CHAPTER02ARTICLE03

Fluid mechanics / Write conservation laws

FLUID MECHANICS · 02–03 / BASIC

Bernoulli equation

Formulate the relation among pressure, velocity and elevation along a streamline as conservation of mechanical energy.

11 min read2026-08-13Sources checkedJA version

Abstract

Bernoulli's equation expresses conservation of mechanical energy for steady, incompressible and inviscid flow. Along a streamline, pressure, velocity and elevation contributions interchange while their sum remains constant.

02. Understand in 30 seconds

Bernoulli's equation is conservation of mechanical energy for a fluid

It compares pressure energy, kinetic energy and gravitational potential energy along the same streamline. In a level pipe with no loss, for example, an increase in speed is accompanied by a decrease in static pressure.

PressureAbility of pressure to do work

\dfrac{p}{\rho}

SpeedKinetic energy of the flow

\dfrac{V^2}{2}

ElevationGravitational potential energy

gz

03. Principle and equations

For steady, inviscid, incompressible flow, compare points 1 and 2 on the same streamline. The mechanical energy per unit mass is

Bernoulli's equation per unit mass

\frac{p}{\rho}+\frac{V^2}{2}+gz=\mathrm{const.}

All three terms are energy per unit mass, with units

\mathrm{J\,kg^{-1}}

, equivalent to

\mathrm{m^2\,s^{-2}}

Two-point form

\frac{p_1}{\rho}+\frac{V_1^2}{2}+gz_1=\frac{p_2}{\rho}+\frac{V_2^2}{2}+gz_2

The energy present at point 1 appears at point 2 as a different combination of pressure, speed and elevation.

Head form

Dividing the equation by $g$ expresses every term as an equivalent fluid-column height.

\frac{p}{\rho g}+\frac{V^2}{2g}+z=\mathrm{const.}

The terms are pressure head, velocity head and elevation head. Each has units of metres.

04. Symbols and units

Symbol	Quantity	SI unit	Meaning
$p$	Static pressure	$\mathrm{Pa}$	Pressure exerted by the fluid on its surroundings
$\rho$	Density	$\mathrm{kg\,m^{-3}}$	Mass per unit volume
$V$	Flow speed	$\mathrm{m\,s^{-1}}$	Magnitude of velocity along the streamline
$g$	Gravitational acceleration	$\mathrm{m\,s^{-2}}$	Standard gravity is $9.80665\ \mathrm{m\,s^{-2}}$
$z$	Elevation above datum	$\mathrm{m}$	Elevation used for potential energy
$q$	Dynamic pressure	$\mathrm{Pa}$	$q=\frac{1}{2}\rho V^2$

Absolute or gauge pressure may be used, provided the same pressure reference is used at both points.

05. Derivation

Apply the equation of motion along a streamline to a fluid element with negligible viscosity. This is Euler's equation projected in the streamline direction.

Force balance along the streamline
The pressure gradient and gravity accelerate the fluid element.
$\frac{1}{\rho}\,\mathrm{d}p+V\,\mathrm{d}V+g\,\mathrm{d}z=0$
Integrate at constant density
Integrate each term from point 1 to point 2.
$\int_1^2\frac{\mathrm{d}p}{\rho}+\int_1^2 V\,\mathrm{d}V+\int_1^2 g\,\mathrm{d}z=0$
Evaluate the terms
Combining the pressure, kinetic and elevation terms gives Bernoulli's equation.
$\frac{p}{\rho}+\frac{V^2}{2}+gz=C$

Important: the derivation does not say that pressure alone directly determines fluid speed.

The boundary conditions and equations of motion determine the flow field; pressure, speed and elevation along a streamline then satisfy this relation.

06. Conditions of use

✓ Conditions for the basic form

• Steady flow

• Inviscid flow, or negligible loss

• Nearly constant density

• Two points on the same streamline

• No pump or turbine work between the points

! When the basic form is insufficient

• Long or narrow pipes with appreciable friction

• Pumps, fans or turbines

• High-speed gas flow with large density changes

• Strongly unsteady flow

• Comparison across streamlines in rotational flow

Engineering calculations include losses and machine work

Pipe calculations include pump head $h_{\mathrm{p}}$ and head loss $h_{\mathrm{L}}$ in the mechanical energy equation.

\frac{p_1}{\rho g}+\frac{V_1^2}{2g}+z_1+h_{\mathrm{p}}=\frac{p_2}{\rho g}+\frac{V_2^2}{2g}+z_2+h_{\mathrm{L}}

In a real flow, viscosity dissipates mechanical energy as heat, so head loss is normally greater than zero.

07. Worked example

Find the downstream pressure in a horizontal contraction

Water of density $1000\ \mathrm{kg\,m^{-3}}$ flows through a horizontal pipe. At point 1, the static pressure is $200\ \mathrm{kPa}$ and the mean speed is $2.0\ \mathrm{m\,s^{-1}}$ . At point 2, the mean speed is $8.0\ \mathrm{m\,s^{-1}}$ . Neglect losses and find the static pressure at point 2.

\rho=1000\ \mathrm{kg\,m^{-3}}

p_1=200\ \mathrm{kPa}

V_1=2.0\ \mathrm{m\,s^{-1}}

V_2=8.0\ \mathrm{m\,s^{-1}}

Cancel the elevation terms
The pipe is horizontal, so $z_1=z_2$ .
Rearrange for the pressure at point 2 $p_2=p_1+\frac{\rho}{2}\left(V_1^2-V_2^2\right)$
Substitute the values $p_2=200000+\frac{1000}{2}\left(2.0^2-8.0^2\right)=170000\ \mathrm{Pa}$

p_2=170\ \mathrm{kPa}

Check: the increase in speed is accompanied by a 30 kPa decrease in static pressure. In a real pipe, friction adds a loss, so for the same inlet conditions the downstream pressure would be lower than this ideal result.

08. Calculator

Calculate the static pressure at point 2

This calculator uses the steady, inviscid, incompressible form. Use a consistent pressure reference and a common elevation datum at both points.

Density ρ

\mathrm{kg\,m^{-3}}

Static pressure p₁

\mathrm{kPa}

Speed V₁

\mathrm{m\,s^{-1}}

Speed V₂

\mathrm{m\,s^{-1}}

Elevation z₁

\mathrm{m}

Elevation z₂

\mathrm{m}

Static pressure p₂170.000 kPa

Static-pressure decrease p₁ − p₂30.000 kPa

Verified against the worked example: 170 kPa. Friction loss, pump work and compressibility are not included.

09. Common mistakes

Faster always means lower pressure

The relation requires two points on a streamline with elevation, losses and machine work properly accounted for. Unrelated flows cannot be compared from speed alone.

It is the continuity equation

Continuity expresses conservation of mass; Bernoulli expresses idealised conservation of mechanical energy. A contraction problem normally uses both.

It always works in pipes

Viscous losses are often important in long or narrow pipes. The mechanical energy equation with head loss is then required.

One constant for every streamline

In general, the Bernoulli constant may differ between streamlines. It is common to all streamlines only when the flow is also irrotational.

It alone explains lift

Bernoulli's equation relates local pressure and speed around an airfoil, but does not by itself determine why that velocity distribution exists or set the circulation. The equal-transit-time explanation is incorrect.

10. Applications

Venturi meterPitot-static tubeNozzleSiphonTank dischargeAirfoil pressure distribution

A common engineering workflow is to obtain speeds from continuity, then determine pressures or required pump head from Bernoulli's equation or the mechanical energy equation with losses.

11. Prerequisites and related topics

Prerequisites

Continuity equationConservation of mass connecting area and speedPressure and densityStatic, absolute and gauge pressure; density

Where to go next

Euler equationLocal equation of motion for inviscid flowPipe friction and pressure lossAdd viscous losses to real pipe calculationsNavier–Stokes equationsGeneral fluid motion including viscosityPitot and Venturi metersUse pressure differences to measure speed or flow rate

Sources

Primary sourceFrank M. White, Fluid Mechanics, 9th ed., McGraw Hill, 2021. Used to verify the equation, derivation and conditions of use.
University notesMIT OpenCourseWare, “Bernoulli Equation, Pitot-Static Tube, Airspeed Measurement, and Pressure Nondimensionalization.” Used to cross-check the equation and measurement applications.
Supporting guideNASA Glenn Research Center, “Bernoulli’s Equation.” Used to check the beginner explanation, dynamic pressure and limitations.
Units and styleBIPM, The International System of Units (SI Brochure), 9th ed., revised 2026. Used for Pa, m, s, kg and SI notation.

Editorial note

The primary source was cross-checked against independent university notes, a public-agency guide and the SI standard. No independent review by an external fluid-mechanics specialist has yet been performed.

Abstract

0202. Understand in 30 seconds

0303. Principle and equations