34. Motion symmetrical about an Axis.—When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function ψ can be found analogous to that employed in plane motion, such that the flux across the surface generated by the revolution of any curve AP from A to P is the same, and represented by 2π (ψ − ψ0); and, as before, if dψ is the increase in ψ due to a displacement of P to P′, then k the component of velocity normal to the surface swept out by PP′ is such that 2πdψ = 2πyk·PP′; and taking PP′ parallel to Oy and Ox,
u = −dψ/ydy, v = dψ/ydx,
(1)
and ψ is called after the inventor, “Stokes’s stream or current function,” as it is constant along a stream line (Trans. Camb. Phil. Soc., 1842; “Stokes’s Current Function,” R. A. Sampson, Phil. Trans., 1892); and dψ/yds is the component velocity across ds in a direction turned through a right angle forward.
In this symmetrical motion
| ξ = 0, η = 0, 2ζ = | d | ( | 1 | dψ | ) + | d | ( | 1 | dψ | ) | ||
| dx | y | dx | dy | y | dy |
| = | 1 | ( | d2ψ | + | d2ψ | − | 1 | dψ | ) = − | 1 | ∇2ψ, | |
| y | dx2 | dy2 | y | dy | y |
(2)
suppose; and in steady motion,
| dH | + | 1 | dψ | ∇2ψ = 0, | dH | + | 1 | dψ | ∇2ψ = 0, | ||
| dx | y2 | dx | dy | y2 | dy |