Rotational versus irrotational flow—relation to angular velocity A flow is termed rotational if the circulation per unit area (vorticity) equals which multiple of the local angular velocity vector?

Introduction / Context:
Rotationality in fluid flow is characterized by non-zero vorticity, which reflects local spin of fluid elements. Understanding the relationship between angular velocity and vorticity is fundamental for classifying flows and applying potential-flow theory appropriately.

Given Data / Assumptions:

• Continuum fluid with differentiable velocity field.
• Two- or three-dimensional flow.
• Classical definition of vorticity as the curl of velocity.

Concept / Approach:

The vorticity vector ω (bold) is defined as ω = ∇ × V. For a rigid-body rotation, the local angular velocity vector Ω relates to vorticity by ω = 2 Ω. Thus, a non-zero vorticity indicates rotational flow; irrotational flow has ω = 0 and therefore Ω = 0 for fluid elements.

Step-by-Step Solution:

Verification / Alternative check:

Consider a solid-body rotation V = Ω × r. Then ∇ × V = 2 Ω, confirming the factor of two between vorticity and angular velocity.

Why Other Options Are Wrong:

(a) and (c) assign incorrect proportionality; (d) is incorrect because a precise relation exists.

Common Pitfalls:

Confusing circulation Γ around a finite loop with local vorticity; assuming turbulent flow is always “rotational” in the mathematical sense—classification is local and based on ω.

Final Answer:

twice the angular velocity vector