(1)
so that we can put
u = −dψ/dy, v = dψ/dx,
(2)
where ψ is a function of x, y, called the stream- or current-function; interpreted physically, ψ − ψ0, the difference of the value of ψ at a fixed point A and a variable point P is the flow, in ft.3/second, across any curved line AP from A to P, this being the same for all lines in accordance with the continuity.
Thus if dψ is the increase of ψ due to a displacement from P to P′, and k is the component of velocity normal to PP′, the flow across PP′ is dψ = k·PP′; and taking PP′ parallel to Ox, dψ = v dx; and similarly dψ= −u dy with PP′ parallel to Oy; and generally dψ/ds is the velocity across ds, in a direction turned through a right angle forward, against the clock.
In the equations of uniplanar motion
| 2ζ = | dv | − | du | = | d2ψ | + | d2ψ | = −∇2ψ, suppose, |
| dx | dy | dx2 | dy2 |
(3)