Equation of Force Vortex Flow

v = ωr

Equation of vortex motion is given by

dp = ρv² / r dr-ρg dz

Substituting value of v in equation

dp = ρω²r² / r dr- ρgdz

dp = ρ ω²r dr- ρgdz

Consider two points 1 and 2 in the fluid having force vortex flow as

∫ dp = ∫ ρ ω² r dr - ∫ ρ g dz

[p]₁² = ρω² [ r² / 2 ]² - ρ g [Z]₁²

P₂ - P₁ = Pω² ( r₂² - r₁² ) - ρ g (z₂ - z₁)

P₂ - P₁ = ρ / 2 (V₂² - V₁² ) - ρG  (z₂ - z₁)

V₁ = ω₁r₁ and v₂ = ω₂ r₂

When the points 1 and 2 lies on the free surface of the liquid, then p₁ = p₂ and the equation becomes,

0 = ρ / 2 (v₂² - v₁²) - pg (z₂ - z₁)

z₂ - z₁ = v₂² - v₁² / 2g

when the point 1 lies on the axis of rotation, then

v₁ = ωr₁ = ω x 0 = 0

z₂ - z₁ = v₂² / 2g

Let     z₂ - z₁ = Z, then

Z = v₂² / 2g = ω² r₂² / 2g

Thus z varies with square of r. hence the equation of parabola which means that the free surface of liquid is a parabola.

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