Type of vortex motion: when the whole fluid mass rotates without external torque (due to gravity/pressure or imparted rotation), the motion is

Introduction / Context:
Vortical motions are categorized by the presence or absence of external torque. Classifying a flow as a free or forced vortex guides the expected velocity distribution, pressure field, and free-surface shape in rotating fluids (e.g., draining sinks, cyclone separators, rotating tanks).

Given Data / Assumptions:

• No external torque input after initiation (no continuous stirring).
• Viscous dissipation is negligible over the short observation period.
• Axisymmetric swirl about a vertical axis.

Concept / Approach:

In a free vortex, angular momentum is conserved along streamlines, giving v_θ * r = constant and v_θ ∝ 1/r. The free surface assumes a depressed center (paraboloid under gravity), and pressure distribution satisfies radial equilibrium. In a forced vortex, continuous torque (e.g., a rotating container) imposes solid-body rotation, v_θ ∝ r, with a different free-surface shape and pressure field.

Step-by-Step Solution:

Verification / Alternative check:

Everyday example: water draining from a basin forms a free vortex after initial stirring ceases. Contrastingly, a laboratory mixer creates a forced vortex while power is applied.

Why Other Options Are Wrong:

(b) Requires continuous torque; (c) is vague; (d) describes forced, not free, motion; (e) is not vortical.

Common Pitfalls:

Confusing transient spin-up (forced) with sustained swirl after torque removal (free); assuming v_θ is uniform across radius.

Final Answer:

Free vortex