Diffusion and advection of heat with a single space variable

Abstract

Consider a medium in which both the thermal conductivity K and the thermal capacity per unit mass C vary with temperature. Its density ϱ varies with temperature and pressure. The thermal conductivity is assumed to be isotropic, an assumption which, even in the case of very anisotropic rocks, is not too bad. The temperature at a point with fixed coordinates is T(x, y, z, t). The heat transfered by conductivity per unit time (in Js^-1m^-2, in S.I. units) is then, according to Fourier law:

$$ \phi_c = - K\,\,\mathbf{grad}\,\,T $$