Vortex Motion

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

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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 center of rotation is called a singular point, where velocity approaches to infinite. An example of free vortex motion is whirling mass of liquid in a washbasin, 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.

Forced Vortex
Fig. Forced Vortex

v = r × ω

h = ω²r²/2g

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

The volume of paraboloid = 1/2 π ×r² h

                                        = 1/2 of the 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 a rotating cylinder and flow inside the centrifugal pump.

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Variation of Pressure

Pressure Variation vortex motion
Fig. 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 FlowForced 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 = constant or v × r = constant (mass is constant)

ω = constant
v/r = constant

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