This Worked Problem considers steady-state heat conduction into an anisotropic media made up by two different materials of different thermal conductivities modelling a contraction and a subsequent channel. We impose uniform (unity) flux at the inlet; fixed temperature at the outlet; and zero flux (insulated) on the top and bottom walls. Our interest is in the temperature distribution and in particular in the average temperature. The geometrical parametrization of a curved domain (representing the thermal contraction) illustrates a new feature for a broader class of problems with many possible applications in different engineering fields related with shape optimization, variable-geometries and sizing.
We describe the notion of (i) a (variable-area) restriction thermal resistance, (ii) anisotropic conductivities, (iii) the development of 1D models, (iv) bounding techniques, (v) St. Venant's principle, and (vi) the asymptotic incorporation in building blocks. The physical principles illustrated in this example deal with steady heat transfer in an anisotropic media made up by two different materials (with different physical properties).