## Vortex Motions

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# Vortex Motions

• They are two types viz (i) forced vortex flow (ii), free vortex flow.

**Forced Vortex Flow:**

• When some force or torque is required to rotate a fluid mass is called as forced vortex flow.

• In this flow, fluid mass is rotate with a constant angular velocity ω .

ω = V/r = constant

• The velocity increases with increases in distance from the vortex centre.

E.g (1) Flow through central core of a mixer

(2)Flow through the runner of a turbine.

(3)Flow of liquid inside the impeller of a centrifugal pump.

(4)Vertical cylinder containing liquid and rotating about central axis with a constant angular velocity.

## Free Vortex Flow:

• When no external torque is required to rotate the fluid mass, such a flow is called as free vortex flow.• The fluid mass rotates either due to fluid pressure itself or the gravity or due to rotation previously imparted.

• It is also called as potential vortex or irrigational vortex.

• Example.

(1) A Whirlpool in a river.

(2) Flow around a circular bend in a pipe.

(3) Flow by fluid in centrifugal pump casing

(4) Flow of a liquid through a hole provided at the bottom of a container. (e.g. bath tube, wash basin etc.)

(5) The relation between velocity and radius is obtained by putting the value of external torque is equal to zero or the rate of change of angular momentum (moment of momentum) must be zero.

Consider a particle of mass ‘m’ at a radial distance r form the axis of rotation, having tangential velocity v. Then

Angular momentum = mass x velocity = m. v

Moment of momentum = momentum x r = m v r

Time rate of change of momentum = ∂ / ∂t (mvr)

But for free vortex

∂ / ∂t (mvr) = 0 Integrating

mvr = constant

v. r = Constant / m

v. r = Constant = C

Where C is a constant and is known as strength of vortex.

v = C/r v ∝ 1/r

i.e. tangential velocity is inversely proportional to distance r.

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