20 HYDRODYNAMICAL NOTES
Solutions of interest are afforded in the case n = 1. The C-sol
cos
sn2
-^ = !?•£ (cos \Q — cos ^-(9) = —
vanishing when 6 = TT, as well as when 9 = 0, 0 = 2-n-, and for n admissible value of 6. The values of ty are reversed when we write for 6. As expressed, this value is negative from 0 to IT and positi TT to 2?r. The minimum, occurs when 6 = 109° 28'. Every stream-lin enters the circle (r= 1) on the left of this radius leaves it on the righ
The velocities, represented by d^r/dr and r~ld-ty/d0, are infinite origin.
For the D-solution we may take
= r sna
Here ^ retains its value unaltered when 2?r — Q is substituted for 0. r is given, i/r increases continuously from 6 — 0 to 6 = nr. On the lin the motion is entirely transverse to it. This is an interesting exampl flow of viscous fluid round a sharp corner. In the application to ai plate -fy represents the displacement at any point of the plate, suppose clamped along 0 = 0, and otherwise free from force within the regi sidered. The following table exhibits corresponding values of r and as to make -»^= 1 in (15) :
i
e r 0 r
180° 1-00 60° 64-0
150° T23 20° 104x3'65
120° 2-37 10° 10° x 2-28
90° 8-00 0° oo
When n = 2, (12) appears to have no significance.
When n = 3, the dependence on 0 is the same as when n = 1. Tl: and (15) may be generalized :
cos Q sin2
For example, we could satisfy either of the conditions ^ = 0, or the circle r=I.
For n=4s the JD-solution becomes nugatory; but for the G'-solu have
The wall (or in the elastic plate problem the clamping) along Q = 0 without effect.nts an irrotational motion.