Page:EB1911 - Volume 14.djvu/136

The components of the liquid velocity q, in the direction of the normal of the ellipse η and hyperbola ξ, are

−mJ⁻¹sh(η−α)cos(ξ−β), mJ⁻¹ch(η−α)sin(ξ−β).

(10)

The velocity q is zero in a corner where the hyperbola β cuts the ellipse α; and round the ellipse α the velocity q reaches a maximum when the tangent has turned through a right angle, and then

q = Qea	√(ch 2α−cos 2β)	;
	sh 2α

(11)

and the condition can be inferred when cavitation begins.

With β = 0, the stream is parallel to x₀, and

φ = m ch (η − α) cos ξ
= −Uc ch (η − α) sh η cos ξ/sh (η − α)

(12)

over the cylinder η, and as in (12) § 29,

φ₁ = −Ux = −Uc ch η cos ξ,

(13)

for liquid filling the cylinder; and

φ	=	th η	,
φ1		th (η − α)

(14)

over the surface of η; so that parallel to Ox, the effective inertia of the cylinder η, displacing M′ liquid, is increased by M′th η/th(η−α), reducing when α = ∞ to M′ th η = M′ (b/a).

Similarly, parallel to Oy, the increase of effective inertia is M′/th η th (η − α), reducing to M′/th η = M′ (a/b), when α = ∞, and the liquid extends to infinity.

32. Next consider the motion given by

φ = m ch 2(η − α) sin 2ξ, ψ = −m sh 2(η − α)cos 2ξ;

(1)

in which ψ = 0 over the ellipse α, and

ψ′ = ψ + 1/2R (x² + y ²)
= [ −m sh 2(η − α) + 1/4Rc² ]cos 2ξ + 1/4Rc² ch 2η,

(2)

which is constant over the ellipse η if

1/4 Rc² = m sh 2(η − α);

(3)

so that this ellipse can be rotating with this angular velocity R for an instant without distortion, the ellipse α being fixed.

For the liquid filling the interior of a rotating elliptic cylinder of cross section

x²/a² + y ²/b² = 1,

(4)

ψ₁′ = m₁ (x²/a² + y ²/b²)

(5)

with

∇²ψ₁′ = −2R = −2m₁ (1/a² + 1/b²),

ψ₁ = m₁ (x²/a² + y ²/b²) − 1/2R (x² + y ²)
= −1/2R (x² − y ²) (a² − b²) / (a² + b²),

(6)

φ₁ = Rxy(a² − b²) / (a² + b²),

 w₁ = φ₁ + ψ₁i = −1/2iR (x + yi)² (a² − b²) / (a² + b²).

The velocity of a liquid particle is thus (a² − b²)/(a² + b²) of what it would be if the liquid was frozen and rotating bodily with the ellipse; and so the effective angular inertia of the liquid is (a² − b²)²/(a² + b²)² of the solid; and the effective radius of gyration, solid and liquid, is given by

k² = 1/4(a² + b²), and 1/4(a² − b²)² / (a² + b²).

(7)

For the liquid in the interspace between α and η,

φ	=	m ch 2(η − α) sin 2ξ
φ1		1/4 Rc2 sh 2η sin 2ξ (a2 − b2) / (a2 + b2)

= 1/th 2(η−α)th 2η;

(8)

and the effective k² of the liquid is reduced to

1/4 c²/th 2(η−α)sh 2η,

(9)

which becomes 1/4 c²/sh 2η = 1/8 (a² − b²)/ab, when α = ∞, and the liquid surrounds the ellipse η to infinity.

An angular velocity R, which gives components −Ry, Rx of velocity to a body, can be resolved into two shearing velocities, −R parallel to Ox, and R parallel to Oy; and then ψ is resolved into ψ₁ + ψ₂, such that ψ₁ + 1/2Rx² and ψ₂ + 1/2Ry ² is constant over the boundary.

Inside a cylinder

φ₁ + ψ₁i = −1/2iR(x+yi)²a²/(a²+b²),

(10)

φ₂ + ψ₂i =  1/2iR(x+yi)²b²/(a²+b²),

(11)

and for the interspace, the ellipse α being fixed, and α₁ revolving with angular velocity R

φ₁ + ψ₁i = −1/8 iRc² sh 2(η − α + ξi) (ch 2α + 1)/sh 2(α₁ − α),

(12)

φ₂ + ψ₂i = 1/8 iRc² sh 2(η − α + ξi) (ch 2α − 1)/sh 2(α₁ − α),

(13)

satisfying the condition that ψ₁ and ψ₂ are zero over η = α, and over η = α₁

ψ₁ + 1/2 Rx² = 1/8 Rc² (ch 2α₁ + 1),

(14)

ψ₂ + 1/2 Ry ² = 1/8 Rc² (ch 2α₁ − 1),

(15)

constant values.

In a similar way the more general state of motion may be analysed, given by

w = m ch 2(ζ−γ), γ = α+βi,

(16)

as giving a homogeneous strain velocity to the confocal system; to which may be added a circulation, represented by an additional term mζ in w.

Similarly, with

x + yi = c√[ sin (ξ + ηi) ]

(17)

the function

ψ = Qc sh1/2(η − α) sin 1/2 (ξ − β)

(18)

will give motion streaming past the fixed cylinder η = α, and dividing along ξ = β; and then

x² − y ² = c² sin ξ ch η, 2xy = c² cos ξ sh η.

(19)

In particular, with sh α = 1, the cross-section of η = α is

x⁴ + 6x²y ² + y ⁴ = 2c⁴, or x⁴ + y ⁴ = c⁴

(20)

when the axes are turned through 45°.

33. Example 3.—Analysing in this way the rotation of a rectangle filled with liquid into the two components of shear, the stream function ψ₁ is to be made to satisfy the conditions

(i.) ∇²ψ₁ = 0,
(ii.) ψ₁ + 1/2Rx² = 1/2Ra², or ψ₁ = 0 when x = ± a,
(iii.) ψ₁ + 1/2Rx² = 1/2Ra², ψ₁ = 1/2R (a² − x²), when y = ± b.

Expanded in a Fourier series,

a2 − x2 =	32	a2 $\sum$	cos (2n + 1) 1/2 πx/a	,
	π3		(2n + 1)3

(1)

so that

ψ1 = R	16	a2 $\sum$	cos (2n + 1) 1/2πx/a · ch (2n + 1) 1/2πy/a)	,
	π3		(2n + 1)3 · ch (2n + 1) 1/2πb/a

w1 = φ1 + ψ1i = iR	16	a2 $\sum$	cos (2n + 1) 1/2πz/a	,
	π3		(2n + 1)3 ch (2n + 1) 1/2πb/a

(2)

an elliptic-function Fourier series; with a similar expression for ψ₂ with x and y, a and b interchanged; and thence ψ = ψ₁ + ψ₂.

Example 4.—Parabolic cylinder, axial advance, and liquid streaming past.

The polar equation of the cross-section being

r ^1/2 cos 1/2θ = a^1/2, or r + x = 2a,

(3)

the conditions are satisfied by

ψ′ = Ur sin θ − 2Ua^1/2r ^1/2 sin 1/2θ = 2Ur ^1/2 sin 1/2θ (r ^1/2 cos 1/2θ − a^1/2),

(4)

ψ = 2Ua^1/2r ^1/2 sin 1/2θ = −U √ [ 2a(r − x) ],

(5)

w = −2Ua^1/2z^1/2,

(6)

and the resistance of the liquid is 2πρaV²/2g.

A relative stream line, along which ψ′ = Uc, is the quartic curve

y − c = √ [ 2a(r − x) ],   x =	(4a2y 2 − (y − c)4	,   r =	4a2y 2 + (y − c)4	,
	4a(y − c)2		4a(y − c)2

(7)

and in the absolute space curve given by ψ,

dy	= −	(y − c)2	, x =	2ac	− 2a log (y − c).
dx		2ay		y − c

(8)

34. Motion symmetrical about an Axis.—When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function ψ can be found analogous to that employed in plane motion, such that the flux across the surface generated by the revolution of any curve AP from A to P is the same, and represented by 2π (ψ − ψ₀); and, as before, if dψ is the increase in ψ due to a displacement of P to P′, then k the component of velocity normal to the surface swept out by PP′ is such that 2πdψ = 2πyk·PP′; and taking PP′ parallel to Oy and Ox,

u = −dψ/ydy,   v = dψ/ydx,

(1)

and ψ is called after the inventor, “Stokes’s stream or current function,” as it is constant along a stream line (Trans. Camb. Phil. Soc., 1842; “Stokes’s Current Function,” R. A. Sampson, Phil. Trans., 1892); and dψ/yds is the component velocity across ds in a direction turned through a right angle forward.

In this symmetrical motion

ξ = 0, η = 0, 2ζ =

d

(

1

dψ

) +

d

(

1

dψ

)

dx

y

dx

dy

y

dy

=

1

(

d 2ψ

+

d 2ψ

−

1

dψ

) = −

1

∇2ψ,

y

dx2

dy 2

y

dy

y

(2)

suppose; and in steady motion,

dH	+	1		dψ	∇2ψ = 0,	dH	+	1		dψ	∇2ψ = 0,
dx		y 2		dx		dy		y 2		dy

(3)

so that

2ζ/y = −y ⁻²∇²ψ = dH/dψ

(4)

is a function of ψ, say ƒ′(ψ), and constant along a stream line;

dH/dv = 2qζ,   H − ƒ(ψ) = constant,

(5)

throughout the liquid.

When the motion is irrotational,

ζ = 0,  u = −

dφ

= −

1

dψ

,  v = −

dφ

=

1

dψ

,

dx

y

dy

dy

y

dx

(6)

∇2ψ = 0, or	d 2ψ	+	d 2ψ	−	1		dψ	= 0.
	dx2		dy 2		y		dy

(7)