## Vortex Motion

Contents

- A
or*whirling*mass of fluid is called vortex flow.**rotating** - The flow of the fluid in a curved path.

### Free Vortex Flow

In this flow fluid mass rotates due to the conservation of angular momentum. The velocity profile is inversely proportional to the radius.

*v × r = constant*

The point at the centre of rotation is called singular point, where velocity approaches to infinite. An example of free vortex motion is whirling mass of liquid in a wash basin, whirlpool in the river etc.

**Points to remember:**

- No
torque or energy is required.*external* - In free vortex flow,
can be applied.*Bernoulli’s equation*

### Forced Vortex Flow

When a fluid is rotated about a vertical axis at a constant speed, such every particle of it has the same angular velocity, the motion is known as the forced vortex.

*v = r × ω*

*h = ω ^{2}r^{2}/2g*

Where ‘h’ is a height of paraboloid, and r is the radius of the cylinder.

Volume of paraboloid = *1/2 π × r ^{2 }h*

= 1/2 of volume of circumscribing cylinder

**Points to Remember**

- The surface profile of a forced vortex flow is parabolic.
- Forced vortex requires a constant supply of external energy or torque.

An example of forced vortex flow is rotating cylinder and flow inside the centrifugal pump.

**Variation of Pressure**

*dp = (ρv ^{2}/r) × dr – ρg dz*

*z is in the upward direction*

**Equation of Free Vortex Flow**

*We know for free vortex v × r = constant = k*

and *dp = (ρv ^{2}/r) × dr – ρg dz*

dp = [{(ρ × k^{2})/r×r^{2} }× dr] + ρg dz (putting *v* = k/r)

*On Integration for point 1 to point 2*

∫dp = ∫[{(ρ×k^{2})/r^{3} }× dr] – ρg ∫dz

P_{2} – P_{1} = ρ/2 (v_{1}^{2} – v_{2}^{2}) – ρg(z_{2} – z_{1}) (Again putting value of k )

P_{1} + ½ ρv_{1}^{2} + ρgz_{1} = P_{2} + ½ ρv_{2}^{2} + ρgz_{2} *(Bernoulli’s Equation is applicable in Free Vortex)*

**Equation of Forced Vortex Flow**

*We know for free vortex v/r = ω = constant*

dp = [(ρ×ω^{2}r^{2})/r] × dr + ρg dz

*On Integration for point 1 to point 2*

∫dp = ρ×ω^{2} ∫r × dr + ρg ∫dz

P_{2} – P_{1} = ρω^{2}/2 (r_{2}^{2} – r_{1}^{2}) – ρg(z_{2} – z_{1})

*On Putting v = ω × r*

P_{2} – P_{1} = ρ/2 (v_{2}^{2} – v_{1}^{2}) – ρg(z_{2} – z_{1})

P_{1} – ½ ρv_{1}^{2} + ρgh_{1} = P_{2} – ½ ρv_{2}^{2} + ρgz_{2} *(Bernoulli’s Equation Not Applicable)*

### Differences between Free and Forced Vortex Flow

Free Vortex Flow |
Forced Vortex Flow |

No external torque i.e. torque required to rotate is zero. | External torque is required to rotate the fluid mass. |

We know T = d (mvr)/dt and since T = 0,So, mvr = constantor v × r = constant (mass is constant) |
ω = constantv/r = constant |