Questions: Rotating Flow, Vortex Motion, and Circulation

Question 1

A fluid parcel near a bathtub drain is traveling in a circular path around the drain. A student argues: 'Any particle moving in a circle must be rotating — it has angular velocity around the center, so the vorticity must be nonzero.' What is wrong with this reasoning?

• A. Fluid parcels near a drain do not actually move in circles — they spiral inward radially and never complete a full orbit
• B. Vorticity measures the spin of a fluid parcel about its own center, not the curvature of its path around a fixed point — in a free vortex, fluid parcels orbit without spinning about themselves
• C. The student is correct; free vortex flow has high vorticity near the center where V_θ is largest
• D. Vorticity is only defined for forced vortices; free vortex flow is described by circulation rather than vorticity

Answer: (B)

Vorticity ω = ∇ × V measures local spin — rotation of a fluid parcel about its own center of mass. In a free vortex (V_θ = Γ/2πr), the curl of the velocity field is zero everywhere except the singular center: ∇ × V = 0. Fluid parcels travel in circles (orbital motion) but do not spin about themselves. The analogy: a car driving in a circle at a roundabout is not necessarily rotating (spinning) about its own center — it is translating along a curved path. A floating cork in a bathtub vortex orbits the drain but maintains the same orientation, never rotating about its own axis. This irrotational property of the free vortex is why it fits within potential flow theory.

Question 2

In a free vortex (V_θ = Γ/2πr), pressure increases toward the center. In a forced vortex (V_θ = ωr, solid-body rotation), pressure also increases toward the outside. What accounts for the different pressure distribution in each case?

• A. Forced vortices require continuous energy input, which manifests as elevated pressure; free vortices are passive and maintain lower pressure throughout
• B. Both vortex types have identical pressure distributions; they differ only in their tangential velocity profiles
• C. In both cases ∂P/∂r = ρV_θ²/r > 0, so pressure increases outward; but in the free vortex the steep rise of V_θ toward the center creates a very low-pressure core, while in the forced vortex the pressure increases monotonically from center outward with no such extreme minimum
• D. The pressure difference is an artifact of cylindrical coordinates; in Cartesian coordinates both vortices have identical pressure fields

Answer: (C)

The radial momentum equation ∂P/∂r = ρV_θ²/r applies to both — pressure always increases outward (centripetal acceleration requires inward pressure gradient). In a forced vortex (V_θ = ωr), the pressure rises smoothly from the center outward: P ∝ r². In a free vortex (V_θ = Γ/2πr), V_θ increases steeply as r → 0, and integrating the pressure gradient inward gives a dramatic pressure drop toward the singular center — the low-pressure vortex core of a tornado or drain. The centrifuge exploits the forced vortex pressure rise (denser material goes outward); the tornado's destructive suction exploits the free vortex pressure drop (the core is far below atmospheric).

Question 3

According to Kelvin's circulation theorem, when a wing develops bound circulation to generate lift, an equal and opposite starting vortex must be shed into the wake to conserve total circulation.

• T. True
• F. False

Answer: True

Kelvin's theorem: for an inviscid fluid, the circulation around any material loop is conserved over time. Before the wing starts moving, total circulation is zero everywhere. When the wing develops bound circulation Γ_wing (which generates lift via the Kutta-Joukowski theorem), an equal and opposite starting vortex Γ_start = −Γ_wing must be shed — so that total circulation remains zero. This starting vortex is deposited in the wake and can briefly be observed during takeoff. The bound circulation on the wing and the trailing vortices shed at the wingtips are the direct consequence of Kelvin's conservation law. Jet aircraft trailing vortices — hazardous to following planes for minutes after passage — are the real-world manifestation of this theorem.

Question 4

In a free vortex, fluid farther from the center moves faster than fluid close to the center, because the larger radius gives fluid parcels more rotational momentum.

• T. True
• F. False

Answer: False

In a free vortex, V_θ = Γ/(2πr) — velocity INCREASES as radius DECREASES. Fluid closest to the center moves fastest. Angular momentum per unit mass (r × V_θ = Γ/2π) is conserved, so as a parcel moves inward (r decreases), its tangential speed must increase proportionally — just as a figure skater spins faster when pulling in her arms. The rapid increase in velocity near the center creates the low-pressure core characteristic of drains and tornadoes. This is opposite to a forced vortex (solid-body rotation, V_θ = ωr), where fluid farther from the center does move faster. Distinguishing these two cases is fundamental to understanding rotating flow.

Question 5

Explain why the free vortex flow field is described as 'irrotational' even though fluid parcels travel in circular paths. What physical picture clarifies the distinction between the parcel's orbital motion and its local rotation?

Model answer: Irrotational means each fluid parcel does not spin about its own center of mass — its vorticity (∇ × V) is zero. In a free vortex, parcels orbit the center but each parcel maintains a fixed orientation, never rotating about itself. Mathematically, computing ∇ × V for V_θ = Γ/(2πr) gives zero everywhere outside the singular center. The physical picture: imagine a small cork floating in a bathtub drain vortex. The cork orbits the drain but always faces the same wall — it translates along a curved path without spinning. In a forced vortex (solid-body rotation, V_θ = ωr), the cork would spin like a pinwheel, completing one self-rotation per orbit. Vorticity measures only that local spin, not the curvature of the path. The free vortex's irrotational character places it within potential flow theory, making it analytically tractable and directly applicable to aerodynamic lift calculations.

The distinction between orbital motion and self-rotation is the central conceptual trap of vortex mechanics. Fluid dynamics students who conflate 'circular path' with 'rotation' will incorrectly classify free vortex flow as rotational. The cork thought experiment gives a concrete, physical way to test: does the parcel spin about itself? If not, vorticity is zero regardless of path curvature.