Hüdro- ja aeromehaanika
flow can be write in terms of the streamfunction .
Viscosity and the Navier-Stokes equations a two-dimensional vorticity field
= ( z , r , o )) = ( (r ),0,0) . The idea is to prevent the vorticity from diffusing by placing it in
a steady irrotational flow field of the form (U z , U r , U o ) = (z,-r / 2,0) . Thus the full velocity
field has the form (U z ,U r , U o ) = (z ,-r / 2,U 0 (r , t ))
Now the z-component of the vortiity equation is, with
r 1 1 ru
- - = r , =
t 2 r r r r r r
First note that if = 0, so that there is no diffusion of , we my solve the equation to obtain
= e t F ( r 2 e t ) ,
( )
where F r is the initial value of . This solution exhibits the exponential growth of vorticity
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