Bernoulli’s Principle: Law, Derivation, Formula, Limitations

Bernoulli’s principle is based on the law of conservation of energy and applies to ideal fluids. Bernoulli’s equation relates the speed of a fluid at a point, the pressure at that point and the height of that point above a reference level. It is just the application of the work-energy theorem in the case of fluid flow.

Key Terms: Bernoulli’s Theorem, Theory, Proof, Potential Energy, Kinetic Energy

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Bernoulli’s principle 

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Bernoulli's principle states that “the sum of pressure energy, kinetic energy and potential energy per unit volume of an incompressible, non-viscous fluid in a streamlined irrotational flow remains constant along a streamline”.

Mathematically, it can be expressed as

Bernoulli’s Theorem Video Lecture

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Bernoulli’s Principle Proof 

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Proof: Consider a non-viscous and incompressible fluid flowing steadily between the sections A and B of a pipe of varying cross-section. Let a, be the area of cross-section at A, v, the fluid velocity, P, the fluid pressure, and h, the mean height above the ground level.

Let a, v2?, P? and h? be the values of the corresponding quantities at B.

Bernoulli’s Principle

Let p be the density of the fluid. As the fluid is incompressible, so whatever mass of fluid enters the pipe at section A in time At, an equal mass of fluid flows out at section B in time At. This mass is given by

m= Volume x density

= Area of cross-section x length x density

or, m = a₂ v₂ Δt ρ = a₂ v₂ Δt ρ ……(1)

or, a₁v₁ = a₂v₂ ……(2)

Therefore, Change in K.E. of the Fluid

 = K.E. at B – K.E. at A

= 1/2m (v₂² - v₁²) = 12 a₁ v₁ Δt ρ (v₂² - v₁²)

[Using (1)]

Change in P.E. of the fluid

 = P.E. at B – P.E at A

 = mg (h₂ – h₁) = a₁ v₁ Δt ρ g (h₂ – h₁) [Using (1)]

Net work done on the fluid 

= Work done on the fluid at A – Work done by fluid at B

= P₁ a₁ x v₁ Δt – P₂ a₂  x v₂ Δt

= P₁ a₁ v₁ Δt - P₂ a₁ v₁ Δt [Using (2)]

= a₁ v₁ Δt (P₁ - P₂)

By conservation of energy,

Net work done by the fluid 

= Change in K.E. of the fluid + Change in P.E. of the fluid

Therefore, a₁  v₁ Δt (P₁ - P₂)

= 1/2 a₁ v₁ Δt ρ (v₂² - v₁²) + a₁ v₁ Δt ρ g (h₂ – h₁)

Dividing both the sides by a1  v1 Δt, we get

P₁ - P₂ = 1/2 ρ v₁² - 1/2 ρ v₁² + ρ gh₂ - ρ gh₁

or, P₁ + 1/2 ρ v₁² + ρgh₁ = P₂ + 1/2 ρ v₁² + ρgh₂

or, P + 12 ρv₂ + ρgh = constant ……..(3)

This proves Bernoulli’s principle according to which the total energy per unit volume remains constant.

Equation (3) can also be written as

Pg + 1/2 v₂g + h = constant

This is another form of Bernoulli’s principle according to which- “the sum of pressure head, velocity head and gravitational head remains constant in the streamline flow of an ideal fluid.”

Also Read:

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Limitations of Bernoulli’s equation 

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• Bernoulli’s equation ideally applies to fluids with zero viscosity or non-viscous fluids. In case of viscous fluids, we need to take into account the work done against viscous drag.
• Bernoulli’s equation has been derived on the assumption that there is no loss of energy due to friction. But in practice, when fluids flow, some of their kinetic energy gets converted into heat due to work done against the internal forces of friction or viscous forces.
• Bernoulli’s equation is applicable only to incompressible fluids because it does not take into account the elastic energy of fluids.
• Bernoulli’s equation is applicable only to streamline flow of a fluid and not when the flow is turbulent.
• Bernoulli’s equation does not take into consideration the angular momentum of the fluid. So it cannot be applied when the fluid flows along a curved path.

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Things to Remember

1. Bernoulli’s principle is a fundamental principle of fluid dynamics based on the law of conservation of energy.
2. In Bernoulli’s equation : P + 1/2 ρv₂ + ρ gh = constant, the term (P + ρ gh) is called static pressure, because it is the pressure of the fluid even if it is at rest, and the term 12 ρv2 is the dynamic pressure of the fluid which is the pressure by virtue of its velocity, v. So Bernoulli’s equation can be written as

Static pressure + Dynamic pressure = constant

3. If a liquid is flowing through a horizontal tube, h remains constant and we can write 

 

This shows that if v increases, P decreases and vice versa. Thus for the streamline flow of an ideal liquid flowing horizontally, the pressure decreases when velocity increases and vice versa. This is an important aspect of Bernoulli’s principle which finds many applications.

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