# Bernoulli’s Principle

Bernoulli’s principle formulated by Daniel Bernoulli states that as the speed of a moving fluid increases (liquid or gas), the pressure within the fluid decreases. Although Bernoulli deduced the law, it was Leonhard Euler who derived Bernoulli’s equation in its usual form in the year 1752.

Bernoulli’s principle can be derived from the principle of conservation of energy. This states that the total mechanical energy of the moving fluid comprising the gravitational potential energy of elevation, the energy associated with the fluid pressure and the kinetic energy of the fluid motion, remains constant.

## Bernoulli’s Principle Formula

\(p+\frac{1}{2}\rho V^{2}+\rho gh=constant\) |

### Notations Used In The Bernoulli’s Principle Formula

- p is the pressure
- ρ is the density
- V is the velocity
- h is the elevation
- g is the gravitational acceleration
- p is the pressure head
- \(\frac{1}{2}\rho V^{2}\) is the velocity head
- ρgh is the gravitational head

Bernoulli’s equation gives great insight into the balance between pressure, velocity, and elevation.

When we are standing on a railway station and a train comes we tend to fall towards the train. This can be explained using Bernoulli’s principle as the train goes past, the velocity of air between the train and us increases.

Hence, from the equation, we can say that the pressure decreases so the pressure from behind pushes us towards the train. The working of an airplane is also based on Bernoulli’s effect.

It is designed such that the speed of air above the wing is faster than the speed beneath the wing and hence the pressure beneath is more which lifts the airplane. Another principle is the Principle of continuity.

### Principle of Continuity

According to this if the fluid is in streamline flow and is in-compressible then we can say that mass of fluid passing through different cross sections are equal.

From the above situation, we can say the mass of liquid inside the container remains the same.

The rate of mass entering = Rate of mass leaving

The rate of mass entering = ρA_{1}V_{1}Δt—– (1)

The rate of mass entering = ρA_{2}V_{2}Δt—– (2)

Using the above equations,

ρA_{1}V_{1}=ρA_{2}V_{2}

This equation is known as the Principle of continuity.

Suppose we need to calculate the speed of efflux for the following setup.

Using Bernoulli’s

Equation at point 1 and point 2, \(p+\frac{1}{2}\rho v_{1}^{2}+\rho gh=p_{0}+\frac{1}{2}\rho v_{2}^{2}\)\(v_{2}^{2}=v_{1}^{2}+2p-\frac{p_{0}}{\rho }+2gh\)

Generally, A_{2} is much smaller than A_{1}; in this case, v_{1}^{2} is very much smaller than v_{2}^{2} and can be neglected. We then find, \(v_{2}^{2}=2\frac{p-p_{0}}{\rho }+2gh\)

Assuming A_{2}<<A_{1},

We get, v_{2}=\(\sqrt{2gh}\)

Hence the velocity of efflux is \(\sqrt{2gh}\) In fluid dynamics, for a non – viscous, incompressible, non – conducting fluid, an increase in the velocity of the fluid during its flow (laminar), results in a simultaneous decrease in pressure (or potential energy) of the fluid. The converse of this is also true.”

**How is it based on the principle of conservation of energy?**

Bernoulli’s equation is based on Bernoulli’s Principle for fluid flow. According to Bernoulli’s equation for a fluid, we can see that the total energy of the system is always constant. It is expressed as per the relation is given below.

\(p+\frac{1}{2}\rho V^{2}+\rho gh=constant\) |

### Notations Used To Express Bernoulli’s Principle In Terms Of Total Energy Of The System

- P is the Pressure Energy
- \(\frac{1}{2}\rho V^{2}\) is the Kinetic Energy of the system per unit volume
- ρgh is the Potential Energy per unit volume
- Look at the image given below. It shows us the flow in a venturi tube.

During this kind of flow, at the point where the pressure across the tube decreases, the velocity increases. Similarly, when the pressure increases at the tube (because of the increase in diameter), the velocity at this point will decrease. That is why if you partially cover over the outlet of the water pipe with your thumb, the water comes out with a greater force. You reduce the area of the outlet, it makes up for it by increasing the velocity.

**Let us try the following application to understand this concept.**

Say water is flowing through a pipe. You have the following information at hand.

**Profile at one of the pipe**: Point A**Pipe pressure**= 150,000 Pa**Speed**= 5 m/s**Height**= 0 m**Profile at other ends of pipe**: Point B**Speed**= 10 m/s**Height**= 2 m

The density of water is 1000 kg/m^{3}. Your mission should you chose to accept it, is to figure out the pressure at the other end of the pipe (point B). How are you going to solve this?

**Hint**: Based on the conservation of energy and Bernoulli’s principle, if the total energy of the system is constant then \(P_{1}+\frac{1}{2}\rho v_{1}^{2}+\rho gh_{1}=P_{2}+\frac{2}{2}\rho v_{1}^{2}+\rho gh_{2}\)