In considering the effect of area variation on flow properties in isentropic flow, we shall determine the effect on the velocity V and the pressure p .
From Eq . (39.11), we can write
Dividing by 
, we obtain
 | (40.8) |
A convenient differential form of the continuity equation can be obtained from Eq. (39.6) as
Substituting from Eq. (40.8),
 | (40.9) |
Invoking the relation (39.3b) for isentropic process in Eq. (40.9), we get
 | (40.10) |
From Eq.(40.10), we see that for Ma<1 an area change causes a pressure change of the same sign, i.e. positive dA means positive dp for Ma<1 . For Ma>1 , an area change causes a pressure change of opposite sign.
Again, substituting from Eq. (40.8) into Eq. (40.10), we obtain
 | (40.11) |
From Eq. (40.11) we see that Ma<1 an area change causes a velocity change of opposite sign, i.e. positive dA means negative dV for Ma<1 . For Ma>1 an area change causes a velocity change of same sign.
These results can be summarized in fig 40.2. Equations (40.10) and (40.11) lead to the following important conclusions about compressible flows:
At subsonic speeds(Ma<1) a decrease in area increases the speed of flow. A subsonic nozzle should have a convergent profile and a subsonic diffuser should possess a divergent profile. The flow behaviour in the regime of Ma<1 is therefore qualitatively the same as in incompressible flows.
In supersonic flows (Ma>1) the effect of area changes are different. According to Eq. (40.11), a supersonic nozzle must be built with an increasing area in the flow direction. A supersonic diffuser must be a converging channel. Divergent nozzles are used to produce supersonic flow in missiles and launch vehicles.
Fig 40.2 Shapes of nozzles and diffusers in subsonic and supersonic regimes
