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Heat equation

The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. In the special case of a heat propagation in an isotropic and homogeneous medium, this equation is

<math>u_t = k ( u_{xx} + u_{yy} + u_{zz} ) \quad <math>

where:

• u(t,x,y,z) is temperature as a function of time and space

• <math>u_t<math> is the rate of change of temperature at a point over time

• <math>u_{xx}<math>, <math>u_{yy}<math>, and <math>u_{zz}<math> are the second spatial derivatives (thermal conductions) of temperature in the x, y, and z directions, respectively

• k is a material-specific constant (thermal diffusivity)

To solve the heat equation, we also need to specify boundary conditions for u.

Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of heat from warmer to colder areas of an object.

The heat equation is the prototipic example of a parabolic differential equation

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

<math> q_t(V) = \int_V Q(t,x) d x \quad <math>

• 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 d S and with unit normal vector n is

<math> \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS <math>

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

<math> q_t(V)= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS <math>

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

<math> \mathbf{H}(x) = -\mathbf{A}(x) \cdot [\operatorname{grad}(u)] (x) <math>

where A(x) is a 3 x 3 real matrix, which in fact is symmetric and non-negative.

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

<math> q_t(V) = - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS <math>

<math> = \int_{\partial V} \mathbf{A}(x) \cdot [\operatorname{grad}(u)] (x) \cdot \mathbf{n}(x) \, dS <math>

<math> = \int_V \sum_{i, j} \partial_{x_i} a_{i j} \partial_{x_j} u (t,x) d x <math>

• 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

<math> \partial_t u(t,x) = \kappa(x) Q(t,x) d x<math>

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

<math> \partial_t u(t,x) = \kappa(x) \sum_{i, j} \partial_{x_i} a_{i j} \partial_{x_j} u (t,x) <math>

Remarks.

• The constant κ(x) is the inverse of specific heat of the substance at x &times density of the substance at x.

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