Bernoulli’s Theorem and Its Applications

Bernoulli’s theorem deals with fluid dynamics and how the pressure of fluid changes depending on the velocity of flow. It was named after Daniel Bernoulli, a Swiss mathematician (1700-1782). Although Daniel Bernoulli discovered the principle, Leonard Euler was the one who developed it into the Bernoulli equation we use today. 

The principle finds its applications in diverse fields like aerodynamics, hydrotherapy, and many more. The aim of this article is to provide an insight into Bernoulli’s theorem and its applications in everyday life.

What is Bernoulli’s Principle?

Bernoulli’s principle refers to a relationship between the pressure exerted by an incompressible fluid on an object and its velocity. In essence, the theorem states that when the speed of an object moving through a fluid increases, it creates a decrease in the pressure exerted by the fluid on the object. The reverse is also true: if the speed of an object moving through a fluid decreases, this results in an increase in the pressure exerted on it by the fluid. 

Bernoulli’s theorem and its application can be derived from the principle of conservation of energy. Bernoulli’s principle use can be seen in venturi tubes, thermo-compressors, aspirators and other devices where fluids move at high velocities.

Bernoulli’s Equation

Bernoulli’s equation is a mathematical expression of the relationship between pressure, velocity, and total energy in an incompressible fluid flow that is derived from Newton’s second law for fluids. Bernoulli’s equation may be used to predict how changes in fluid flow velocity affect pressure variations.

It can be given as

p+12v2+gh=constant

Where,

• p – the pressure that the fluid is exerting
• v – velocity of the fluid
• ρ – density of the fluid
• h – the height of the container
• g – acceleration due to gravity.

The equation takes into account variables such as density, height and velocity within fluids. It can be used to describe many different phenomena, including subsonic and supersonic flows, incompressible and compressible flows, laminar flow, turbulent flow and flows with rotating components. It should be noted that this equation of Bernoulli’s principle is only applicable to incompressible fluids (e.g., liquids or gases moving at lower mach number). 

Principle of Continuity

The principle of continuity claims that the mass of fluid travelling through different cross-sections is equal if the fluid is in a streamlined flow and is incompressible. Simply put, the mass of a fluid when it enters into a volume V1 in a defined time T1 will be equal to the mass of the fluid that exits from volume V2 at time T2. The principle of continuity is a result of the law of mass conservation.

We can say  that

The rate of mass entering = Rate of mass leaving

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

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

From (1) and (2), we can derive,

ρA1V1= ρA2V2 

This equation is nothing but the Principle of Continuity. 

Bernoulli’s Principle Use

There are numerous applications in modern-day physics and life that can be successfully explained by Bernoulli’s principle. Some of those are: 

• The aerofoil shape of aircraft wings is designed in such a way that the velocity of air passing over the wing would be higher than the velocity of air passing under the wing. It causes the pressure above the wing to decrease and the pressure under the wing to increase. This is how aircraft attain lift. This phenomenon is the best example of Bernoulli’s principle use.
• When we are at a railway station and a train arrives, we tend to get pulled towards it. This is explained by Bernoulli’s principle, which states that as the train passes, the velocity of the air between the train and us increases. As a result of the equation, we can conclude that the pressure decreases, causing the pressure from behind to push us towards the train. This is based on Bernoulli’s principle.
• A venturi is used in many reciprocating engines to create a region of low pressure in order to draw fuel into the carburetor and thoroughly mix it with the incoming air. Bernoulli’s principle can explain the low pressure in a venturi’s throat; in the narrow throat, the air is moving at its fastest speed and thus at its lowest pressure.

Conclusion

Bernoulli’s principle sheds light on a number of applications, making it not only an important building block in the world of physics, but in other branches of science as well. Bernoulli’s theorem and its applications are extensive and diverse. The best way to understand these different uses is through an open discussion of the principle itself.