What is Bernoulli’s theorem?

Short Answer:

Bernoulli’s theorem states that in a steady and incompressible flow of an ideal fluid, the total energy per unit weight remains constant along a streamline. The total energy includes pressure energy, kinetic energy, and potential energy. It means that when the speed of the fluid increases, its pressure decreases, and vice versa.

This theorem explains the relationship between the velocity and pressure of a moving fluid. It is widely used in fluid mechanics to design pumps, turbines, airplanes, and pipelines where fluid flow plays a major role.

Detailed Explanation :

Bernoulli’s Theorem

Bernoulli’s theorem was given by Daniel Bernoulli in the 18th century. It is one of the most important principles in fluid mechanics and deals with the conservation of energy in fluid flow. The theorem states that in a steady, incompressible, and frictionless flow of an ideal fluid, the sum of the pressure energy, kinetic energy, and potential energy per unit weight remains constant along a streamline.

Mathematically, Bernoulli’s theorem is expressed as:

Where,

  • P = Pressure energy (N/m²)
  • ρ = Density of the fluid (kg/m³)
  • g = Acceleration due to gravity (m/s²)
  • V = Velocity of the fluid (m/s)
  • Z = Height of the point from the reference level (m)

Here,
= Pressure head
= Velocity head
= Potential head or elevation head

The theorem simply shows that the total head or energy of the fluid remains constant if there is no loss due to friction or other factors.

Explanation of Terms

  1. Pressure Energy:
    This is the energy possessed by the fluid due to its pressure. It allows the fluid to do work when moving from one point to another.
  2. Kinetic Energy:
    This is the energy possessed by the fluid due to its motion. It depends on the velocity of the fluid particles.
  3. Potential Energy:
    This is the energy due to the height of the fluid above a reference point or level. It is also called elevation energy.

According to Bernoulli’s theorem, when the velocity of the fluid increases, the pressure decreases, and when the velocity decreases, the pressure increases. This relationship is the basis of many engineering applications.

Derivation of Bernoulli’s Theorem

To derive Bernoulli’s theorem, consider a fluid flowing through a pipe of varying cross-section. The flow is steady, incompressible, and frictionless.

  • Let the pressure, velocity, and height at the first section be , , and .
  • Let the same parameters at the second section be , , and .

According to the principle of conservation of energy, the total energy at section 1 must be equal to the total energy at section 2.

This equation shows that the total energy (pressure head + velocity head + potential head) remains constant along the streamline.

Assumptions of Bernoulli’s Theorem

  • The fluid flow is steady (does not change with time).
  • The fluid is incompressible (density is constant).
  • The flow is frictionless (no energy loss due to friction).
  • The flow takes place along a streamline.
  • The fluid behaves as an ideal fluid.

These assumptions make Bernoulli’s theorem applicable for ideal conditions, but in real situations, some corrections are needed to account for frictional losses.

Applications of Bernoulli’s Theorem

  1. Venturimeter:
    It measures the flow rate of a fluid through a pipe. Based on Bernoulli’s theorem, the difference in pressure between two points helps to calculate velocity and discharge.
  2. Airplane Wings:
    The shape of an airplane wing is such that air moves faster above the wing and slower below it. According to Bernoulli’s theorem, the pressure on top is lower, creating lift and helping the plane to fly.
  3. Carburetors in Engines:
    Bernoulli’s principle helps in mixing air and fuel properly by using pressure differences created by air velocity.
  4. Piton Tube:
    Used to measure the velocity of fluid flow in pipes or open channels.
  5. Hydraulic Machines:
    Used in pumps and turbines where the conversion of kinetic energy to pressure energy or vice versa takes place.

Limitations

  • The theorem is applicable only to ideal fluids and not to viscous or compressible fluids.
  • It ignores frictional losses and turbulence.
  • It cannot be applied in cases where heat transfer or mechanical work is done by external forces.

Despite these limitations, it gives a very close estimation for practical engineering problems when corrections are applied.

Examples

  • When water flows through a narrowing pipe, its velocity increases and pressure decreases — as per Bernoulli’s theorem.
  • When a fast-moving vehicle passes by, nearby lightweight objects get pulled toward it because of the drop in pressure.
  • In atomizers and sprayers, fluid velocity increases at the nozzle, reducing pressure and causing liquid to spray.
Conclusion

Bernoulli’s theorem is a fundamental principle in fluid mechanics that explains the relationship between pressure, velocity, and height in a flowing fluid. It is based on the law of conservation of energy, which states that energy cannot be created or destroyed but only converted from one form to another. This theorem is the backbone of many engineering devices like Venturimeters, pumps, turbines, and aircraft wings. It helps engineers design systems that efficiently control and utilize fluid flow in various machines and industrial applications.