## Rotation

**Rotation** Assignment Help | **Rotation** Homework Help

# Rotation

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.

## Rotational flow:

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

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

ω

Perpendicularly angular displacement at AD about Z-axis,

ω

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

ω

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

ω

ω

ω

Rotational vector, ωx = ω

Condition for rotational flow,

ω

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

Rotational components,

ω

= 1/2 [ -∂

Perpendicularly, ω

ω

If Φ is continuous function,

[ -∂

For more help in

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 / ∂xPerpendicularly angular displacement at AD about Z-axis,

ω

_{AD}= ∂u / ∂yThe 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 vCondition for rotational flow,

ω

_{x}= 0 ω_{y}= 0 ω_{z }= 0Substituting 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 [ -∂

^{2}Φ / ∂x .∂y + ∂^{2}Φ / ∂y. ∂xPerpendicularly, ω

_{z}= 1/2 [ -∂^{2}Φ / ∂y .∂z + ∂^{2}Φ / ∂z. ∂y ]ω

_{z}= 1/2 [ -∂^{2}Φ / ∂z .∂x + ∂^{2}Φ / ∂x. ∂z ]If Φ is continuous function,

[ -∂

^{2}Φ / ∂y .∂z + ∂^{2}Φ / ∂z. ∂y ] , [ -∂^{2}Φ / ∂z .∂x + ∂^{2}Φ / ∂x . ∂z ] , [ -∂^{2}Φ / ∂x .∂y + ∂^{2}Φ / ∂y. ∂x ]For more help in

**Rotation**click the button below to submit your homework assignment