|Section 9: Vorticity and circulation|
|9.1||Vorticity and its physical significance. Vortex lines and vortex tubes. The solenoidal nature of vortex lines. Examples.|
|9.2||Circulation G: its definition and relationship to vorticity. Vorticity as a "source" of circulation (Biot-Savart's law).|
theorem on circulation in barotropic flows in irrotational force fields:
For inviscid flows:
Examples: steady sink vortex, tornado or cyclone, bathtub vortex.
Three vortex theorems for inviscid, barotropic flow in an irrotational force field (corollaries of Kelvin's theorem):
(a) Vortex lines move with the fluid.
Examples: behavior of vortex rings; instability of shear layer or vortex sheet; secondary flow induced in bends.
(b) Once irrotational( ), a fluid particle will remain so forever.
Consequence: Potential flows.
Example: solution of steady sink vortex based on constraint that .
(c) For vortex "tube",
Example: accelerating inviscid flow with transverse velocity gradient.
|9.5||Vorticity transport equation in differential form. The effect on vorticity of vortex line stretching and turning; the role of kinematic viscosity as the diffusivity of vorticity.|
Fay, pp 271-276
|Problem Set Section 9 (from Shapiro and Sonin)|