The heat conduction equation for bodies with constant material properties

2 The heat conduction equation for bodies with constant material properties 

In the derivation of the heat conduction equation in (2.8) we presumed an incompressible body, g = const. The temperature dependence of both the thermal conductivity A and the specific heat capacity c was also neglected. These assumptions have to be made if a mathematical solution to the heat conduction equation is to be obtained. This type of closed solution is commonly known as the exact solution. The solution possibilities for a material which has temperature dependent properties will be discussed in section 2.1.4. 

With constant thermal conductivity the differential operator div [A(d) gradd] in (2.8) becomes the Laplace operator 

The constant which appears here is the thermal diffusivity 

In the two most important coordinate systems, cartesian coordinates x, y, z, and cylindrical coordinates r, (p, z the heat conduction equation takes the form 

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