A uniform isotropic solid of density

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Tensors and differential geometry

Invariance, Ch Index notation and Cartesian tensors

Problem

A uniform isotropic solid of density ρ\rho, specific heat capacity cc and thermal conductivity tensor κij\kappa_{ij} carries a heat flux given by Fourier's law qi=−κij∂jTq_i=-\kappa_{ij}\partial_jT and obeys the local energy balance ρc ∂T/∂t=−∂iqi\rho c\,\partial T/\partial t=-\partial_iq_i, with no internal heat sources. ((a)) Explain, citing the isotropic tensor result for rank two, why isotropy forces κij=κ δij\kappa_{ij}=\kappa\,\delta_{ij} for a single scalar κ\kappa. ((b)) Show, using the symmetry of ∂j∂kT\partial_j\partial_kT under j↔kj\leftrightarrow k together with the antisymmetry of ϵijk\epsilon_{ijk}, that ϵijk∂j∂kT=0\epsilon_{ijk}\partial_j\partial_kT=0 for every temperature field with continuous second partial derivatives T(x)T(\mathbf x), and hence that ∇×q=0\nabla\times\mathbf q=\mathbf 0 in this solid. ((c)) Derive the heat equation ∂T/∂t=D∇2T\partial T/\partial t=D\nabla^2T and give DD in terms of κ,ρ,c\kappa,\rho,c.

Browse the problem library