# Isothermal Expansion of Supersonic Flow

• I
Hi,

In a standard converging-diverging rocket nozzle, we have (ex.) the relation:$$(1-M^2)\frac{dV}{V}=-\frac{dA}{A}$$By substituting in definitions, we can obtain$$\left(1-\frac{V^2}{\gamma R T}\right)\frac{dV}{V} = -\frac{dA}{A}$$This shows the dependence on temperature.

The relation assumes that the gas expands, accelerates, and cools in an adiabatic process. I would like to know what would happen if the temperature were instead held constant (i.e., by adding energy as the gas expands and accelerates)--but I have been utterly unable to find appropriate equations to replace the isentropic flow relations in the derivation.

What equations apply in this situation?

Ian