Short Answer:

Bernoulli’s equation states that for an ideal, incompressible, and non-viscous fluid flowing in a streamline, the total energy per unit weight of the fluid remains constant. It means that the sum of the pressure energy, kinetic energy, and potential energy of a flowing fluid remains the same along a streamline.

In simple words, Bernoulli’s equation shows that if the speed of a fluid increases, its pressure decreases, and vice versa. It is based on the law of conservation of energy and helps in understanding how pressure and velocity are related in fluid motion.

Detailed Explanation:

Bernoulli’s Equation

Bernoulli’s equation is one of the most fundamental principles in fluid mechanics. It is derived from the law of conservation of energy, which states that energy can neither be created nor destroyed, only transformed from one form to another. In the case of fluid flow, energy exists in three main forms — pressure energy, kinetic energy, and potential energy.

Bernoulli’s equation states that the total mechanical energy of the fluid remains constant as it flows along a streamline, provided there is no energy loss due to friction or other factors. It is mainly applicable to steady, incompressible, and non-viscous flows.

1. Statement of Bernoulli’s Equation

Bernoulli’s equation can be stated as:

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

Mathematically, the equation is expressed as:

where,

•  = pressure 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 or elevation head (m)

Each term in the equation represents a specific type of energy per unit weight of the fluid.

2. Explanation of Each Term

1. Pressure Head ( ):
   This term represents the potential energy due to the fluid pressure. It shows the height of a fluid column that would exert the same pressure at a point in the fluid.
2. Velocity Head ( ):
   This term represents the kinetic energy of the fluid due to its motion. It indicates the height through which a fluid particle would have to fall freely under gravity to acquire the same velocity.
3. Elevation Head ( ):
   This term represents the potential energy of the fluid due to its height above a reference level. It shows the gravitational potential energy of the fluid.

The sum of these three heads gives the total head or total energy per unit weight of the fluid.

3. Derivation of Bernoulli’s Equation

Bernoulli’s equation is derived from the energy principle for a small fluid element moving along a streamline.

Consider a steady flow of incompressible, frictionless fluid through a pipe that changes its cross-sectional area and elevation. Let the two sections of the pipe be at points 1 and 2.

At point 1:

• Pressure = , velocity = , elevation =

At point 2:

• Pressure = , velocity = , elevation =

Step 1: Work done by pressure forces
When fluid moves from section 1 to 2, the work done per second by pressure forces is:

Step 2: Change in potential energy
The change in potential energy per second is:

Step 3: Change in kinetic energy
The change in kinetic energy per second is:

By the principle of conservation of energy,

 

Dividing the entire equation by  :

This is the Bernoulli’s equation.

4. Assumptions in Bernoulli’s Equation

Bernoulli’s equation is based on the following assumptions:

1. The flow is steady (does not change with time).
2. The fluid is incompressible (density remains constant).
3. The fluid is non-viscous (no frictional losses).
4. The flow is along a streamline.
5. The gravitational field is uniform.

These assumptions make the equation ideal but useful for many engineering applications with negligible energy losses.

5. Applications of Bernoulli’s Equation

1. Venturi Meter:
   Used to measure the rate of flow in a pipe by using pressure difference.
2. Orifice Meter:
   Used for measuring discharge through small openings in tanks or pipes.
3. Pitot Tube:
   Used to measure the velocity of a fluid by converting kinetic energy into pressure energy.
4. Aircraft Wing (Lift Generation):
   Air moving faster over the upper surface of the wing creates lower pressure, generating lift.
5. Nozzle and Diffuser Design:
   Used in turbines, jet engines, and rocket propulsion systems for efficient energy conversion.
6. Flow through Pipes:
   Helps in analyzing pressure and velocity variations in pipe systems.

6. Limitations of Bernoulli’s Equation

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

1. It assumes ideal flow, neglecting viscosity.
2. It cannot be applied to compressible fluids (like gases at high velocity).
3. It does not account for energy losses due to friction or turbulence.
4. It is valid only along a streamline, not for the entire flow field.

In real-world applications, modifications are made by adding head loss terms to account for friction and turbulence.

Conclusion

In conclusion, Bernoulli’s equation is a fundamental principle in fluid dynamics that expresses the conservation of energy for a flowing fluid. It states that the total energy per unit weight (pressure head, velocity head, and elevation head) remains constant along a streamline. Despite its assumptions and limitations, Bernoulli’s equation is widely used in engineering to analyze fluid motion, measure flow rates, and design hydraulic and aerodynamic systems efficiently. It remains one of the most powerful and practical tools in fluid mechanics.