A cube of side and uniform conductivity contains a steady internal heat-source…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Partial differential equations and boundary-value problems
Conditions of Validity, Ch Gauss, Stokes and Green proved, and the first inversion of a differential operator
Problem
A cube of side L=0.40m and uniform conductivity k contains a steady internal heat-source density f(r) (in Wm−3) and reaches a steady temperature u(r), twice continuously differentiable on the closed cube, satisfying ∇2u=−f/k.
((a)) The two pairs of faces perpendicular to the y and z axes are held at prescribed fixed temperatures (Dirichlet); the remaining two opposite faces, x=0 and x=L, are perfectly insulated, ∂u/∂n=0 there (Neumann). Using Green's first identity and the energy argument, prove that this mixed problem has at most one solution, saying which faces remove which part of the boundary term.
((b)) Now insulate all six faces. Show that the same argument still makes the difference of two solutions constant, but no longer zero. Then apply the divergence theorem to ∇u to find the condition on f without which no steady temperature exists at all.