Bernoulli’s Theorem is one of the most important theorems that provides the relation between velocity, pressure and viscosity. It was first derived by the Swiss Mathematician Daniel Bernoulli. Although the law was deduced by Bernoulli, the equation in its used form was derived by Leonhard Eurel in 1752. 

Statement

According to Bernoulli’s principle, the total energy at any point in the subsurface per unit mass of a flowing fluid remains constant and is equal to the sum of the potential, kinetic and pressure head of the fluid. In simpler terms it is the energy conservation principle for ideal fluids in steady state or streamline motion and forms the basis of a lot of hydraulic applications. We will look at the different forms of energy associated with fluids:

1. Pressure Head: It is defined as the static pressure that the fluid applies to the walls of the container when it is static. 
2. Kinetic Energy: It is the most common form of energy that is associated with any mass in motion. The energy exists due to the motion of the fluid and can be termed as dynamic pressure. 
3. Potential Energy: The gravitational potential energy from the elevation of the fluid. This accumulated potential energy can also be termed as hydrostatic pressure. 

P+1/2ρv2+ρgh=constant

P= The absolute pressure
V= Velocity of the fluid
h = Height above the reference point for the calculation of the potential energy
g = Acceleration due to gravity

If we follow a small volume of fluid along its path, the individual values of the energies change but the overall sum remains constant. 

P1+1/2ρv12+ρgh1=P2+1/2ρv22+ρgh2

Bernoulli’s equation is a form of the energy conservation principle. If we carefully notice, the second and third terms are similar to the kinetic and potential energy formula where the mass ‘m’ is replaced by density ‘ρ’. Density is mass/volume and we can replace the kinetic energy term in the above equation, we get: 

1/2ρv2=1/2mv2/V = KE/V

So 1/2ρv2 is the kinetic energy per unit volume. Making the same substitution into the third term in the equation, we find

ρgh=mgh/V=PEg/V

so ρgh is the gravitational potential energy per unit volume. Note that pressure P has units of energy per unit volume, too. Since P = F/A, its units are N/m2. If we multiply these by m/m, we obtain N ⋅ m/m3 = J/m3, or energy per unit volume. Bernoulli’s equation is, in fact, just a convenient statement of conservation of energy for an incompressible fluid in the absence of friction.

Continuity Equation: 

According to the principle of continuity, the overall mass remains constant when an incompressible fluid that is in streamline motion passes through different cross-sections. 

By the energy conservation principle of incompressible fluids:

The rate of Mass entering the cross-section = the rate of mass leaving the cross-section 

The rate of mass entering = ρA1V1Δt—– (1)

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

Using the above equations,

ρA1V1=ρA2V2

This equation is known as the principle of continuity.

Application of Bernoulli’s Theorem

1. Aeroplane Wings: The aeroplane wing is the most exquisite application of the principle in action. Observe the characteristics of the wing below: The wing is tilted above at a small angle and the length is slightly longer causing the air to flow faster over it. This reduces the pressure on the top of the wing creating a lift.
2. Crashing Ships: A historic tale, two ships crashing into each other when travelling parallel to each other is a classic example of Bernoulli’s Theorem at play. Imagine, there are two boats sailing at equal velocities, parallel to each other. The middle gap acts as a funnel squeezed pipe for the water between the two boats. This causes the area to increase and decrease causing the velocity to increase. However, with the increase in the velocity, the pressure between the boats decreases. As a result, in order to stabilize this decrease in pressure, the water on the outside of the boat causes the boats to move closer to each other and thereby crashing. 
3. Venturi Meter: It is a simple mechanical device that is based on Bernoulli’s Theorem and very frequently employed to measure the rate of flow of fluid through pipes. 

Summary: 

1. Bernoulli’s Theorem is based on the principle of energy conservation where the entire energy throughout the ideal fluid in streamline motion remains constant and is equal to the sum of Hydrostatic Pressure, Kinetic Energy & Potential Energy due to height. 
2. The principle finds application in multiple important applications such as wing lift, hydraulic jacks, hydraulic lifts, venturi meters etc.