---
title: "Circulation, vorticity and rotational flow"
subject: "Fluid Mechanics"
chapter: 3
key: "semester_4/fluid_mechanics_mechanical:circulation-vorticity-rotational-flow"
source: https://www.bctengineer.com/q/semester_4/fluid_mechanics_mechanical/circulation-vorticity-rotational-flow
site: BCT Engineering
---

# Circulation, vorticity and rotational flow

Fluid Mechanics · Chapter 3: Kinematics of Fluid Flow

- Asked in: year not given, 6 marks
- Video: [Introductory Fluid Mechanics L13 p8 - Vorticity and Circulation](https://www.youtube.com/watch?v=VTQaU6l-VPQ) — Ron Hugo
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/fluid_mechanics_mechanical/circulation-vorticity-rotational-flow
- Whole chapter: https://www.bctengineer.com/md/semester_4/fluid_mechanics_mechanical/chapter-3

## Question

Define circulation and vorticity. Differentiate between rotational and irrotational flow. For a two-dimensional flow, show that the circulation around an elementary rectangle equals the vorticity multiplied by its area, and state the condition of irrotationality in terms of velocity components.

## Short answer

Circulation Γ is the line integral of tangential velocity around a closed curve; vorticity ζ = ∇×V is twice the fluid angular velocity. In rotational flow ζ ≠ 0; in irrotational flow ζ =0, so ∂v/∂x = ∂u/∂y in 2D. For an elementary rectangle, dΓ = (∂v/∂x – ∂u/∂y)dxdy = ζ_z dA.

## Answer

#### Definitions
**Circulation** $\Gamma$ is the line integral of the tangential velocity component around a closed curve:

$$
\Gamma = \oint \vec V \cdot d\vec s
$$

**Vorticity** $\vec\zeta$ is twice the angular velocity of a fluid particle, $\vec\zeta = 2\vec\omega = \nabla \times \vec V$. For 2D flow in the $xy$-plane:

$$
\zeta_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}
$$

#### Circulation round a rectangle
Take an element $dx \times dy$ with velocity components $u$, $v$ at the lower left corner, and go anticlockwise.

- Bottom side (to the right): $u\,dx$
- Right side (up): $\left(v + \dfrac{\partial v}{\partial x}dx\right)dy$
- Top side (to the left): $-\left(u + \dfrac{\partial u}{\partial y}dy\right)dx$
- Left side (down): $-v\,dy$

Adding:

$$
d\Gamma = \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right) dx\,dy = \zeta_z\, dA
$$

So circulation per unit area equals vorticity. This is the basis of Stokes' theorem, $\Gamma = \iint \zeta\, dA$.

#### Rotational and irrotational flow
| Basis | Rotational flow | Irrotational flow |
|---|---|---|
| Particle rotation | Particles rotate about their own axes | No rotation of particles |
| Vorticity | $\nabla\times\vec V \neq 0$ | $\nabla\times\vec V = 0$ |
| Condition in 2D | $\partial v/\partial x \neq \partial u/\partial y$ | $\partial v/\partial x = \partial u/\partial y$ |
| Velocity potential | Does not exist | Exists |
| Example | Forced vortex, flow near a wall (boundary layer) | Free vortex (except at centre), ideal-fluid flow |

For irrotational flow a velocity potential $\phi$ exists, with $u = \partial\phi/\partial x$ and $v = \partial\phi/\partial y$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Vorticity is defined as:
   A. the line integral of velocity around a closed curve
   B. twice the angular velocity of a fluid particle, curl V
   C. the divergence of velocity
   D. half the angular velocity of a fluid particle
   Answer: B. Vorticity zeta = 2 omega = curl V; in 2D, zeta_z = dv/dx - du/dy. (Key point: Vorticity definition)

2. Summing the four sides of the elementary rectangle gives circulation dGamma equal to:
   A. (du/dx + dv/dy) dx dy
   B. (dv/dx - du/dy) dx dy = vorticity x area
   C. (du/dy - dv/dx) dx dy
   D. u dx + v dy
   Answer: B. The terms u dx and v dy cancel, leaving (dv/dx - du/dy) dx dy = zeta dA, the basis of Stokes' theorem. (Key point: Circulation around elementary rectangle)

3. A 2D flow is irrotational when:
   A. du/dx + dv/dy = 0
   B. dv/dx = du/dy
   C. u = v
   D. dv/dx = -du/dy
   Answer: B. Zero vorticity means dv/dx - du/dy = 0, and then a velocity potential exists. (Key point: Condition of irrotationality)

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