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U72 =+'n2+i2, (12) a vortex See also:line is defined to be such that the tangent is in the direction of w, the resultant of , n, 3-, called the components of molecular rotation. A small See also:sphere of the fluid, if frozen suddenly, would retain this angular velocity. If w vanishes throughout the fluid at any instant, See also:equation (It) shows that it will always be zero, and the fluid See also:motion is then called irrotational; and a See also:function ¢ exists, called the velocity function, such that udx+vdy+wdz =-d,y, (13) and then the velocity in any direction is the space-decrease or downward gradient of ¢. 25. But in the most See also:general See also:case it is possible to have three functions ,, >', m of x, y, z, such that udx+vdy+wdz =-See also:dip-md(1) as A. Clebsch has shown, from purely See also:analytical considerations (Crelle, lvi.) ; and then ,d(,,&, m) _ad(¢, m) m) =2d(y,z)' -2d(z, x)' 2=3d(x,y and Eax++3 dZ=o, Eaz+nom+3'=0, (3) so that, at any instant, the surfaces over which ,y and in are See also:constant intersect in the vortex lines. Putting do di'_ H-d-mw -K, the equations of motion (4), (5), (6) § 24 can be written dx-2u3'+2wr~- d z )) =o, and therefore dK+I+d- =o. (6) Equation (5) becomes, by a rearrangement, ..., (7) dK di/,(dm dm dm dm dx-ddt+udx+tidy+ud ) +dx (a~+dx+dy+d ) =0,..., dK dO Dm dm Dt' (8) ,dx-dx dt +dx d t =o, ...' ..., and as we prove subsequently (§ 37) that the vortex lines are composed of the same fluid particles throughout the motion, the See also:surface m and ,y satisfies the See also:condition of (6) § 23; so that K is See also:uniform throughout the fluid at any instant, and changes with the See also:time only, and so may be replaced by F(t). 26. When the motion is steady, that is, when the velocity at any point of space does not See also:change with the time, ~K-22T+2wn=o, .. .. (1) dK+ndydz dK+3d —=o, udK+vdK+wdj=o, (2) dx dy dz and td +dV dH dz_Q2 p dv dv - dv dv r the normal See also:acceleration. The osculating See also:plane of a stream line in steady motion contains the resultant acceleration, the direction ratios of which are du du du d;q2 d2Q2 dH (7) udx+v~+wdz= dx -ZVS+2wn°~- d, ... , and when q is stationary, the acceleration is normal to the surface H =constant, and the stream line is a geodesic. Calling the sum of the pressure and potential See also:head the statical head, surfaces of constant statical and dynamical head intersect in lines on H, and the three surfaces See also:touch where the velocity is stationary. Equation (3) is called See also:Bernoulli's equation, and may be interpreted as the See also:balance-See also:sheet of the See also:energy which enters and leaves a given See also:tube of flow. If homogeneous liquid is See also:drawn off from a See also:vessel so large that the motion at the See also:free surface at a distance may be neglected, then Bernoulli's equation may be written H = p/p+z+q2/2g = P/p+h, (8) where P denotes the atmospheric pressure and h the height of the free surface, a fundamental equation in See also:hydraulics; a return has been made here to the See also:gravitation unit of See also:hydrostatics, and Oz is taken vertically upward. - q2/2g=h-z, (9) or the velocity of the See also:jet is due to the head k-z of the still free surface above the orifice; this is See also:Torricelli's theorem (1643), the See also:foundation of the See also:science of See also:hydrodynamics. 27. Uniplanar Motion.—In the uniplanar motion of a homogeneous liquid the equation of continuity reduces to du dv dx+dy=o, so that we can put u= -d,&/dy, v=d>G/dx, (2) (2) (3) (4) (5) (6) we have (9) (10) (I1) (2) (4) (5) K =fdplp+V + sq2 =H (3) is constant along a vortex line, and a stream line, the path of a fluid particle, so that the fluid is traversed by a See also:series of H surfaces, each covered by a network of stream lines and vortex lines; and if the motion is irrotational H is a constant throughout the fluid. Taking the See also:axis of x for an instant in the normal through a point on the surface H =constant, this makes u = o, E=o; and in steady motion the equations reduce to dH/dv =2v'-2wn=2qw sine, (4) where a is the See also:angle between the stream line and vortex line; and this holds for their See also:projection on any plane to which dv is drawn perpendicular. In plane motion (4) reduces to dH'g-g(a+, if r denotes the See also:radius of curvature of the stream line, so that (5) (6) In particular, for a jet issuing into the See also:atmosphere, where p=P, (I) I22 where 4 is a function of x, y, called the stream- or current-function; interpreted physically, 4-1Go, the difference of the value of 4, 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 d4, is the increase of 4, due to a displacement from P to P', and k is the component of velocity normal to PI"; the flow across PP' is d4, = k.PP ; and taking PP' parallel to Ox, d4, = vdx; . and similarly d4,= -udy with PP' parallel to Oy; and generally d¢(ds is the velocity across ds, in a direction turned through a right angle forward, against the See also:clock. In the equation§ of uniplanar motion 2r =ax ay = ax + ay = -v 4,, suppose, (3) so that in steady motion dH dJ+v~G_0, dH o +60=o, (4) dx dy and v',p must be a function of ¢. If the motion is irrotational,
u='-dz dy v=-- =dx so that 4 and O. are conjugate functions of x and,y,
4+4i=f(x+yi), v2'I'=o, V'$3 o; (6) or putting
0+44=w, x+yi=z, w=f(z)•
The curves o=constant and 4,=constant See also:form an orthogonal See also:system; and the interchange of 4, and 4, will give a new See also:state of uniplanar motion," in which the velocity at every point is turned through a right angle without alteration of magnitude.
For instance, in a uniplanar flow, radially inward towards 0, the flow across any circle of radius r being the same and denoted by 2irm, the velocity must be m/r, and
4,=m See also:log r, 4, =m6, 0+4,i=m log reie, w=m log z. (7) Interchanging these values
'G=m log r, =mO, J.,+0i=m log reie (8) gives a state of vortex motion, circulating See also:round Oz, called a straight or columnar vortex.
A single vertex will remain at See also:rest, and cause a velocity at any point inversely as the distance from the axis and perpendicular. to its direction; analogous to the magnetic See also: Example s.—Liquid motion past a circular cylinder. Consider the motion given by w=U(z+a'/z), (1) See also:a2 ¢=U (r+ra2) cosh =U (I +~2)x, 4 U (r - a2)'.sin o = U (1 -;2) y. Then 4, =o over the cylinder r = a, which may be considered a fixed See also:post; and a stream line past it along which tG=Uc, a constant, is the See also:curve ( r sin o = c, (x2+y2)(Y - c) - a2y = o, a cubic curve (C,). Over a concentric cylinder, See also:external or See also:internal, of radius r=b, +{'=#+Uiy=(U ( 1- b~) + Ui]y, and 4/ is zero if Ui/U = (a' -b')/b' ; (5) so that the cylinder may swim for an instant in the liquid without distortion, with this velocity Ui ; and w in (I) will give the liquid motion in the,interspace between the fixed cylinder r=a and the concentric cylinder r = b, moving with velocity U,. When b=o, U, ; and when b=oo, U1=s-U, so that at infinity the liquid is streaming in the direction xO with velocity U. If the liquid is reduced to rest at infinity by the superposition of an opposite stream given by w= -Ur, we are See also:left with w =Ua2/z, (6) ¢ = U(a2/r) See also:cos b = Ua3x/(x2+y2), (7) 4'= -U(a'/r) sin b= --Ua'y/(x'+y2), (8) giving the motion due to the passage of the Cylinder r=a with velocity U through the origin 0 in the direction Ox. If the direction of motion makes an angle b' with Ox, tan B' = aye _ x 2 = tan 20, 0 =, arc and the Velocity is Ua2/r'. Along the path of a particle, defined by the C3 of (3), (5) spiral r sin n8=na. (II) A single vortex in a circular cylinder of radius a at a distance e from the centre will move with the velocity due to an equal opposite image at a distance a'/c, and so describe a circle with velocity. me/(a'-r.2)in the periodic time 2sr(a'-c2)/m. (12) (7) (8) so that (2) (3) (4) (9) sin' '-b'= Y2 _y(Y-c) y2 — a2 ' sin ow _ _ 2 a2 _ aP, (It) on the radius of curvature is 10/(y-1c), which shows that the curve is an Elastica or Lintearia. (J. C. See also:Maxwell, Collected See also:Works, ii. 208.). If Oi denotes the velocity function of the liquid filling the cylinder, r=b, and moving bodily with it with velocity (tI=-Uix, (t2) and over the separating surface r=b ( a2\ a2+b' T TL $i +b' and this, by § 36, is also the ratio of the kinetic energy in the See also:annular interspace between the two cylinders to the kinetic energy of the liquid moving bodily inside r=b. Consequently the inertia tc overcome in moving the cylinder r=--b, solid or liquid, is its own inertia, increased by the inertia of liquid (a'+b')/(at...b') times the See also:volume of the cylinder r=b; this See also:total inertia is called the effective inertia of the cylinders = b, at the instant the two cylinders are concentric. Conjugate functions can be employed also for the motion of liquid in a thin sheet between two concentric spherical surfaces; the components of velocity along the See also:meridian and parallel in colatitude: b and See also:longitude X can be written do = I dip I d1, dl, (13) sib sin -6 dX' sin 0 dT = de' and then 4,+Gi=F(tan O. e1i)• (14) 28. Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it.—A stream-function 4' must be determined to satisfy the conditions v',/.,=o, throughout the liquid; (I) w=constant, over any fixed boundary; (2) d+k/ds=normal velocity reversed over a solid boundary, (3) se that, if the solid is moving with velocity U in the direction Ox, d4y/ds = — Udy/ds, or 4,+Uy =constant over the moving cylinder; With liquid of See also:density p, this gives rise to a kinetic reaction to acceleration dU/dt, given by a' +b' dU 'rpb2a=- ~ -a-t- az ±ba M' dt ' if M' denotes the See also:mass of liquid displaced by 'unit length of the cylinder r = b. In particular, when a = co , the extra inertia is M'. When the cylinder r=a is moved with velocity U and r=b with velocity U, along Ox, = Ubza2at, (7+r) cos0-U,b2bza2 (r+-) cos 0, st = - Ui- a2a2 f\ - r) sin B - U1Ez a2 (r See also:r2) sin 0; (16) and similarly, with velocity components V and Vi along Oy Vb.a^a2(r2+r)sine-V, 2 (r+a2)sin v, a2 b^- 32 a2, = V62 (r—r)cos+Vt(- rcosB; (18) and then for the resultant motion z b' w= (U2 + V2)b2a2a"L7+Vi + 622 0=U z V? 1)' a'b2 -(Ul2+Vi2)b_,2n-i2+. -/3-32 a2Ut zVti. The resultant impulse of the.liquid on the cylinder is given by the component, over r=a (§ 36), X = f pb cos &a& =,rpa2 (U 2±a2 U,62b 2 a2) ; and over r=b X,=fpocos6.bdo=7rpb2(U62¢a2-U1 a2), and the difference X-X, is the component momentum of the liquid in the interspace; with similar expressions for Y and V,. Then, if the outside cylinder is free to move Xi=o, U V 2a' b' ` b2+a2' X=7rpa2U +ar (22) But if the outside cylinder is moved with velocity U,, and the inside cylinder is solid or filled with liquid of density a, X = 7raa. U, U t = p(b2+a2) +o (b2-a2)' U- ( p-a)(b3-(12) Ui p(b2+az) +o(brr- a2)' and the inside cylinder starts forward or backward with respect to the outside cylinder, according as p> or <a. 30. The expression for w in (1) § 29 may be increased by the addition of the See also:term See also:im log z =-m8 + im log r, (1) representing vortex motion circulating round the annulus of liquid. Considered by itself, with the cylinders held fixed, the vortex sets up a circumferential velocity m/r on a radius r, so that the angular momentum of a circular filament of annular See also:cross See also:section dA is pmdA, and of the whole vortex is pm-r(b2-a2). Any circular filament can be started from rest by the application of a circumferential impulse 7rpmdr at each end. of a See also:diameter; so that a mechanism attached to the cylinders, which can set up a uniform distributed impulse rpm across the two parts of a diameter in the liquid, will generate the vortex motion, and react on the cylinder with an impulse couple-pmara2 and pmlrb2, having resultant pmr(b2and this couple is See also:infinite when b=co, as the angular momentum of the vortex is infinite. Round the cylinder r =a held fixed in the U current the liquid streams past with velocity q' =2U sin 0+m/a; (2) and the loss of head due to this increase of velocity from U to q' is U2 = (2U sin 8+m/a)2 - U2 (3) 2g 2g ' so that cavitation will take See also:place, sinless the head at a See also:great distance exceeds this loss. The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a -resultant thrust on the whole cylinder is 2mU sin 9/ga, and its thrust is 27rpmU See also:absolute See also:units in the direction Cy, to be counteracted by a support at, the centre C; the liquid is streaming past r=a with velocity U reversed, and the cylinder is surrounded by a vortex. Similarly, the streaming velocity V reversed will give rise to a thrust 27rpmV in the direction xC. Now if the cylinder is released, and the components U and V are reversed so as to become the velocity of the cylinder with respectto space filled with liquid, and at rest at infinity, the cylinder will experience components of force per unit length (i.) -27rpmV, 27rprU, due to the vortex motion; (ii.) -apa d , -xpa2T, due to the kinetic reaction of the liquid; (iii.) o, lY-7r(a-p)a2g, due to gravity, taking Oy vertically upward; and denoting the density of the cylinder by a; so that the equations of motion are • True— = -7rpa-w— 2r mV, dt dt p 7raa2 dt = -lrpa2 dt +2xpmV - -A-(0-- p)a-g, or, putting m=a2w, so that the vortex velocity is due to an angular velocity ce at a radius a, (a+p)dU/dt+2pwV =o, (6) (a+p)dV/dt-2pwU+(a-p)g=o. (7) Thus with g=o, the cylinder will describe a circle with angular velocity 2pw/(a; }-p), so that the radius is (a+p)v/2pw, if the velocity is v.. With a¢o, the angular velocity of the cylinder is 2w; in this way the velocity may be calculated of the See also:propagation of ripples and waves on the surface of a See also:vertical whirlpool in a sink. Restoring a will make the path of the cylinder a trochoid; and so the swerve can be explained of the See also:ball in See also:tennis, See also:cricket, See also:base-ball, or See also:golf. Another explanation may be given of the sidelong force, arising from the velocity of liquid past a cylinder, which is encircled by a vortex. Taking two planes x= =b, and considering the increase of momentum in the liquid between' them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P' at opposite ends of the diameter PP', is pdy (U - Ua2r-2 cos 20+mr-'sine) (Ua2r-2 sin 2B+mr'tcose) + pdy (-U+Ua2r' cos20+mr-' sin0)(Ua2rr2 sin20-mrt cosO) =2pdym>Jrt(cos 0-a2r-cos36), (8) - and with y =b tan B, r = b sec 0, this is (9) 2pmUdO(1-a2b-2cos 30 cos 0), and integrating between the limits 0= =qtr, the resultant, as before, is 27rpmU. 31. Example 2.—Confocal Elliptic Cylinders.—Employ the elliptic coordinates n, i;, and i'=n+ii, such that z=cch x=cchrlcos ,y=cshnsine; (1) then the curves for which n and i; are constant are confocal ellipses and hyperbolas, and J=d(n- c?(ch'n-cose ) 1c2(ch2n-cos2E) =rtr2=0D2, (2) if OD is the semi-diameter conjugate to OP, and ri, r2 the See also:focal distances, ri,r2 = c (ch n = cos Z) ; r¢=x2+y2=c2(See also:ch2, sine ) = 1-c2(ch 21)+COs 2E). Consider the streaming motion given by w=mch(i.-'t), Y=(5) ¢=mch(n-a)cos(i;-¢=msh(n-a)sin(t-0). (6) Then >G = o over the See also:ellipse n = a, and the See also:hyperbola l; = 0, so that these may be taken as fixed boundaries; and ¢ is a constant on a C4. Over any ellipse n, moving with components U and V of velocity, =¢+Uy=[msh(n-a)coss+Ucshn]sint [msh(n-a) sin0+Vcchn]cos ; (7) so that jt' o, if u = m sh(n - a)cos R, sh(n - 0 a) . c sh n V c mch n ,sin ' hawing a resultant in the direction PO, where P is the intersection of an ellipse n with the hyperbola 0; and with this velocity the ellipse n can be See also:swimming in the liquid, without distortion for an instant. At infinity U = -m -e-'cos ft = - a m bcos , V --e 'sin = —sin ft, (9) a and b denoting the semi-axes of the ellipse a; so that the liquid is streaming at infinity with velocity Q =m/(a+b) in the direction of the asymptote of the hyperbola p. An ellipse interior to n = a will move in a direction opposite to the exterior current ; and when n =o, U = co , but V= (m/c) sh a sin f. Negative values of n must be interpreted by a streaming motion on a, parallel plane at a level slightly different, as on a See also:double See also:Riemann sheet, the 'tream passing from One sheet to the other across a cut SS' joining the foci S, S'. A See also:diagram has been drawn by See also:Col. R. L. Hippisley. (14) (15) ~= (17) ('9) (2O). (21) (23) (4) (5) (3) (4) (8) The components of the liquid velocity q, in the direction of the normal of the ellipse n and hyperbola E, are -mJ 1sh(n-a)cos(E--p),mJ-1ch(n-a)sin(E-ii). (to) The velocity q is zero in a corner where the hyperbola i4 cuts the ellipse a; and round the ellipse a the velocity q reaches a maximum when the tangent has turned through a right angle, and then '' q - (ch 2a-cos 219) ' (I I) Qea sh 2a and the condition can be inferred when cavitation begins. With t3=o, the stream is parallel to xo, and 14 =m ch (•1--a) cos t Uc ch (7)-a) sh n cos E/sh (n-a) (12) over the cylinder n, and as in (12) § 29, =-Ux=-UcchncosE, (13) for liquid filling the cylinder; and th n th (n-a)' over the surface of n; so that parallel to Ox, the effective inertia of the cylinder n, displacing M' liquid, is increased by M'thn/th(n-a), reducing when a=ao to M' th n=M'(b/a). Similarly, parallel to Oy, the increase of effective inertia is M'/th n th(n-a), reducing to With n=M'(a/b), when a=oo, and the liquid extends to infinity. 32. Next consider the motion given by m ch 2(n-a)sin 2E, ¢=-m sh 2(71-a)cos 2E; in which ,y=o over the ellipse a, and =>G+4R(x2+y2) =[-m sh 2 (n - a) + iRc2]cos 2E+ IRO ch 2n, which is constant over the ellipse n if Rc2 = m sh 2 (n- 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 with V i'=-2R=-2m1(I/a2+I/b2,, #1 = m1(x2/ a2+y2/b2) - 1R (x2 +y2) =-J R (x2-y2) (a2 - b2)/ (a2+b2), =Rxy(a2-b2)/(a2+b2), wt =ti1+1P1i = - See also:aiR(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 a and n, t= mch2(n-a)sin2E (171 See also:Rc'sh 2n sin 2E(a2-b2)/(a2+b2) = i/th 2(n-a)th 271; (8) and the effective k2 of the liquid is reduced to (9) ;c2/th 2(n-a)sh 2n, which becomes `c2/sh2n=g(a2—b2)/ab, when a=oo, and the liquid surrounds the ellipse n 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 ,y is resolved into such that ¢1+4Rx2 and 1y2+4Ry2 is constant over the boundary. Inside a cylinder = - )iR(x+yi)2a2/(a2+b2), (to) 9z+1G2i= iiR(x+yi)2b2/(a2+b2), (II) and for the interspace, the ellipse a being fixed, and a1 revolving with angular velocity R 01-f-¢1i= - BiRc2sh 2(n-a+ti)(ch 2a+1)/sh 2(a1-a), (12) 4'2++1'2i= BiRc2sh 2(n-a+>;i)(ch 2a-I)/sh 2(a1-a), (13) satisfying the condition that 01 and y2 are zero over n=a, and over n=a1 constant values. In a similar way the more general state of motion may be analysed, given by w=mch2(r-y), y=a+pi, (16) as giving a homogeneous See also:strain velocity to the confocal system; to which may be added a circulation, represented by an additional term mf in V.x+yi=c,/[sin(+ni)] (17) =Qc sh4(n-a)sina (E-19) (18) will give motion streaming past the fixed cylinder n=a, and dividing along t={9; and then x2 -y2 = c2 sin t ch n, 2xy = c2 cos sh n. In particular, with sh a= I, the cross-section of n = a is x4+6x2y2+y4 =2c4, or x4+y4 =c4 when the axes are turned through 450. 33. Example 3.—Analysing in this way the rotation of a rectangle filled with " liquid into the two components of shear, the stream function '71 is to be made to satisfy the conditions (i.) V';I'l/i = o, (ii.) 7r1+SRx2=4Ra2, or %pi=owhenx= *a, (iii.) ¢1+4Rx2=4Ra2, y1=4R(a2-x2), when y= =b See also:Expanded in a See also:Fourier series, a2- x— 2_32 112\cos(2(n2n (++I)rx/a 7r 1)Z3 , #1=R— 16 2Xcos(2n+I)4rx/a.ch(zn+I)a2ry/a a. (2n+1)3.ch(2n+I)Zxb/a w1=01+0=iRI3a2 . n+cos) I(2nch(+I2n)i+xI)z/4arb/a, (2 an elliptic-function Fourier series; with a similar '/expression for }'z with x and y, a and b interchanged; and thence 1P =4,1+02. Example 4.—Parabolic cylinder, axial advance, and liquid streaming past. The polar equation of the cross-section being rl cos 28 =al, or r + x =2a, (3) the conditions are satisfied by 4'=Ur sin 8-2Ualri sin aB=2Uri sin ZB(ri cos 40- ai), (4) 1G=2Ualri sin 40=-U~f [2a(r-x)], (5) W =-2Uaizl, (6) and the resistance of the liquid is 2rpaV2/2g. A relative stream line, along which 0' = Uc, is the quartic curve y-c=i[2a(r-x)], x=4a24alyy2-(yc ) -2)4 r= 4a24ayz(y-+(cy)2 04, (7) and in the absolute space curve given by ¢, (y-c)2 rx= See also:lac -2a log (y-c)• (8) 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 ,y can be found analogous to that employed in plane motion, such that the 'See also:flux across the surface generated by the revolution of any curve AP from A to P is the same, and represented by 2T(4'-¢o); and, as before, if di, 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 2rdy=2ryk.PP'; and taking PP' parallel to Oy and Ox, u=-dy/ydy, v=d¢/ydx, (1) and +y is called after the inventor, " See also:Stokes's stream or current function," as it is constant along a stream line (Trans. Camb. Phil. See also:Soc., 1842; " Stokes's Current Function," R. A. See also:Sampson, Phil. Trans., 1892); and See also:day/yds is the component velocity across ds in a direction turned through a right angle forward. In this symmetrical motion =o, n=o, 2f=a (y dx d (y =1 ((d2>G d2¢ _14) _ I y \dx2+dye y' dy suppose; and in steady motion, dH 1 (LP dH 1 d4y dx+y2dxv2,y=o, ay+y~o24'=0, so that 2i-/y=-y-217211=dH/d'y is a function of ¢, say f'(¢), and constant along a stream line; dH/dv =2qi', H- f(4') =constant, throughout the liquid. When the motion is irrotational, d~ _ 1 d+y dA I d4 =o, u= ---y iy, v=- dy=ydx,. G214=o d?4d?~ 4 I or dx2+ dye - y dY =o. (14) (1) (2) x2/~a"2+y2/b2 =1, (4) 'y1' = m1(x2/a2+y2/b2) (5) (6) ¢1+4Rx2=BRc2(ch2a1+I), (14) 4'z+4Ry2=BRc2(ch 2a1- I), (15) Similarly, with the function (19) (20) (I) (2) (2) so that (4) (5) (6) (7) HYDRODYNAMICS] Changing to polar coordinates, x =r cos B, y=r sin B, the equation (2) becomes, with cos B=µ, r2 d +(1-u')(-a' =2 f'r3 sin 0, (8) of which asolution, when i-=o, is • = (Ar' +B) (1-µ2)dd = (Ar"'i++2) y2r (9) 4) =[(n+I)Ar nBr-^-'}R„ (io) where P. denotes the zonal See also:harmonic of the nth See also:order; also, in the exceptional case of '=Ao cos B, 4.=Ao/r; 'G=Bor, 4)=-Bo log tan 10 = - lBo sh-3x/y (II) Thus cos 0 is the Stokes' function of a point source at 0, and PA- PB of a line source AB. The stream function 4, of the liquid motion set up by the passage of a solid of revolution, moving with axial velocity U, is such that yds=-Ud', +zUy2 =constant, (12) over the surface of the solid; and ' must be replaced bye'=&-} 3Uy2 in the general equations of steady motion above to obtain the steady relative motion of the liquid past the solid. For instance, with n =1 in equation (9), the relative stream function is obtained for a sphere of radius a, by making it =4,+zUy2=1U(r2-a3/r) sing 0, 4, =-1Ua3 sing air ; and then =Ux(i+aa3/r2), = Ua3 cos 0/r2, tti d ~ = U d 3 cos 0, r = 1U 3 sin 0, so that, if the direction of motion makes an angle 4, with Ox, tan 4- O) =I tan 0, tan g=3 tan 0/(2-tan' 0). (i6) Along the path of a liquid particle ;I' is constant, and putting it equal to ZUcs, (r2-a3/r) sin' 0=c2, sin' 0=c2r/(r3-a3), the polar equation; or y2= c2r3/(r3 —a3), r3= a3y2/(y2 —c2), a curve of the loth degree (Clo). In the absolute path in space cos = (2 -3 sin2B)/J (4-sin20), and sin30 = (y3-c2y)/aa which leads to no See also:simple relation. The velocity past the surface of the sphere is I r ddr,-' = (2r +2r2)rsisnin2BB = lU sin B, when r = a; (20) sinB so that the loss of head is (g sin2B-i)U2/2g, having a maximum tU2/2g, (21) which must be less than the head at infinite distance to avoid cavitation at the surface of the sphere. With n=2, astate of motion is given by y = — yUy2a4µ/r4, i' = 3Uy2(I — See also:a4µ/r4), =Ux+4), ek =- U(a4/r3)P2, P2=$µ2-z, representing a stream past the surface r4=a4µ. 35• A circular vortex, such as a See also:smoke See also:ring, will set up motion symmetrical about an axis, and provide an See also:illustration; a See also:half vortex ring can be generated in See also:water by See also:drawing a semicircular blade a See also:short distance forward, the tip of a See also:spoon for instance. The vortex advances with a certain velocity; and if an equal circular vortex is generated coaxially with the first, the mutual See also:influence can be observed. The first vortex dilates and moves slower, while the second contracts and shoots through the first; after which the motion is reversed periodically, as if in a See also:game of leap-See also:frog. Projected perpendicularly against a plane boundary, the motion is determined by an equal opposite vortex ring, the optical image; the vortex ring spreads out and moves more slowly as it approaches the See also:wall; at the same time the molecular rotation, inversely as the cross-section of the vortex, is seen to increase. The analytical treatment of such vortex rings is the same as for the electro-magnetic effect of a current circulating in each ring. 36. Irrotational Motion in General.—Liquid originally at rest in a singly-connected space cannot be set in motion by a field of force due to'a single-valued potential function; any motion set up in the liquid must be due to a See also:movement of the boundary,.and the motion will be irrotational; for any small spherical See also:element of the liquid may be considered a smooth solid sphere for a moment, and the normal pressure of the surrounding liquid cannot impart to it any rotation. The kinetic energy of the liquid inside a surface S due velocity function 4) is given by r l r l l T=IPJJJ L\d )2+( )2+(d~/ ]dxdydz, cld) =spf f d-LdS by See also:Green's transformation, de denoting an elementary step along the nor'inal to the exterior of the surface; so that d¢/dv=o over the surface makes T =o, and then (da l 2 + lay1 z+ 1 2= - 1 0, -=0, -=0, -2 =0. (2) If the actual motion at any instant is supposed to be generated instantaneously from rest by the application of pressure impulse over the surface, or suddenly reduced to rest again, then, since no natural forces can See also:act impulsively throughout the liquid, the pressure impulse m satisfies the equations I do, tits pdx = - n, - =—v, P-=-n, (3) n=p4-{-a constant, (4) and the constant may be ignored; and Green's transformation of the energy T amounts to the theorem that the See also:work done by an impulse is the product of the impulse and See also:average velocity, or half the velocity from rest. In a multiply connected space, like a ring, with a multiply valued velocity function 41, the liquid can circulate in the circuits independently of any motion of the surface; thus, for example, = mB =m tan—ry/x (5) will.give motion to the liquid, circulating in any ring-shaped figure of revolution round Oz. To 1 id the kinetic energy of such motion in a multiply connected space, the channels must be supposed barred, and the space made acyclic by a membrane, moving with the velocity of the liquid; and then if k denotes the cyclic constant of 4) in any See also:circuit, or the value by which 4) has increased in completing the circuit, the values of 4, on the two sides of the membrane are taken as differing by k, so that the integral over the membrane f f d dS =kf f ddS, (6) and this term is to be added to the terms in (I) to obtain the additional See also:part in the kinetic energy; the continuity shows that the integral is See also:independent of the shape of the barrier membrane, and its position. Thus, in (5), the cyclic constant k=2irm. In planer motion the kinetic energy per unit length parallel to Oz T = PJ f L \dx/ 2+ \dy/ 2] dxdy = 2Pf J L ( \dx) 2+ `dy) ']dxdy = 3P f d ds zP f ‘p w; ds. For example, in the equilateral triangle of (8) § 28, referred to co-See also:ordinate axes made by the base and height, 11,' = -2Ralgy/h = - IRy[(h-y)2-3x2]/h (8) 4) =4)' - aR[(3h.-y)2+x2] = - zR[ h3+ih2y+h)x2-y2) -3x2y+y3]/l3 (9) and over the base y =o, dx/dv=-dx/dy=+R(1h2= -12R(-h2+x2). (io) Integrating over the base, to obtain one-third of the kinetic energy T, h/J 3 IT=~P f h/J3`R2(3x4-zrh4)dx/h =PR2h4/135 J 3 (II) so that the effective k2 of the liquid filling the triangle is given by k2 = T/zpR2A = 2h2/45 =§ (radius of the inscribed circle)2, (12) or two-fifths of the k2 for the solid triangle. Again, since d4,/dv =d¢/ds, d¢/ds = - d,//dv, (13) T= iPJ4)d4) = - ipI'Gd4)• (14) With the Stokes' function 4, for motion symmetrical about an axis. T = P f - 2lryds=2PIOd4). (15) yds 37. Flow. Circulation, and Vortex Motion.-The line integral of the tangential velocity along a curve from one point to another, defined by 1( uds+ v di+was) ds=f(udx+vdy+zdz), (1) is called the " flux " along the curve from the first to the second point; and if the curve closes in on itself the line integral round the curve is called the " circulation " in the curve. With a velocity function 0, the flow Id$ ( =See also:roe — ~2r (2) (13) (14) (15) (17) (18) (19) (22) (23) 125 to the (I) (7) so that the flow is independent of the curve for all curves mutually reconcilable; and the circulation round a closed curve is zero, if the curve can be reduced to a point without leaving a region for which 0 is single valued. If through every point of a small closed curve the vortex lines are drawn, a tube is obtained, and the fluid'contained is called a vortex filament. By See also:analogy with the spin of a rigid body, the component spin of the fluid in any plane at apoint is defined as the circulation round a small See also:area in the plane enclosing the point, divided by twice the area. For in a rigid body, rotating about Oz with angular velocity f, the circulation round a curve in the plane xy is f \ .dv dx xi -yds) ds = j times twice the area. In a fluid, the circulation round an elementary area dxdy is equal to dv du uud.v + (v+axdx) dy- (u+dydy) dx-vdy= (dx dy) dxdy, (4). so that the component spin is z \dx dy) =f, (5) in the previous notation of § 24; so also for the other two .com- ponents and ,t. Since the circulation round any triangular area of given aspect is the sum of the circulation round the projections of the area on the coordinate planes, the See also:composition of the components of spin, p, ;-, is according to the vector See also:law. Hence in any infinitesimal part of the fluid the circulation is zero round every small plane curve passing through the vortex line; and consequently the circulation round any curve drawn on the surface of a vortex filament is zero. If at any two points of a vortex line the cross-section See also:ABC, A'B'C' is drawn of the vortex filament, joined by the vortex line AA', then, since the flbw in AA' is taken in opposite directions in the See also:complete circuit ABC AA'B'C' A'A, the resultant flow in AA' cancels, and the circulation in ABC, A'B'C' is the same; this is expressed by saying that at all points of a vortex filament we is constant where a is the cross-section of the filament and w the resultant spin (\V. K. See also:Clifford, Kinematic, See also:book iii.). So far these theorems on vortex motion are kinematical; but introducing the equations of motion of § 22, D+Q=o, D+ =o, See also:Dw+dg=9, Q = fdp/p+V, and taking dx, dy, dz in the direction of u, v, w, and dx: dy: dz=u: v: w, d (audx+vdy+wdz) =D~dx+u~+... = -dQ+zd42, and integrating round a closed curve dtD f (udx+vdy+wdz) =o, and the circulation in any circuit composed of the same fluid particles is constant; and if the motion is See also:differential irrotational and due to a velocity function, the circulation is zero round all reconcilable paths. Interpreted dynamically the normal pressure of the surrounding fluid on a tube cannot create any circulation in the tube. The circulation being always zero round a small plane curve passing through the axis of spin in vortical motion, it follows conversely that a vortex filament is composed always of the same fluid particles; and since the circulation round a cross-section of a vortex filament is constant, not changing with the time, it follows from the previous kinematical theorem that aw is constant for all time, and the same for every cross-section of the vortex filament. A vortex filament must See also:close on itself, or end on a bounding surface, as seen when the tip of a spoon is drawn through the surface of water. Denoting the cross-section a of a filament by dS and its mass by dm, the quantity wdS/dm is called the vorticity; this is the same at all points of a filament, and it does not change during the motion; and the vorticity is given by w cos EdS/dm, if dS is the oblique section of which the normal makes an angle E with the filament, while the aggregate vorticity of a mass M inside a surface S is M''fw cos edS. Employing the equation of continuity when the liquid is homogeneous. di- dn d d' d (i 2 dy dz C2=dx2 dw dz.' which is expressed by p2(u, v, w) a.i2 curl (E, n, c), (::, n, ~) } curl (u,'v, w). (ti) 38. Moving Axes in Fl ydrodynamics.—In many problems, such as the motion of a solid in liquid, it is convenient to take edordinate axes fixed to the solid and moving with it as the movable trihedron See also:frame of reference. The components of velocity of the moving dl ='- dm nP- lR, dn do in R n Q_lQ-mP ; (4) ~ = at at so that dtk=l (dt -vR+wQ+u'dy+v'dy+w'dx) +m(...)+n(..) Idp Idp Idp =l(X-pdx)+m(Y-Pay)+n(z-Paz), (5) for all values of 1, m, n, leading to the equations of motion with moving axes. When the motion is such that
do d4' do d,j, d~ this
u= -dx-md' v= -dy-may, w= - dz-m dz, (6)
as in § 25 (I), a first integral of the equations in (5) may be written
J dP +V+ II, -dt-md +(u-u') (dx+max)
+(v-v') (dy+may) + (w-w') (--+maa) =F(t), in which
do(uu')do (v-v')dydo (w-w'
-di - )
dx dz
= dt-(U-yR+zQ)az-(V-zP+xR)dy-(W-xQ+yP)dz (8)
is the time-See also:rate of change of o at a point fixed in space, which is left behind with velocity components u-u', v-v', w-w'.
In the case of a steady motion of homogeneous liquid symmetrical about Ox, where 0 is advancing with velocity U, the equation (5) of §34
p/p+V + -f ('l'') = constant
becomes transformed into
p +V + z42- ydY + zU2-f(& +41.42) =constant, (To)
+~'=G+4UY2r (II)
subject to the condition, from (4) § 34,
y2v2,' = y-2v2+k= -f'('+4Uy2). (12)
Thus, for example, with
%I,' = a Uy2 (r2a-2 - I), r2 = x2 +y2, (13)
for the space inside the sphere r=a, compared with the value of ¢: in § 34 (13) for the space outside, there is no discontinuity of the velocity in See also:crossing the surface.
Inside the sphere
d r d>G' d I dtG' _15u y = Tx (v dx 1 dy ( dy) --2- a2'
so that § 34 (4) is satisfied, with
f'4') =15Ua-2, f(IP') =!51.Wa-2; and (to) reduces to
12 P +V 8 a2 -2U (x. -I) (d 2
s i) = constant; (16)
this gives the state of motion in M. J. M. See also: If u', v', w' denote the components of the velocity of the liquid relative to the axes, z a u'u+yR—zQ=a22+.brzoaY c:+a2gaz, 2112 2112 v'=v+zP—xR= 2Shz—ai+ Siax, 2c2 2c2 w' = w+xQ —yP =c2+a2$Lax —+SLY, P=Sh+t, Q=c2+n, R=Zia+r. (7) u'x—+v'b2+w'ca =o , (8) so that a liquid particle remains always on a similar ellipsoid. The hydrodynamical equations with moving axes, taking into See also:account the mutual gravitation. of the liquid, become dI du du du du dx+4,rpAx+dt _vR+wQ+u'dx+v'ay+w'dz=o, (9) where abcdX A, B, C,=I (a2+X b2+X,c2+),)P P2=4(42+x)(b+a)(c2--x). (Jo) With the values above of u, v, w, u', v', w', the equations become of the form p d +4lrpAx+ax+hy+gz =0, (II) P +4apBy+hx+Ry+fz= (12) P d +4TpCz+gx+fy+7z=0, (13) and integrating PP-' + 2Tp (Ax2 + Bye + Cs') +li (ax2+Ry2+yz2+afya±2gzx+2hxy) R See also:coast., so that the surfaces of equal pressure are similar See also:quadric surfaces, which, symmetry and dynamical considerations show, must be coaxial surfaces; and f, g, h vanish., as follows also by algebraical reduction; and 4c2(c2—a2) y (c2—La2rr11\2. a= (c2+a2)2 n2 c2+a2>v2—~ -4a~+b2)22a2 { bzStc31 (15) with similar equations for p and T. 1 if we can make - (4rpA+a)x2 = (4irpB+$)b2 = (4xpC+1)c2, (16) the surfaces of equal pressure are similar to the external case, which can then be removed .vithout'affecting the motion, provided a, d, -y remain constant. This is so when the axis of revolution is a See also:principal, axis, say 0z; when ih=o, li?=4, =o, n=o. (17) If R, = o or B, = f in addition; we obtain the See also:solution of See also:Jacobi's ellipsoid of liquid of three unequal axes, rotating bodily about the least axis; and putting•a=b, See also:Maclaurin's solution is obtained of the rotating See also:spheroid. In the general motion again of the liquid filling a case, when a=b, S2, may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to dt 2c2 do 2a2 d; dt _ —a—2=—Fe's ' 2c2 dt =a'+ez'''i =a2+ ex(Slit-ii+1) a2+c2 d5k a2+c-2 dt =~l+ax_c2)ll dt-Sh3—a2—cxti'; a( which three integrals are +q2 =1- - ~al~, (20) +ti = Y' 2c2 (za c2c 2) (21) iht+SknN = +a- , ; (22) and then 3 (at) s =(as+c2)2(ftxt—Stan)? 2+ ca c (a :)g[(t2+n2) (al +al) - +(fh#--Sian)2] a 4ca t M—]!4?+ { L (42+62 c2)2 M--Nat+x2}r2 (al +62) ` ` 2c (a —c2) c 2c (a?+cs) (9as-c?) r6ct(a2-c2) J =Z, (23) where Z is a quadratic in , so that is an elliptic function of 1, except when c=a, or 3a. Put Sti =See also:f2 cos ¢, _ —S2 sin , 02—= d1~ia =02r—a±c(fht+5220)i•, (24) a2+c2 a2+c2:N+ 4c, S s a2—c2 (a2+c2)2 y~. M +2G2 (a2 _ c2) 02+c2 // (rdr a2+c' + 462 '0_ 7-2—a'—c2) M+ (a2+c2)2 , Z, c2 2 2'- .(a c) (26) which, as Z is a quadratic function of r2, are non-elliptic integrals; See also:sea also for #, where t=w cosip, = —w sin In a state of steady motion An experiment was devised by See also:Lord See also:Kelvin for demonstrating this, in which the difference of steadiness was shown of a See also:copper See also:shell filled with liquid and spun gyroscopically, according as the shell was slightly oblate or prolate. According to the theory above the stability is regained when the length is more than three diameters, so that a See also:modern projectile with a cavity more than three diameters See also:long should See also:fly steadily when filled with water; See also: If there are more B corners than one, either on xA or x'A', the expression for is the product of corresponding factors, such as in (5). Restricting the See also:attention to a single corner B, r = (Q) *(cos ne+i sin ne) =J (b-a'.u-a)+J (b-a.u-a') (6) Al (a-a .u-b) ch nft =ch log ()"cos ne+i sh log (2) *sin ne = z(3"'+r") =1 1b~' its-a a-a u -=I' sh 1112= sh log (2) cos ne +i ch log (gq) *sin nO =#(e-3-") ~b -a lu-a' a-a za 0o >a> b> o> a'> - co ; and then dit t Al (b-a.b-a') dw__ du 2n(u-b)J(u-a.u-a')' du iu' to) the formulas by which the conformal representation is obtained. For the El polygon has a right angle at u = a, a', and a zero angle at u=b, where 0 changes from o to zar/n and ft increases by zi,r/n;so that dit_ A where A = -J(b-a.b-a') (II) du-(u-it(u-a.u-a,)' 2n And the w polygon has a zero angle at u =o, oo , where 1,G changes from o to m and back again, so that w changes by imi and u=B where B=-A. (I2) Along the stream line xBAPJ, =o, u=ae-r4l'"; and over the jet surface J PA, where the skin velocity is (iAt. = -q= -Q, u=ae~nQlm=ae,rsic, Ts denoting the arc AP by s, starting at u=a; ch nf2=cos ne= Nib- a',\lu=a a-a u-b' \la -b a Vu-b' co > u = aeA"> a, (17) and this gives the See also:intrinsic equation of the jet, and then the radius of curvature ds Idd, idw_Idw See also:Ida I' =—de—Qde_-QT Q-3-OT c u - bA/ (u-a.u-a') =A. 2n -- u J (a-b.b-a') , not requiring the integration of (II) and (12)If e = a across the end J J' of the jet, where u = oo ,q= Q, chnil=cosna= ashnfl=I sinner=i Q (19) Then cos 2na—cos 2110 = = 2 , 2 sing 211aa - - a-a.u-b sin 2n9 2 u-b J (a-b.b-a')J (u-a.u-b') =sin 2na a-u-a .ub--b ll (u--a,.ua') ' (20) 2nc ( b \,/(a-b.b—a') a p— I+u bJ Al (u—a.u—a') a—a' +(a+a') cos 2na—[a+a'+(a—a') cos 2na]cos 2119 (a-a') sins 2na X COS 2na—cos 2119 sin 2110 Along the wall AB, cos nO =0, sin nO =1, a> u> b, (22) ch nft=i sh tog \ " b=¢~ a-u (23) r4I Va-a ~-b' ( shnf2=ichlog(()=i~¢-b Nlu-a' a-a u-b' ds ds dq, _ in dud4, dt-aqu-or qu (27) ch nS2 = ch log O _ ~a a (28) ` 'a-b.~ Iu-a' sh nf2=sh log O v¢-a' Jb-u' At x where 4'=oo, u =o, and q=qo, ' * b-a'¢ a + a-b =a' qo) - \t a - ~b ~a a q (30) In crossing to the line of flow x'A'P'J', ¢ changes from o to m, so that with q=Q across Jr, while across xx' the velocity is qo, so that m=qo.xx'=Q.JJ' (31) _r1 ib-a' .\ter-b (3 xx' -Q L v a b- a-a' ~ b j 2) giving the contraction of the jet compared with the initial breadth of the stream. Along the line of flow x'A'P'J', t'=m, u=a'e-"d' I"', and from x' to A', cos nO =I, sin See also:pie =o; ¢' ch nft=ch log (Q) = —,\Ia-u , a-a b-u sh to * ¢ bu-a', g(q) V¢-a Va o> u> a'. Along the jet surface A'J', q=Q, b-a' a-u ch nft=cos 118= a a b-u' -u sh 1112=i sin ne=i' Ja aa-b \lba'-u' a'>u=a'e,r1.r> -oo giving the intrinsic equation. 41. The first problem of this See also:kind, worked out by H. v. See also:Helmholtz, of the efflux of a jet between two edges A and Al in an infinite wall, is obtained by the symmetrical duplication of the above, with n=1,b=0, a'=-oo,asin fig. 5, chft=u,shft= V u¢; and along the jet APJ, oo > u =ere*'/'> a, 2 shl2=isin9=i u=ie , oo k a -in' is c PM=f sin0ds= fe ds=t-e =sine, (3) (7) (8) . (9) (13) Q, (14) (15) (16) (2I) (24) (25) AB 1" du Ac = bqu J(a-b)J(u-a')-HI (b-¢')J(a-u) '/"du J (a-a')d (u-b) u . 26) Along the wall Bx, cos nO =I, sin nO = o, b>u>o (29) sh nfl (33) (34) (35) (36) (37) (38) (1) (2) HYDRODYNAMICS] so that PT=c/ZZr, and the curve AP is the tractrix; and the co-efficient of contraction, or breadth of the jet a breadth of the orifice =2r+2' A change of it and B into nit and nO will give the solution for two walls converging symmetrically to the orifice AAI at an angle a/n. With n=, the re-entrant walls are given of See also:Borda's See also:mouthpiece, and the coefficient of contraction becomes Generally, by making a'= —co, the line x'A' may be taken as a straight stream line of infinite length, forming an axis of symmetry; and then by duplica- tion the result can be ob- tained, with assigned n, a, A and b, of the efflux from a symmetrical converging mouthpiece, or of the flow of water through the See also:arches of a bridge, with See also:wedge-shaped piers to See also:divide the stream. 42. Other arrangements of the constants n, a, b, a' will give the results of See also:special problems considered by J. M. See also:Michell, Phil. Trans. 189o. Thus with a' =o, a stream is split symmetrically by a wedge of angle a/n as in Bobyleff's problem ; and, by making a = , the wedge extends to infinity; then ch nil= 11b b u, sh nil=NInu Over the jet surface 4,=m, q=Q, u= —e rr4l'^=—be'r'l°, I I en. /~ ch it= cos nB=1ie,r,/,+I, sh sin nB=iV1e2r,/7+1, (2) el"'/'=tan nO bir ds 2n (3) c 3 sin 2220. For a jet impinging normally on an infinite plane, as in fig. 6, n=1, elg'/0=tan 0, ch (irs/c) sin 20 =I, (4) sh i,rx/c = cot B, sh 'wry/c = tan B, sh airx/c sh I7ry/c=1, eln<:+r>k=el*=/~+el*y/(5) With n =1, the jet is reversed in direction, and the See also:profile is the See also:catenary of equal strength. In Bobyleff's problem of the wedge of finite breadth, ch nil = a Nu —b, sh nit = Nib a as Nu u b, (6) Ib a—b cos na =1 l a, sin na = ¢ , and along the free surface APJ, q=Q, 4,=o, u=e-,r4,I=ae'r8/c, e'r' l' - i cos nB=cos na e,,,/,_cos2na' en, c cos2na sin2na (8) sin2na — sin2na' the intrinsic equation, the other free surface A'P'J' being given by See also:ems,/~ = cos2na sin2na sin2na —sin2na' (9) Putting n =I gives the case of a stream of finite breadth disturbed by a transverse plane, a particular case of Fig. 7. When a=b, a=o, and the stream is very broad compared with the wedge or lamina; so, putting w =w' (a —b)/a in the penultimate case, and u=ae-'.1 a—(a—b)w', w'+1 ch nit =, , sh nil= T w in which we may write w' =4)+0. (12) Along the stream line xABPJ, ¢=o; and along the jet surface APJ, —I>4'>—oo; and putting 4) =—as/c—1, the intrinsic equation is irs/c = cot'nO, (13) which for n=1 is the evolute of a catenary. 43. When the barrier AA' is held oblique to the current, the stream line xB is curved to the See also:branch point B on AA' (fig. 7), and so must be excluded from the boundary of u; the conformal re- C presentation is made now with V (b—a.b—a') du= (u—b)J (u—a.u—a') dw m I m' I du r u—j— a u—j' m+m' u—b _— ir u -1.0 , mj'+m'j taking u = co at the source where FIG.7. 4=co, u=b at the branch point B, u=j, j' at the end of the two diverging streams where c l)= -co ; while 4, =o along the stream line which divides at B and passes through A, A'; and 4, =m, —m' along the outside boundaries, so that m/Q, m'/Q is the final breadth of the jets, and (m+m')/Q is the initial breadth, c, of the impinging stream. Then b—a' u—a lb—a u—a' chi=~a—aa'\u—b' sh See also:pit=a ¢ Nu—b, ch it= 2b—a—a' N a-a' —u—b' sh 2=v NV (2 . a —u .0 —al) u—b N=za—b.b—a' a—a' , Along a jet surface, q =Q, and chil =See also:cosa=cosa—'3 sin5a(a—a')/(u—b), (5) if B=a at the source x of the jet xB, where u=co ; and supposing B =0, p' at the end of the streams where u = j, j', u—b i sin 2a u— 1 cos 0—cos p —cos a—cos 0' ~ = 5 sin Ya (cos a —cos ^$) (cos a —cos BY u—j' 2 cosa —cos,B' (6) a—a =2 sin a (cos a—cos /3') (cos a—cos B)' and ¢ being constant along a stream line d~dw ds d¢ _dw du du= ~' cT,O =dB —duda' irQ ds ir ds (cos a-cos Vii) (cos a—cos/') sin B m+m` dB TO = (cos a -cos 0) (cos 0 —cos /3) (cos B —cos r1)' sin B cos a—cos $' sin 0 =cosa—cos B+cos /3—cos 0'. cos B—cos /3 cos a—cos l3 sine cos (3—cos 13''cos B —cos," (7) giving the intrinsic equation of the surface of a jet, with proper attention to the sign. From A to B, a>u>b, B=o, ch Sl=ch log Q=cos a—z sin 2aa=a' sh =sh log ?_V (a u-b—a')sin a (u—b) cos a—'i(a—a') sin2a+J (a—u.u—a')sin a q u—b ds ds ai> _ dw Qdu=Qd¢du— q du =m +m' (u—b) cos a — ' (a —a') sin2a+II (a—u.u—a') sin a (9) r j—u.u—j' AB _ f'(2b—a—a')(u-b)—2(a-b)(b-a')+21f (a—b.b—a'.a—u.u—a')du, ( ) c J b a—a'.j-u.u—j' IO with a similar expression for BA'. The motion of a jet impinging on an infinite barrier is obtained by putting j = a, j' = a' ; duplicated on the other side of the barrier, the motion reversed will represent the See also:direct collision of two jets of unequal breadth and equal velocity. When the barrier is small compared with the jet, a=0=13', and G. See also:Kirchhoff's solution is obtained of a barrier placed obliquely in an infinite stream. Two corners BI and B2 in the wall xA, with a' = —co , and n =I, will give the solution, by duplication, of a jet issuing by a reentrant mouthpiece placed symmetrically in the end wall of the channel; or else of the channel blocked partially by a See also:diaphragm across the See also:middle, with edges turned back symmetrically, problems discussed by J. H. Michell, A. E. H. Love and M. R6thy. 9 (4) 0 A f ;1 A B x (I) (7) (10) (II) b=m+m" (I) (2) (3) (4) (8) ellipsoid; and the liquid flows over an ellipsoid along a line of slope with respect to Ox, treated as the vertical. Along the normal itself df, — >y-}-2(a2+x)d, 1, (to) so that over. the surfac 0f an ellipsoid where a and 4' are constant, the.normnal velocity.,is the same as that of the ellipsoid itself, moving as a solid with velocity parallel to Ox U =.—O — 2 (a2+x) d~' 0 i ) and so the boundary condition is satisfied; moreover, any ellipsoidal surface X may be supposed moving as if rigid with the velocity in (11), without disturbing the liquid motion for the moment. The continuity is secured if the liquid between two ellipsoids X and Xi, . moving with the velocity U and Ui of equation (II), is squeezed'out or sucked in across the plane x =o at a rate equal to the integral flow of the velocity 4' across the annular area al—a of the two ellipsioida niade,by x= o; or if aU—aiUi= J i'4' dXdX, (12) a= irJ (b2+X . c2+X). (13) Expressed as a differential relation, with the value of ,U from (II), d.dx [04+2 (0.2+x) ac ki _,da so 3a 'l-?(a2-,-lT(a—) =o' When the polygon is closed by the walls joining, instead of reaching back to infinity at xx', the liquid motion ''must be' due'toa source, and this modification has been worked out by B'. See also:Hopkinson in the Proc. Load. Math. Soc., 1808. Michell has discussed also the hollow 'vottex stationary inside a polygon (Phil. Trans., 189o) ; the solution is given by ch nc=snw, shnt2=icnw (Is) so that, round' the boundary of the polygon, . '=K', sin n8 =o; and on the surface of the vortex 0=o, q = Q, and cosnO=snn0=za-am.s/c, (12) the intrinsic equation of the curve. This is a closed See also:Sumner line for n= 1, when the boundary consists of two parallel walls; and n = fi gives an Elastica. 44. The Motion of a Solid through a Liquids—An important problem in the motion of a liquid is the determination of the state of velocity set up by the passage of a solid through it; and thence of the pressure and reaction of the liquid on the surface of the solid, by which its motion is influenced when it is free. Beginning with a single body in liquid extending to infinity'; and denoting by U, V, W, P, Q, R the components of linear and ahgular velocity with respect to axes fixed in the body, the velocity function takes the form =U¢i+V02-FW~3i-Px1+Qx2+Rx3, (1) where the apd x's are functions of, 'x, y, z,"depending on the shape of the body ; interpreted dynamically, C —p¢ represents the impulsive pressure required to stop the motion, or C+p¢ to start it again from rest. The terms of ¢ may be determined one at a time, and this problem is purely kinematical; thus to determine 01, the component U alone is taken to exist, and then 1, m, n, denoting the direction cosines of the normal of the surface drawn into the exterior liquid, the function 01 must be determined to satisfy the conditions (i.) V2spi=o, throughout the liquid; (ii.) d' = —1, the gradient of ¢ down the normal at the surface of the moving solid ; (iii.) dcbi over a fixed boundary, or at infinity; similarly for 412 and ¢3. To determine xi the angular velocity P alone is introduced, and the conditions to be satisfied are (i.) v2xt=o, throughout the liquid; dxl (ii.) dv = mz — ny, at the surface of the moving body, but zero over a fixed surface, and at infinity; the same for xs and x3. For a cavity filled with liquid in the interior of the body, since the liquid-inside moves bodily for a motion of tfanslation only, ~'7=—x, -y, ''3= —z;::.. (2) but a rotation will stir up the liquid in the cavity, so that the x's depend on the shape of the surface. The ellipsoid was the shape first worked out, by See also:George Green, in his See also:Research on the Vibration of a Pendulum in a Fluid See also:Medium (1833) ; the See also:extension to any other surface will form an important step in this subject. A system of confocal ellipsoids is taken 1,2 z't a2+X+T----X+c2+X= I' and a velocity function of the form cp = 4, (4) where 4' is a function of X only; so that 4. is constant over an ellipsoid ; and we seek to determine the motion set up, and the form of 4' which will satisfy the equation of continuity. Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane, = py = ~z 1a2+,pX' m—b2- X' n c2+X p2x2 p2J fAz2 1 = (a2+),)2+ (b2+X)2 (0+102' p2=(.22-W12+ x)m2+ (c2-rx)n3, -a'12+b2m2-1 c2n2+X, 2D -_-. Qs as' Thence d4' "d~+xd,p _ tG+2(a2- a) ld ,. i+o that'the velocity of the liquid Inayy be resolved into 1i coinpodent --0 parallel to Ox, and -2(a2+X)ldi/.,1dX along the normal' of the so that we may put MdX = J (17) P2-4(22"f x)(b2+x)(c.2+x), (18) where M denotes a constant; so that is an elliptic integral of the second kind. The quiescent ellipsoidal surface, over which the motion is entirely tangential; is the one for which 2(a2+x)da 0-9, (19) and this is the infinite boundary ellipsoid if we make the upper limit Xi=, The velocity of the ellipsoid defined by =o is then ll=—2a'dw—'o M (°° MdX abc-lo (a2+X)P abc(1—Ric(' AorA abcdX A= a (a2+X))F _ — 2abc1J f.9, (21) M l oxA =cxA= A, 01 =1—Ao, (22) in (I) for an ellipsoid. The impulse required to setup the motion in liquid of density p is the resultant of an impulsive pressure po over the surface S of the ellipsoid, and is therefore aW'U =4'0W' (24) 4'o Ao (25) °_—U _ 1—As' in this way the air See also:drag was calculated by Green for an ellipsoidal pendulum. Similarly, the inertia parallel to Oy and Oz is $W'=1 _Bo ow,, rW'=1 CCoW'11 ° abcdX BA' Ca=J A (b'+X, c2+X)P; A+B+C =abc/iP, Ao+Bo+C.o =1. For a sphere a=b=c,As=Bo=C5=jf, a=137=13, and (3) (9) (14) (15) and integrating (a2+X)s'2a =a constant, (16) (20) with the notation so that in (4) f f p4ldS = p>Go f (xldS =pipo (volume of the ellipsoid) =00W', (23) where W' denotes the See also:weight of liquid displaced. Denoting the effective inertia of the liquid parallel to Ox by aW', the momentum (26) (27) (28) (29) is s cos PSx, and of the source H and line oink OH is s,(a/f).cos PHx and - (µ/a) (PO - PH) ; so that = /2 (cos PSx+1cosPHx-'PO a PH-) ' (4) and 4, = —µ, 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 ilnage system; to realize physically the condition of zero flow across the sphere, an equal sink must he introduced at some other internal point S'. When S and S' See also:lie on the same radius, taken along Ox, the Stokes' function can be written down; 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 cos PSx = m y m pr, ; and for its image at H the Stokes' function is cosPHx=m f,pHi; so that for the See also:combination =my2(f pH3- ) =mn ( a_ ), (7) and this vanishes over the surface of the sphere. There is no Stokes' function when the axis of the doublet at S does not pass through 0; the image system will consist of an inclined doublet at H, making an equal angle with OS as the doublet S, and of a parallel negative line doublet, extending from H to 0, of moment varying as the distance from Q. A See also:distribution of See also:sources and doublets over a moving surface will enable an expression to be obtained for the velocity function of a body moving in the presence of a fixed sphere, or inside it. The method of See also:electrical images will enable the stream function 0' to be inferred from a distribution of doublets, finite in number when the surface is composed of two See also:spheres intersecting at au angle it/m, where m is an integer (R. A. Herman, Quart. Jour. of Math. xxii.). Thus for m=2, the spheres are orthogonal, and it can be verified that = ifJy' (I - a13 - a23 a3) , (8) Ia .123 +y3 P+X with a' =o over the surface of the paraboloid; and then 0'= U[y2 sl -F12)+pxl (9) =-2Up[ J (x2+y2)-xl ; (10) 4)=-IUP log (x'+y2)+x]. (II) The relative path of a liquid particle is along a stream line =;Uc', a constant, (12) p2y2 - (y2 - c2)2 2 2 _ p'y'+(y i2)2 x= 2p(y2-c') J{x +y) - 2p(y2 -02) (13) a Cs; while the absolute path of a particle in space will he given by dy,-rrx=-y2-c2 dx y 2py y' 46. Between two concentric spheres, with +x=r', d'+ ,=al'; A=B=C=a'/3r', as as as as 7+2 3 3 = 3Ux 1-as See also:ass' = aUy2 L---as a,3' (2) and the effective inertia of the liquid in the interspace is Ao+2.Ai al' +2a' We. (3) 2Ao - 2AIW = s al' - as When the spheres are•not concentric, an expression for the effective inertia can be found by the -method of images (W. M. See also:Hicks, Phil.' Trans., 188o). The image of a source of strength u at S outside a sphere of radius a is a source of strength .paff at H, where OS=f, OH=a'/f, and a line sink reaching from the image H to the centre 0 of line strength - µ/a; this combination will be found to produce no flow across the surface of the sphere. Taking Ox along OS, the Stokes' function at P for the source S 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 See also:parabola with vertical acceleration W+ sW,g• (30) Thus a spherical air bubble, in which W/W' is insensible, will begin to rise in water with acceleration 2g. 45. When the liquid is bounded externally by the fixed ellipsoid a=x,, a slight extension will give the velocity function 4, of the liquid in the interspace as the ellipsoid 7,=o is passing with velocity U through the confocal position; ¢ must now take the formx(0+N), and will satisfy the conditions in the shape abc CMI abcdX A+Bi+C, _ abcl+.l a(a2+x)P = UXBo+Cs- -CI - Ux Ao+B,+CI W'. (2) Bo+Co - Bi - Ci (3) If the ellipsoid is of revolution, with b=t, -UxBo --2 and the Stokes' current function 4, can be written down (4) S - iUy2Bo - BiI; reducing, when the liquid extends to infinity and B, =o, to (5) 0=31.14, 4, -5Uy2Bo; abc (a' abcdX ' (1) ai bi ci - J o (a2 + x) P and any confocal ellipsoid defined by internal or external to a=a,, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox UBs+Ca -BI- C, Bs+Co - BI - Cs' Since - Ux is the velocity function for the liquid W' filling the ellipsoid a=o, and moving bodily with it, the effective inertia of the liquid in the interspace is (5) (6) so that in the relative motion past the body, as when fixed in the current U parallel to xO, #'_ ux(1+g), 0'=1 Uy2(1 - ). (6) Changing the origin from the centre to the See also:focus of a prolate I spheroid, then putting b' = pa, _ A'a, and proceeding to the limit where a= so , we find for a paraboloid of revolution B=;p, Bo-p (7) (8) where al, 112, a = ala2/V (See also:ale+az2) is the radius of the spheres and their circle of intersection, and rl, r2r r the distances of,a point from their centres. The corresponding expression for two orthogonal cylinders will be 042 2 =Uy(I- x- +4). (9) With as = oo , these reduce to' 6 / 4 x =3Uy'(1-7 a,x orUy1I--i far a sphere or cylinder, and a/ diametral pplane. Two equal spheres, intersecting at 120e, will require` 0'=1Uy2 [-- a33+a9(a-fi )+ a33 a*(s+a x)1 (I11 a 2r, 2rI 2r3 2rs 11 with .a similar expression for cylinders; so that the plane x=o may be introduced as a boundary, cutting the surface at 6o°: The motion of these cylinders across the line of centres is the See also: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 =xyx (1) for a rotation R about Oz; and a similar See also:procedure shows,that an ellipsoidal surface X may be in rotation about Oz without disturbing the motion if ( (2) R = a + ) x+2dX I/(b'+x)- I/(a'+a) , and that the continuity of the liquid is secured if ,dx (3) (a'+X)3t'(b'+X)372(c'+A) =constant, f NA N Bs-AA (4) x=Ja (a2+7t)(b'+a)P=Zc' a'-b' ; and at the surface x=o, (5) I (az+bI: ) N Bo-Av ' } abc a --b' abc n' R - — 1/a' ' N = R 1/b".- 1/o' . (6) I 1 I Bo — A, - (See also:ate a' b2) a' - b' = R. 2 b2h (as- b')/(a2+2)-(Bo-Av)' The velocity function of the liquid inside the ellipsoid =o due to the same angular velocity will be ml = Rxy(a2-b2)/(a2+b2), and on the surface outside N Bo-Ao 4io = XYxo = xYabc a2 - bt' so that the ratio of the exterior and interior value of di at the surface is d'o Bo-Ao = (9) (Pi (a2—b2)/(a2+b2)-(Bo-Ao)' and this is the ratio of the effective angular inertia of the liquid, outside and inside the ellipsoid X=o. The extension to the case where the liquid is bounded externally by a fixed ellipsoid X= XI is made in a similar manner, by putting =xY(x+M), (to) and the ratio of the effective angular inertia in (9) is changed to a, - bb1,2 (Bo -Ai) - (Bi - Ai) + a1x+2 aibab,ci a2-b2 b,2 abc a2 +b2 ,2 a -(Bo-Ao)+(BI-AI) Make c = so for confocal elliptic cylinders; and then AA = bi+X ,1 (a2+a)J (4 a2+a.b2d-a -a2a bz \I - /Va2-h3)' (12) A 2 BA= aQ ( a -I) CA=o; and then as above in § 31, with a=cch a, b=csh a, a1= (a2+X)=cch al, b1=csh a1 the ratio in (II) agrees with § 31 (6). As before in § 31, the rotation may be resolved into a shear-pair, in planes perpendicular to Ox and Oy. A torsion of the ellipsoidal surface will give rise to a velocity function of the form 4) =xyzft, where it can be expressed by the elliptic integrals AA, BA, CA, in a similar manner, since co fl = L da/P2. 48. The determination of the 0's and x's is a kinematical problem, solved as yet only for a few cases, such as those discussed above. But supposing them determined for the motion of a body through a liquid, the kinetic energy T of the system, liquid and body, is expressible as a quadratic function of the components U, V, W, P, Q, R. The partial differential coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or.angular, Which cortesponds. Conversely, if the kinetic energy, T is expressed as a quadratic function of x1, x2, x2,y1, y2, y2, the components of momentum, the partial differential coefficient with respect to a momentum com- ponent will give the component of velocity to correspond. '. These theorems, which hold for the motion of a single rigid body, are true generally for a flexible system, such•a'considered here for a liquid, with one or more rigid bodies swimming in it_; and they ex-See also:press the statement that the work done by an impulse is the product of the impulse and the See also:arithmetic mean of the initial and final velocity; so that the kinetic energy is the work done by the impulse in starting the motion from rest. Thus if T is expressed as a quadratic function of U, V, W, P, Q, R', the components of momentum corresponding are x1= ' x2=TT xa= ' (1) dT dT dT y1= Y2= ya=TR; but when it is expressed,as a quadratic function of Y2, Ys, dT v=dW=de, dT dT R.-=35-3• dT P Q=-, The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; aid thence the dynamical equations follow X-dx, x dT +xadT, Y= ..., Z= ..., dt -ay, iiy2 I- = d, -Y2dY -r X -{-xaT-P M = ..., N - ..., (4) where X, Y, Z, L, M, N denote components of external applied force on the body. These equations are proved by taking a line fixed in space, whose direction cosines are 1, m, n, then dl=mR-nQ, dm=nP-lR, dtdn=lQ-mP. dt dt If P denotes the resultant linear impulse or momentum direction P = lx1+mxx+nxa, d dP =—X1+ .dm x2+ ~~ xa for all values of 1, m, n. Next, taking a fixed origin 12 and axes parallel to Ox, by, Oz through 0, and denoting by x, y, z the coordinates of 0, and by G the component angular momentum about Sl in the direction (1, m, n) G = l (y1-x2z+xay) +m(y2-x2x+x1z) +n (Ya-xiy+xax). (8) Differentiating with respect to t, and afterwards moving the fixed origin up to the moving origin 0, so that x=y=z=o, but dt =U' dt=V, T =W, dG ay, dt =1(dt -y2R+yzQ-x2W+xaV) +m (t -yaP+YIR-xaU+xiW) +n (d -t-v' y1Q+y2P-xiV +x2U ) =1L+mM+nN, for all values of 1, m, n. When no external force acts, the case which we shall consider, there are three integrals of the equations of motion (i.) T =constant, (ii.) x,2-Fx22+xa2=F2, a constant, (iii.) xjy1+x2y2+xaya=n=GF, a constant; and the dynamical equations in (3) See also:express the fact that x1, x2, xa are the components of a constant vector having a fixed direction; while (4) shows that the vector resultant of y1, y2, ya moves as if subject to a couple of components x2W-x3V, xiU-x1W, x1V-x2U, (Io) and the resultant couple is therefore perpendicular to F, the resultant of x1, x2, x2, so that the component along OF is constant, as expressed by (iii). If a See also:fourth integral is obtainable, the solution is reducible to a See also:quadrature, but this is not possible except in a limited series of cases, investigated by H. See also:Weber, F. Kotter, R. Lionville, Caspary, Jukovsky, Liapounoff, Kolosoff and others, chiefly See also:Russian mathematicians; and the general solution requires the double-theta hyperelliptic function. 49. In the motion which can be solved by the elliptic function, the most general expression of the kinetic energy was shown by A. Clebsch to take the form T=3 (x12+x22)+ap'xa2 + (x1Y1+x2y2) +11 q'xayi +ir(Y12+Y22)+3r'ya2 (I) so that a fourth integral is given by dya/dt=o, yi=constant ; (2) See also:delta=x1 (xz r 2 -x2(gxt+ryl 4 + Y) ) = r(xlyz - xxY1), (3) i2 (102=(x12+x22)(y12+y22) — (xlyl +x2y2)2 = (x12 +x22) (y12+y22)-(FG—x3ya)2 = (x12+x22) (y12+y22+y32—G2)-(Gxa"FYa)2, (4) x22 +x22=F2-xa2, xiy1+x2y2 FG-xaya, (5) r(y12 +y22) = 2T- p (x12+x22)—p'xa2 -2q(xlyi +xsy2)-2q'xayr-r'ya2 i=2T-pF 2gFaG-rxya4 g')xsYa in -l-mu (7) r2 (02_X2, l8) where Xi is a quartic function of xi, and thus t is given by an elliptic (7) (8) (13) x1, x2, Xi, Yl, (2) (3) +1ddt +m at21-n dta, =1(dt1-x2R+x3Q) +m (dt2-x2P+xiR) +n (del -x,Q =1X +mY+nZ, (5) in this (6) (7) (9) in which so that integral of the first kind; and by See also:inversion x3 is in elliptic function of the time t. Now (xi -xzi) (yi+y2i) = xiyi+xeyi+i(xiyi -x2Yi) = FG -xaya+id Xa, yi+Yzi _ FG -xaya+is/ Xi xi+xzi- xiz+xzz , dt(x1+x2i) = -i[(q'-q)xa+r'Y,]+irxa(yi+Yzi), d log(xl+x2i) =-(q'-q)xa-r'ya+rx3 F FG-2—x32JXa dti ~ / - ' d" log' V x~-xsi= -(q'-q)xa-(r'-r)ya-FrFF3-xaxa' requiring the elliptic integral of the third kind; thence the expression of x,+x2i and yi+yzi. Introducing See also:Euler's angles 0, ¢, ', x, =F sin 0 sin 0), Xi =F sin B cos 0, xi+xzi=iF sinBe'4'i, xa=F cos0; sin = P sin ¢+Q cos 0, (ow F sine 0— =--x~+d-rx1 dt dy, dy, _ (qxi+ryi)xi+(qx2+ry2)x2 = q(x,2+x22) +r(x,y,+xzy2) = qF2 sine 0 +r (FG -xaya) , Ft= I FG-x3Ya Frdx3 4,-q Fz-xas X2' elliptic integrals of the third kind. Employing G. Additional information and CommentsThere are no comments yet for this article.
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