A theoretical analysis of heat transfer in fully-developed laminar Row in a concentric annulus with peripherally varying and axially constant heat flux has been made. To determine the temperature, superposition of the temperature solutions for the average axisymmetric heat-flux component and the perturbation hear-flux component (the boundary integral of which is zero) have been used. The axisymmetric temperature solution is already known, and the perturbation temperature solution has been obtained numerically using a point-matching method. A number of examples are presented to demonstrate the method, and to show typical wall temperature and heat transfer coefficient distributions around the boundary of the duct. The use of the average heat transfer coefficient in the case of peripherally varying boundary conditions has been critically examined.