Page:EB1911 - Volume 14.djvu/142

When the polygon is closed by the walls joining, instead of reaching back to infinity at xx′, the liquid motion must be due to a source, and this modification has been worked out by B. Hopkinson in the Proc. Lond. Math. Soc., 1898.

Michell has discussed also the hollow vortex stationary inside a polygon (Phil. Trans., 1890); the solution is given by

ch nΩ = sn w, sh nΩ = i cn w

(11)

so that, round the boundary of the polygon, ψ = K′, sin nθ = 0; and on the surface of the vortex ψ = 0, q = Q, and

cos nθ = sn φ, nθ = 1/2π − am s/c,

(12)

the intrinsic equation of the curve.

This is a closed Sumner line for n = 1, when the boundary consists of two parallel walls; and n = 1/2 gives an Elastica.

44. The Motion of a Solid through a Liquid.—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 angular velocity with respect to axes fixed in the body, the velocity function takes the form

φ = U_φ1 + V_φ2 + W_φ3 + P_χ1 + Q_χ2 + R_χ3,

(1)

where the φ’s and χ’s are functions of x, y, z, depending on the shape of the body; interpreted dynamically, C − ρφ represents the impulsive pressure required to stop the motion, or C + ρφ to start it again from rest.

The terms of φ may be determined one at a time, and this problem is purely kinematical; thus to determine φ₁, the component U alone is taken to exist, and then l, m, n, denoting the direction cosines of the normal of the surface drawn into the exterior liquid, the function φ₁ must be determined to satisfy the conditions

(i.) ∇²φ₁ = 0. throughout the liquid;

(ii.) dφ₁/dν = −l, the gradient of φ down the normal at the surface of the moving solid;

(iii.) dφ₁/dν = 0, over a fixed boundary, or at infinity;

   similarly for φ₂ and φ₃.

To determine χ₁ the angular velocity P alone is introduced, and the conditions to be satisfied are

(i.) ∇²χ₁ = 0, throughout the liquid;

(ii.) dχ₁/dν = mz − ny, at the surface of the moving body, but zero over a fixed surface, and at infinity; the same for χ₂ and χ₃.

For a cavity filled with liquid in the interior of the body, since the liquid inside moves bodily for a motion of translation only,

φ₁ = −x, φ₂ = −y, φ₃ = −z;

(2)

but a rotation will stir up the liquid in the cavity, so that the χ’s depend on the shape of the surface.

The ellipsoid was the shape first worked out, by George Green, in his Research on the Vibration of a Pendulum in a Fluid Medium (1833); the extension to any other surface will form an important step in this subject.

A system of confocal ellipsoids is taken

x2	+	y2	+	z2	= 1
a2 + λ		b2 + λ		c2 + λ

(3)

and a velocity function of the form

φ = xψ,

(4)

where ψ is a function of λ only, so that ψ is constant over an ellipsoid; and we seek to determine the motion set up, and the form of ψ which will satisfy the equation of continuity.

Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane,

l =	px	,   m =	py	,   n =	pz
	a2 + λ		b2 + λ		c2 + λ

(5)

1 =	p2x2	+	p2y2	+	p2z2	,
	(a2 + λ)2		(b2 + λ)2		(c2 + λ)2

(6)

p² = (a² + λ) l² + (b² + λ) m² + (c² + λ) n²,
 = a²l² + b²m² + c²n² + λ,

(7)

2p	dp	=	dλ	;
	ds		ds

(8)

Thence

dφ	=	dx	ψ + x	dψ
ds		ds		ds

=	dx	ψ + 2 (a2 + λ)	dψ	l	dp	,
	ds		dλ		ds

(9)

so that the velocity of the liquid may be resolved into a component -ψ parallel to Ox, and −2(a² + λ)ldψ/dλ along the normal of the 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

dφ	{ ψ + 2(a2 + λ)	dψ	} l,
ds		dλ

(10)

so that over the surface of an ellipsoid where λ and ψ are constant, the normal velocity is the same as that of the ellipsoid itself, moving as a solid with velocity parallel to Ox

U = −ψ − 2 (a2 + λ)	dψ	,
	dλ

(11)

and so the boundary condition is satisfied; moreover, any ellipsoidal surface λ 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 λ and λ₁, moving with the velocity U and U₁ of equation (11), is squeezed out or sucked in across the plane x = 0 at a rate equal to the integral flow of the velocity ψ across the annular area α₁ − α of the two ellipsoids made by x = 0; or if

αU − α1U1 = $\int _{\lambda }^{\lambda _{1}}$ ψ	dα	dλ,
	dλ

(12)

α = π√ (b² + λ.c² + λ).

(13)

Expressed as a differential relation, with the value of U from (11),

d	[ αψ + 2 (a2 + λ) α	dψ	] − ψ	dα	= 0,
dλ		dλ		dλ

(14)

3α	dψ	+ 2 (a2 + λ)	d	( α	dψ	) = 0,
	dλ		dλ		dλ

(15)

and integrating

(a2 + λ)3/2 α	dψ	= a constant,
	dλ

(16)

so that we may put

ψ = $\int$	M dλ	,
	(a2 + λ) P

(17)

P² = 4 (a² + λ) (b² + λ) (c² + λ),

(18)

where M denotes a constant; so that ψ 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 + λ)	dψ	+ ψ = 0,
	dλ

(19)

and this is the infinite boundary ellipsoid if we make the upper limit λ₁ = ∞.

The velocity of the ellipsoid defined by λ = 0 is then

U = −2a2	dψ0	− ψ0
	dλ

⁠=	M	− $\int _{0}^{\infty }$	M dλ
	abc		(a2 + λ)P

=	M	(1 − A0),
	abc

(20)

with the notation

A or Aλ = $\int _{\lambda }^{\infty }$	abc dλ
	(a2 + λ) P

= −2abc	d	$\int _{\lambda }^{\infty }$	dλ	,
	da2		P

(21)

so that in (4)

φ =	M	xA =	UxA	,   φ1 =	xAλ	,
	abc		1 − A0		1 − A0

(22)

in (1) for an ellipsoid.

The impulse required to set up the motion in liquid of density ρ is the resultant of an impulsive pressure ρφ over the surface S of the ellipsoid, and is therefore

∬ ρφldS = ρψ₀ ∬ xldS⁠
⁠= ρψ₀ (volume of the ellipsoid) = ψ₀W′,

(23)

where W′ denotes the weight of liquid displaced.

Denoting the effective inertia of the liquid parallel to Ox by αW′. the momentum

αW′U = ψ₀W′

(24)

α =	ψ0	=	A0	;
	U		1 − A0

(25)

in this way the air drag was calculated by Green for an ellipsoidal pendulum.

Similarly, the inertia parallel to Oy and Oz is

βW′ =	B0	W′,   γW′ =	C0	W′,
	1 − B0		1 − C0

(26)

Bλ, Cλ = $\int _{\lambda }^{\infty }$	abc dλ	;
	(b2 + λ, c2 + λ) P

(27)

and

A + B + C = abc/1/2P,   A₀ + B₀ + C₀ = 1.

(28)

For a sphere

a = b = c,   A₀ = B₀ = C₀ = 1/3,   α = β = γ = 1/2,

(29)