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FLUID MECHANICS - CASE STUDY SOLUTION


Free Vortex

 

The velocity components of the free vortex are given by

     vr = 0   and    vθ = K/r   (assumed vz = 0)

The vorticity can be determined by the curl of the velocity as follows:

     

Since the vorticity is zero, Tom has learned that the given free vortex is indeed irrotational.

     

Equipotential Lines
 

The velocity potential associated with the free vortex can then be determined as follows:

     

Since Vr = ∂φ/∂r, the velocity potential is not a function of r, and f(r) reduces to an arbitrary constant. For simplicity, set f(r) = 0 (the value of an individual velocity potential is irrelevant) to yield

     φ = Kθ

Note that when a flow is referred to as irrotational, it simply implies that the individual fluid elements do not rotate. Although the fluid elements of the given free vortex follow a circular path, it is still irrotational flow.

     
   
 
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