Hi Chris thanks for your reply , I don't understand the open and closed distinction, researching it seems both energy balances can have a change in the kinetic term while $dQ$ and $P.dV$ are zero

Thanks for your reply Chris that’s exactly what I would expect but I also keep getting referred by some very smart people to the work/energy theorem (applied to Bernoullis incompressible adiabatic) saying that $dU=dQ+dW =0$ and therefore if $dU=0$, $dT=0$. This does not account for the conversion along a streamline of Static enthalpy $H$ to Dynamic enthalpy $v$ , which as you say for constant total enthalpy $H_T$ must mean a reduction in Static enthalpy $H$ If $PdV=0$ this must come from static $U$ which explains how the temperature changes with no $dQ$ or $P.dV$ Which one is right ?

Thanks for your very good reply Peter , a few questions I always understood that the radial force of the pressure gradient is what keeps the flow in uniform rotation against the inertial centrifugal force . Are you saying that in the core the centrifugal if greater than the centripetal and this is what allows the expansion? Also when we allow expansion in the core we must have also compression outside the core? So outside the core heats up as the core cools down?

@frederico I have asked a follow up question to your answer , hope you can help