A nonlocal problem arising from heat radiation on non-convex surfaces

Abstract

We consider both stationary and time-dependent heat equations for a non-convex body or a collection of disjoint conducting bodies with Stefan-Boltzmann radiation conditions on the surface. The main novelty of the resulting problem is the non-locality of the boundary condition due to self-illuminating radiation on the surface. Moreover, the problem is nonlinear and in the general case also non-coercive. We show that the non-local boundary value problem admits a maximum principle. Hence, we can prove the existence of a weak solution assuming the existence of upper and lower solutions. This result is then applied to prove existence under some hypotheses that guarantee the existence of sub- and supersolutions. Some special cases where the problem is coercive are also discussed. Finally, the analysis is extended to cases with nonlinear material properties.

Steady-State Flows of Thermal Viscous Incompressible Fluids With Convective-Radiation Effects

“…However, industrial devices such as furnaces with surfaces which permit emitted as well as incident radiation require convective-radiation effect coupled with a nonlocal boundary condition. 2,11,18 The constitutive law of the stress-tensor for the class of non-Newtonian fluids considered here as dependent on the temperature is σ = −πI + ν(·, θ)τ (D(u)), (1.1) where π denotes the pressure, I the identity matrix, ν the viscosity, θ the temperature, u the velocity of the fluid and D(u) = 1 2 (∇u+ ∇u T ) the symmetrized velocity gradient, and τ is the deviator stress-tensor defined such that τ : M n×n sym → M n×n sym is a continuous function which satisfies the conditions of p-coercivity…”

“…2 When the surface Γ is not convex, the outgoing radiation is a combination of emission and reflected fraction of incoming radiation which receives radiation from other parts of itself (see Ref. 18 and the references therein)…”

Existence for a time-dependent heat equation with non-local radiation terms

“…The question of the existence and uniqueness of a stationary solution of the above-mentioned heat conducting problem with surface radiation has already been discussed in [5,6,8,9]. The time-dependent case however has not been treated very Fig.…”

“…In situations shown in the right picture the radiation can escape thoroughly. Tiihonen considered the parabolic case in [8]. For his existence result, he assumed that there exist bounded sub-and supersolutions for the problem.…”

Integro-differential equation modelling heat transfer in conducting, radiating and semitransparent materials

“…Apart from papers dealing either with heat equation and convex radiating surfaces or with pure radiative transfer, the mathematical community seems to ignore the wealth of interesting problems arising from heat radiation. In fact, in addition to our previous work [13,14], the only mathematically oriented works coupling radiation and conduction we are aware of are those of Saldanha da Gama [8,9].…”