Page:EB1911 - Volume 14.djvu/138

so that the flow is independent of the curve for all curves mutually reconcilable; and the circulation round a closed curve is zero, if the curve can be reduced to a point without leaving a region for which φ is single valued.

If through every point of a small closed curve the vortex lines are drawn, a tube is obtained, and the fluid contained is called a vortex filament.

By analogy with the spin of a rigid body, the component spin of the fluid in any plane at a point is defined as the circulation round a small area in the plane enclosing the point, divided by twice the area. For in a rigid body, rotating about Oz with angular velocity ζ, the circulation round a curve in the plane xy is

$\int$ ζ ( x	dy	− y	dx	) ds = ζ times twice the area.
	ds		ds

(3)

In a fluid, the circulation round an elementary area dxdy is equal to

udx + ( v +	dv	dx ) dy − ( u +	du	dy ) dx − vdy = (	dv	−	du	) dx dy,
	dx		dy		dx		dy

(4)

so that the component spin is

1/2 (	dv	−	du	) = ζ,
	dx		dy

(5)

in the previous notation of § 24; so also for the other two components ξ and η.

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, ξ, η, ζ, 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 ωα is constant where α is the cross-section of the filament and ω 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	= 0,	Dv	+	dQ	= 0,	Dw	+	dQ	= 0,
dt		dx		dt		dy		dt		dz

(6)

Q = ∫dp/ρ + V,

(7)

and taking dx, dy, dz in the direction of u, v, w, and

dx : dy : dz = u : v : w,

D	( udx + vdy + wdz ) =	Du	dx + u	Ddx	+ . . . = −dQ + 1/2 dq2,
dt		dt		dt

(8)

and integrating round a closed curve

D	$\int$ (udx + vdy + wdz) = 0,
dt

(9)

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 αω 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 α of a filament by dS and its mass by dm, the quantity ωdS/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 ω cos εdS/dm, if dS is the oblique section of which the normal makes an angle ε with the filament, while the aggregate vorticity of a mass M inside a surface S is

M⁻¹ ∫ ω cos ε dS.

Employing the equation of continuity when the liquid is homogeneous,

2 (	dζ	−	dη	) = ∇2u, ... , ∇2 = −	d2	−	d2	−	d2	,
	dy		dz		dx2		dy2		dz2

(10)

which is expressed by

∇² (u, v, w) = 2 curl (ξ, η, ζ), (ξ, η, ζ) = 1/2 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, v = V + v′ - zP + xR, w = W + w′ − xQ + yP.

(1)

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	u +	dm	v +	dn	w
dt		dt		dt		dt

+ l (	du	+ u′	du	+ v′	du	+ w′	du	)
	dt		dx		dy		dz

+ m (	dv	+ u′	dv	+ v′	dv	+ w′	dv	)
	dt		dx		dy		dz

+ n (	dw	+ u′	dw	+ v′	dw	+ w′	dw	).
	dt		dx		dy		dz

(3)

But as l, m, n are the direction cosines of a line fixed in space,

dl	= mR − nQ,	dm	= nP − lR,	dn	= lQ − mP;
dt		dt		dt

(4)

so that

Dk	= l (	du	− vR + wQ + u′	du	+ v′	du	+ w′	du	) + m (...) + n (...)
dt		dt		dx		dy		dz

= l ( X −

1

dp

) + m ( Y −

1

dp

) + n ( Z −

1

dp

),

p

dx

p

dy

p

dz

(5)

for all values of l, m, n, leading to the equations of motion with moving axes.

When the motion is such that

u = −

dφ

− m

dψ

, v = −

dφ

− m

dψ

, w = −

dφ

− m

dψ

,

dx

dx

dy

dy

dz

dz

(6)

as in § 25 (1), a first integral of the equations in (5) may be written

$\int$	dp	+ V + 1/2q2 −	dφ	− m	dψ	+ (u − u′) (	dφ	+ m	dψ	)
	ρ		dt		dt		dx		dx

+ (v − v′) (	dφ	+ m	dψ	) + (w − w′) (	dφ	+ m	dψ	) = F(t),
	dy		dy		dz		dz

(7)

in which

dφ	− (u − u′)	dφ	− (v − v′)	dφ	− (w − w′)	dφ
dt		dx		dy		dz

=	dφ	− (U − yR + zQ)	dφ	− (V − zP + xR)	dφ	− (W − xQ + yP)	dφ
	dt		dx		dy		dz

(8)

is the time-rate of change of φ at a point fixed in space, which is left behind with velocity components u − u′, v − v′, w − w′.

In the case of a steady motion of homogeneous liquid symmetrical about Ox, where O is advancing with velocity U, the equation (5) of § 34

p/ρ + V + 1/2q′² − ƒ (ψ′) = constant

(9)

becomes transformed into

p	+ V + 1/2q2 −	U		dψ	+ 1/2U2 − ƒ (ψ + 1/2Uy2) = constant,
ρ		y		dy

(10)

ψ′ = ψ + 1/4U y²,

(11)

subject to the condition, from (4) § 34,

y⁻² ∇²ψ′ = −ƒ′(ψ′),   y⁻² ∇²ψ = −ƒ′ (ψ + 1/2Uy²).

(12)

Thus, for example, with

ψ′ = 3/4U y² (r²a⁻² − 1), r² = x² + y²,

(13)

for the space inside the sphere r = a, compared with the value of ψ′ in § 34 (13) for the space outside, there is no discontinuity of the velocity in crossing the surface.

Inside the sphere

2ζ =

d

(

1

dψ′

) +

d

(

1

dψ′

) =

15

U

y

,

dx

y

dx

dy

y

dy

2

a2

(14)

so that § 34 (4) is satisfied, with

ƒ′ (ψ′) =	15	Ua−2, ƒ (ψ′) =	15	Uψ′ a−2;
	2		2

(15)

and (10) reduces to

p	+ V −	9	U { (	x2	− 1 )	2	− (	y2	− 1/2 )	2	} = constant;
ρ		8		a2				a2

(16)

this gives the state of motion in M. J. M. Hill’s spherical vortex, advancing through the surrounding liquid with uniform velocity.

39. As an application of moving axes, consider the motion of liquid filling the ellipsoidal case

x2	+	y2	+	z2	= 1;
a2		b2		c2

(1)

and first suppose the liquid to be frozen, and the ellipsoid to be