dp/dt, udp/dx terms in static enthalpy equation

CFDtoy

Hello: I am trying to work with static and stagnation enthalpy models and not sure about some terms that arise while moving from stagnation to static enthalpy equation.

H=stagnation enth.

H=CpT+u**2/2

d(rhoH)/dt+div(rhoUH)=div.(tau.U)+grad k gradT

What happens when I move to static?

I see some textbooks with and without dp/dt terms and udp/dx terms in static enthalpy models. Can someone clarify?

Please suggest me some complete derivation sources.

Thanks

CFDtoy