Hot relativistic jets , passing through a background medium with a pressure gradient p \propto r ^ { - \eta } where 2 < \eta \leq 8 / 3 , develop a shocked boundary layer containing a significant fraction of the jet power . In previous work , we developed a self-similar description of the boundary layer assuming isentropic flow , but we found that such models respect global energy conservation only for the special case \eta = 8 / 3 . Here we demonstrate that models with \eta < 8 / 3 can be made self-consistent if we relax the assumption of constant specific entropy . Instead , the entropy must increase with increasing r along the boundary layer , presumably due to multiple shocks driven into the flow as it gradually collimates . The increase in specific entropy slows the acceleration rate of the flow and provides a source of internal energy that could be channeled into radiation . We suggest that this process may be important for determining the radiative characteristics of tidal disruption events and gamma-ray bursts from collapsars .