Understanding Free Vortex Motion

A free vortex is a type of fluid flow where the fluid particles move in concentric circles, and there is no external torque applied to the fluid mass. Examples include the whirl created when water drains from a sink or bathtub, or large-scale atmospheric vortices like hurricanes (though these are more complex). A key characteristic of a free vortex is that its circulation is constant everywhere in the flow field (except possibly at the central singularity). Circulation ($\Gamma$) is defined as the line integral of the velocity component along a closed contour divided by the area enclosed by the contour, or more generally, the integral of velocity along any closed loop.

Velocity Distribution in a Free Vortex

In a free vortex, the flow is considered irrotational everywhere except possibly at the very center. For an irrotational flow, the velocity components can often be related to a potential function. For a free vortex, the tangential velocity ($v_\theta$) is the dominant velocity component and it varies inversely with the radial distance ($r$) from the center of the vortex. The radial velocity ($v_r$) is typically zero in an ideal free vortex.

The relationship between the tangential velocity ($v_\theta$) and the radius ($r$) in a free vortex is given by the formula:

\( v_\theta = \frac{C}{r} \)

where \( C \) is a constant related to the strength of the vortex (or circulation). This constant \( C \) is equal to \(\frac{\Gamma}{2\pi}\), where \(\Gamma\) is the circulation. So, we can also write:

\( v_\theta = \frac{\Gamma}{2\pi r} \)

This equation clearly shows an inverse relationship between the tangential velocity ($v_\theta$) and the radius ($r$).

Analyzing the Options

Let's consider the relationship \( v_\theta = \frac{C}{r} \).

• If the radius ($r$) increases, the value of \(\frac{C}{r}\) decreases (assuming \(C\) is a positive constant for a given vortex).
• If the radius ($r$) decreases, the value of \(\frac{C}{r}\) increases.

This means that as you move further away from the center of the free vortex (increasing radius), the velocity of the fluid particles decreases. Conversely, as you move closer to the center (decreasing radius), the velocity increases significantly. At the theoretical center ($r = 0$), the velocity becomes infinite, which is why real-world free vortices always have a core region where viscous effects become important, and the free vortex model breaks down.

Now let's evaluate the given options based on this understanding:

• Option 1:

  decreases with radius

  - According to our analysis, as the radius increases, the velocity decreases. This matches the behavior described by the formula \( v_\theta \propto \frac{1}{r} \).
• Option 2:

  increases with radius

  - This is the opposite of what we observed. Velocity decreases as radius increases.
• Option 3:

  is constant

  - The velocity clearly depends on the radius ($r$), so it is not constant.
• Option 4:

  none of the above

  - Since Option 1 correctly describes the behavior, this option is incorrect.

Conclusion on Free Vortex Velocity

Based on the fundamental principles and equations governing an ideal free vortex flow, the velocity is inversely proportional to the radial distance from the center. Therefore, as the radius increases, the velocity decreases.

Revision Table: Key Free Vortex Concepts

Additional Information: Free vs. Forced Vortex

It's useful to compare a free vortex with a forced vortex to highlight their differences, particularly in velocity distribution.

• Free Vortex:
  • No external torque.
  • Circulation is constant (\(\Gamma = \text{constant}\)).
  • Velocity \( v_\theta \propto 1/r \).
  • Irrotational flow (mostly).
  • Angular velocity is not constant; varies with radius.
• Forced Vortex:
  • Created by an external torque (e.g., stirring).
  • Fluid rotates like a rigid body.
  • Velocity \( v_\theta \propto r \) (proportional to radius).
  • Rotational flow.
  • Angular velocity is constant (\(\omega = \text{constant}\)).

Understanding this distinction is crucial in fluid mechanics problems involving rotational flow patterns.