What is Bernoulli’s theorem?

Short Answer:

Bernoulli’s theorem states that in a steady, incompressible, and frictionless flow of a fluid, the total energy per unit weight of the fluid remains constant along a streamline. This total energy includes the pressure energy, kinetic energy, and potential energy of the fluid.

In simple words, Bernoulli’s theorem explains the relationship between pressure, velocity, and height of a flowing fluid. When the speed of a fluid increases, its pressure decreases, and vice versa. This principle is widely used in engineering applications such as in aircraft wings, nozzles, and Venturimeters.

Detailed Explanation :

Bernoulli’s Theorem

Bernoulli’s theorem is one of the most important principles in fluid dynamics, named after the Swiss scientist Daniel Bernoulli, who discovered it in 1738. The theorem describes the conservation of energy in a flowing fluid. It helps to understand how pressure, velocity, and elevation (height) of a fluid are related to each other.

According to Bernoulli’s theorem:

“For an incompressible, non-viscous, and steady flow of fluid, the total energy per unit weight of the fluid remains constant along a streamline.”

This means that the sum of pressure energy, kinetic energy, and potential energy in a fluid flow remains the same if there is no energy loss due to friction or other resistances.

Mathematical Expression of Bernoulli’s Theorem

For a unit weight of fluid, Bernoulli’s theorem can be expressed as:

Where,
= Pressure intensity of the fluid (N/m²)
= Density of the fluid (kg/m³)
= Velocity of the fluid (m/s)
= Acceleration due to gravity (9.81 m/s²)
= Height of the fluid above a reference level (m)

Each term in this equation represents a type of energy per unit weight of the fluid:

  •  → Pressure head (pressure energy)
  •  → Velocity head (kinetic energy)
  •  → Potential head (potential energy due to elevation)

The sum of these three heads remains constant for any two points along a streamline, which means:

Explanation of Bernoulli’s Theorem

Bernoulli’s theorem is based on the principle of conservation of energy, which means energy can neither be created nor destroyed; it can only change from one form to another.

When a fluid flows through a pipe or a passage, it has three types of energy:

  1. Pressure energy – energy due to pressure in the fluid.
  2. Kinetic energy – energy due to motion of the fluid.
  3. Potential energy – energy due to height of the fluid above a reference level.

As the fluid moves, these forms of energy can convert from one to another, but their total sum remains constant. For example:

  • If the velocity of fluid increases, its pressure decreases.
  • If the height increases, velocity or pressure may reduce to maintain total energy balance.

This principle helps engineers predict how fluids behave when moving through different pipe diameters, bends, and elevations.

Assumptions Made in Bernoulli’s Theorem

Bernoulli’s theorem is valid only under the following conditions:

  1. The flow of fluid is steady (does not change with time).
  2. The fluid is incompressible (its density remains constant).
  3. The flow is non-viscous (no energy loss due to friction).
  4. The flow occurs along a streamline (smooth path of flow).
  5. The flow is irrotational (no spinning motion of fluid particles).

These assumptions are ideal, but in real situations, small losses occur due to viscosity and turbulence. Still, Bernoulli’s theorem provides a good approximation for most engineering problems.

Applications of Bernoulli’s Theorem

Bernoulli’s theorem has several practical applications in mechanical, civil, and aerospace engineering. Some of the most common applications include:

  1. Venturimeter:
    A device used to measure the rate of flow of a fluid in a pipe. It works on Bernoulli’s principle by measuring pressure difference between two sections of different diameters.
  2. Orifice Meter:
    Similar to a Venturimeter, it measures discharge through a pipe by using pressure difference at a sharp-edged opening.
  3. Pitot Tube:
    Used to measure fluid velocity, especially in aircraft and wind tunnels. It compares static and dynamic pressures to find the flow speed.
  4. Airplane Wings (Lift of an Aircraft):
    The shape of an airplane wing is such that air flows faster over the upper surface and slower below, creating a pressure difference that lifts the aircraft upward.
  5. Atomizers and Sprayers:
    Devices such as perfume sprayers and carburetors use pressure difference to draw liquid into an airstream and convert it into fine droplets.
  6. Flow through Nozzles:
    Bernoulli’s theorem is applied to design nozzles where high-velocity jets are needed, like in turbines and rockets.

Limitations of Bernoulli’s Theorem

Although Bernoulli’s theorem is very useful, it has some limitations:

  • It does not consider frictional losses or viscosity of the fluid.
  • It is not valid for compressible fluids like gases at high pressure.
  • It applies only along a streamline, not for turbulent flows.
  • It neglects energy losses due to heat, vibration, or turbulence.

To overcome these limitations, engineers often include correction factors or use modified equations in practical fluid mechanics.

Conclusion

Bernoulli’s theorem explains that the total energy per unit weight of a fluid — consisting of pressure energy, kinetic energy, and potential energy — remains constant along a streamline for a steady, incompressible, and frictionless flow. It is one of the most important principles in fluid mechanics and forms the foundation for many hydraulic and aerodynamic devices. From measuring flow rate to designing airplanes and turbines, Bernoulli’s theorem plays a vital role in modern engineering applications.