Ideal Fluids in Motion — Question 5

Question 5

Circular streamlines do not by themselves imply local fluid rotation

Two planar flows have azimuthal velocity $v_\theta$ and zero radial velocity: A has $v_\theta=\Omega r$; B has $v_\theta=K/r$ for $r>0$. For this geometry the local vorticity is supplied as $\omega_z=(1/r)d(rv_\theta)/dr$. Half the vorticity is the local rigid-rotation rate of a small fluid element.

1. Compute the vorticity of each flow and compare with the angular speed $v_\theta/r$ of a particle about the origin.

2. Can B have circular paths and still be locally irrotational? State why the origin must be excluded from its idealized domain.

Question 5 - Solution

Circular streamlines do not by themselves imply local fluid rotation

(a) Derivation and calculation

Using the supplied vorticity formula,$$A:\quad\omega_z=\frac1r\frac{d(\Omega r^2)}{dr}=\boxed{2\Omega},\qquad v_\theta/r=\Omega.$$$$B:\quad\omega_z=\frac1r\frac{dK}{dr}=\boxed0,\qquad v_\theta/r=K/r^2.$$

(b) Result and interpretation

$$\text{local spin rate}=\omega_z/2,\quad\text{orbital rate}=v_\theta/r.$$B has circular paths while remaining locally irrotational for $r>0$.$$r\to0^+:\quad v_\theta=K/r\to\infty.$$The origin must be excluded or replaced by a different core model.

Physical check

Local rotation of a small element is distinct from its motion around a distant center.