dH / dQ q \
----- = 2q[zeta] = q ( -- + --- ), (5)
d[nu] \ dv r /
if r denotes the radius of curvature of the stream line, so that
1 dp dV dH d½q² q²
----- ----- + ----- = ----- - ----- = ---, (6)
[rho] d[nu] d[nu] d[nu] d[nu] r
the normal acceleration.
The osculating plane of a stream line in steady motion contains the
resultant acceleration, the direction ratios of which are
du du du d½q² d½q² dH
u-- + v-- + w-- = ---- - 2v[zeta] + 2w[eta] = ---- - --, ..., (7)
dx dy dz dx dx dx
and when q is stationary, the acceleration is normal to the surface H
= constant, and the stream line is a geodesic.
Calling the sum of the pressure and potential head the statical head,
surfaces of constant statical and dynamical head intersect in lines on
H, and the three surfaces touch where the velocity is stationary.
Equation (3) is called Bernoulli's equation, and may be interpreted as
the balance-sheet of the energy which enters and leaves a given tube
of flow.
If homogeneous liquid is drawn off from a vessel so large that the
motion at the free surface at a distance may be neglected, then
Bernoulli's equation may be written