where P denotes the atmospheric pressure and h the height of the free
surface, a fundamental equation in hydraulics; a return has been made
here to the gravitation unit of hydrostatics, and Oz is taken
vertically upward.
In particular, for a jet issuing into the atmosphere, where p = P,
q²/2g = h - z, (9)
or the velocity of the jet is due to the head k - z of the still free
surface above the orifice; this is Torricelli's theorem (1643), the
foundation of the science of hydrodynamics.
27. _Uniplanar Motion._--In the uniplanar motion of a homogeneous
liquid the equation of continuity reduces to
du dv
-- + -- = 0, (1)
dx dy
so that we can put
u = -d[psi]/dy, v = d[psi]/dx, (2)
where [psi] is a function of x, y, called the stream- or
current-function; interpreted physically, [psi] - [psi]0, the
difference of the value of [psi] at a fixed point A and a variable
point P is the flow, in ft.³/second, across any curved line AP from A
to P, this being the same for all lines in accordance with the
continuity.
Thus if d[psi] is the increase of [psi] due to a displacement from P
to P´, and k is the component of velocity normal to PP´, the flow
across PP´ is d[psi] = k·PP´; and taking PP´ parallel to Ox, d[psi] =
vdx; and similarly d[psi]= -udy with PP´ parallel to Oy; and generally
d[psi]/ds is the velocity across ds, in a direction turned through a
right angle forward, against the clock.