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 movement of the boundary, and the motion will be irrotational; for any small spherical 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 to the velocity function φ is given byT = ½ρ∫ ∫ ∫ [ (dφ)2+(dφ)2+(dφ)2]dx dy dz,dxdydz= ½ρ∫ ∫φdφdSdν(1)by Green’s transformation, dν denoting an elementary step along the normal to the exterior of the surface; so that dφ/dν = 0 over the surface makes T = 0, and then(dφ)2+(dφ)2+(dφ)2= 0,dφ= 0,dφ= 0,dφ= 0.dxdydzdxdydz(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 act impulsively throughout the liquid, the pressure impulse ῶ satisfies the equations1dῶ= −u,1dῶ= −v,1dῶ= ῶ,ρdxρdyρdz(3)ῶ = ρφ + a constant,(4)and the constant may be ignored; and Green’s transformation of the energy T amounts to the theorem that the work done by an impulse is the product of the impulse and average velocity, or half the velocity from rest.In a multiply connected space, like a ring, with a multiply valued velocity function φ, the liquid can circulate in the circuits independently of any motion of the surface; thus, for example,φ = mθ = m tan−1y/x(5)will give motion to the liquid, circulating in any ring-shaped figure of revolution round Oz.To find 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 φ in any circuit, or the value by which φ has increased in completing the circuit, the values of φ on the two sides of the membrane are taken as differing by k, so that the integral over the membrane∫ ∫φdφdS = k∫ ∫dφdS,dνdν(6)and this term is to be added to the terms in (1) to obtain the additional part in the kinetic energy; the continuity shows that the integral is independent of the shape of the barrier membrane, and its position. Thus, in (5), the cyclic constant k = 2πm.In plane motion the kinetic energy per unit length parallel to OzT = ½ρ∫ ∫ [ (dφ)2+(dφ)2]dx dy = ½ρ∫ ∫ [ (dψ)2+(dψ)2]dx dydxdydxdy= ½ρ∫φdφds = ½ρ∫ψdψds.dνdν(7)For example, in the equilateral triangle of (8) § 28, referred to coordinate axes made by the base and height,ψ′ = −2Rαβγ/h = −½ Ry [ (h − y)2− 3x2] /h(8)ψ = ψ′ − ½R [ (1⁄3h − y)2+ x2]= −½R [ ½h3+1⁄3h2y + h) (x2− y2) − 3x2y + y3] /h(9)and over the base y = 0,dx/dν = −dx/dy = + ½R (1⁄3h2− 3x2) / h, ψ = −½R (1⁄9h2+ x2).(10)Integrating over the base, to obtain one-third of the kinetic energy T,1⁄3T = ½ρ∫h / √3¼R2(3x4−1⁄27h4) dx/h = ρR2h4/ 135 √3−h / √3(11)so that the effective k2of the liquid filling the triangle is given byk2= T / ½ρR2A = 2h2/ 45=2⁄5(radius of the inscribed circle)2,(12)or two-fifths of the k2for the solid triangle.Again, sincedφ/dν = dψ/ds, dφ/ds = −dψ/dν,(13)T = ½ρ ∫ φ dψ = −½ρ ∫ ψ dφ.(14)With the Stokes’ function ψ for motion symmetrical about an axis.T = ½ρ∫φdψ2πy ds = πρ ∫ φ dψ.y ds(15)37.Flow, Circulation, and Vortex Motion.—The line integral of the tangential velocity along a curve from one point to another, defined by∫ (udx+ vdy+ wdz)ds = ∫ (u dx + v dy + z dz),dsdsds(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 φ, the flow−∫ dφ = φ1− φ2,(2)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 φ 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 avortex filament.By analogy with the spin of a rigid body, the component spin of the fluid in any plane at a point is defined as the circulation round a small area in the plane enclosing the point, divided by twice the area. For in a rigid body, rotating about Oz with angular velocity ζ, the circulation round a curve in the plane xy is∫ζ(xdy− ydx)ds = ζ times twice the area.dsds(3)In a fluid, the circulation round an elementary area dxdy is equal tou dx +(v +dvdx)dy −(u +dudy)dx − vdy =(dv−du)dx dy,dxdydxdy(4)so that the component spin is½(dv−du)= ζ,dxdy(5)in the previous notation of § 24; so also for the other two components ξ and η.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 composition of the components of spin, ξ, η, ζ, is according to the vector 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 ABC, A′B′C′ is drawn of the vortex filament, joined by the vortex line AA′, then, since the flow in AA′ is taken in opposite directions in the 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 ωα is constant where α is the cross-section of the filament and ω the resultant spin (W. K. Clifford,Kinematic, book iii.).So far these theorems on vortex motion are kinematical; but introducing the equations of motion of § 22,Du+dQ= 0,Dv+dQ= 0,Dw+dQ= 0,dtdxdtdydtdz(6)Q = ∫ dp/ρ + V,(7)and taking dx, dy, dz in the direction of u, v, w, anddx : dy : dz = u : v : w,D(u dx + v dy + w dz)=Dudx + uD dx+ ... = −dQ + ½ dq2,dtdtdt(8)and integrating round a closed curveD∫(u dx + v dy + w dz) = 0,dt(9)and the circulation in any circuit composed of the same fluid particles is constant; and if the motion is 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 αω is constant for all time, and the same for every cross-section of the vortex filament.A vortex filament must 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 α of a filament by dS and its mass by dm, the quantity ωdS/dm is called thevorticity; this is the same at all points of a filament, and it does not change during the motion; and the vorticity is given by ω cosεdS/dm, if dS is the oblique section of which the normal makes an angle ε with the filament, while the aggregate vorticity of a mass M inside a surface S isM−1∫ ω cos ε dS.Employing the equation of continuity when the liquid is homogeneous,2(dζ−dη)= ∇2u, ... , ∇2= −d2−d2−d2,dydzdx2dy2dz2(10)which is expressed by∇2(u, v, w) = 2 curl (ξ, η, ζ), (ξ, η, ζ) = ½ curl (u, v, w).(11)38.Moving Axes in Hydrodynamics.—In many problems, such as the motion of a solid in liquid, it is convenient to take coordinate axes fixed to the solid and moving with it as the movable trihedron frame of reference. The components of velocity of the moving origin are denoted by U, V, W, and the components of angular velocity of the frame of reference by P, Q, R; and then if u, v, w denote the components of fluid velocity in space, and u′, v′, w′ the components relative to the axes at a point (x, y, z) fixed to the frame of reference, we haveu = U + u′ − yR + zQ,v = V + v′- zP + xR,w = W + w′ − xQ + yP.(1)Now if k denotes the component of absolute velocity in a direction fixed in space whose direction cosines are l, m, n,k = lu + mv + nw;(2)and in the infinitesimal element of time dt, the coordinates of the fluid particle at (x, y, z) will have changed by (u′, v′, w′)dt; so thatDk=dlu +dmv +dnwdtdtdtdt+ l(du+ u′du+ v′du+ w′du)dtdxdydz+ m(dv+ u′dv+ v′dv+ w′dv)dtdxdydz+ n(dw+ u′dw+ v′dw+ w′dw).dtdxdydz(3)But as l, m, n are the direction cosines of a line fixed in space,dl= mR − nQ,dm= nP − lR,dn= lQ − mP;dtdtdt(4)so thatDk= l(du− vR + wQ + u′du+ v′du+ w′du)+ m (...) + n (...)dtdtdxdydz= l(X −1dp)+ m(Y −1dp)+ n(Z −1dp),pdxpdypdz(5)for all values of l, m, n, leading to the equations of motion with moving axes.When the motion is such thatu = −dφ− mdψ, v = −dφ− mdψ, w = −dφ− mdψ,dxdxdydydzdz(6)as in § 25 (1), a first integral of the equations in (5) may be written∫dp+ V + ½q2−dφ− mdψ+ (u − u′)(dφ+ mdψ)ρdtdtdxdx+ (v − v′)(dφ+ mdψ)+ (w − w′)(dφ+ mdψ)= F(t),dydydzdz(7)in whichdφ− (u − u′)dφ− (v − v′)dφ− (w − w′)dφdtdxdydz=dφ− (U − yR + zQ)dφ− (V − zP + xR)dφ− (W − xQ + yP)dφdtdxdydz(8)is the time-rate of change of φ 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 O is advancing with velocity U, the equation (5) of § 34p/ρ + V + ½q′2− ƒ (ψ′) = constant(9)becomes transformed intop+ V + ½q2−Udψ+ ½U2− ƒ (ψ + ½Uy2) = constant,ρydy(10)ψ′ = ψ + ¼U y2,(11)subject to the condition, from (4) § 34,y−2∇2ψ′ = −ƒ′(ψ′), y−2∇2ψ = −ƒ′ (ψ + ½Uy2).(12)Thus, for example, withψ′ = ¾U y2(r2a−2− 1), r2= x2+ y2,(13)for the space inside the sphere r = a, compared with the value of ψ′ in § 34 (13) for the space outside, there is no discontinuity of the velocity in crossing the surface.Inside the sphere2ζ =d(1dψ′)+d(1dψ′)=15Uy,dxydxdyydy2a2(14)so that § 34 (4) is satisfied, withƒ′ (ψ′) =15Ua−2, ƒ (ψ′) =15Uψ′ a−2;22(15)and (10) reduces top+ V −9U{(x2− 1)2−(y2− ½)2}= constant;ρ8a2a2(16)this gives the state of motion in M. J. M. Hill’s spherical vortex, advancing through the surrounding liquid with uniform velocity.39. As an application of moving axes, consider the motion of liquid filling the ellipsoidal casex2+y2+z2= 1;a2b2c2(1)and first suppose the liquid to be frozen, and the ellipsoid to berotating about the centre with components of angular velocity ξ, η, ζ; thenu = − yζ + zη, v = − zξ + xζ, w = − xη + yξ.(2)Now suppose the liquid to be melted, and additional components of angular velocity Ω1, Ω2, Ω3communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-functionφ = − Ω1b2− c2yz − Ω2c2− a2zx − Ω3a2− b2xy,b2+ c2c2+ a2a2+ b2(3)as may be verified by considering one term at a time.If u′, v′, w′ denote the components of the velocity of the liquid relative to the axes,u′ = u + yR − zQ =2a2Ω3y −2a2Ω2z,a2+ b2c2+ a2(4)v′ = v + zP − xR =2b2Ω1z −2b2Ω3x,b2+ c2a2+ b2(5)w′ = w + xQ − yP =2c2Ω2x −2c2Ω1y,c2+ a2b2+ c2(6)P = Ω1+ ξ, Q = Ω2+ η, R = Ω3+ ζ.(7)Thusu′x+ v′y+ w′z= 0,a2b2c2(8)so that a liquid particle remains always on a similar ellipsoid.The hydrodynamical equations with moving axes, taking into account the mutual gravitation of the liquid, become1dp+ 4πρAx +du− vR + wQ + u′du+ v′du+ w′du= 0, ... , ... ,ρdxdtdxdydz(9)whereA, B, C =∫∞0abcdλ(a2+ λ, b2+ λ, c2+ λ) PP2= 4 (a2+ λ) (b2+ λ) (c2+ λ).(10)With the values above of u, v, w, u′, v′, w′, the equations become of the form1dp+ 4πρ Ax + αx + hy + gz = 0,ρdx(11)1dp+ 4πρBy + hx + βy + fz = 0,ρdy(12)1dp+ 4πρCz + gx + fy + γz = 0,ρdz(13)and integratingpρ−1+ 2πρ (Ax2+ By2+ Cz2)+ ½ (αx2+ βy2+ γz2+ 2fyz + 2gzx + 2hxy) = const.,(14)so that the surfaces of equal pressure are similar quadric surfaces, which, symmetry and dynamical considerations show, must be coaxial surfaces; and f, g, h vanish, as follows also by algebraical reduction; andα =4c2(c2− a2)Ω22−(c2− a2Ω2− η)2(c2+ a2)2c2+ a2−4b2(a2− b2)Ω32−(a2− b2Ω3− ζ)2,(a2+ b2)2a2+ b2(15)with similar equations for β and γ.If we can make(4πρA + α) x2= (4πρB + β) b2= (4πρC + γ) c2,(16)the surfaces of equal pressure are similar to the external case, which can then be removed without affecting the motion, provided α, β, γ remain constant.This is so when the axis of revolution is a principal axis, say Oz; whenΩ1= 0, Ω2= 0, ξ = 0, η = 0.(17)If Ω3= 0 or θ3= ζ in addition, we obtain the solution of Jacobi’s ellipsoid of liquid of three unequal axes, rotating bodily about the least axis; and putting a = b, Maclaurin’s solution is obtained of the rotating spheroid.In the general motion again of the liquid filling a case, when a = b, Ω3may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce todξ= −2c2Ω2ζ,dη=2a2Ω1ζ,dζ=2c2(Ω2ξ − Ω2η)dta2+ c2dta2+ c2dta2+ c2(18)dΩ1= Ω2ζ +a2+ c2ηζ,dΩ2= −Ω1ζ −a2+ c2ξζ;dta2− c2dta2− c2(19)of which three integrals areξ2+ η2= L −a2ζ2,c2(20)Ω12+ Ω22= M +(a2+ c2)2ζ2,2c2(a2− c2)(21)Ω1ξ + Ω2ηN = +a2+ c2ζ2;4c2(22)and then(dζ)2=4c4(Ω2ξ − Ω12η)2dt(a2+ c2)=4c4[ (ξ2+ η2) (Ω12+ Ω22) − (Ω1ξ + Ω2η)2](a2+ c2)2=4c4[LM − N2+{(a2+ c2)2− Ma2− Na2+ c2}ζ2(a2+ c2)22c2(a2+ c2)c22c2−(a2+ c2) (9a2− c2)ζ4]= Z,16c4(a2− c2)(23)where Z is a quadratic in ζ2, so that ζ is an elliptic function of t, except when c = a, or 3a.Put Ω1= Ω cos φ, Ω2= −Ω sin φ,Ω2dφ=dΩ1Ω2− Ω1dΩ2= Ω2ζ −(a2+ c2)(Ω1ξ + Ω2η) ζ,dtdtdt(a2− c2)(24)dφ= ζ −(a2+ c2)·N +a2+ c24c2,dt(a2− c2)M +(a2+ c2)2ζ22c2(a2− c2)(25)φ =∫ζ dζ−a2+ c2∫N +a2+ c24c2·ζ dζ,√Za2− c2M +(a2+ c2)2ζ22c2(a2− c2)√Z(26)which, as Z is a quadratic function of ζ2, are non-elliptic integrals; so also for ψ, where ξ = ω cos ψ, η = −ω sin ψ.In a state of steady motiondζ= 0,Ω1=Ω2,dtξη(27)φ = ψ = nt, suppose,(28)Ω1ξ + Ω2η = Ωω,(29)dφ= ζ −a2+ c2ωζ,dta2− c2Ω(30)dψ= −2a2Ωζ,dta2+ c2ω(31)1 −a2+ c2ω= −2a2Ω,a2− c2Ωa2+ c2ω(32)(ω− ½a2+ c2)2=(a2− c2) (9a2− c2),Ωa2− c24 (a2+ c2)(33)and a state of steady motion is impossible when 3a > c > a.
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 movement of the boundary, and the motion will be irrotational; for any small spherical 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 to the velocity function φ is given by
(1)
by Green’s transformation, dν denoting an elementary step along the normal to the exterior of the surface; so that dφ/dν = 0 over the surface makes T = 0, and then
(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 act impulsively throughout the liquid, the pressure impulse ῶ satisfies the equations
(3)
ῶ = ρφ + a constant,
(4)
and the constant may be ignored; and Green’s transformation of the energy T amounts to the theorem that the work done by an impulse is the product of the impulse and average velocity, or half the velocity from rest.
In a multiply connected space, like a ring, with a multiply valued velocity function φ, the liquid can circulate in the circuits independently of any motion of the surface; thus, for example,
φ = mθ = m tan−1y/x
(5)
will give motion to the liquid, circulating in any ring-shaped figure of revolution round Oz.
To find 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 φ in any circuit, or the value by which φ has increased in completing the circuit, the values of φ on the two sides of the membrane are taken as differing by k, so that the integral over the membrane
(6)
and this term is to be added to the terms in (1) to obtain the additional part in the kinetic energy; the continuity shows that the integral is independent of the shape of the barrier membrane, and its position. Thus, in (5), the cyclic constant k = 2πm.
In plane motion the kinetic energy per unit length parallel to Oz
(7)
For example, in the equilateral triangle of (8) § 28, referred to coordinate axes made by the base and height,
ψ′ = −2Rαβγ/h = −½ Ry [ (h − y)2− 3x2] /h
(8)
ψ = ψ′ − ½R [ (1⁄3h − y)2+ x2]= −½R [ ½h3+1⁄3h2y + h) (x2− y2) − 3x2y + y3] /h
ψ = ψ′ − ½R [ (1⁄3h − y)2+ x2]
= −½R [ ½h3+1⁄3h2y + h) (x2− y2) − 3x2y + y3] /h
(9)
and over the base y = 0,
dx/dν = −dx/dy = + ½R (1⁄3h2− 3x2) / h, ψ = −½R (1⁄9h2+ x2).
(10)
Integrating over the base, to obtain one-third of the kinetic energy T,
(11)
so that the effective k2of the liquid filling the triangle is given by
k2= T / ½ρR2A = 2h2/ 45=2⁄5(radius of the inscribed circle)2,
k2= T / ½ρR2A = 2h2/ 45
=2⁄5(radius of the inscribed circle)2,
(12)
or two-fifths of the k2for the solid triangle.
Again, since
dφ/dν = dψ/ds, dφ/ds = −dψ/dν,
(13)
T = ½ρ ∫ φ dψ = −½ρ ∫ ψ dφ.
(14)
With the Stokes’ function ψ for motion symmetrical about an axis.
(15)
37.Flow, Circulation, and Vortex Motion.—The line integral of the tangential velocity along a curve from one point to another, defined by
(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 φ, the flow
−∫ dφ = φ1− φ2,
(2)
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 φ 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 avortex filament.
By analogy with the spin of a rigid body, the component spin of the fluid in any plane at a point is defined as the circulation round a small area in the plane enclosing the point, divided by twice the area. For in a rigid body, rotating about Oz with angular velocity ζ, the circulation round a curve in the plane xy is
(3)
In a fluid, the circulation round an elementary area dxdy is equal to
(4)
so that the component spin is
(5)
in the previous notation of § 24; so also for the other two components ξ and η.
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 composition of the components of spin, ξ, η, ζ, is according to the vector 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 ABC, A′B′C′ is drawn of the vortex filament, joined by the vortex line AA′, then, since the flow in AA′ is taken in opposite directions in the 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 ωα is constant where α is the cross-section of the filament and ω the resultant spin (W. K. Clifford,Kinematic, book iii.).
So far these theorems on vortex motion are kinematical; but introducing the equations of motion of § 22,
(6)
Q = ∫ dp/ρ + V,
(7)
and taking dx, dy, dz in the direction of u, v, w, and
dx : dy : dz = u : v : w,
(8)
and integrating round a closed curve
(9)
and the circulation in any circuit composed of the same fluid particles is constant; and if the motion is 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 αω is constant for all time, and the same for every cross-section of the vortex filament.
A vortex filament must 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 α of a filament by dS and its mass by dm, the quantity ωdS/dm is called thevorticity; this is the same at all points of a filament, and it does not change during the motion; and the vorticity is given by ω cosεdS/dm, if dS is the oblique section of which the normal makes an angle ε with the filament, while the aggregate vorticity of a mass M inside a surface S is
M−1∫ ω cos ε dS.
Employing the equation of continuity when the liquid is homogeneous,
(10)
which is expressed by
∇2(u, v, w) = 2 curl (ξ, η, ζ), (ξ, η, ζ) = ½ curl (u, v, w).
(11)
38.Moving Axes in Hydrodynamics.—In many problems, such as the motion of a solid in liquid, it is convenient to take coordinate axes fixed to the solid and moving with it as the movable trihedron frame of reference. The components of velocity of the moving origin are denoted by U, V, W, and the components of angular velocity of the frame of reference by P, Q, R; and then if u, v, w denote the components of fluid velocity in space, and u′, v′, w′ the components relative to the axes at a point (x, y, z) fixed to the frame of reference, we have
u = U + u′ − yR + zQ,v = V + v′- zP + xR,w = W + w′ − xQ + yP.
u = U + u′ − yR + zQ,
v = V + v′- zP + xR,
w = W + w′ − xQ + yP.
(1)
Now if k denotes the component of absolute velocity in a direction fixed in space whose direction cosines are l, m, n,
k = lu + mv + nw;
(2)
and in the infinitesimal element of time dt, the coordinates of the fluid particle at (x, y, z) will have changed by (u′, v′, w′)dt; so that
(3)
But as l, m, n are the direction cosines of a line fixed in space,
(4)
so that
(5)
for all values of l, m, n, leading to the equations of motion with moving axes.
When the motion is such that
(6)
as in § 25 (1), a first integral of the equations in (5) may be written
(7)
in which
(8)
is the time-rate of change of φ 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 O is advancing with velocity U, the equation (5) of § 34
p/ρ + V + ½q′2− ƒ (ψ′) = constant
(9)
becomes transformed into
(10)
ψ′ = ψ + ¼U y2,
(11)
subject to the condition, from (4) § 34,
y−2∇2ψ′ = −ƒ′(ψ′), y−2∇2ψ = −ƒ′ (ψ + ½Uy2).
(12)
Thus, for example, with
ψ′ = ¾U y2(r2a−2− 1), r2= x2+ y2,
(13)
for the space inside the sphere r = a, compared with the value of ψ′ in § 34 (13) for the space outside, there is no discontinuity of the velocity in crossing the surface.
Inside the sphere
(14)
so that § 34 (4) is satisfied, with
(15)
and (10) reduces to
(16)
this gives the state of motion in M. J. M. Hill’s spherical vortex, advancing through the surrounding liquid with uniform velocity.
39. As an application of moving axes, consider the motion of liquid filling the ellipsoidal case
(1)
and first suppose the liquid to be frozen, and the ellipsoid to berotating about the centre with components of angular velocity ξ, η, ζ; then
u = − yζ + zη, v = − zξ + xζ, w = − xη + yξ.
(2)
Now suppose the liquid to be melted, and additional components of angular velocity Ω1, Ω2, Ω3communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function
(3)
as may be verified by considering one term at a time.
If u′, v′, w′ denote the components of the velocity of the liquid relative to the axes,
(4)
(5)
(6)
P = Ω1+ ξ, Q = Ω2+ η, R = Ω3+ ζ.
(7)
Thus
(8)
so that a liquid particle remains always on a similar ellipsoid.
The hydrodynamical equations with moving axes, taking into account the mutual gravitation of the liquid, become
(9)
where
P2= 4 (a2+ λ) (b2+ λ) (c2+ λ).
(10)
With the values above of u, v, w, u′, v′, w′, the equations become of the form
(11)
(12)
(13)
and integrating
pρ−1+ 2πρ (Ax2+ By2+ Cz2)+ ½ (αx2+ βy2+ γz2+ 2fyz + 2gzx + 2hxy) = const.,
(14)
so that the surfaces of equal pressure are similar quadric surfaces, which, symmetry and dynamical considerations show, must be coaxial surfaces; and f, g, h vanish, as follows also by algebraical reduction; and
(15)
with similar equations for β and γ.
If we can make
(4πρA + α) x2= (4πρB + β) b2= (4πρC + γ) c2,
(16)
the surfaces of equal pressure are similar to the external case, which can then be removed without affecting the motion, provided α, β, γ remain constant.
This is so when the axis of revolution is a principal axis, say Oz; when
Ω1= 0, Ω2= 0, ξ = 0, η = 0.
(17)
If Ω3= 0 or θ3= ζ in addition, we obtain the solution of Jacobi’s ellipsoid of liquid of three unequal axes, rotating bodily about the least axis; and putting a = b, Maclaurin’s solution is obtained of the rotating spheroid.
In the general motion again of the liquid filling a case, when a = b, Ω3may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to
(18)
(19)
of which three integrals are
(20)
(21)
(22)
and then
(23)
where Z is a quadratic in ζ2, so that ζ is an elliptic function of t, except when c = a, or 3a.
Put Ω1= Ω cos φ, Ω2= −Ω sin φ,
(24)
(25)
(26)
which, as Z is a quadratic function of ζ2, are non-elliptic integrals; so also for ψ, where ξ = ω cos ψ, η = −ω sin ψ.
In a state of steady motion
(27)
φ = ψ = nt, suppose,
(28)
Ω1ξ + Ω2η = Ωω,
(29)
(30)
(31)
(32)
(33)
and a state of steady motion is impossible when 3a > c > a.
An experiment was devised by Lord Kelvin for demonstrating this, in which the difference of steadiness was shown of a copper 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 modern projectile with a cavity more than three diameters long should fly steadily when filled with water; while the old-fashioned type, not so elongated, would be highly unsteady; and for the same reason the gas bags of a dirigible balloon should be over rather than under three diameters long.
40.A Liquid Jet.—By the use of the complex variable and its conjugate functions, an attempt can be made to give a mathematical interpretation of problems such as the efflux of water in a jet or of smoke from a chimney, the discharge through a weir, the flow of water through the piers of a bridge, or past the side of a ship, the wind blowing on a sail or aeroplane, or against a wall, or impinging jets of gas or water; cases where a surface of discontinuity is observable, more or less distinct, which separates the running stream from the dead water or air.