This paper finds by an exact method the temperature distribution in a viscous fluid undergoing Poiseuille flow between two infinite parallel plates. The upstream portion of the plates is assumed to be at temperature T = T and downstream portion of the plates to be at temperature T = T+. Results are obtained numerically for the “entrance region” temperature profiles. These tend rapidly to zero away from the point where the wall temperature changes abruptly. Using asymptotic expansions for the functions involved estimates are made for the number of terms required to achieve a given level of accuracy for a given Peclet number (Pe). Typical results for Pe = 1 and 10 are shown graphically.

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