# Bernoulli equation derivation with examples and applications

## Bernoulli equation definition

Bernoulli’s equation is defined as:”The sum of pressure,the kinetic and potential energies per unit volume in a steady flow of an in compressible and non viscous fluid remains constant at every point of its path.”Mathematically it is expressed as:

## Bernoulli’s principle formula

**P+ ρgh +1/2 ρv ^{2} =constant**

Bernoulli’s equation,which is a fundamental relation in fluid mechanics,is not a new principle but is derivable from the basic laws of Newtonian mechanics.We find it convenient to derive it from the work energy theorem,for it is essentially a statement of the work energy theorem for fluid flow.

## Bernoulli equation derivation

Consider the steady,in compressible ,non viscous,and irrptational flow of a fluid through the pipe line or tube of the flow as shown in figure.The portion of pipe shown in figure has a uniform cross section **A _{1}**,at the left.It is horizontal there at an elevation

**y**above some reference level.It gradually widens and rises and at the right has a uniform cross section

_{1}**A**.It is horizontal there at an elevation

_{2}**y**. At all points in the narrow apart of the pipe the pressure is

_{2}**p**and the speed

_{1}**v**;at all points in the wide portion the pressure is

_{1}**p**and the speed is

_{2}**v**.

_{2}The work energy theorem states:The work done by the resultant force acting on a system is equal to the change in kinetic energy of the system.In figure the forces that do work on the system,assuming that we can neglect viscous forces ,are the pressure forces **p _{1}A_{1 }**and

**p**that act on the left and right hand ends of the system,respectively,and the force of gravity.As fluid flows through the pipe the net effect.We can find the work

_{2}A_{2 }**W**done on the system by the resultant force as follows:

- The work done on the system by the pressure force
**p**and_{1}A_{1 }**p**._{1}A_{1}Δl_{1} - The work done on the system by the pressure force
**p**is_{2}A_{2 }**p**_{2}A_{2}Δl_{2 }. Note that it is negative ,because the force acts in a direction opposite to the horizontal displacement. - The work done on the system by the gravity is associated with lifting the fluid element from height
**y**to height_{1 }**y**and is_{2}**-Δmg(y**in which_{2}– y_{1})**Δm**is the mass of fluid in either area.This contribution is also negative because the gravitational force acts in a direction opposite to the vertical displacement.

The net work W done on the system by all the forces is found by adding these three terms,or

**W = p _{1}A_{1}Δl_{1 }– p_{2}A_{2}Δl_{2} -Δmg(y_{2} – y_{1})**

Now **A _{1}Δl_{1}** and

**A**is the volume of the fluid element which we can write as

_{2}Δl_{2}**Δm**

_{/ρ}_{ ,}in which

**ρ**

_{ }is the constant fluid density.Recall that the two fluid elements have the same mass,so that in setting:

**A _{1}Δl_{1} = A_{2}Δl_{2 }**

we have assumed the fluid to be incompressible .With this assumption we have

**W=(p _{1} –p_{2})(Δm_{/ρ }– Δmg(y_{2}-y_{2}) ………….(1)**

The change in kinetic energy of the fluid element is:

From the work energy theorem,W=Δk,we then have:

Which after canceling the common factor of m,we can rearranged to read:

Since the subscript 1 and 2 refer to any two locations along the pipe line,we can drop the subscript and write:

Equation (4) is called the bernoulli’s equation for steady,incompressible,non viscous,and irrotational flow.It was first presented by Daniel Bernoulli in his Hydrodynamica in 1738.

## Applications of Bernoulli’s principle

- venturi tube
- Baseball Curve
- Sailboats
- Torricelli’s theorem
- Blood Flow

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