Bernoulli's Principle Equation, Derivation & Rearrangements

Core idea

Overview

Bernoulli's Principle is a fundamental expression of the conservation of energy for flowing fluids, relating pressure, velocity, and elevation. It dictates that in a steady flow of an incompressible, frictionless fluid, an increase in speed occurs simultaneously with a decrease in static pressure or potential energy.

When to use: Apply this equation to steady, incompressible, and inviscid flows along a streamline where friction and heat transfer are negligible. It is primarily used to analyze fluid behavior in closed conduits, calculate flow through orifices, or determine lift on aerodynamic surfaces.

Why it matters: This principle is the cornerstone of aerodynamics and hydraulics, explaining how aircraft wings generate lift and how venturi meters measure flow rates. It allows engineers to predict pressure changes in complex piping networks and design efficient fluid transport systems.

Symbols

Variables

H = Total Pressure, P = Static Pressure, $ρ$ = Density, v = Velocity, g = Gravity

H

Total Pressure

Pa

P

Static Pressure

Pa

ρ

Density

k g / m^{3}

v

Velocity

m/s

g

Gravity

m / s^{2}

h

Height

m

Walkthrough

Derivation

Understanding Bernoulli's Equation

Bernoulli’s equation applies conservation of energy to fluid flow, relating pressure, speed, and height along a streamline.

Fluid is incompressible and inviscid (negligible viscosity).
Flow is steady and along a streamline.

1

State Bernoulli’s Equation (Along a Streamline):

Static pressure, kinetic energy per volume, and gravitational potential energy per volume sum to a constant along a streamline.

P + \frac{1}{2} ρ v^{2} + ρ g h = constant

2

Apply Between Two Points:

If speed increases in a constriction, pressure tends to decrease to keep the total energy per volume constant (when assumptions hold).

P_{1} + \frac{1}{2} ρ v_{1}^{2} + ρ g h_{1} = P_{2} + \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2}

Result

P_{1} + \frac{1}{2} ρ v_{1}^{2} + ρ g h_{1} = P_{2} + \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2}

Source: Standard curriculum — A-Level Fluid Mechanics

Visual intuition

Graph

Graph unavailable for this formula.

The graph of Total Pressure (H) against an independent variable like velocity (v) or height (h) is parabolic or linear depending on the chosen variable. Plotting against velocity results in a parabolic curve due to the squared term, while plotting against height produces a linear relationship with a constant slope of density times gravity.

Graph type: parabolic

Why it behaves this way

Intuition

Imagine water flowing steadily through a twisting pipe that changes both its diameter and its vertical height; Bernoulli's principle shows how the water's speed, internal pressure, and height adjust to keep its total

H

The total mechanical energy per unit volume of the fluid along a streamline.

Represents the constant sum of static pressure, dynamic pressure, and hydrostatic pressure, reflecting the conservation of energy in an ideal fluid flow.

P

Static pressure, the thermodynamic pressure of the fluid exerted equally in all directions.

The internal pressure of the fluid, which decreases when the fluid speeds up to maintain constant total energy.

\frac{1}{2} ρ v^{2}

Dynamic pressure, representing the kinetic energy per unit volume of the fluid due to its motion.

This term captures the energy associated with fluid movement; it increases significantly with fluid speed.

ρ g h

Hydrostatic pressure, representing the potential energy per unit volume of the fluid due to its elevation.

Accounts for the energy stored or released as fluid moves vertically against gravity.

ρ

Fluid density, the mass per unit volume of the fluid.

A measure of how much 'stuff' is packed into a given volume, directly influencing kinetic and potential energy terms.

v

Fluid velocity, the speed of the fluid flow along the streamline.

The primary driver of the dynamic pressure term; higher velocity means more kinetic energy.

g

Acceleration due to gravity.

The fundamental constant determining the strength of the gravitational potential energy.

h

Elevation or height of the fluid element above a reference datum.

The vertical position that dictates the fluid's gravitational potential energy.

Free study cues

Insight

Canonical usage

This equation requires all terms to have consistent units of pressure (or energy per unit volume) for dimensional homogeneity, typically in Pascals (Pa) in the SI system or pounds per square inch (psi)

Common confusion

Students frequently confuse the 'pressure form' of Bernoulli's equation with the 'head form', leading to incorrect unit assignments or calculations.

Unit systems

$H$ Pa · Represents the total mechanical energy per unit volume of the fluid, often referred to as total pressure.

$P$ Pa · The static pressure of the fluid at a point along the streamline.

$ρ$ kg/m^3 · The density of the fluid, assumed to be incompressible.

$v$ m/s · The fluid velocity at the point along the streamline.

$g$ m/s^2 · The acceleration due to gravity. A standard value of 9.80665 m/s^2 is often used unless specific local gravity is known.

$h$ m · The elevation or vertical height of the fluid relative to a chosen datum.

One free problem

Practice Problem

A horizontal water pipe has a total energy head H of 200000 Pa. If the water (density 1000 kg/m³) flows at 4 m/s at an elevation of 5 meters, determine the static pressure P within the pipe using g = 9.81 m/s².

Total Pressure200000 Pa

Density1000 kg/m^3

Velocity4 m/s

Gravity9.81 m/s^2

Height5 m

Solve for: $H$

Hint: Rearrange the formula to P = H - 0.5ρv² - ρgh.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When estimating pressure drop when pipe speed increases, Bernoulli's Principle is used to calculate Total Pressure from Static Pressure, Density, and Velocity. The result matters because it helps size components, compare operating conditions, or check a design margin.

Study smarter

Tips

Ensure all units are consistent, typically using Pascals for pressure, kg/m³ for density, and m/s for velocity.
The total head (H) remains constant only along a single streamline in the absence of energy-adding devices like pumps.
Verify that the fluid density (rho) does not change significantly, as this principle assumes incompressibility.

Avoid these traps

Common Mistakes

Ignoring energy losses in real pipes.
Mixing m and cm for height.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Bernoulli’s equation applies conservation of energy to fluid flow, relating pressure, speed, and height along a streamline.

Apply this equation to steady, incompressible, and inviscid flows along a streamline where friction and heat transfer are negligible. It is primarily used to analyze fluid behavior in closed conduits, calculate flow through orifices, or determine lift on aerodynamic surfaces.

This principle is the cornerstone of aerodynamics and hydraulics, explaining how aircraft wings generate lift and how venturi meters measure flow rates. It allows engineers to predict pressure changes in complex piping networks and design efficient fluid transport systems.

Ignoring energy losses in real pipes. Mixing m and cm for height.