Short Answer:

The assumptions in Bernoulli’s theorem are made to simplify real fluid motion and derive an ideal relationship between pressure, velocity, and elevation in a fluid flow. The main assumptions are that the flow is steady, incompressible, non-viscous, and takes place along a streamline. It also assumes that the gravitational field is uniform and no energy is added or lost due to friction or heat.

In simple words, Bernoulli’s theorem works only under ideal conditions where the fluid flows smoothly without turbulence or viscosity. These assumptions help apply the law of conservation of energy to a moving fluid and derive Bernoulli’s equation.

Detailed Explanation:

Assumptions in Bernoulli’s Theorem

Bernoulli’s theorem is one of the most important principles in fluid mechanics. It is derived from the law of conservation of energy and states that for a fluid flowing in a streamline, the sum of its pressure energy, kinetic energy, and potential energy per unit weight remains constant.

However, this theorem is based on several ideal assumptions about the nature of the flow and the fluid itself. In real-world applications, these assumptions are not always fully valid, but they simplify complex fluid flow behavior and make the equation usable for practical engineering analysis.

1. Steady Flow Assumption

Bernoulli’s theorem assumes that the fluid flow is steady, meaning that the velocity of the fluid particles at any point does not change with time.

Mathematically,

In steady flow, the flow parameters such as velocity, pressure, and density remain constant at a given point. This assumption ensures that the rate of flow and energy at any section do not change with time.

Example:
Flow of water in a uniform pipe at a constant rate is a steady flow. If the rate of flow changes with time, such as in pulsating or unsteady flow, Bernoulli’s theorem cannot be applied directly.

2. Incompressible Fluid Assumption

The fluid is assumed to be incompressible, meaning that its density remains constant throughout the flow.

Mathematically,

This assumption is valid for liquids like water and oil, whose density does not change significantly with pressure. However, for gases, which are compressible, Bernoulli’s theorem can only be applied under limited conditions — such as at low speeds where density changes are negligible.

Example:
In water flow through pipes or channels, density is practically constant, so this assumption holds true.

3. Non-Viscous or Ideal Fluid Assumption

Bernoulli’s theorem assumes that the fluid is non-viscous, meaning it has no internal friction (viscosity).

In real fluids, viscosity causes energy loss due to friction between fluid layers and with the walls of the pipe or channel. But Bernoulli’s equation neglects these losses to simplify calculations.

This assumption is reasonable when viscous effects are small compared to inertial and pressure forces, such as in high-speed flows or streamlined (laminar) conditions.

Example:
Airflow around an airplane wing is often treated as inviscid (non-viscous) for simplicity in aerodynamic calculations.

4. Flow Along a Streamline

Bernoulli’s theorem is derived for flow along a streamline, which means the fluid particles follow well-defined paths that do not cross each other.

The pressure, velocity, and elevation are considered only along that particular streamline. The total energy remains constant for a single streamline but may differ between neighboring streamlines.

If the flow is irrotational, Bernoulli’s equation can be applied to the entire flow field; otherwise, it applies only to individual streamlines.

Example:
In the smooth flow of water around a curved obstacle, the Bernoulli equation applies along each curved streamline.

5. Gravitational Field is Uniform

It is assumed that the acceleration due to gravity ( ) is constant throughout the region of flow.

This assumption allows potential energy per unit weight to be expressed as  , where   is the height above a chosen reference level.

For most engineering applications near the Earth’s surface, gravity is approximately uniform, so this assumption is accurate.

6. No Energy Losses

Bernoulli’s theorem assumes that there are no losses of energy in the flow due to friction, turbulence, heat transfer, or viscous dissipation.

This means all the energy in the fluid (pressure, kinetic, and potential) is conserved as it moves along a streamline.

In real flows, some energy is always lost due to viscosity and other factors. For practical purposes, engineers use a modified Bernoulli’s equation that includes a head loss term ( ) to account for these losses:

7. No Shaft Work or Heat Transfer

Bernoulli’s theorem assumes that no mechanical work (such as pump or turbine action) or thermal energy (heat addition or removal) is involved between the two points considered.

In other words, energy is not added to or extracted from the fluid as it flows.

If energy is added (e.g., by a pump) or removed (e.g., by a turbine), the Bernoulli equation is modified to include the work terms:

where   = head added by pump, and   = head removed by turbine.

8. Flow is Irrotational (Optional Assumption)

In some cases, Bernoulli’s equation assumes irrotational flow, meaning that the fluid elements do not rotate about their own axis.

When this condition is satisfied, Bernoulli’s equation can be applied not only along a streamline but throughout the entire fluid flow region.

This is an idealization often used in potential flow theory to simplify calculations.

Summary of Assumptions

To summarize, Bernoulli’s theorem is based on the following key assumptions:

1. Flow is steady.
2. Fluid is incompressible.
3. Fluid is non-viscous (ideal).
4. Flow occurs along a streamline.
5. The gravitational field is uniform.
6. There is no energy loss due to friction.
7. No external work (like pumps or turbines) is done on or by the fluid.
8. Flow is sometimes assumed to be irrotational (for full-field application).

These assumptions together ensure that the total energy of the fluid remains constant along a streamline.

Conclusion

In conclusion, Bernoulli’s theorem is derived under a set of ideal assumptions that simplify real fluid behavior into an easy-to-use mathematical relationship between pressure, velocity, and height. These assumptions—steady, incompressible, non-viscous flow along a streamline without energy loss—make Bernoulli’s theorem a fundamental tool for analyzing ideal fluid flow. Although real fluids always have viscosity and energy losses, the theorem provides a very good approximation for many engineering applications such as pipe flow, airfoils, and nozzle design.