# Fluid Mechanics Applications/B-34: Sudden contraction

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## Introduction[edit]

The sudden contraction in the area/diameter of a ﬂuid jet after it emerges from a circular aperture in a pressurized reservoir is called as vena contracta. Coefficient of contraction is the ratio of the cross sectional area of the jet at the vena contracta to the area of the orifice. The typical value may be taken as .64

## Checking the vena contracta[edit]

**Assumption**
P_{1}A_{1}v_{1} be the Pressure , Area and velocity of fluid in the tank,

P_{2}A_{2}v_{2} be the Pressure , Area and velocity of fluid in the orifice,

P_{3}A_{3}v_{3} be the Pressure , Area and velocity of fluid at the vena contracta,

Vena contractaplays a very important role in the minor losses in pipes.

The diameter of the vena contracta is nearly equal to .64 times the diameter of the original hole . Assume fluid is incompressible

using continuity equation

v_{1}A_{1}=v_{2}A_{2}(1) and v_{1}<< v_{2}(2)

Ignoring energy losses due to viscosity,Bernoullis' holds for points along streamlines

P_{1} +1/2 d v_{12} = P_{2} +1/2dv_{22} (3)

where d is the(constant) density of fluid and P is the pressure using equation (2) and (3)

v_{2}^{2}=2( P_{1}- P_{2})/d (4)

Considering momentum in the system,
the mass flux =dvA
so, momentum flux = dv^{2}A
The net flux bounded by area A_{1} and A_{2}

dp/dt = d(v_{2}^{2}A_{2}-v_{1}^{2}A_{1})=dv_{2}^{2}A_{2}(5)

F ≈ P_{1}A_{1} - [P_{1}(A_{1} - A_{2})] = (P_{1} - P_{2})A_{2} (6)

equating to equation (5)

v_{2}^{2}= (P_{1}-P_{2})/d (7)

This is the contradiction with equation (4) based on conservation of energy.
Then according to Torricelli
this contradiction is resolved in nature by a contraction of fluid to area A_{3} after it passes through A_{2}
The momentum flux is actually

dp/dt = d(v_{3}^{2}A_{3}-v_{1}^{2}A_{1}) ≈ dv_{3}^{2}A_{3}≈ 2P_{1}A_{3}(8)

According to Bernoulli's equation in the limit that P_{3} << P_{2} The force that causes this change is now

F ≈ P_{1}A_{1}- [P_{1}(A_{1}- A_{2}) + P_{3}A_{3}] = (P_{1}A_{2}- P_{3}A_{3}) ≈ P_{1}A_{2}(9)

we estimate vena contracta to be A_{3}= A_{2}/2

** ^{[2]}**
http://www.physics.princeton.edu/~mcdonald/examples/vena_contracta.pdf

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