d[phi]/d[nu] = d[psi]/ds, d[phi]/ds = -d[psi]/d[nu], (13)
_ _
/ /
T = ½[rho] | [phi] d[psi] = -½[rho] | [psi] d[phi]. (14)
_/ _/
With the Stokes' function [psi] for motion symmetrical about an
axis.
_ _
/ d[psi] /
T = ½[rho] | [phi] ------ 2[pi]y ds = [pi][rho] | [phi] d[psi]. (15)
_/ yds _/
37. _Flow, Circulation, and Vortex Motion._--The line integral of the
tangential velocity along a curve from one point to another, defined
by
_ _
/ / dx dy dz \ /
| ( u-- + v-- + w-- ) ds = | (u dx + v dy + z dz), (1)
_/ \ ds ds ds / _/
is called the "flux" along the curve from the first to the second
point; and if the curve closes in on itself the line integral round
the curve is called the "circulation" in the curve.
With a velocity function [phi], the flow
_
/
- | d[phi] = [phi]1 - [phi]2, (2)
_/
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
[phi] is single valued.
If through every point of a small closed curve the vortex lines are
drawn, a tube is obtained, and the fluid contained is called a _vortex
filament_.
By analogy with the spin of a rigid body, the component spin of the
fluid in any plane at a point is defined as the circulation round a
small area in the plane enclosing the point, divided by twice the
area. For in a rigid body, rotating about Oz with angular velocity
[zeta], the circulation round a curve in the plane xy is
_
/ / dy dx \
| [zeta] ( x -- - y -- ) ds = [zeta] times twice the area. (3)
_/ \ ds ds /