

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 (BiotSavart'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 271276


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