Page:EB1911 - Volume 14.djvu/136

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

−mJ−1sh(−)cos(−), mJ−1ch(−)sin(−). (10)

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

(11)

and the condition can be inferred when cavitation begins.

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

= m ch ( − ) cos

= −Uc ch ( − ) sh cos /sh ( − ) (12)

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

1 = −Ux = −Uc ch cos , (13)

for liquid filling the cylinder; and

(14)

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

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

32. Next consider the motion given by

= m ch 2( − ) sin 2, = −m sh 2( − )cos 2; (1)

in which = 0 over the ellipse, and

′ = + R (x2 + y&#8202;2)

= [ −m sh 2( − ) + Rc2 ]cos 2 + Rc2 ch 2, (2)

which is constant over the ellipse if

Rc2 = m sh 2( − ); (3)

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

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

x2/a2 + y&#8202;2/b2 = 1, (4)

1′ = m1 (x2/a2 + y&#8202;2/b2) (5)

with

∇21′ = −2R = −2m1 (1/a2 + 1/b2),

1 = m1 (x2/a2 + y&#8202;2/b2) − R (x2 + y&#8202;2)

= −R (x2 − y&#8202;2) (a2 − b2) / (a2 + b2), (6)

1 = Rxy(a2 − b2) / (a2 + b2),

&emsp;w1 = 1 + 1i = −iR (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 = (a2 + b2), and (a2 − b2)2 / (a2 + b2). (7)

For the liquid in the interspace between and ,

= 1/th 2(−)th 2; (8)

and the effective k2 of the liquid is reduced to

c2/th 2(−)sh 2, (9)

which becomes c2/sh 2 =  (a2 − b2)/ab, when  = ∞, and the liquid surrounds the ellipse 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 is resolved into 1 + 2, such that 1 + Rx2 and 2 + Ry&#8202;2 is constant over the boundary.

Inside a cylinder

1 + 1i = −iR(x+yi)2a2 / (a2+b2), (10)

2 + 2i = iR(x+yi)2b2 / (a2+b2), (11)

and for the interspace, the ellipse being fixed, and 1 revolving with angular velocity R

1 + 1i = − iRc2 sh 2( − + i) (ch 2 + 1) / sh 2(1 − ), (12)

2 + 2i = iRc2 sh 2( −  + i) (ch 2 − 1) / sh 2(1 − ), (13)

satisfying the condition that 1 and 2 are zero over =, and over = 1

1 + Rx2 =  Rc2 (ch 21 + 1), (14)

2 + Ry&#8202;2 =  Rc2 (ch 21 − 1), (15)

constant values.

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

w = m ch 2(−), = +i, (16)

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

Similarly, with

x + yi = c√[ sin ( + i) ] (17)

the function

= Qc sh( − ) sin ( − ) (18)

will give motion streaming past the fixed cylinder =, and dividing along = ; and then

x2 − y&#8202;2 = c2 sin ch, 2xy = c2 cos  sh. (19)

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

x4 + 6x2y&#8202;2 + y&#8202;4 = 2c4, or x4 + y&#8202;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 1 is to be made to satisfy the conditions (i.) ∇21 = 0, (ii.) 1 + Rx2 = Ra2, or 1 = 0 when x = ± a, (iii.) 1 + Rx2 = Ra2, 1 = R (a2 − x2), when y = ± b.

Expanded in a Fourier series,

(1)

so that

(2)

an elliptic-function Fourier series; with a similar expression for 2 with x and y, a and b interchanged; and thence = 1 + 2.

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

The polar equation of the cross-section being

r&#8202;undefined cos = aundefined, or r + x = 2a, (3)

the conditions are satisfied by

′ = Ur sin − 2Uaundefinedr&#8202;undefined sin  = 2Ur&#8202;undefined sin  (r&#8202;undefined cos  − aundefined), (4)

= 2Uaundefinedr&#8202;undefined sin = −U √ [ 2a(r − x) ], (5)

w = −2Uaundefinedzundefined, (6)

and the resistance of the liquid is 2aV2/2g.

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

(7)

and in the absolute space curve given by ,

(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 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 2 ( − 0); and, as before, if d is the increase in 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 2d = 2yk·PP′; and taking PP′ parallel to Oy and Ox,

u = −d/ydy, &emsp; v = d/ydx, (1)

and 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 d/yds is the component velocity across ds in a direction turned through a right angle forward.

In this symmetrical motion

(2)

suppose; and in steady motion,

(3)

so that

2 / y = −y&#8202;−2∇2 = dH / d (4)

is a function of, say &fnof;′, and constant along a stream line;

dH/dv = 2q, &emsp; H − &fnof; = constant, (5)

throughout the liquid.

When the motion is irrotational,

(6)

(7)