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