Temperature field and heat flow around an elliptical lava tube

We study the temperature distribution around a cylindrical lava tube with an elliptical cross section. The steady-state heat equation is solved by assuming an unbounded medium and a uniform temperature of the tube wall. An analytical solution for the temperature field is obtained by the conformal mapping technique. It is found that the isothermal lines on the planes perpendicular to the tube are confocal ellipses. The heat flow in the medium surrounding the tube has hyperbolic field lines, while the curves on which the magnitude of heat flow is constant are Cassini ovals. At the tube wall, the heat flow density is proportional to the cubic root of the curvature of the wall. A solution is also obtained for a tube embedded in a half-space, when the size of the tube is small with respect to its depth. Formulae are given relating the heat flow at the surface of the half-space to the eccentricity and to the orientation of the major axis of the tube cross section, as well as to the temperature and the depth of the tube. For eccentricity equal to zero, all the formulae are reduced to those for a circular tube.