Vortex Motion

  • A whirling  or rotating mass of fluid is called vortex flow.
  • 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 external torque or energy is required.
  • In free vortex flow, Bernoulli’s equation can be applied.

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 = ω2r2/2g

Vortex Motion

Fig. 1- Forced Vortex Motion

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

Volume of paraboloid = 1/2 π × r2 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

Vortex Motion

Fig. 2- Pressure Variation

dp = (ρv2/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 = (ρv2/r) × dr – ρg dz

dp = [{(ρ × k2)/r×r2 }× dr] + ρg dz (putting v = k/r)

On Integration for point 1 to point 2

∫dp = ∫[{(ρ×k2)/r3 }× dr] – ρg ∫dz

P2 – P1 = ρ/2 (v12 – v22) – ρg(z2 – z1) (Again putting value of k )

P1 + ½ ρv12 + ρgz1 = P2 + ½ ρv22 + ρgz2 (Bernoulli’s Equation is applicable in Free Vortex)

Equation of Forced Vortex Flow

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

dp = [(ρ×ω2r2)/r] × dr + ρg dz

On Integration for point 1 to point 2

∫dp = ρ×ω2 ∫r × dr + ρg ∫dz

P2 – P1 = ρω2/2 (r22 – r12) – ρg(z2 – z1)

On Putting v = ω × r

P2 – P1 = ρ/2 (v22 – v12) – ρg(z2 – z1)

P1 – ½ ρv12 + ρgh1 = P2 – ½ ρv22 + ρgz2 (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 = constant
v/r = constant