applications in heat conduction problems with nonuniform boundary conditions, 42 geophysical flows, 43 and acoustic scattering 44 . In computational fluid dynamics (CFD) studies of blood flow problems, integral boundary conditions provide the means to consider an arbitrary shaped cross‐section of blood vessels 45 and help to regularize ill‐posed parabolic backward problems (bacterial self‐regularization model 46 ) For details and applications in engineering problems, see previous studies 47–49 …”