---
title: "Bernoulli's equation and four applications"
subject: "Fluid Mechanics"
chapter: 5
key: "semester_3/fluid_mechanics:bernoulli-from-euler-applications"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/bernoulli-from-euler-applications
site: BCT Engineering
---

# Bernoulli's equation and four applications

Fluid Mechanics · Chapter 5: Hydrodynamics

- Asked in: 2072 Chaitra, 2+2 marks
- Video: [Bernoulli's Equation for Fluid Mechanics in 10 Minutes!](https://www.youtube.com/watch?v=Eh_XbMBDgy0) — Less Boring Lectures
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/bernoulli-from-euler-applications
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-5

## Question

Develop Bernoulli's equation based on Euler's equation of motion. Explain the four applications of this principle in engineering.

## Short answer

Bernoulli's equation p/ρ + V²/2 + gz = constant is obtained by integrating Euler's equation dp/ρ + V dV + g dz = 0 along a streamline for steady, incompressible, inviscid flow. Engineering applications include venturimeter for flow measurement, Pitot tube for velocity, orifice flow (Torricelli's theorem V = √(2gH)), and siphon analysis for discharge and crest pressure.

## Answer

#### Bernoulli's equation from Euler's equation

Consider a small fluid element of length $ds$ and area $dA$ along a streamline, in steady flow of an ideal fluid. Newton's second law along $s$ (pressure and the weight component, no friction):

$$
-\frac{\partial p}{\partial s}\,ds\,dA - \rho g\,ds\,dA\cos\theta = \rho\,ds\,dA\,V\frac{\partial V}{\partial s}
$$

With $\cos\theta = dz/ds$, this gives **Euler's equation**:

$$
\frac{dp}{\rho} + V\,dV + g\,dz = 0
$$

For incompressible flow, integrate along the streamline:

$$
\frac{p}{\rho} + \frac{V^2}{2} + gz = \text{constant}\quad\Rightarrow\quad\frac{p}{\gamma} + \frac{V^2}{2g} + z = \text{constant}
$$

Assumptions: steady, incompressible, frictionless (inviscid) flow along a streamline, with no energy added or removed.

#### Four applications

1. **Venturimeter:** a pipe contraction raises the velocity and lowers the pressure. From the pressure difference, $Q = C_d\dfrac{A_1A_2}{\sqrt{A_1^2 - A_2^2}}\sqrt{2g\,h}$. It is used to measure discharge in pipelines.
2. **Pitot tube:** at the stagnation point the velocity head is converted into pressure; $V = \sqrt{2g\,h}$ gives the velocity in pipes and channels, and the airspeed in aircraft airspeed indicators.
3. **Orifice / Torricelli's theorem:** water issuing from a tank under head $H$ has velocity $V = \sqrt{2gH}$; this is used for orifices, sluice gates and discharge estimates.
4. **Siphon:** Bernoulli's equation gives the discharge velocity $\sqrt{2gh}$ and the pressure at the crest. The crest pressure must stay above the vapour pressure (maximum crest height about 7 to 8 m of water at sea level) to avoid cavitation and breaking of the flow.

Other uses include pump and turbine head calculations, flow nozzles and orifice meters, aerofoil lift and spray nozzles.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which forces act on the element?
   A. Viscous and surface tension
   B. Only pressure
   C. Pressure and weight component
   D. Only friction
   Answer: C. Ideal fluid has no viscous force. (Key point: Newton's law)

2. Which is not an assumption?
   A. Steady
   B. Incompressible
   C. Frictionless
   D. Compressible flow
   Answer: D. Bernoulli assumes incompressible flow. (Key point: Assumptions)

3. Principle of the venturimeter:
   A. Contraction raises velocity and lowers pressure
   B. Contraction raises pressure
   C. Friction measures flow
   D. Mass is lost
   Answer: A. Pressure difference gives Q. (Key point: Venturimeter)

4. Velocity from tank under head H:
   A. 2gH
   B. sqrt(2gH)
   C. gH
   D. H/2g
   Answer: B. Bernoulli gives V = sqrt(2gH). (Key point: Torricelli)

## Related in Fluid Mechanics

- [Bernoulli from Euler with head losses](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aeuler-to-bernoulli-with-losses) — Fluid Mechanics, chapter 5
- [Navier-Stokes and Euler equations in 2D](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Anavier-stokes-euler-2d-applications) — Fluid Mechanics, chapter 5
- [Navier-Stokes and Bernoulli equations with terms](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Anavier-stokes-bernoulli-terms) — Fluid Mechanics, chapter 5

## Related in other subjects

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