Hot relativistic jets, passing through a background medium with a pressure gradient p propto r^{-eta} where 2 < eta <= 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.
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