In kinematics and mechanics, we deal with the various types of motion of a body. Among them, the freely falling body, uniform velocity motion etc., are described properly by using only Newton’s laws of motion and Newton’s equation for motion. But when we deal with a body’s motion in a fluid medium, these above laws are not sufficient to describe the motion. Like in a fluid medium, pressure, velocity, etc., comes into the picture. So to describe the motion and the mechanism of a body moving in a fluid, Bernoulli’s Principle is useful. It gives the relation between pressure, velocity and potential head.
Equation of continuity
Before going to Bernoulli’s principle, we must have some idea about the equation of the continuity. We know that for a control volume system, the mass entered in the system is equal to the mass out. Hence the continuity is
Where A1 is area of inlet valve
V1 is velocity of the entering fluid
A2 is area of outlet valve
V2 is the velocity of the fluid that exits.
Statement: “when a nonviscous and incompressible flows stately in a streamline, the sum of the pressure energy, kinetic energy and potential energy per unit mass at any point in stream flows remain constant.”
In symbol the statement can be written as
Where P is the pressure which stands for pressure energy, v is the velocity of the fluid, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height of the container.
Derivation of Bernoulli’s formula
Consider a nonviscous, incompressible fluid flowing through a pipe, then the energies associated with the streamline are:
- Potential energy
- Kinetic energy
- Pressure energy
The potential energy of a liquid of mass m at a height h above the ground is mgh, potential energy per unit mass is gh and potential energy per unit volume is gh, where is the density of mass per unit volume.
The kinetic energy of a Liquid of mass moving with a velocity v is given by12mv2.
Per unit volume the KE is given by 12v2.
When a fluid flows through a pipe, According to the equation of continuity, the volume of the fluid entering in a time is equal to the volume of the liquid leaving at the same time.
So if we calculate the work done by the pressure on the liquid, then
Where A is the area of the cross-section of the pipe
And dx = vΔt
Hence the energy associated with the pressure is
W = PA ⋅ vΔt
= P ΔV
where ∆V is the volume that passes through the region through the cross-section.
Then the pressure energy per unit volume is P
So as we discussed earlier, according to the conservation of the energy theorem, the summation of all energy remains constant.
Hence P+12v2+gh=constant ,
This is Bernoulli’s equation.
Application of Bernoulli’s Principle
In engineering and science, there are several applications of Bernoulli’s theorem. Let us discuss some of the applications.
- Venture metre
It is a device that is used to measure the flow of speed in a pipe.
The speed formula is given by v1=2gh(a1a2)2-1
Where a1a2 is the ratio of surface area of inlet and outlet valve
v1= velocity of fluid in the pipe
h= Height of Venture mater
g=acceleration due to gravity
This formula is an application of Bernoulli’s Principle
- Pitot’s tube
It is a device used for measuring the velocity of the liquid or gases through the pipe.
The working of this device is totally based on Bernoulli’s Principle.
The speed is given by v=2hdg
Where H is the height of the tube and d is the density of the liquid.
- Lift of an Aeroplane
Aeroplane, a medium of transportation in the atmosphere, operates on the principle of Bernoulli’s theorem. When the atmospheric air passes through a high velocity on the top surface of the wings of the aeroplane, then according to Bernoulli’s theorem, the pressure of the upper surface of the wings decreases. So due to the pressure difference, an upward thrust force is acting on the plane, which helps fly the aeroplane. So it is an application of Bernoulli’s Principle.
- Bunsen Burner
In a bunsen burner, the gases enter through the base and come out through the nozzle.
As the pressure at the nozzle decreases, the gases flow through the base towards the nozzle, which is an application of the principle of Bernoulli’s theorem.
Limitations of Bernoulli’s Principle
- It is only valid for the stream line flows.
- It only holds for the incompressible liquid.
- It does not take viscosity into consideration.
In science and engineering, Bernoulli’s principle has several uses as it gives the relation between pressure, velocity and height or potential gradient of the fluid. In both horizontal and vertical, Bernoulli’s principle holds good. This study material helps in the preparation of IIT JEE. Bernoulli’s principle and its application are properly described in this study material.