Encyclopaedia Britannica > Volume 3, Athens-BOI

44
ATOM
continuous in space and time, and if we follow any portion
of it as it moves, tlie mass of that portion remains invari¬
able. These properties it shares with all material sub¬
stances. In the next place, it is incompressible. The
form of a given portion of the fluid may change, but its
volume remains invariable; in other words, the density of
the fluid remains the same during its motion. Besides this,
the fluid is homogeneous, or the density of all parts of the
fluid is the same. It is also continuous, so that the mass
of the fluid contained within any closed surface is always
exactly proportional to the volume contained within that
surface. This is equivalent to asserting that the fluid is
not made up of molecules ; for, if it were, the mass would
vary in a discontinuous manner as the volume increases
continuously, because first one and then another molecule
would be included within the closed surface. Lastly, it is
a perfect fluid, or, in other words, the stress between one
portion and a contiguous portion is always normal to the
surface which separates these portions, and this whether
the fluid is at rest or in motion.
We have seen that in a molecular fluid the interdiffusion
of the molecules causes an interdiffusion of motion of
different parts of the fluid, so that the action between
contiguous parts is no longer normal but in a direction
tending to diminish their relative motion. Hence the
perfect fluid cannot be molecular.
All that is necessary in order to form a correct mathe¬
matical theory of a material system is that its properties
shall be clearly defined and shall be consistent with each
other. This is essential; but whether a substance having
such properties actually exists is a question which comes to
be considered only when we propose to make some practi¬
cal application of the results of the mathematical theory.
The properties of our perfect liquid are clearly defined and
consistent with each other, and from the mathematical
theory we can deduce remarkable results, some of which
may be illustrated in a rough way by means of fluids
which are by no means perfect in the sense of not being
viscous, such, for instance, as air and water.
The motion of a fluid is said to be irrotational when it
is such that if a spherical portion of the fluid were sud¬
denly solidified, the solid sphere so formed would not be
rotating about any axis. When the motion of the fluid is
rotational the axis and angular velocity of the rotation of
any small part of the fluid are those of a small spherical
portion suddenly solidified.
The mathematical expression of these definitions is as
follows:—Let u, v, w be the components of the velocity of
the fluid at the point (#, y, z), and let
dv dw o _^w du _du dv ....
a dz dy ’ P dx dz ’ ^ dy dx ' ’
then a, (3, y are the components of the velocity of rotation
of the fluid at the point (x, y, z). The axis of rotation
is in the direction of the resultant of a, (3, and y, and
the velocity of rotation, w, is measured by this resultant.
A line drawn in the fluid, so that at every point of the
line
\ dx 1 ch/ _ 1 cfe _ 1
a, ds (5 ds y ds a ' ’
where s is the length of the line up to the point x, y, z, is
called a vortex line. Its direction coincides at every point
with that of the axis of rotation of the fluid.
We may now prove the theorem of Helmholtz, that the
points of the fluid which at any instant lie in the same
vortex line continue to lie in the same vortex line during
the whole motion of the fluid.
The equations of motion of a fluid are of the form
hi dp dV
P It + dx^~ ^ dx
when p is the density, which in the case of our homogene¬
ous incompressible fluid we may assume to be unity, the
g
operator — represents the rate of variation of the symbol to
which it is prefixed at a point which is carried forward
with the fluid, so that
hi du du du du
Yt = dt + udx+%+wd^
(4),
p is the pressure, and V is the potential of external forces.
There are two other equations of similar form in y and z.
Differentiating the equation in y with respect to z, and
that in z with respect to y, and subtracting the second from
the first, we find
d hv dhw_ /g,
dz It dy It
Performing the differentiations and remembering equa¬
tions (1) and also the condition of incompressibility,
we find
du dv dw
dx dy dz
hz du r, du du
= “-+(3^ + 7
(6),
(7).
at ^ dx ' ^ dy ' ' dz
Now, let us suppose a vortex line drawn in the fluid so
as always to begin at the same particle of the fluid. The
components of the velocity of this point are u, v, w. Let
us find those of a point on the moving vortex line at a
distance ds from this point where
ds = wdar . . . (8).
The co-ordinates of this point are
x + adar , y + /3'io- , z + ydcr . . (9),
and the components of its velocity are
(3),
§06 7
u+ui,J
S/3 ,
Sy 7
W+Ytd<T
Consider the first of these components,
equation (7) we may write it
du 7 du n. du j
U^"dx a<*°" '77
dy1
du dy
dz
(10).
In virtue of
• (11),
du dx , du dy , , du dz ,,
u +-T-— do-+ y-do-+-p-~r da . (12),
dx da dy da dz da
or
u + ^do- (13).
da
But this represents the value of the component u of the
velocity of the fluid itself at the same point, and the same
thing may be proved of the other components.
Hence the velocity of the second point on the vortex line
is identical with that of the fluid at that point. In other
words, the vortex line swims along with the fluid, and is
always formed of the same row of fluid particles. The
vortex line is therefore no mere mathematical symbol, but
has a physical existence continuous in time and space.
By differentiating equations (1) with respect to x, y, and
z respectively, and adding the results, we obtain the equa¬
tion—
(h* + d/3 + ^y = 0
dx dy dz
(14).
This is an equation of the same form with (6), which
expresses the condition of flow of a fluid of invariable
density. Hence, if we imagine a fluid, quite independent
of the original fluid, whose components of velocity are a,
(3, y, this imaginary fluid will flow without altering its
density.
Now, consider a closed curve in space, and let vortex