Mechanics of Fluids

458 Flow with a free surface
Fig. 10.40
increase of h’. If, in fact, h decreases, then δh is negative and will be seen to
be so in the final result.)
If it is assumed that the streamlines in the flow are sensibly straight and
parallel and that the slope of the bed is small so that the variation of pressure
with depth is hydrostatic, then the total head above the datum level is h +
sδl + αu 2 /2g at the first section. As in Section 10.3, α is the kinetic energy
correction factor accounting for non-uniformity of velocity over the crosssection.
Its value is not normally known: for simplicity we here assume that
it differs only slightly from unity so that, without appreciable error, it may
be omitted as a factor. Similarly, the total head above datum at the second
section is h + δh + (u + δu) 2 /2g. If the head loss gradient (i.e. the loss of
head in friction divided by the length along the channel) is i, then
h + sδl + u2
(u + δu)2
− iδl = h + δh +
2g 2g
(10.38)
Rearrangement gives δh = (s−i)δl−u δu/g, the term in (δu) 2 being neglected.
Therefore in the limit as δl → 0
dh
dl
u du
= (s − i) −
g dl
(10.39)
To eliminate du/dl we make use of the one-dimensional continuity equation
which, when differentiated with respect to l, yields
so
Au = constant (10.40)
Adu/dl + udA/dl = 0
du
dl =−u
dA
A dl
(10.41)
To evaluate dA/dl we assume that, although the cross-section may be of any
shape whatever, the channel is prismatic, that is, its shape and alignment do
not vary with l. Thus A changes only as a result of a change in h. Figure 10.20
shows that δA then equals Bδh where B denotes the surface width of the
cross-section. So putting dA/dl = Bdh/dl in eqn 10.41 and then substituting