(98)
where h_o is a constant along any streamline and referred to as the total or stagnation enthalpy per unit mass. Since all streamlines originate in the stagnation source, where h=h_o, the constant is the same throughout the expansion. Enthalpy is the relevant quantity instead of internal energy, because the flow is driven by a pressure gradient that accomplishes the flow work, P/rho. Enthalpy decreases and the mean velocity increases as the gas expands and cools. For ideal gases,
(99) 
and, using equation (98), we can write the energy balance relating velocity to temperature as:
(100) 
If
is constant, over the range of interest, T_o to T, then equation (100) becomes:
(101) 
With
(102) 
equation (101) becomes:
(103) 
If the gas is cooled substantially during the expansion, T<<T_o, we obtain the maximum or terminal velocity:
(104) 
for an ideal gas. For ideal gas mixtures W and C_p are replaced with their molar averages in equation (101):
(105) 
and 
The mean velocity of each specie tends to be the same, in the continuum limit. Therefore each specie in a mixture may acquire energies well above or well below the average, depending on their individual masses. Hence a specie can be accelerated by diluting it in a lighter gas or decelerated by diluting it in a heavier gas. When noncontinuum conditions prevail before the terminal velocity, V_infinity, is reached, velocity slip occurs, as discussed later. It is important to remember that C_p and gamma may not be independent of temperature.
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