2.25: Advanced Fluid Dynamics
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  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).
  9.3 Kelvin's theorem on circulation in barotropic flows in irrotational force fields:

                                   

For inviscid flows:

                                   

Examples: steady sink vortex, tornado or cyclone, bathtub vortex.
  9.4

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.
 
  Reading
 

Fay, pp 271-276
e.g Potter & Foss pp. 179-184, 250-262, 390-392

 
  Problem Set Section 9 (from Shapiro and Sonin)
  Problem 10.3
  Problem 10.4
  Problem 10.5
  Problem 10.8
  Problem 10.11