§ 153. When a stream of fluid in steady motion impinges on a solid
surface, it presses on the surface with a force equal and opposite to
that by which the velocity and direction of motion of the fluid are
changed. Generally, in problems on the impact of fluids, it is
necessary to neglect the effect of friction between the fluid and the
surface on which it moves.
_During Impact the Velocity of the Fluid relatively to the Surface on
which it impinges remains unchanged in Magnitude._--Consider a mass of
fluid flowing in contact with a solid surface also in motion, the
motion of both fluid and solid being estimated relatively to the
earth. Then the motion of the fluid may be resolved into two parts,
one a motion equal to that of the solid, and in the same direction,
the other a motion relatively to the solid. The motion which the fluid
has in common with the solid cannot at all be influenced by the
contact. The relative component of the motion of the fluid can only be
altered in direction, but not in magnitude. The fluid moving in
contact with the surface can only have a relative motion parallel to
the surface, while the pressure between the fluid and solid, if
friction is neglected, is normal to the surface. The pressure
therefore can only deviate the fluid, without altering the magnitude
of the relative velocity. The unchanged common component and, combined
with it, the deviated relative component give the resultant final
velocity, which may differ greatly in magnitude and direction from the
initial velocity.
From the principle of momentum, the impulse of any mass of fluid
reaching the surface in any given time is equal to the change of
momentum estimated in the same direction. The pressure between the
fluid and surface, in any direction, is equal to the change of
momentum in that direction of so much fluid as reaches the surface in
one second. If P_a is the pressure in any direction, m the mass of
fluid impinging per second, v_a the change of velocity in the
direction of P_a due to impact, then
P_a = mv_a.
If v1 (fig. 152) is the velocity and direction of motion before
impact, v2 that after impact, then v is the total change of motion due
to impact. The resultant pressure of the fluid on the surface is in
the direction of v, and is equal to v multiplied by the mass impinging
per second. That is, putting P for the resultant pressure,
P = mv.
Let P be resolved into two components, N and T, normal and tangential
to the direction of motion of the solid on which the fluid impinges.
Then N is a lateral force producing a pressure on the supports of the
solid, T is an effort which does work on the solid. If u is the
velocity of the solid, Tu is the work done per second by the fluid in
moving the solid surface.
[Illustration: FIG. 152.]
Let Q be the volume, and GQ the weight of the fluid impinging per
second, and let v1 be the initial velocity of the fluid before
striking the surface. Then GQv1²/2g is the original kinetic energy of
Q cub. ft. of fluid, and the efficiency of the stream considered as an
arrangement for moving the solid surface is
[eta] = Tu/(GQv1²/2g).
§ 154. _Jet deviated entirely in one Direction.--Geometrical Solution_
(fig. 153).--Suppose a jet of water impinges on a surface ac with a
velocity ab, and let it be wholly deviated in planes parallel to the
figure. Also let ae be the velocity and direction of motion of the
surface. Join eb; then the water moves with respect to the surface in
the direction and with the velocity eb. As this relative velocity is
unaltered by contact with the surface, take cd = eb, tangent to the
surface at c, then cd is the relative motion of the water with respect
to the surface at c. Take df equal and parallel to ae. Then fc
(obtained by compounding the relative motion of water to surface and
common velocity of water and surface) is the absolute velocity and
direction of the water leaving the surface. Take ag equal and parallel
to fc. Then, since ab is the initial and ag the final velocity and
direction of motion, gb is the total change of motion of the water.
The resultant pressure on the plane is in the direction gb. Join eg.
In the triangle gae, ae is equal and parallel to df, and ag to fc.
Hence eg is equal and parallel to cd. But cd = eb = relative motion of
water and surface. Hence the change of motion of the water is
represented in magnitude and direction by the third side of an
isosceles triangle, of which the other sides are equal to the relative
velocity of the water and surface, and parallel to the initial and
final directions of relative motion.
[Illustration: FIG. 153.]