The combination of Bernoulli’s equation and the law of conservation of energy demonstrate that the total mechanical energy consisting of the moving fluid is correlated with the fluid pressure, the changing gravitational potential energy, and kinetic energy of the moving fluid continues steadily. Based on this principle, there are many applications in the thermodynamic field that are constantly emerging. This article will help to understand and apply these important components of physics.

Bernoulli’s Equation

Bernoulli’s equation states a direct relationship between the velocity of any fluid and the pressure exerted by its intermolecular forces. This phenomenon was initially studied by Daniel Bernoulli, a Swiss mathematician, in 1738. However, the derivation for this principle was derived by Leonhard Euler in 1750. This principle has been utilised in many engineering-related applications such as the working of aeroplanes, sailboats, etc.

Conservation of Energy

The conservation of energy is a crucial doctrine in the field of physics. It states that the amount of energy that arises due to the interaction of bodies and particles in any closed system will remain constant. These principles explained many kinds of energies, such as kinetic energy (energy due to the motion of objects), elastic energy (energy due to the collision between objects), and potential energy (energy due to gravitational effect). Furthermore, the sum of all these energies consisting of any object will remain steady. Energy can be stored or transformed from one object to another. In other words, energy cannot be created or destroyed due to any change, and this is also regarded as the first law of thermodynamics.

Relationship Between Energy Conservation and Bernoulli’s Equation

For any incompressible and abrasion less fluid, the sum of pressure within the fluid and the sum of all kinds of energies, including kinetic energy, potential energy, and thermal energy, will remain constant throughout the pipeline. Hence, the amount of total energy acquired by the fluid of any isolated system is steady. 

The application of the principle of conservation of energy on an abrasion less fluid helps develop a relationship between pressure and velocity of the moving fluid. Daniel Bernoulli mentioned this phenomenon in Hydrodynamica (1738), his book on liquid mobility.

Bernoulli’s theorem states that the total amount of these three energies within a constant moving fluid, i.e., kinetic energy, potential energy, and pressure energy, will remain steady throughout the pipeline. Regardless of these facts, one must understand several factors:

•     The movement of the liquid is inferred to be constant; there is no alteration with respect to pressure, speed, and consistency of the liquid at any point in time. However, in the case of an unstable liquid flow, there will be a change in speed. Hence, this equation can be utilised and does not show an error.
•     The fluid is considered incompressible. As all liquids are incompressible in nature, this principle can be applied to all fluids. However, in the case of gas flow, there should be a constant density, and one can manage with little difference in pressure, velocity and temperature.
•     Another consideration is the irrotational feature of liquids. This means that the overall angular momentum throughout the liquid is zero.
•     For this principle, the type of fluid should be ideal, meaning there should be no loss of energy due to frictional activity. Hence, there is no excess energy production due to internal particle friction.

Bernoulli’s Equation

Bernoulli’s equation is expressed below in relation to the law of conservation of energy for the constant flow of any fluid.

p+1 ⁄ 2v²+pgh=constant

It also demonstrates the law of conservation of energy.

In this equation, the variables are explained below:

P represents the fluid pressure,

ρ represents the fluid density,

v represents the fluid velocity,

g represents the acceleration due to gravity, and

z represents the elevation of the fluid above a fixed reference point.

Applications of the combination of Bernoulli’s Equation and Conservation of Energy

Some applications of the combination of Bernoulli’s equation and the conservation of energy are

Venturi meter – An instrument to measure the rate of flow of fluids through tube lines is based on Bernoulli’s principle.

The working of aeroplanes – The most common example of this principle is the structure of an aeroplane. The variation in airspeed is evaluated and calculated through Bernoulli’s principle and develops a pressure variation. This pressure lifts the aeroplane and supports the plane in flying.

People falling towards fast-moving trains – A person standing near a fast-moving train is likely to fall towards it. This can be understood with the help of Bernoulli’s principle, which suggests that in fast-moving trains, the pressure reduces and increases the pressure pushing us toward the train.

The working of sailboats – Bernoulli’s principle is largely associated with the working of sailboats. The two parts of sailboats, the sail and the keel, are in opposite directions. With the increasing speed of the air, a difference in pressure is created, which supports lifting and stimulates the boat’s sailing towards the moving water.

Conclusion 

The combination of Bernoulli’s principle and the law of conservation of energy state that for any incompressible and abrasion less fluid, the sum of pressure within the fluid and sum of all kinds of energies, including kinetic, potential, and thermal energy, will remain constant throughout the pipeline. Although there is widespread use of these principles in the field of fluid mechanics, they are only applicable to steady fluids (liquids and gases).