---
title: "Bernoulli from Euler with head losses"
subject: "Fluid Mechanics"
chapter: 5
key: "semester_3/fluid_mechanics:euler-to-bernoulli-with-losses"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/euler-to-bernoulli-with-losses
site: BCT Engineering
---

# Bernoulli from Euler with head losses

Fluid Mechanics · Chapter 5: Hydrodynamics

- Asked in: 2073 Shrawan, 2+2 marks
- Video: [NE 3221   M04   07 Extended Bernoulli Equation](https://www.youtube.com/watch?v=yh4tyctloZE) — Joshua Schlegel
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/euler-to-bernoulli-with-losses
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-5

## Question

Integrate Euler's equation along a streamline and obtain Bernoulli's equation (No derivation of Euler equation required). What will be the Bernoulli's equation between two points where there are head losses, work done by a machine (turbine) and energy supplied by the machine (pump) between those points.

## Short answer

Integrating Euler's equation dp/ρ + V dV + g dz = 0 for steady incompressible inviscid flow along a streamline yields p/γ + V²/(2g) + z = constant. With head losses hL, pump head Hp, and turbine head HT between points 1 and 2, the equation becomes p1/γ + V1²/(2g) + z1 + Hp - HT - hL = p2/γ + V2²/(2g) + z2.

## Answer

#### Euler's equation to Bernoulli's equation

Euler's equation of motion along a streamline $s$ for a steady flow of an ideal fluid:

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

Integrating along the streamline for an incompressible fluid ($\rho$ constant):

$$
\frac{p}{\rho} + \frac{V^2}{2} + gz = \text{constant}
$$

Dividing by $g$:

$$
\frac{p}{\gamma} + \frac{V^2}{2g} + z = H = \text{constant}
$$

This is **Bernoulli's equation**: the sum of pressure head, velocity head and elevation head is constant along a streamline (steady, incompressible, frictionless flow).

#### With head loss, turbine and pump

Between points 1 and 2, with energy supplied by a pump $H_p$ (added to the fluid), energy extracted by a turbine $H_T$ and head loss $h_L$:

$$
\frac{p_1}{\gamma} + \frac{V_1^2}{2g} + z_1 + H_p - H_T - h_L = \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + z_2
$$

- $h_L$ = head lost to friction and minor losses between 1 and 2 (always positive).
- $H_p$ = head added to the flow by the pump (positive).
- $H_T$ = head taken from the flow by the turbine (positive).

Power: pump $P = \gamma Q H_p/\eta_p$ (input), and turbine $P = \eta_T\gamma Q H_T$ (output).

## Check yourself

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

1. Euler's equation along a streamline:
   A. dp + rho V dV = 0
   B. dp/rho = g dz
   C. dp/rho + V dV + g dz = 0
   D. V dV = mu d2V
   Answer: C. Integrating gives Bernoulli. (Key point: Euler equation)

2. After integrating for incompressible flow:
   A. p + rho V = const
   B. pV = const
   C. p/rho = const
   D. p/rho + V^2/2 + gz = const
   Answer: D. Divide by g to get heads. (Key point: Integration)

3. Where does the pump head appear?
   A. Added on the upstream side
   B. Subtracted
   C. Added to the downstream side
   D. Multiplied
   Answer: A. Hp is added with the inlet; HT and hL subtracted. (Key point: Energy equation)

4. Turbine head HT is:
   A. Head added
   B. Head taken from the flow
   C. A loss
   D. Pressure
   Answer: B. Turbine extracts energy. (Key point: Turbine)

## Related in Fluid Mechanics

- [Bernoulli's equation and four applications](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Abernoulli-from-euler-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
- [Acceleration and streamline curvature, u = Axy](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aacceleration-radius-curvature-streamline) — Fluid Mechanics, chapter 4
- [Streamline and time, particle x = 1 to 2](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Astreamline-time-ax-ayx2-x1-2) — Fluid Mechanics, chapter 4
- [Streamline and time in $A/x$, $Ay/x^2$ field](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Astreamline-time-ax-ayx2-x1-3) — Fluid Mechanics, chapter 4
- [Pump horsepower, TEL and HGL](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Apump-horsepower-tel-hgl) — Fluid Mechanics, chapter 5

## Related in other subjects

- [Derivation of Euler's turbine equation](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Aeuler-turbine-equation-derivation) — Fluid Machines, chapter 1
- [Deriving Bernoulli's equation from Euler's equation](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aeuler-bernoulli-derivation) — Hydro Power, chapter 1
- [Euler equation, Bernoulli and Pitot tube](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Aeuler-equation-pitot-numerical) — Applied Computational Fluid Dynamics, chapter 2
- [Euler work equation for turbomachines](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Aeuler-turbine-equation-derivation) — Turbo Machines, chapter 2