The stream function, y of the liquid motion set up by the passage of a solid of revolution, moving with axial velocity U, is such that y Glib = - 15 42, iI ' + Uy 2 =cons t ant, (12) over the surface of the solid; and 4, must be replaced by41' =1l.-1-1-Uy2 in the general equations of steady motion above to obtain the steady relative motion of the liquid past the solid.
With a velocity function 49, the flow -f d 4 = 4)142, (2) (9) (to) (6) (22) Z Uy (I -a4,ic /r4), so that the flow is independent of the curve for all curves mutually reconcilable; and the circulation round a closed curve is zero, if the curve can be reduced to a point without leaving a region for which 4 is single valued.
The corresponding expression for two orthogonal cylinders will be With a 2 = co, these reduce to / y /, = Uy (I ra 2 p22 +-C24)..
(9) a 5 ` = 2 (I -) y a x, or Uy (1- - 2 Y a 4 4) a, (io) for a sphere or cylinder, and a diametral plane.
If u be the acceleration at unit distance, the component accelerations parallel to axes of x and y through 0 as origin will be ux, uy, whence ~ = ~sy.