/ dv \ / du \ / dv du \
udx + ( v + --dx )dy - ( u + --dy )dx - vdy = ( -- - -- )dx dy, (4)
\ dx / \ dy / \ dx dy /
so that the component spin is
/ dv du \
½ ( -- - -- ) = [zeta], (5)
\ dx dy /
in the previous notation of § 24; so also for the other two components
[xi] and [eta].
Since the circulation round any triangular area of given aspect is the
sum of the circulation round the projections of the area on the
coordinate planes, the composition of the components of spin, [xi],
[eta], [zeta], is according to the vector law. Hence in any
infinitesimal part of the fluid the circulation is zero round every
small plane curve passing through the vortex line; and consequently
the circulation round any curve drawn on the surface of a vortex
filament is zero.
If at any two points of a vortex line the cross-section ABC, A´B´C´ is
drawn of the vortex filament, joined by the vortex line AA´, then,
since the flow in AA´ is taken in opposite directions in the complete
circuit ABC AA´B´C´ A´A, the resultant flow in AA´ cancels, and the
circulation in ABC, A´B´C´ is the same; this is expressed by saying
that at all points of a vortex filament [omega][alpha] is constant
where [alpha] is the cross-section of the filament and [omega] the
resultant spin (W. K. Clifford, _Kinematic_, book iii.).
So far these theorems on vortex motion are kinematical; but
introducing the equations of motion of § 22,
Du dQ Dv dQ Dw dQ
-- + -- = 0, -- + -- = 0, -- + -- = 0, (6)
dt dx dt dy dt dz
_
/
Q = | dp/[rho] + V, (7)
_/
and taking dx, dy, dz in the direction of u, v, w, and
dx : dy : dz = u : v : w,
D / \ Du D dx
-- (u dx + v dy + w dz ) = -- dx + u ---- + ... = -dQ + ½dq², (8)
dt \ / dt dt
and integrating round a closed curve
_
D /
-- | (u dx + v dy + w dz) = 0, (9)
dt _/
and the circulation in any circuit composed of the same fluid
particles is constant; and if the motion is differential irrotational
and due to a velocity function, the circulation is zero round all
reconcilable paths. Interpreted dynamically the normal pressure of the
surrounding fluid on a tube cannot create any circulation in the tube.
The circulation being always zero round a small plane curve passing
through the axis of spin in vortical motion, it follows conversely
that a vortex filament is composed always of the same fluid particles;
and since the circulation round a cross-section of a vortex filament
is constant, not changing with the time, it follows from the previous
kinematical theorem that [alpha][omega] is constant for all time, and
the same for every cross-section of the vortex filament.
A vortex filament must close on itself, or end on a bounding surface,
as seen when the tip of a spoon is drawn through the surface of water.
Denoting the cross-section [alpha] of a filament by dS and its mass by
dm, the quantity [omega]dS/dm is called the _vorticity_; this is the
same at all points of a filament, and it does not change during the
motion; and the vorticity is given by [omega] cos[epsilon]dS/dm, if dS
is the oblique section of which the normal makes an angle [epsilon]
with the filament, while the aggregate vorticity of a mass M inside a
surface S is
_
/
M^(-1) | [omega] cos [epsilon] dS.
_/
Employing the equation of continuity when the liquid is homogeneous,
/ d[zeta] d[eta]\ d² d² d²
2( ------ - ------ ) = [nabla]²u, ... , [nabla]² = - --- - --- - ---, (10)
\ dy dz / dx² dy² dz²
which is expressed by
[nabla]²(u,v,w) = 2 curl ([xi], [eta], [zeta]),
([xi], [eta], [zeta]) = ½ curl (u, v, w). (11)
38. _Moving Axes in Hydrodynamics._--In many problems, such as the
motion of a solid in liquid, it is convenient to take coordinate axes
fixed to the solid and moving with it as the movable trihedron frame
of reference. The components of velocity of the moving origin are
denoted by U, V, W, and the components of angular velocity of the
frame of reference by P, Q, R; and then if u, v, w denote the
components of fluid velocity in space, and u´, v´, w´ the components
relative to the axes at a point (x, y, z) fixed to the frame of
reference, we have
u = U + u´ - yR + zQ, (1)
v = V + v´- zP + xR,
w = W + w´ - xQ + yP.
Now if k denotes the component of absolute velocity in a direction
fixed in space whose direction cosines are l, m, n,
k = lu + mv + nw; (2)
and in the infinitesimal element of time dt, the coordinates of the
fluid particle at (x, y, z) will have changed by (u´, v´, w´)dt; so
that
Dk dl dm dn
-- = --u + --v + --w
dt dt dt dt
/ du du du du \
+ l( -- + u´-- + v´-- + w´-- )
\ dt dx dy dz /
/ dv dv dv dv \
+ m( -- + u´-- + v´-- + w´-- )
\ dt dx dy dz /
/ dw dw dw dw \
+ n( -- + u´-- + v´-- + w´-- ). (3)
\ dt dx dy dz /
But as l, m, n are the direction cosines of a line fixed in space,
dl dm dn
-- = mR - nQ, -- = nP - lR, -- = lQ - mP; (4)
dt dt dt
so that
Dk / du du du du \
-- = l( -- - vR + wQ + u´-- + v´-- + w´-- ) + m(...) + n(...)
dt \ dt dx dy dz /
/ 1 dp \ / 1 dp \ / 1 dp \
= l( X- --- -- ) + m( Y - --- -- ) + n( Z - --- -- ), (5)
\ p dx / \ p dy / \ p dz /
for all values of l, m, n, leading to the equations of motion with
moving axes.