1911] ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID 31
motion of liquid past a solid obstacle of any form limited in all directions, which satisfies the necessary conditions both at the surface of the obstacle and at infinity, and farther that the force required to hold the solid is finite. But if the obstacle be an infinite cylind-er of any cross-section, no such steady motion is possible, and the force required to hold the cylinder in position continually diminishes as the motion continues.
§ 3. For further developments the simplest case is that of a material plane, coinciding with the coordinate plane oc = 0 and moving parallel to y in a fluid originally at rest. The component velocities u, w are then zero; and the third velocity v satisfies (even though its square be not neglected) the general equation
dv cfiv ...
in which v, equal to pfp, represents the kinematic viscosity. In § 7 of his memoir Stokes considers periodic oscillations of the plane. Thus in (4) if v be proportional to eint, we have on the positive side
v = Aeint e~x^ {in/v) ............................... (5)
When 55=0, (5) must coincide with the velocity ( F) of the plane. If this be Vneint, we have A — Vn; so that in real quantities
v = Vne~x^ (n''iv] cos {nt - x VO/2")} .................. (6)
corresponds with V — }rn cos nt .............................. (7)
for the plane itself.
In order to find the tangential force (— 1\) exercised upon the plane, we have from (5) when x = 0
and T3=~-/j, (dv/das\ = p Vn eint V( inv)
......... (9)
giving the force per unit area due to the reaction of the fluid upon one side. " The force expressed by the first of these terms tends to dimmish the amplitude of- the oscillations of the plane. The force expressed by the second has the same effect as increasing the inertia of the plane." It will be observed that if Vn be given, the force diminishes without limit with n.
In note B Stokes resumes the problem of § 7 : instead of the motion of the plane being periodic, he supposes that the plane and fluid are initially at rest, and that the plane is then (t = 0) moved with a constant velocity F.. The stream-function (ty) for this motion satisfies the same differential equation as does the transverse displacement (w'} of a plane elastic plate. And a surface on which the fiuid remains at rest (i/r = 0, d^/dn = 0) corresponds to a curve along which the elastic plate is clamped.