Rotation is defined the movement of a fluid element in such a way that both of its horizontal as well as vertical axes rotate in the same direction.

Flow is said to be rotation, when every fluid element rotates at its axis which is perpendicular to the plane of motion.

Consider a rectangular fluid element ABCD at a certain time in a two-dimensional x-y plane.


The velocity components x-direction at A and D are,

u and  [ u + ∂u / ∂y . dy ] respectively.

The velocity component in y-direction at B and D are.

v and [ v + ∂v / ∂x . dx ] respectively.

Since these velocities are different, there will be angular velocity developed for linear element AD and AB respectively.

In time interval ‘dt’ the elements AB and AD would move relative to point A, the fluid element takes position AB’C’D’.

Consider anti-clockwise rotation as positive, the angular velocity of element AB at

z-axis is,

We know,  L = r θ and ω = dθ / dt

ω_AB  = Angular displacement of AB / time

= [ ( v + ∂v / ∂x . dx ) - v ] dt / dx = v dt / dx + ( ∂v / ∂x .dx ) dt / dx / dt

ω_AB  =  ∂v / ∂x

Perpendicularly angular displacement at AD about Z-axis,

ω_AD =  ∂u / ∂y

The average of angular velocities of two mutually perpendicular elements gives the rotation about Z-axis.

ω_AD  =  1/2 [ ∂u / ∂X - ∂u / ∂y ]

For these dimensional fluid, there-rotational components is given by,

ω_x  = 1/2 [ ∂w / ∂y - ∂v / ∂z ]

ω_y  = 1/2 [ ∂u / ∂z - ∂w / ∂x ]

ω_z  = 1/2 [ ωx / ∂x - ∂u / ∂y ]

Rotational vector,   ωx  =  ω_x ⁱ + ω_y ^j + ω_z k = 1/2 Curl v

Condition for rotational flow,

ω_x  =  0              ω_y  = 0       ω_z  =  0

Substituting u = - ∂Φ / ∂x and v = ∂Φ / ∂y in equations

Rotational components,

ω_z  =  1/2 [ ∂v / ∂x - ∂u / ∂y ] = 1/2 [ ∂ / ∂x (-∂Φ / ∂y) - ∂ / ∂y (-∂Φ / ∂x) ]

=  1/2 [ -∂²Φ / ∂x .∂y + ∂²Φ / ∂y. ∂x

Perpendicularly,   ω_z = 1/2 [ -∂²Φ / ∂y .∂z + ∂²Φ / ∂z. ∂y ]

ω_z = 1/2 [ -∂²Φ / ∂z .∂x + ∂²Φ / ∂x. ∂z ]

If Φ is continuous function,

 [ -∂²Φ / ∂y .∂z + ∂²Φ / ∂z. ∂y ] ,    [ -∂²Φ / ∂z .∂x + ∂²Φ / ∂x . ∂z ] ,   [ -∂²Φ / ∂x .∂y + ∂²Φ / ∂y. ∂x ]

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