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HYDRODYNAIMCS]
131
HYDROMECHANICS


so that the effective inertia of a sphere is increased by half the weight of liquid displaced; and in frictionless air or liquid the sphere, of weight W, will describe a parabola with vertical acceleration V-Y, 'l,5%g no

Thus a spherical air bubble, in which W/W' is insensible, will beg-in to rise in water with acceleration 2g. 45. When the liquid is bounded externally by the fixed ellipsoid )=>, , a slight extension will give the velocity function ¢ of the liquid in the inters ace as the ellipsoid)=o is passing with velocity U through the conffxcal position; ¢» must now take the form x(//~{-N), 5 and will satisfy the conditions in the shape Q B abc ' M abcdk

A+ 1'f'C1, = abc)(a2'l'))P

¢“ U"B., +C..-B.-C. U'°x db; ' m, abcdx ' (0 615161 0 (012 'i')>P

and any confocal ellipsoid defined bi '), internal or external to. ).=), , may be supposed to swim wit the liquid for an instant, without distortion or rotation, with velocity along Ox 2 Bx+C.-B1-Ci s?

4 U BTW# C., ~ BT Ci

Since - Ux is the velocity function for the liquid W' filling thel ellipsoid A-=o, and moving bodily with it, the effective mertiaof the liquid in the inter space isA..+B.+c. Bo'l'Eo " Ei " C1W, ' (2) I

If the ellipsoid is of revolution, with b=C, j A +281

45" Bl:

and the Stokes' current function up can be written down ¢= - iUy2§ § €-3. <4>

reducing, when the liquid extends to 'infinity and B1=o, to ¢=iUx§ . ¢= -rUy=-gi. <5> =

so that in the relative motion past the body, as when fixed in the g current U parallel to xO, f

¢'==sUx(1+§ ';,). ¢'=sUf (1 -§ ;) 4 <61 E Changing] the origin from the centre to the-focus of a prolatel spheroi, t en putting b“=pa, J=>'a, and proceeding to thelimité where a =oo, we find for a paraboloid of revolution 1, =; = 1L EL: ' l ~ T 1 ~ ' 2

A;, %f5/=1>+>~'-2x.- ' .<8> §

with X =o over the surface of the paraboloid; and then ¢'=%Uly'-in/(r*+y?)'f12x1; oi, ¢ =-%U1>l/ (x'+;v')-xl; (10) 5

¢=-%Ui> log l-i(x'+y')+#l» (H)

The relative path of a liquid particle is along a stream line A 4/= § Uc', a constant, (12)

S 2y2 (2 c2)2 g = , 62)2 ' 1 K

" L=T><y=' L”-ci) ' *'<"'+y') ==z>o= -is "9 a Ci; while the absolute path of a particle in space'will be given by d z 2

if-if ='%1>i» <'4>

y'-c'=a'e"/P. (15)

46. Between two concentric spheres, with G2+X=f2, U2+7|=a12;

A==B =C =a°/3r°,

I-a' ai” 2 1-ar' af"

and the effective inertia of the liquidin the inter space An-i-2A, °+ 3,

2A0 2A1W = ixatils -2:25 W '

When the spheres are not concentric, an expression for the effective inertia can be found by the method of'*irnag|es (W. M. Hicks, Phil, Trans., 1880). »-The

image of a 'source of strcn h /4 at S outside a s here of radius a is a source of strength pa/gat H, where OS =f, OpH=a'/f, and a line sink reaching from the image H to the centre O of line strength - ii/a; this combination 'wil be found to produce no How across the surface of the sphere. v Taking Ox along OS, the Stokes' function at P for the source S is it cos PSx, and of the source H and line sink OH is »;¢(a, [f).cos Pl-fx and - (/1/a)(PO - PH); so that, , ip = p (cos P'5x+?cosPHx-'QQ-E-E, (4) and il/ = -p, a constant, over the surface of the sphere, so that there is no flow across.

When the source S is inside the sphere and H outside, the line sink must extend from H to infinity in the ipaage system; to realize physically the condition of zero How across the sphere, an equal sink must be introduced at some other internal point S'. When S and S' lie on the same radius, taken along Ox, the Stokes function caube written clown; and when S and S' coalesce a doublet is produced, with a doublet' image at H. ' ' For a doublet at S, of moment m, the Stokes' function is' “ d ., y2

im 3] cos PSx= f m%; i (5)

and for its imageat H the Stokes' function is m Iéjcos PHx =m%!-Fifi; (6)

so that for the combination .. ., s g 2 3, :s

~°='"y" i'1rm"1%=1)='"f'§ ('i§ n@°ifs=i~ <7> and this vanishes oven the surface of 'the sphere. There is no- Stokes' Lunction when the axis of the doublet at S does not pass through O; the imager system will, consist of an inclined doublet at H, making an equa angle with OS as the doublet S, and of a parallel negative line doublet, extending from H to O, of moment varying as the distance from O. ' ', A distribution of sources and doublets over' a moving surface will enable an expression to be obtained. for the velocity function of a body movinfg in the presence of a. fixed sphere, or, inside it. The method o electrical images will enable the stream function ¢ to be inferred from a distribution of doublets, finite in number when 'the surface is composed of two, spheres intersecting at an angle wr/m, where m is an integer (R. A. Herman, Quart. Jour. of Math. xxii.).

hThus for -m=2, the spheres are orthogonal, and it can be' verified t at -' i ' »

a 3 a as

¢, =§ .Uy2 1 -is-':¢;+;, , (8)

the radius pf the spheres, and

rl, rs, r the' distances of, a point where, qi, wi. afaiaz/V (ai'+a¢.*') is their clrcle of intersection, and from their centres.

The correspondiugexpression for two orthogonal cylinders will be ~ ' ' '2 ' '2' y

¢'=Uy "ft-7é+%=, <9>.;

With a, =°o, these 'reduce to'- - - i, 'I 5, i », 5 ' .

u 'P'=%Uy”<1-gs)§ ».01' U95 (f gl A (lo) for a sphere or cylinder, and a diarnetral #plane 'O ' Two equal spheres, intersecting at Iso, will require O ..¢g 02 a“(a-2.9 313-, a'(¢%l-fx), ip " a. 2n3'|T 2715 +2723 2736 r '(I with fa similar expression for cylinders; so that the plane x=o rnayibe introduced as a boundary, cutting the surface at 6092 The motion of these cylinders across the line .of centres is the equivalent of a line doublet along each axis. 47. The extension of Green's solution to a rotation of the ellipsoid was made by A. Clebsch, by taking a velocity function; . (~) f 4>=J¢x., A

for a rotation R about Oz; and ai similar procedure shows, that an ellipsoidal surface A may be in rotation about Oz without disturbing the motion if

I I — dx

R 3 a2 +A' +b2+>.> *HEX

W ' I/(b"'l°?)” I/(1124-X)

and that the continuity of the liquid is secured 'if (2)

(a'+7)3/2(b2+})3/'(¢2+})%2§ =constant, A (3) °°, Nd> N B.-A,

X- i 'f'X ""P<f»+ ><1f+»> Tb? WT' 4 and at the surface }=0, : j V '-N, B>A- Q, .

R Gi'2+7;§ >El'i ao?-b“0 a.bcifiF?', - " rib*- 1/Ei, , ' (5)

N I/l>* - I/Q'

T=R ' .

° 's for (a+i=)*-”..2 1.=° <9

a2 b2' z/(a2+ bs)

“Rm ~ 2>/<a2+1»2> -c 0-Ao