This thesis considers the two-dimensional transient heat conduction problems due to a circular cylinder held at one constant temperature and immersed in a conducting medium with its axis parallel to a plane surface held at another constant temperature. In the conducting medium each phase is assumed to have constant properties.

Approximate perturbation solutions are obtained for (i) the “simple" problem, in which the boundary temperatures are such that no change-of-phase takes place in the medium and (ii) the non-linear "two-phase" problem where change-of-phase occurs at a temperature between the boundary temperatures.

These analyses are mathematical idealisations of the physical problem of, say, a pipe containing a “warm" liquid flowing through a “cold" environment (or its’ converse) and, in case (ii), melting the frozen material near the pipe.

Although numerical methods of solution for both of these cases are available, the convenience and usefulness of simple analytical approximations is obvious. In the case of the non-linear problem, the accuracy of the methods employed is questionabl