We consider the rarefied longitudinal cylindrical Couette problem. We find the analytical expressions for the surface temperature ratios at which the free molecular energy fluxes to the surfaces reach the conditional maximums if the temperatures of the opposite surfaces remain constant. We show that the dependencies of the maximum energy fluxes on the ratio of the radii of the cylinders in the longitudinal Couette problem are essentially different from the analogous dependencies for the heat transfer problem. We find that, in contrast to the heat transfer problem, in the longitudinal Couette problem the heat capacity and the molecular mass affect the positions of the maxima of the energy fluxes. For the transient mode, we investigate the conditional maximums of the energy fluxes at different Knudsen numbers by the Direct Simulation Monte Carlo method. (C) 2017 Elsevier Ltd. All rights reserved.