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 questionable
