An analytical study was carried out to determine the effects of eccentricity on both local and average heat-transfer coefficients for turbulent flow of liquid metals through eccentric annuli. The study was based on the conditions of 1) heat transfer from the inner wall only, 2) heat flux, at a given circumferential angle, independent of length, 3) inner-wall temperature, at a given axial position, independent of circumferential angle, and 4) fully developed velocity and temperature profiles. This study is a sequel to an earlier one, which described a similar case, except that the heat flux in that case %as uniform in all directions. The scopes of the two studies were identical, as far as parameter ranges are concerned. In general, the effects of eccentricity were found to be much less in the present case. For a given radius ratio r2/r1, and a given eccentricity, the circumferential variation of the local heat-transfer coefficient and the reduction in the average heat-transfer coefficient were both much less. Moreover, the reduction in the average heat-transfer coefficient, caused by a given degree of eccentricity, was found to be only slightly dependent on the radius ratio, in the present study. This is also in sharp contrast with the results of the previous study. It was further found that circumferential variation of the normalized local heat flux q/q̄, and, therefore, that of the normalized local heat-transfer coefficient also, remained the same over the large range of Peclet numbers investigated, for a given radius ratio and a given degree of eccentricity.