Free Vortex — Speed–Radius Relationship For a free vortex (no external torque), the tangential velocity of a water particle varies with distance r from the centre as:

Introduction:
Vortical motions are classified as free (irrotational, no forced swirl) or forced (solid-body rotation). The velocity–radius law distinguishes them and determines pressure distribution and surface shape.

Given Data / Assumptions:

• Free vortex in incompressible, inviscid approximation.
• No external torque; angular momentum of each fluid particle is conserved.
• Steady, axisymmetric flow.

Concept / Approach:
For a free vortex, tangential momentum mv_thetar is constant along a streamline (neglecting viscosity). Therefore v_theta * r = constant ⇒ v_theta ∝ 1/r. The flow is irrotational except at the core.

Step-by-Step Solution:
Use angular momentum conservation: v_theta*r = K (constant).Rearrange to get v_theta = K/r.Hence as r increases, v_theta decreases proportionally to 1/r.

Verification / Alternative check:
Measured velocities in drain whirlpools and tornado outer regions follow approximately v ∝ 1/r, unlike forced vortices where v ∝ r.

Why Other Options Are Wrong:
v ∝ r: describes forced vortex (solid-body) rotation.
v ∝ r^2 or v ∝ 1/r^2: do not conserve angular momentum for free vortices.

Common Pitfalls:
Confusing free with forced vortex; assuming solid-body rotation where none exists; ignoring viscous core where the ideal relation breaks down.

Final Answer:
inversely proportional to r