Discussion
The relation between pressure (p) and velocity (V) for a steady, isentropic flow at two points along a streamline is, (c is a constant)
Answer: C
The relation is \(c(p_{2}^{(\gamma-1)/\gamma} - p_{1}^{(\gamma-1)/\gamma})\) = \(\frac{V_{1}^{2}}{2} - \frac{V_{2}^{2}}{2}\).
Along a streamline in steady inviscid flow, Euler's equation is \(dp/\rho + V\,dV = 0\). For isentropic flow \(p/\rho^{\gamma} = K\), so \(\rho = (p/K)^{1/\gamma}\) and
\[ \begin{aligned} \int \frac{dp}{\rho} &= K^{1/\gamma}\int p^{-1/\gamma}\,dp \\ &= \frac{\gamma}{\gamma - 1}K^{1/\gamma} \\ &\quad \times p^{(\gamma-1)/\gamma} \end{aligned} \]
Integrating Euler's equation from point 1 to point 2 then gives the relation above, with \(c = \frac{\gamma}{\gamma-1}K^{1/\gamma}\). It is the compressible form of Bernoulli's equation.
The exponents \(\gamma\), \(\gamma - 1\) and \(\gamma/(\gamma - 1)\) in the other options do not come out of the integration: the power of p is fixed at \(1 - 1/\gamma = (\gamma - 1)/\gamma\) by the isentropic relation.
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