Question 4.9: Determination of Rotationality of a Line Sink A simple two-d...
Determination of Rotationality of a Line Sink
A simple two-dimensional velocity field called a line sink is often used to simulate fluid being sucked into a line along the z-axis. Suppose the volume flow rate per unit length along the z-axis, \dot{V}/L , is known, where \dot{V} is a negative quantity. In two dimensions in the r𝜃-plane,
Line sink: u_r = \frac{\dot{V} }{2\pi L}\frac{1}{r} and u_\theta = 0 (1)
Draw several streamlines of the flow and calculate the vorticity. Is this flow rotational or irrotational?
Streamlines of the given flow field are to be sketched and the rotationality of the flow is to be determined.
Analysis Since there is only radial flow and no tangential flow, we know immediately that all streamlines must be rays into the origin. Several streamlines are sketched in Fig. 4–53. The vorticity is calculated from Eq. 4–33:
\vec{\zeta } = \frac{1}{r} \left(\frac{∂(ru_\theta )}{∂r} – \frac{∂}{∂\theta } u_r\right)\vec{k} = \frac{1}{r}\left(0 – \frac{∂}{∂\theta }\left(\frac{\dot{V} }{2\pi L}\frac{1}{r} \right) \right)\vec{k} = 0 (2)
Since the vorticity vector is everywhere zero, this flow field is irrotational.
Discussion Many practical flow fields involving suction, such as flow into inlets and hoods, can be approximated quite accurately by assuming irrotational flow (Heinsohn and Cimbala, 2003).
