Heat Equation - Heat Conduction in Non-homogeneous Anisotropic Media

Heat Conduction in Non-homogeneous Anisotropic Media

In general, the study of heat conduction is based on several principles. Heat flow is a form of energy flow, and as such it is meaningful to speak of the time rate of flow of heat into a region of space.

• The time rate of heat flow into a region V is given by a time-dependent quantity q_t(V). We assume q has a density, so that

• Heat flow is a time-dependent vector function H(x) characterized as follows: the time rate of heat flowing through an infinitesimal surface element with area dS and with unit normal vector n is

Thus the rate of heat flow into V is also given by the surface integral

where n(x) is the outward pointing normal vector at x.

• The Fourier law states that heat energy flow has the following linear dependence on the temperature gradient

where A(x) is a 3 × 3 real matrix that is symmetric and positive definite.

By Green's theorem, the previous surface integral for heat flow into V can be transformed into the volume integral

• The time rate of temperature change at x is proportional to the heat flowing into an infinitesimal volume element, where the constant of proportionality is dependent on a constant κ

Putting these equations together gives the general equation of heat flow:

Remarks.

• The coefficient κ(x) is the inverse of specific heat of the substance at x × density of the substance at x.

• In the case of an isotropic medium, the matrix A is a scalar matrix equal to thermal conductivity.

• In the anisotropic case where the coefficient matrix A is not scalar (i.e., if it depends on x), then an explicit formula for the solution of the heat equation can seldom be written down. Though, it is usually possible to consider the associated abstract Cauchy problem and show that it is a well-posed problem and/or to show some qualitative properties (like preservation of positive initial data, infinite speed of propagation, convergence toward an equilibrium, smoothing properties). This is usually done by one-parameter semigroups theory: for instance, if A is a symmetric matrix, then the elliptic operator defined by

is self-adjoint and dissipative, thus by the spectral theorem it generates a one-parameter semigroup.