as in a stream with rigid boundaries.
[Illustration: FIG. 8.]
In the case of compressible fluids the variation of volume due to the
difference of pressure at the two sections must be taken into account.
If the motion is steady the weight of fluid between two cross sections
of a stream must remain constant. Hence the weight flowing in must be
the same as the weight flowing out. Let p1, p2 be the pressures, v1,
v2 the velocities, G1, G2 the weight per cubic foot of fluid, at cross
sections of a stream of areas A1, A2. The volumes of inflow and
outflow are
A1v1 and A2v2,
and, if the weights of these are the same,
G1A1v1 = G2A2v2;
and hence, from (5a) § 9, if the temperature is constant,
p1A1v1 = p2A2v2. (3)
§ 15. _Stream Lines._--The characteristic of a perfect fluid, that is,
a fluid free from viscosity, is that the pressure between any two
parts into which it is divided by a plane must be normal to the plane.
One consequence of this is that the particles can have no rotation
impressed upon them, and the motion of such a fluid is irrotational. A
stream line is the line, straight or curved, traced by a particle in a
current of fluid in irrotational movement. In a steady current each
stream line preserves its figure and position unchanged, and marks the
track of a stream of particles forming a fluid filament or elementary
stream. A current in steady irrotational movement may be conceived to
be divided by insensibly thin partitions following the course of the
stream lines into a number of elementary streams. If the positions of
these partitions are so adjusted that the volumes of flow in all the
elementary streams are equal, they represent to the mind the velocity
as well as the direction of motion of the particles in different parts
of the current, for the velocities are inversely proportional to the
cross sections of the elementary streams. No actual fluid is devoid of
viscosity, and the effect of viscosity is to render the motion of a
fluid sinuous, or rotational or eddying under most ordinary
conditions. At very low velocities in a tube of moderate size the
motion of water may be nearly pure stream line motion. But at some
velocity, smaller as the diameter of the tube is greater, the motion
suddenly becomes tumultuous. The laws of simple stream line motion
have hitherto been investigated theoretically, and from mathematical
difficulties have only been determined for certain simple cases.
Professor H. S. Hele Shaw has found means of exhibiting stream line
motion in a number of very interesting cases experimentally. Generally
in these experiments a thin sheet of fluid is caused to flow between
two parallel plates of glass. In the earlier experiments streams of
very small air bubbles introduced into the water current rendered
visible the motions of the water. By the use of a lantern the image of
a portion of the current can be shown on a screen or photographed. In
later experiments streams of coloured liquid at regular distances were
introduced into the sheet and these much more clearly marked out the
forms of the stream lines. With a fluid sheet 0.02 in. thick, the
stream lines were found to be stable at almost any required velocity.
For certain simple cases Professor Hele Shaw has shown that the
experimental stream lines of a viscous fluid are so far as can be
measured identical with the calculated stream lines of a perfect
fluid. Sir G. G. Stokes pointed out that in this case, either from the
thinness of the stream between its glass walls, or the slowness of the
motion, or the high viscosity of the liquid, or from a combination of
all these, the flow is regular, and the effects of inertia disappear,
the viscosity dominating everything. Glycerine gives the stream lines
very satisfactorily.
[Illustration: FIG. 9.]
[Illustration: FIG. 10.]
[Illustration: FIG. 11.]
[Illustration: FIG. 12.]
[Illustration: FIG. 13.]
Fig. 9 shows the stream lines of a sheet of fluid passing a fairly
shipshape body such as a screwshaft strut. The arrow shows the
direction of motion of the fluid. Fig. 10 shows the stream lines for a
very thin glycerine sheet passing a non-shipshape body, the stream
lines being practically perfect. Fig. 11 shows one of the earlier
air-bubble experiments with a thicker sheet of water. In this case the
stream lines break up behind the obstruction, forming an eddying wake.
Fig. 12 shows the stream lines of a fluid passing a sudden contraction
or sudden enlargement of a pipe. Lastly, fig. 13 shows the stream
lines of a current passing an oblique plane. H. S. Hele Shaw,
"Experiments on the Nature of the Surface Resistance in Pipes and on
Ships," _Trans. Inst. Naval Arch._ (1897). "Investigation of Stream
Line Motion under certain Experimental Conditions," _Trans. Inst.
Naval Arch._ (1898); "Stream Line Motion of a Viscous Fluid," _Report
of British Association_ (1898).
III. PHENOMENA OF THE DISCHARGE OF LIQUIDS FROM ORIFICES AS
ASCERTAINABLE BY EXPERIMENTS
§ 16. When a liquid issues vertically from a small orifice, it forms a
jet which rises nearly to the level of the free surface of the liquid
in the vessel from which it flows. The difference of level h_r (fig.
14) is so small that it may be at once suspected to be due either to
air resistance on the surface of the jet or to the viscosity of the
liquid or to friction against the sides of the orifice. Neglecting for
the moment this small quantity, we may infer, from the elevation of
the jet, that each molecule on leaving the orifice possessed the
velocity required to lift it against gravity to the height h. From
ordinary dynamics, the relation between the velocity and height of
projection is given by the equation
v = [root](2gh). (1)
As this velocity is nearly reached in the flow from well-formed
orifices, it is sometimes called the theoretical velocity of
discharge. This relation was first obtained by Torricelli.
[Illustration: FIG. 14.]
If the orifice is of a suitable conoidal form, the water issues in
filaments normal to the plane of the orifice. Let [omega] be the area
of the orifice, then the discharge per second must be, from eq. (1),