An adiabatic process is one in which no heat enters or leaves the system.
The thermdynamic first law takes the form
dU=−PdV,
since dQ=0.
Note that Cv=(∂U/∂T)V , but the internal energy of an ideal gas depends only on the temperature and is independent of the volume (because there are no intermolecular forces). Thus for an ideal gas, we have
dU=CvdT=PdV
PV=RT=(Cp−Cv)T
γ=Cp/Cv
Then we have
TVγ−1=const
Then we have
TVγ−1=const
dln(T)/dt=−(γ−1)∇⋅u
where we have use
1VdVdt=∇⋅u
where u is the flow speed.
Since KBT∝E, here E is thermal energy. We have
dln(E)/dt=−(γ−1)∇⋅u
For non-relativistic particle, γ=5/3, E=p2/m.
For relativistic particle, γ=4/3, E=pc,
we have
dpdt=−p3∇⋅u
The bulk of particles lose energy by adiabatic cooling due to the work that protons exert on the expanding material. Particles (pressure ?) do (positive) work on the expanding medium cause their bulk energy reduce.
Reference:
https://github.com/bolverk/theoretical_physics_digest
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