Analytical solutions of solid‐liquid conjugated Graetz problems with applications in the analysis of heat transfer in the capillary die are presented. In these problems, temperatures in adjacent phases are coupled through the conjugated conditions, which require that the flux and the rate of transfer are continuous. An analytical solution is obtained by using the direct eigenfunction expansion technique. The resulting generalized Sturm‐Liouville problem is not of the conventional type due to the discontinuity of the coefficient which arises from the conjugated conditions at the phase interface. An efficient numerical algorithm is developed to find the eigenvalues, eigenfunctions and the related coefficients. Sturm's equi‐oscillation theorem is applied to overcome the difficulty due to the discontinuity of the coefficients. The problem of heat removal from a heated solid cylinder by a surrounding annular flow fluid is solved. Results are in good agreement with existing solutions. The same technique is used to analyze the heat transfer in a capillary die. Results obtained show that wall resistance does have a significant effect on the temperature profile of the fluid.