---
title: "Deriving Bernoulli's equation from Euler's equation"
subject: "Hydro Power"
chapter: 1
key: "semester_6/hydro_power:euler-bernoulli-derivation"
source: https://www.bctengineer.com/q/semester_6/hydro_power/euler-bernoulli-derivation
site: BCT Engineering
---

# Deriving Bernoulli's equation from Euler's equation

Hydro Power · Chapter 1: Introduction to Basic Fluid Mechanics and Hydraulics

- Asked in: 2082 Chaitra, 4 marks
- Video: [21) Bernoulli's Equation from Euler's Equation - Derivation || FlUID MECHANICS ~ Hindi](https://www.youtube.com/watch?v=91Ef1_IQWjE) — Defence and SSC Exam
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/euler-bernoulli-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-1

## Question

Using Euler's equation of motion along a streamline, derive the Bernoulli's equation of flow.

## Short answer

Euler's equation dp/ρ + g dz + V dV = 0 is derived from Newton's second law on a fluid element. Integrating for incompressible flow gives Bernoulli's equation: p/ρg + V²/2g + z = constant along a streamline. Assumptions: ideal, steady, incompressible, irrotational flow.

## Answer

**Euler's equation of motion** is Newton's second law applied to a fluid element moving along a streamline, considering only pressure and gravity forces. Integrating it gives Bernoulli's equation.

```text
              (p + dp) dA
                 <---
            ____________
   p dA --> |  element  |  ds, along streamline
            |___________|   (inclined at theta)
                 |
                 v  weight = rho g dA ds
```

#### Forces on the element
Consider an element of area $dA$ and length $ds$ along a streamline inclined at $\theta$ to the horizontal.
1. Pressure force in the direction of flow: $p\,dA$.
2. Pressure force opposite to flow: $(p + dp)\,dA$.
3. Weight component along $s$: $-\rho g\, dA\, ds\sin\theta$, where $\sin\theta = dz/ds$.

#### Newton's second law
Mass × acceleration $= \rho\, dA\, ds\, a_s$, where for steady flow $a_s = V\dfrac{dV}{ds}$:
$$
p\,dA - (p + dp)\,dA - \rho g\,dA\,ds\frac{dz}{ds} = \rho\,dA\,ds\, V\frac{dV}{ds}
$$
This simplifies to $-dp\,dA - \rho g\,dA\,dz = \rho\,dA\,V\,dV$. Dividing by $\rho\, dA$:
$$
\frac{dp}{\rho} + g\,dz + V\,dV = 0
$$
This is **Euler's equation of motion**.

#### Integration (Bernoulli's equation)
For incompressible flow ($\rho$ constant):
$$
\int \frac{dp}{\rho} + \int g\,dz + \int V\,dV = \text{constant}
$$
$$
\frac{p}{\rho} + gz + \frac{V^2}{2} = \text{constant}
$$
Dividing by $g$:
$$
\frac{p}{\rho g} + \frac{V^2}{2g} + z = \text{constant}
$$
This is **Bernoulli's equation**: pressure head + velocity head + datum head is constant along a streamline.

Assumptions: ideal (frictionless) fluid, steady, incompressible and irrotational flow along a streamline.

## Check yourself

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

1. What principle gives Euler's equation of motion along a streamline?
   A. Conservation of mass
   B. Conservation of momentum with friction
   C. Newton's second law with pressure and gravity forces
   D. First law of thermodynamics
   Answer: C. Euler's equation applies F = ma to a fluid element considering only pressure and gravity (no viscosity). (Key point: What Euler's equation is)

2. Which is Euler's equation as derived?
   A. dp/rho - g dz + V dV = 0
   B. p/rho + V = constant
   C. dp/rho + g dz + V dV = 0
   D. dp + rho g dz = 0
   Answer: C. Dividing the force balance by rho dA ds gives dp/rho + g dz + V dV = 0. (Key point: Euler's equation form)

3. What step turns Euler's equation into Bernoulli's equation?
   A. Set V = 0
   B. Multiply by area
   C. Integrate along the streamline for constant rho
   D. Differentiate again
   Answer: C. Integration for incompressible flow gives p/rho + gz + V^2/2 = constant; dividing by g gives heads. (Key point: Step to Bernoulli)

4. Which set of assumptions is needed?
   A. Turbulent and rotational
   B. Uniform pressure everywhere
   C. Ideal, steady, incompressible, irrotational along a streamline
   D. Viscous, unsteady, compressible
   Answer: C. Bernoulli's equation needs frictionless, steady, incompressible flow along a streamline. (Key point: Assumptions)

## Related in Hydro Power

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- [Bernoulli's equation limitations and application](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Abernoulli-limitations-application) — Hydro Power, chapter 1
- [Continuity equation basis and Bernoulli equation](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Acontinuity-and-bernoulli) — Hydro Power, chapter 1
- [Bernoulli's equation in real practice](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Abernoulli-real-practice) — Hydro Power, chapter 1
- [Types of heads and Bernoulli's assumptions](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aheads-bernoulli-assumptions) — Hydro Power, chapter 1, asked 2×

## Related in other subjects

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