A steady, isentropic, quasi-one-dimensional flow of a compressible fluid through a…

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Problem

A steady, isentropic, quasi-one-dimensional flow of a compressible fluid through a channel of slowly varying area

A(x)

obeys mass conservation

\rho u A = \dot m

(constant), the frictionless momentum equation

\rho u\,\mathrm{d}u = -\mathrm{d}p

, and, because the walls are smooth and there is no heat exchange,

\mathrm{d}p = c^2\,\mathrm{d}\rho

, with

c

the local sound speed and

M = u/c

. ((a)) Starting from these three relations, derive

\dfrac{\mathrm{d}A}{A} = (M^2-1)\dfrac{\mathrm{d}u}{u}

. ((b)) State in one sentence why

M=1

can occur only where

\mathrm{d}A = 0

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