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The problem of heat transfer in a duct or tube for large values of the Péclet number has traditionally been solved by assuming that diffusion in the axial direction is negligible. This approach was used by Graetz [2] for the circular tube and by Prins et al [5] for the flat duct to obtain a series solution for downstream temperature field.
Since these series converge very slowly in the neighbourhood of the origin, some other approach is necessary in the thermal entrance region. This was supplied by Lévêque [3] and extended by Mercer [4] who matched the Lévêque solution to the eigenfunction expansion.
In all these solutions it was assumed that the axial diffusion of heat was negligible, but this assumption is invalid close to the discontinuity, since in this region the axial temperature gradient is large and the fluid velocity is small, so that axial diffusion plays an important role.
In this paper, the assumptions implicit in Lévêque's solution are re-examined, and the correct approximation in the neighbourhood of the discontinuity as well as the solution which matches this into Lévêque's solution are presented. In the first of these solutions, diffusion is the only heat-transfer mechanism, while in the matching solution diffusion and convection are in balance.
The corresponding solutions for the case of prescribed flux on the boundary are also considered.
