The components of the liquid velocity q, in the direction of the
normal of the ellipse η and hyperbola ξ, are
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 |
and the condition can be inferred when cavitation begins.
With β = 0, the stream is parallel to x0, and
= −Uc ch (η − α) sh η cos ξ/sh (η − α)
over the cylinder η, and as in (12) § 29,
for liquid filling the cylinder; and
φ | = | th η | , |
φ1 | th (η − α) |
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
in which ψ = 0 over the ellipse α, and
= [ −m sh 2(η − α) + 14Rc2 ]cos 2ξ + 14Rc2 ch 2η,
which is constant over the ellipse η if
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
with
= −12R (x2 − y 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
For the liquid in the interspace between α and η,
φ | = | m ch 2(η − α) sin 2ξ |
φ1 | 14 Rc2 sh 2η sin 2ξ (a2 − b2) / (a2 + b2) |
and the effective k2 of the liquid is reduced to
which becomes 14 c2/sh 2η = 18 (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 + 12Rx2 and ψ2 + 12Ry 2 is constant over the boundary.
Inside a cylinder
and for the interspace, the ellipse α being fixed, and α1 revolving with angular velocity R
satisfying the condition that ψ1 and ψ2 are zero over η = α, and over η = α1
constant values.
In a similar way the more general state of motion may be analysed, given by
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
the function
will give motion streaming past the fixed cylinder η = α, and dividing along ξ = β; and then
In particular, with sh α = 1, the cross-section of η = α is
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 + 12Rx2 = 12Ra2, or ψ1 = 0 when x = ± a,
(iii.) ψ1 + 12Rx2 = 12Ra2, ψ1 = 12R (a2 − x2), when y = ± b.
Expanded in a Fourier series,
a2 − x2 = | 32 | a2 | cos (2n + 1) 12 πx/a | , |
π3 | (2n + 1)3 |
so that
ψ1 = R | 16 | a2 | cos (2n + 1) 12πx/a · ch (2n + 1) 12πy/a) | , |
π3 | (2n + 1)3 · ch (2n + 1) 12πb/a |
w1 = φ1 + ψ1i = iR | 16 | a2 | cos (2n + 1) 12πz/a | , |
π3 | (2n + 1)3 ch (2n + 1) 12πb/a |
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
the conditions are satisfied by
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 |
and in the absolute space curve given by ψ,
dy | = − | (y − c)2 | , x = | 2ac | − 2a log (y − c). |
dx | 2ay | y − c |
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,
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 |
suppose; and in steady motion,
dH | + | 1 | dψ | ∇2ψ = 0, | dH | + | 1 | dψ | ∇2ψ = 0, | ||
dx | y 2 | dx | dy | y 2 | dy |
so that
is a function of ψ, say ƒ′(ψ), and constant along a stream line;
throughout the liquid.
When the motion is irrotational,
ζ = 0, u = − | dφ | = − | 1 | dψ | , v = − | dφ | = | 1 | dψ | , | ||
dx | y | dy | dy | y | dx |
∇2ψ = 0, or | d 2ψ | + | d 2ψ | − | 1 | dψ | = 0. | |
dx2 | dy 2 | y | dy |