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HYDROMECHANICS


where il is a function of x, y, called the stream- or current-function; interpreted physically, ¢~¢0, the difference of the value of it at a fixed point A and a variable point P is the'flow, in ft.'*/ second, across any curved line AP from A to P, this being the same for all lines in accordance with the continuity.

Thus if dip is the increase of up due to a displacement from P to P', and' le is the component of velocity normal to PP", the flow across PP' is dd/=k.PP; and taking P ' parallel to Ox, di;/ = vdxpand similarly dt= -udy with PP' parallel to Oy; and generally dgb/dS is the velocity across ds, in a direction turned throug a right angle forward, against the clock.,

In the equations of uni planar motion ' " 'dv du:Pup d'l/

2i'=E§ .-E; = W +W = -V*¢.SUP1f>0Se, (3) so that in steady motion

i£+v“lf§ 5;-=0. '%I+v'~t%§ = 0. 'g-;+v?»l»=0, (4) and VRD must be a function of it.

If the -motion is irrotational, » ~£'= @ = d¢ § L/'

u-' dx dy' U 35 '° dx' (5)

so that il/ and d> are conjugate functions of x andy, , ¢+I»i=f(x+yi). VW/=0. v'¢=0; (6)

or putting¢+¢1=w.

x+y1=2. w=f(2)-The

curves ¢=constant and i/»=constant form system; and the interchange of 4> and up willxgive a.new state of uni planar motion, ” in which the velocity at every point is turned through a right angle without alteration of magnitude. For instance, in a uni planar flow, radially inward towards 0, the flow across any circle of radius 1' being the' same and denoted by 21rm, the velocity must be m/1, and » ¢='MlOg r, w//=m0, ¢+/»i=m log reiff, w=im log z.' (7) Interchanging these values ' ' ..

¢=m log r, ¢=m0, ¢+¢i==m log rdf (8) gives a state of vortex mqtion, 'circulating round Oz, called a straight or columnar vortex.

A single vortex will remain at rest, and cause a velocity at any point inversely as the distance from the axis and perpendicular to its direcf tion; analogous to the magnetic field of a straight electric current. If other vortices are present, any one may be SUPP(-$¢d, .to, -move Bith the velocity due to the others, the resultant streamsfurlction Qing

¢»¢2m log r=log Ilrfm; (9)

the path of a vortex is obtained by equating the value of it at the vortex to a constant, omitting the rm of the vortex itself. When the liquid is bounded by a cylindrical surface, the motion of a vortex inside may be determined as due to a series of vortex images, so arranged as to make the flow zero across the boundary. For a plane boundary the image is the optical reflection of the vortex. For example, a pair of equal opposite* vortices, molving on a line parallel to a plane boundary, wil have a corresponding pair of images, forming a rectangle of vortices, and the pat of a vortex will be the Cotes spiral

r sin 29==2a, or x'2+y"=af2; ., . (lo) an orthogonal

this is therefore the path of a single vortex in a 'right-angled' corner; and generally. if the angle of the corner is ir/n, the path is the Cotes spiral '

r sin nl) =na. (II)

A single vortex in a circular cylinder of radius ~a at a distance c from the centre will move with the velocity due to an equal. opposite image at aidistance zz*/c, and so describe a circle with velocityhzc/(0)-c2)in- the periodic time'21r(a3-cf)/m. ' (12) Conjugate functions can be employed also' for the motion 'dl' liquid in a thin sheet between two concentric spherical surfaces; the components of velocity along the meridian and parallel in co latitude, -0 and longitude) can be written '

d § .= .£-Q . ..E. .i?== éL[/

tw Sinodx' Sinadi da', 3

and then

¢-l-¢i=F(tan éti. e/i). (14)

28. Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it.-A stream-function it must be determined to satisfy the conditions -,

v'~»//=o, throughout the liquid; ' i (1) I vb ='constant, over any fixed boundary; M (2) dit/ds=»normal velocity reversed over a solid boundary, (3) so that, if the solid is moving with velocity U in the direction Ox, dit/d.s=-Udy/ds, or ¢'l'Uy =constant over the moving cylinder; 0DYNAMlC

andi¢+Uy=sU/"is:the stream function of the relative motion of the liquid past the cylinder, and similarly i//-Vx for the component velocity V along Oy; and generally

V/=!/+Uy-Vx (4)

is the relative stream-function, constant over a solid boundary moving with components U and V of velocity. If the liquid is stirred up by the rotation R of a cylindrical body, dgl/his == normal velocity reversed dx ~ dy

!/+iR(x”+y') =¢/Y U (6)

a constant over the boundary; and it/' is the current-function of the relative motion past the cylinder, but now ~ l, V'=;/+2R=o, (7)

throughout theuliquid. O A

inside an equilateral triangle, for instance, of height h, - it' = r 2R~1I3“//h. (3)

where a, B. 'V are thepenpendiculars on the sides of the triangle. In the general case 4/ =¢+Uy- Vx+a}R(x¢-4-yf) is the relative stream function for velocity compo neg its, U, V, R. 1 29.:Example I.-jluquid motion past a circular cylinder. f Consider the motion given by- »

w=U(2+¢1'/2). (I)

so that i ¢=U (f-Pg) COS 0 =U (1 -rg) N. (2) ¢=U<f- '¥)>sin 6=U<I -lg) y.

Then it =o over the cylinder r =a, which may be considered a fixed post; and a stream line past it along which t=Uc, a constant, is the curve ' '(r

- sin 0=r, (x'+y')(y- c) - a”y=o, (3) a cubic curve (Ca).

Over a concentric cylindexqxexternal or internal, of radius r=b, ~t'=¢+U1y=iU(1 - '§) + Uay, <4>

F

and vp is zero if

Ui/ U = (42 °' bf)/112; (5)

so that the cylinder may swim for an instant in the liquid without distortion, -with this velocity Ui; and w in (1) will givethe liquid motion in thejnterspace between the fixed cylinder r==a and the concentric cylinder r=b, moving with velocity Ui. ' ~-When b=o, Ui=oo; and when l>=oo, U;= -U., so that at infinitely the liquid is streaming in the direction x0 with velocity U. If the liquid is reduced to rest at infinity by the superposition of an opposite stream given by w = -Uz, we are left with w==Ua“/Z, (6)

4> = U(¢1”/f) COS 0 = Uazx/(xf +y'). (7) = -U(“2/t) Sin 0 = -~U¢l'y/(x'+y'). (8) givin the motion due to the passage of the cylinder r=a with velocity U through the' origin O in the direction Ox. If the direction of motion makes an angle 0' with Ox, fl¢ /df# 2x;V

tail 0 =€5l/'~i§ =;:, -;;'2=lZ3.Il 20, 0=fl'0, , and the velocity is Ua'/r'.

Q 'Along the path of a particle, defined by the C3 of (3), 2

SW i0'=xT?iTT==2Q5TQ' . <'°)

dw ~ d ”

= ~ ' %.sinB-a-;=3%-175-3% (ix)

on the radius of curvature is ia'/(y-ic), which shows that the curve is an Elastica or Lintearia. (J. C, Maxwell, Collected Works, ii. 208,) If ¢, d'enotes the velocity function of the liquid filling the cylinder r =b, and moving bodily with it with velocity Ui, ¢l = 'Uxa (lg)

and over the separating surface r=b ¢ U Q az-l-b

" 3;-"Ur'(l Tb2) "a2~' bil A (13)

and this, by § 36, is also the ratio of the kinetic energy in the annular inter space between the two cylinders to the kinetic energy of the liquid moving bodily inside 1=b.

Consequently the inertia to overcome in moving the cylinder 1=b, solid or liquid, is its own inertia, increased by the inertia of liquid (a”+b')/(41/*~b') times the volume of the cylinder r=b; this total inertia is called the effective inertia of the cylinder r=b,

at the instant the two cylinders are concentric.