# VORTICITY AND ROTATION

**VORTICITY AND ROTATION**

## Vorticity use continuity equation to

## tell the existence of flow. Rotation

## helps to tell the flow is rotational

## or irrotational. Rotation is half the

## vorticity.

**Vorticity and Rotation**

#### Before vorticity is discussed, it is important to establish whether the flow exist or not.

#### For flow to exist, equation of continuity must be satisfied.

#### For an in-compressible fluid

#### Firstly ∂u/∂x = 0 for 1- dimensional flow

#### Secondly ∂u/∂x + ∂v/∂y = 0 for a 2- dimensional flow

#### Thirdly ∂u/∂x + ∂v/∂y + ∂w/∂z = 0 for a 3- dimensional flow

#### And for a compressible fluid, condition for flow to exist is

#### ∂(ρu)/∂x + ∂(ρv)/∂y + ∂(ρw)/∂z = 0 for a 3- dimensional flow

#### Vorticity and rotation are applicable to rotational and ir-rotational flows in fluids.

The symbol for vorticity and rotation are Ω and ω respectively.

Rotation is due to shear stress or due to viscous nature of the fluid. Thus, vorticity and rotation are due to viscosity of a fluid. Rotation of the fluid can be about x-axis, y-axis and z-axis respectively.

**Vorticity is difference between two consecutive angular shifts** as given below:

#### Firstly Vorticity about z axis, Ω_{z} = ∂v/∂x –∂u/∂y

Secondly Vorticity about x axis, Ω_{x} = ∂ω/∂y –∂v/∂z

Thirdly Vorticity about y axis, Ω_{y} = ∂u/∂z –∂ω/∂x

#### ∂v/∂x is the angular shift

#### ∂u/∂y is the angular shift

∂ω/∂y is the angular shift

#### ∂v/∂z is the angular shift

∂u/∂z is the angular shift

#### ∂ω/∂x is the angular shift

#### The vorticity of a 2-dimensional flow is always perpendicular to the plane of the flow. Therefore, it is considered a scalar quantity.

**Rotation (angular velocity) is one half the vorticity** and is represented by ω

#### Therefore

#### Firstly rotation about z axis, ω_{z} = 1/2(∂v/∂x –∂u/∂y)

Secondly rotation about x axis, ω_{x} = 1/2(∂ω/∂y –∂v/∂z)

Thirdly rotation about y axis, ω_{y} = 1/2(∂u/∂z –∂ω/∂x)

**Ir-rotational Flow**

#### A flow is ir-rotational when there is no rotation of the fluid elements. Hence mathematically,

#### the flow is ir-rotational when vorticity and rotation are each zero.

It will be true for a non-viscous fluid. If the flow happens to be ir-rotational, analysis becomes simple.

#### Mathematically for an ir-rotational flow

#### (i) ∂v/∂x = ∂u/∂y

#### (ii) ∂ω/∂y = ∂v/∂z

#### (iii) ∂u/∂z = ∂ω/∂x

**Rotational Flow**

#### It will be there for a viscous fluid. There will be a value of Vorticity as well as that of rotation.

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