Chapter II: Kinematics of Fluids

§ 10. Moving fluids as commonly observed are conveniently classified thus:

(1) _Streams_ are moving masses of indefinite length, completely or incompletely bounded laterally by solid boundaries. When the solid boundaries are complete, the flow is said to take place in a pipe. When the solid boundary is incomplete and leaves the upper surface of the fluid free, it is termed a stream bed or channel or canal.

(2) A stream bounded laterally by differently moving fluid of the same kind is termed a _current_.

(3) A _jet_ is a stream bounded by fluid of a different kind.

(4) An _eddy_, _vortex_ or _whirlpool_ is a mass of fluid the particles of which are moving circularly or spirally.

(5) In a stream we may often regard the particles as flowing along definite paths in space. A chain of particles following each other along such a constant path may be termed a fluid filament or elementary stream.

§ 11. _Steady and Unsteady, Uniform and Varying, Motion._--There are
two quite distinct ways of treating hydrodynamical questions. We may
either fix attention on a given mass of fluid and consider its changes
of position and energy under the action of the stresses to which it is
subjected, or we may have regard to a given fixed portion of space,
and consider the volume and energy of the fluid entering and leaving
that space.

If, in following a given path ab (fig. 4), a mass of water a has a
constant velocity, the motion is said to be uniform. The kinetic
energy of the mass a remains unchanged. If the velocity varies from
point to point of the path, the motion is called varying motion. If at
a given point a in space, the particles of water always arrive with
the same velocity and in the same direction, during any given time,
then the motion is termed steady motion. On the contrary, if at the
point a the velocity or direction varies from moment to moment the
motion is termed unsteady. A river which excavates its own bed is in
unsteady motion so long as the slope and form of the bed is changing.
It, however, tends always towards a condition in which the bed ceases
to change, and it is then said to have reached a condition of
permanent regime. No river probably is in absolutely permanent regime,
except perhaps in rocky channels. In other cases the bed is scoured
more or less during the rise of a flood, and silted again during the
subsidence of the flood. But while many streams of a torrential
character change the condition of their bed often and to a large
extent, in others the changes are comparatively small and not easily
observed.

As a stream approaches a condition of steady motion, its regime
becomes permanent. Hence steady motion and permanent regime are
sometimes used as meaning the same thing. The one, however, is a
definite term applicable to the motion of the water, the other a less
definite term applicable in strictness only to the condition of the
stream bed.

§ 12. _Theoretical Notions on the Motion of Water._--The actual motion
of the particles of water is in most cases very complex. To simplify
hydrodynamic problems, simpler modes of motion are assumed, and the
results of theory so obtained are compared experimentally with the
actual motions.

_Motion in Plane Layers._--The simplest kind of motion in a stream is
one in which the particles initially situated in any plane cross
section of the stream continue to be found in plane cross sections
during the subsequent motion. Thus, if the particles in a thin plane
layer ab (fig. 5) are found again in a thin plane layer a´b´ after any
interval of time, the motion is said to be motion in plane layers. In
such motion the internal work in deforming the layer may usually be
disregarded, and the resistance to the motion is confined to the
circumference.

_Laminar Motion._--In the case of streams having solid boundaries, it
is observed that the central parts move faster than the lateral parts.
To take account of these differences of velocity, the stream may be
conceived to be divided into thin laminae, having cross sections
somewhat similar to the solid boundary of the stream, and sliding on
each other. The different laminae can then be treated as having
differing velocities according to any law either observed or deduced
from their mutual friction. A much closer approximation to the real
motion of ordinary streams is thus obtained.

_Stream Line Motion._--In the preceding hypothesis, all the particles
in each lamina have the same velocity at any given cross section of
the stream. If this assumption is abandoned, the cross section of the
stream must be supposed divided into indefinitely small areas, each
representing the section of a fluid filament. Then these filaments may
have any law of variation of velocity assigned to them. If the motion
is steady motion these fluid filaments (or as they are then termed
_stream lines_) will have fixed positions in space.

_Periodic Unsteady Motion._--In ordinary streams with rough
boundaries, it is observed that at any given point the velocity varies
from moment to moment in magnitude and direction, but that the average
velocity for a sensible period (say for 5 or 10 minutes) varies very
little either in magnitude or velocity. It has hence been conceived
that the variations of direction and magnitude of the velocity are
periodic, and that, if for each point of the stream the mean velocity
and direction of motion were substituted for the actual more or less
varying motions, the motion of the stream might be treated as steady
stream line or steady laminar motion.

§ 13. _Volume of Flow._--Let A (fig. 6) be any ideal plane surface, of
area [omega], in a stream, normal to the direction of motion, and let
V be the velocity of the fluid. Then the volume flowing through the
surface A in unit time is

Q = [omega]V. (1)

Thus, if the motion is rectilinear, all the particles at any instant
in the surface A will be found after one second in a similar surface
A´, at a distance V, and as each particle is followed by a continuous
thread of other particles, the volume of flow is the right prism AA´
having a base [omega] and length V.

If the direction of motion makes an angle [theta] with the normal to
the surface, the volume of flow is represented by an oblique prism AA´
(fig. 7), and in that case

Q = [omega]V cos [theta].

If the velocity varies at different points of the surface, let the
surface be divided into very small portions, for each of which the
velocity may be regarded as constant. If d[omega] is the area and v,
or v cos [theta], the normal velocity for this element of the surface,
the volume of flow is
_ _
/ /
Q = | v d[omega], or | v cos [theta] d[omega],
_/ _/

as the case may be.

§ 14. _Principle of Continuity._--If we consider any completely
bounded fixed space in a moving liquid initially and finally filled
continuously with liquid, the inflow must be equal to the outflow.
Expressing the inflow with a positive and the outflow with a negative
sign, and estimating the volume of flow Q for all the boundaries,

[Sigma]Q = 0.

In general the space will remain filled with fluid if the pressure at
every point remains positive. There will be a break of continuity, if
at any point the pressure becomes negative, indicating that the stress
at that point is tensile. In the case of ordinary water this statement
requires modification. Water contains a variable amount of air in
solution, often about one-twentieth of its volume. This air is
disengaged and breaks the continuity of the liquid, if the pressure
falls below a point corresponding to its tension. It is for this
reason that pumps will not draw water to the full height due to
atmospheric pressure.

_Application of the Principle of Continuity to the case of a
Stream._--If A1, A2 are the areas of two normal cross sections of a
stream, and V1, V2 are the velocities of the stream at those sections,
then from the principle of continuity,

V1A1 = V2A2;

V1/V2 = A2/A1 (2)

that is, the normal velocities are inversely as the areas of the cross
sections. This is true of the mean velocities, if at each section the
velocity of the stream varies. In a river of varying slope the
velocity varies with the slope. It is easy therefore to see that in
parts of large cross section the slope is smaller than in parts of
small cross section.

If we conceive a space in a liquid bounded by normal sections at A1,
A2 and between A1, A2 by stream lines (fig. 8), then, as there is no
flow across the stream lines,

V1/V2 = A2/A1,

as in a stream with rigid boundaries.

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.

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).