In case of free vortex flow


We know,

dp  = ρv² / r dr - ρg dr            put v = C/r

dp  = ρ x C² / r³ dr-ρ g dz

Consider two points 1 and 2 in fluid having radius r₁ and r₂ respectively from the central axis and their height being z₁ and z₂ from bottom of the vessel.

Integrating Equation for point 1 and 2.

2         2                       2
∫  dp = ∫ ρ C² / r³ dr - ∫ ρ g dz
1         1                       1
                           2                   2
p₂ - p₁ = ρ C²  ∫ r^-3 dr - ρ g ∫  dz
                           1                   1

= ρ C² ( r^-3+1 / -3+1)₁² - ρ g (Z)₁²  = ρ C² (-1 / 2r² )₁² - ρ g (z₂ - z₁)

=  ρC² / 2 [ 1/2 r₂ 1/2 r₂ ] - ρ g (z₂ - z₁) = - ρ / 2 (v₂² - v₁²) - ρ g (z₂ - z₁)

where      v₂  = c₂ / r₂ and v₁ = c₁ / r₁

P₂ - P₁  = ρ/2 (v₁² - v₂²) - ρ g ( z₂ - z₁)    

dividing both side by ρg

P₂ - P₁ / ρg = v₁² - v₂² / 2g - ( z₂ + z₁)

P₂ / γ - P₁ / γ  = V₁² / 2g - v₂² / 2g - z₂ + z₁

P₁ / γ  + v₁² / 2g + z₁  =  P₂ / γ + v₂² / 2g + z₂

Bernoulli’s equation is applicable for free vortex flow.

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