ch[Omega] = cos [theta] = cos [alpha] - ½sin² [alpha](a - a´)/(u - b), (5)
if [theta] = [alpha] at the source x of the jet xB, where u = [oo];
and supposing [theta] = [beta], [beta]´ at the end of the streams
where u = j, j´,
u - b ½ sin² [alpha] u - j cos[theta] - cos[beta]
----- = -------------------------, ------ = ½ sin² [alpha]-----------------------------------------------------,
a - a´ cos [alpha] - cos [theta] a - a´ (cos [alpha] - cos [beta])(cos [alpha] - cos [theta])
u - j´ cos [theta] - cos [beta]´
----- = ½ sin² [alpha]------------------------------------------------------; (6)
a - a´ (cos [alpha] - cos [beta]´)(cos [alpha] - cos [theta])
and [psi] being constant along a stream line
d[phi] dw ds d[phi] dw du
------ = --, Q -------- = -------- = -- --------,
du du d[theta] d[theta] du d[theta]
[pi]Q ds [pi] ds (cos [alpha] - cos [beta])(cos [alpha] - cos [beta]´)sin[theta]
------ -------- = ---- -------- = ---------------------------------------------------------------------------------,
m + m´ d[theta] c d[theta] (cos [alpha] - cos [theta])(cos [theta] - cos [beta])(cos [theta] - cos [alpha]´)
sin [theta] cos [alpha] - cos [beta]´ sin [theta]
= ------------------------- + ------------------------- · ------------------------
cos [alpha] - cos [theta] cos [beta] - cos [beta]´ cos [theta] - cos [beta]
cos [alpha] - cos [beta] sin [theta]
------------------------ · -------------------------, (7)
cos [beta] - cos [beta]´ cos [theta] - cos [beta]´
giving the intrinsic, equation of the surface of a jet, with proper
attention to the sign.
From A to B, a > u > b, [theta] = 0,
Q a - a´
ch [Omega] = ch log --- = cos [alpha] - ½ sin² [alpha] ------
q a - b
Q [root](a - u·u - a´)
sh [Omega] = sh log --- = -------------------- sin [alpha]
q u - b
Q (u - b) cos [alpha] - ½(a - a´) sin² [alpha] + [root](a - u·u - a´)sin[alpha]
--- = ----------------------------------------------------------------------------- (8)
q u - b
ds ds d[phi] Q dw
Q -- = Q ------ ------ = - --- --
du d[phi] du q du
m + m´ (u - b) cos [alpha] - ½(a - a´) sin² [alpha] + [root](a - u·u - a´) sin [alpha]
= ------ · ------------------------------------------------------------------------------- (9)
[pi] j - u·u - j´
_
AB / a (2b - a - a´)(u - b) - 2(a - b)(b - a´) + 2[root](a - b·b -a´·a - u·u -a´)
[pi]-- = | -------------------------------------------------------------------------- du, (10)
c _/ b a - a´·j - u·u - j´
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 direct collision of two jets of
unequal breadth and equal velocity. When the barrier is small compared
with the jet, [alpha] = [beta] = [beta]´, and G. Kirchhoff's solution
is obtained of a barrier placed obliquely in an infinite stream.
Two corners B1 and B2 in the wall xA, with a´ = -[oo], and n = 1, 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 diaphragm across the
middle, with edges turned back symmetrically, problems discussed by J.
H. Michell, A. E. H. Love and M. Réthy.
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 n[Omega] = sn w, sh n[Omega] = i cn w (11)
so that, round the boundary of the polygon, [psi] = K´, sin n[theta] =
0; and on the surface of the vortex [psi] = 0, q = Q, and
cos n[theta] = sn [phi], n[theta] = ½[pi] - 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 = ½ 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
[phi] = U_[phi]1 + V_[phi]2 + W_[phi]3 + P_[chi]1 + Q_[chi]2 + R_[chi]3, (1)
where the [phi]'s and [chi]'s are functions of x, y, z, depending on
the shape of the body; interpreted dynamically, C - [rho][phi]
represents the impulsive pressure required to stop the motion, or C +
[rho][phi] to start it again from rest.
The terms of [phi] may be determined one at a time, and this problem
is purely kinematical; thus to determine [phi]1, 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
[phi]1 must be determined to satisfy the conditions
(i.) [nabla]²[phi]1 = 0. throughout the liquid;
(ii.) d[ph]1/d[upsilon] = -l, the gradient of [phi] down the normal at
the surface of the moving solid;
(iii.) d[ph]1/d[upsilon] = 0, over a fixed boundary, or at infinity;
similarly for [phi]2 and [phi]3.
To determine [chi]1 the angular velocity P alone is introduced, and
the conditions to be satisfied are
(i.) [nabla]²[chi]1 = 0, throughout the liquid;
(ii.) d[chi]1/d[upsilon] = mz - ny, at the surface of the moving body,
but zero over a fixed surface, and at infinity; the same for [chi]2
and [chi]3.
For a cavity filled with liquid in the interior of the body, since the
liquid inside moves bodily for a motion of translation only,
[phi]1 = -x, [phi]2 = -y, [phi]3 = -z; (2)
but a rotation will stir up the liquid in the cavity, so that the
[chi]'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.