If the pressure is a function of density only, is the flow always isentropic?

Question

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Under (TD2) author says that the flow is always isentropic whenever the pressure is a function of density only. But from the First Law of Thermodynamics (TD1) or (TD2), I can derive the fact.. Do I need more knowledge not in this book? Or can I get the fact just from (TD1) of (TD2) ? :)

Answer 1

I was actually seeing if someone else had figured this out, I am reading the same book now and for anyone else reading the Marsden text, this is my understanding. From TD1 we know that

$$dw = Tds + \frac{1}{\rho}d p $$ on the line when it says that pressure $p$ is a function of density $\rho$, it also mentions that the entropy $s$ is constant, thus $ds = 0$. Since $p$ is a function of $\rho$, then we write $p(\rho)$ and from the chain rule have that

$$dp = \frac{dp}{d\rho}d\rho = p'(\rho)d\rho$$ In the text, Marsden uses $\lambda$ to represent density to avoid the notation. So under these assumptions we find that

$$dw = \frac{p'(\rho)}{\rho} d\rho$$ and the integral follows from this.