# 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.

## What is Bernoulli’s Principle?

Bernoulli’s principle 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 can be derived from the principle of conservation of energy.

## Bernoulli’s Principle Formula

Bernoulli’s equation formula is a relation between pressure, kinetic energy, and gravitational potential energy of a fluid in a container.

The formula for Bernoulli’s principle is given as:

 p + $\frac{1}{2}$ ρ v2 + ρgh =constant

Where,

• p is the pressure exerted by the fluid
• v is the velocity of the fluid
• ρ is the density of the fluid
• h is the height of the container

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

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## Bernoulli’s Equation Derivation

Consider a pipe with varying diameter and height through which an incompressible fluid is flowing. The relationship between the areas of cross sections A, the flow speed v, height from the ground y, and pressure p at two different points 1 and 2 is given in the figure below.

Assumptions:

• The density of the incompressible fluid remains constant at both the points.
• Energy of the fluid is conserved as there are no viscous forces in the fluid.

Therefore, the work done on the fluid is given as:

dW = F1dx1 – F2dx2

dW = p1A1dx1 – p2A2dx2

dW = p1dV – p2dV = (p1 – p2)dV

We know that the work done on the fluid was due to conservation of gravitational force and change in kinetic energy. The change in kinetic energy of the fluid is given as:

$dK = \frac{1}{2}m_{2}v_{2}^{2}-\frac{1}{2}m_{1}v_{1}^{2}=\frac{1}{2}\rho dV(v_{2}^{2}-v_{1}^{2})$

The change in potential energy is given as:

dU = mgy2 – mgy1 = ρdVg(y2 – y1)

Therefore, the energy equation is given as:

dW = dK + dU

(p1 – p2)dV = $\frac{1}{2}\rho dV(v_{2}^{2}-v_{1}^{2})$ + ρdVg(y2 – y1)

(p1 – p2) = $\frac{1}{2}\rho (v_{2}^{2}-v_{1}^{2})$ + ρg(y2 – y1)

Rearranging the above equation, we get

$p_{1}+\frac{1}{2}\rho v_{1}^{2}+\rho gy_{1}=p_{2}+\frac{1}{2}\rho v_{2}^{2}+\rho gy_{2}$

This is Bernoulli’s equation.

## Principle of Continuity

According to principle of continuity

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 = ρA1V1Δt—– (1)

The rate of mass entering = ρA2V2Δt—– (2)

Using the above equations,

ρA1V1=ρA2V2

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, A2 is much smaller than A1; in this case, v12 is very much smaller than v22 and can be neglected. We then find, $v_{2}^{2}=2\frac{p-p_{0}}{\rho }+2gh$

Assuming A2<<A1,

We get, v2=$\sqrt{2gh}$

Hence the velocity of efflux is $\sqrt{2gh}$

## Bernoulli’s Principle Use

Bernoulli’s principle is used for studying the unsteady potential flow which is used in the theory of ocean surface waves and acoustics. It is also used for approximation of parameters like pressure and speed of the fluid.

The other applications of Bernoulli’s principle are:

• Venturimeter: It is a device which is based on Bernoulli’s theorem and is used for measuring the rate of flow of liquid through the pipes. Using Bernoulli’s theorem, Venturimeter formula is given as:
 $V=a_{1}a_{2}\sqrt{\frac{2hg}{a_{1}^{2}-a_{2}^{2}}}$

• Working of an aeroplane: The shape of the wings are such that the air passes at a higher speed over the upper surface than the lower surface. The difference of air speed is calculated using Bernoulli’s principle to create a pressure difference.
• 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. This is based on the Bernoulli’s effect.

### Relation between Conservation of Energy and Bernoulli’s Equation

Conservation of energy is applied to the fluid flow to produce Bernoulli’s equation. The net work done is the result of change in fluid’s kinetic energy and gravitational potential energy. Bernoulli’s equation can be modified depending on the form of energy that is involved. Other forms of energy include dissipation of thermal energy due to fluid viscosity.

### Bernoulli’s Equation at Constant Depth

When the fluid moves at a constant depth that is when h1 = h2, then Bernoulli’s equation is given as:

 $P_{1}+\frac{1}{2}\rho v_{1}^{2}=P_{2}+\frac{1}{2}\rho v_{2}^{2}$

### Bernoulli’s Equation for Static Fluids

When the fluid is static, then v1 = v2 = 0, then Bernoulli’s equation is given as:

 When v1 = v2 = 0 P1 + ρgh1 = P2 + ρgh2 When h2 = 0 P2 = P1 + ρgh1

## Bernoulli’s Principle Example

Q1. Calculate the pressure in the hose whose absolute pressure is 1.01 x 105 N.m-2 if the speed of the water in hose increases from 1.96 m.s-1 to 25.5 m.s-1. Assume that the flow is frictionless and density 103 kg.m-3

Ans: Given,

Pressure at point 2, p2 = 1.01 × 105 N.m-2

Density of the fluid, ρ = 103 kg.m-3

Velocity of the fluid at point 1, v1 = 1.96 m.s-1

Velocity of the fluid at point 2, v2 = 25.5 m.s-1

From Bernoulli’s principle for p1,

$p_{1}=p_{2}\frac{1}{2}\rho v_{2}^{2}-\frac{1}{2}\rho v_{1}^{2}=p_{2}+\frac{1}{2}\rho (v_{2}^{2}-v_{1}^{2})$

Substituting the values in above equation, we get

$p_{1}= (1.01\times 10^{5})+\frac{1}{2}(10^{3})[(25.5)^{2}-(1.96)^{2}]$

p1 = 4.24 × 105 N.m-2

## FAQs

Q1. What is Bernoulli famous for?
Daniel Bernoulli explained how does the speed of a fluid affect the pressure of the fluid which is known as Bernoulli’s effect and also explained the kinetic theory of gases. These two were his greatest contribution to Science and these two concepts made him famous.

According to Bernoulli’s effect, he tried to explain that when a fluid flows through a region where the speed increases, the pressure will decrease. Bernoulli’s effects find many real-life applications such as airplanes wings are used for providing lift to the plane.