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124
[HYDRoDYNAMIcs
HYDROMECHANICS


The components of the liquid velocity q, in the direction of the normal of the ellipse 11 and hyperbola £, are -mJ"Sh(11-f=)C0S(E~l9), m]"Ch(1r-¢)Si11(£-B)~ (10) The velocity q is zero in a corner where the hyperbola B cuts the ellipse a; and round the ellipse a the velocity q reaches a mafriimum when the tangent has turned through a right angle, and then g = QW# (ch;>a.;c:s 28); (I I)

and the condition can be inferred when cavitation begins. With 13=o, the stream is parallel to xo, and 41 = m ch (1r-a)c0sE

= - Uc ch(11-a)sh11cos 2/sh(11~a) (12) over the cylinder 11, and as in (12) § 29, 4>i=-Ux=-Ucch11cos E, (13)

for liquid filling the cylinder; and Q Z th 11 (I)

¢1 th (111) ' 4

over the 'surface of 11; so that parallel to Ox, the effective inertia of the cylinder 11, displacing M' liquid, is increased by M'th11/th(11-a), reducing when a.=oo to I/I'th11=M'(b/a). Similarly, parallel to Oy, the increase of effective inertia is M'/th 11 t (11-11), reducing to M'/th 11=M'(a/b), when a=co, and the liquid extends to infinity.

32. Next consider the motion given by 41=mch2(11-a)sin2£, 1l=-msh2(11-a)cos2£; (1) in which 1]/=-'o over the ellipse o., and //' =l/+%R(x'+;v')

=[-m sh 2(1)-a.)-I-}Rc2]cos 2£+}Rc' ch 211, (2) which is constant over the ellipse 11 if }Rc'=m sh 2(11-a); (3)

so that this ellipse can be rotating with this angular velocity R. for an instant without distortion, the ellipse a. being fixed. For the liquid filling the interior of a rotating elliptic cylinder of cross section

x”/H'+y”/b* = I, (4)

11/1' =m1(x'/U'+y'/b') (5)

with V'¢1'=-2R=-2m1 1 a' 1 bi

(/ + / Q,

  • Pi =m1(x”/¢1'+y'/b') - %R(x'+> )

=~%R(x'-y“) (02 - b')/ (112-l-112). (6) ¢1=Rxy(G'-'b“)/(¢1'+b'),

wi =4>1 +1011 = - %iR(x+yi)“(¢l' ' 52)/(f1'+b')~ 1 The velocity of a liquid particle is thus (a”- bf)/(a*+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 (a'-b')'/(af-l-b')f of the solid; and the effective radius of gyration, solid and liquid, is given by

k' =i(a”+b'), and Ha'-b')'/(a'+b'). (7) For the liquid in the inter space between a and 11, $ m ch 2(11-a) sin 25

41, %R¢:' sh 211 sin 2.E(a2-b*)/la'-1-b') = I/th 2(1)-a)th 211; (8)

and the effective k' of the liquid is reduced to tb'/th 2(11-¢)Sh 211. (9)

which becomes § c'/sh 211=%(a2-bf)/ab, when a=¢o, and the liquid surrounds the ellipse 11 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 -, l is resolved into 1]/, +»l/2, such that 1[q+%Rx2 and 1//1-l-%Ry' is constant over the boundary.

Inside a cylinder

4>1+l/1i= - %iR(x+yi)'11'/(¢1'+b2), (10) ¢1+l/1i= =liR(r+yi)”b'/ (112-l-b°'), (1 I) and for the inter space, the ellipse a being fixed, and ai revolving with angular velocit R

Y

¢1+l/1i= - § iRc'sh 2(11'G+£i) (ch 2a-|-I)/sh 2(a.1-a), (12) 4>2+1l/2i= § iRc'sh 2(-11-a+£i) (ch 2a.- 1)/sh 2(u., - u.), (13) satisfying the condition that gl, and 1//2 are zero over 11 =a., and over fl = 111

»/f1+%Rx'=tRv”(ch 2¢1+1). (14)

»l/1-l-%Ry'= %Rv“(Ch 2111 - 1), (15) constant values.

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

w=m¢h2(I-1). ~/=a+Bi. (16)

as giving a homogeneous strain velocity to the confocal s stem to which may be added a circulation, represented by an additional term mf in w.

Similarly, with

x+;vi =¢~/ lSln(£+111)l (17)

the function

¢=Qv Sh%(11- ¢)S1n%(£-B) (IS)

will give motion streaming past the fixed cylinder 11 =a, and dividing along £=/S; and then

- x2-y'=c' sinfch 11, 2xy=c2 cos£sh 11. (19) In particular, with sh a.= I, the cross-section of 11 = a is x*-I-6x'y'-I-y'=2c“, or x4-I-y4=c' (20) when the axes are turned through 45°. 33. Examlple 3.-Analysing in this way the rotation of a rectangle filled with iquid into the two components of shear, the stream function 1]/1 is to be made to satisfy the conditions (L) V2¢1 = or

(ii.) 1//, +%Rx'= § Ra', or gli =o when x= =*= a, (iii.) ¢, -1-§ Rx' = %Ra', nh = %R(a2-xf), when y = # b» Expanded in a Fourier series,

/

az-.xz=%¢1,2 C—°5(f;'nJQ—3;§ '°°1 a. (1) so that

RE: 2 cos(2n+1)§ 1rx/a ch (2n+1)%1ry/a “ N' ¢2n+1)=* . ¢h(2n+11s1fz»/ti 16 cos(2n+1)l1rz/a

w' =¢l+b" =1R;5a2 (2n+1)3ch(2n-ii I)1}1rbla' (2) an elliptic-function Fourier series; with a similar expression for 1,01 with x and y, a and b interchanged; and thence IP =¢i +1//2. Example 4.—Parabolic cylinder, axial advance, and liquid streaming past.

The polar equation of the cross-section being 1% cos § 0=al, orr + x=2a, (3)

the conditions are satisfied by

gb' = Ur sin 9 -2Ualfi sin £6 =2Url sin § 0(1} cos £0 - ai), (4) ul-=2Ualr} sin § 0= 'U'f[2G(f-0C)], (5) w é-zuaia, (6)

and the resistance of the liquid is 21rpaV2/2g. A relative stream line, along which 1l'=Uc, is the quartic curve 1-¢=v [2110-111. x=4“f, %§ 2f, 1= 4'f, f, § ”“§ ', ,§ ')'. (1) and in the absolute space curve given by ip, QL QQ Q

dx- lay 1, x-y c-2alog (y c). (8) 34. Illotion symmetrical abozlt an Axis.-When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function 11/ can be found analogous to that emplo ed 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 21r(l-gba); and, as before, if dip is the increase in 1/1 due to a displacement of P to P', then k the component of velocit normal to the surface swept out by PP' is such that 21rd¢/=21ryk.¥'P'; and taking PP' parallel to Oy and Ox, , 14 = -di///ydy. v #dv//ydx. (I)

and il/ is called after the inventor, “ Stol»:es'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 dy!//yds is the component velocity across ds in a direction turned through a right angle forward. In this symmetrical motion

LQ i(;d~#)

£'°' "'°' 2'”'dx(ydx) +dy ya?

I dal/ d*1// I dip I

1 =-§ 3;-l-@f§ g§ =-§ V'¢. (2)

suppose; and in steady motion,

!%'l'§ ;%cV2¢=09 '((%I'l"§ , E%¥V2'//=0r » so that

1 2?/y = - y 2V2b =dH/dxf/ ' (4) is a function of 10, say f' (1//), and constant along a stream line; dH/dv=2q§ ', H- f(//) =constant, (5) throughout the liquid.

When the motion is irrotational, f1¢» 1 Q Que#

r'°°' “ dx- ydy' v' dy ydx' (6)

1 Q 'dll L Q./

Vw' 91 01' dxz'l'dyz y dy 9° (7)