VORTICITY AND ROTATION

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https:// www.mesubjects.net/wp-admin/post.php?post=10636&action=edit      FM introduction-2

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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, Ωz = ∂v/∂x –∂u/∂y
Secondly Vorticity about x axis, Ωx = ∂ω/∂y –∂v/∂z
Thirdly Vorticity about y axis, Ωy = ∂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, ωz = 1/2(∂v/∂x –∂u/∂y)
Secondly rotation about x axis, ωx = 1/2(∂ω/∂y –∂v/∂z)
Thirdly rotation about y axis, ωy = 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