Page 429 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 413

In steady flow, the rate of mass flow is constant for all cross-sections. With A denoting the
variable cross-sectional area,p the density and v the velocity, we have

To see the effect of area variation on the flow, we take the total derivative of Eq. (6.29.1), i.e.,

dp(Av)+p(dA)v+pA(dv) = 0
Dividing the above equation bypAv. we obtain

Thus,

Now, for barotropic flow of an ideal gas, we have [see Eq. (6.28.11)]

Thus,

But ^{dp/dp} = c (the speed of sound), thus,

Combining Eqs. (i) and (iii), we get

i.e.,

Eq. (6.29.4) is sometimes known as Hugoniot equation. From this equation, we see that for
subsonic flows (M< 1), an increase in area produces a decrease in velocity, just as in the case
of an incompressible fluid. On the other hand, for supersonic flows (M> I), an increase in area
produces an increase in velocity. Furthermore, the critical velocity (M=l) can only be
obtained at the smallest cross-sectional area where dA = 0.