A proposed model for the thermal conductivity tensor of a glass

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Index notation and Cartesian tensors

Problem

A proposed model for the thermal conductivity tensor κij\kappa_{ij} of a glass, defined by the heat-flux law qi=−κij∂jTq_i = -\kappa_{ij}\partial_jT, is κij=κ0(δij+α ϵijknk),\kappa_{ij} = \kappa_0\left(\delta_{ij} + \alpha\,\epsilon_{ijk}n_k\right), where κ0\kappa_0 and α\alpha are constants and n\mathbf n is a fixed unit vector with components (0,0,1)(0,0,1) in some chosen axes. A glass is isotropic, so its conductivity tensor must be numerically the same array of nine numbers in every set of orthonormal axes, with no direction singled out. Using the fact that ϵijk\epsilon_{ijk} has the same components in every set of rotated axes, and the orthogonality of a rotation matrix R\mathsf R, examine how κij\kappa_{ij} transforms under a rotation of axes, decide whether the proposed form can describe an isotropic glass, and state the conductivity tensor that is actually admissible.

Browse the problem library