Bernoulli’s Principle – Definition, Equation, Examples

Bernoulli’s principle states that as the speed of a fluid increases, its pressure decreases. It is one of the most important principles in fluid dynamics and explains phenomena ranging from airplane lift and perfume atomizers to blood flow and chimney drafts. The principle arises from conservation of energy in moving fluids and connects pressure energy, kinetic energy, and gravitational potential energy.

Bernoulli’s principle is central to physics, engineering, meteorology, medicine, and aerodynamics. It helps explain how fluids behave in pipes, rivers, air currents, and atmospheric systems. Although often introduced using idealized fluids, the principle remains extremely useful in real-world situations when applied carefully and with an understanding of its assumptions and limitations.

Key Takeaways: Bernoulli’s Principle

• Bernoulli’s principle states that faster-moving fluids exert lower pressure.
• The principle comes from conservation of energy in a flowing fluid.
• Bernoulli’s equation relates pressure, velocity, and height in a fluid.
• The principle applies best to incompressible, nonviscous, steady fluid flow.
• Bernoulli’s principle explains airplane lift, Venturi tubes, atomizers, carburetors, and many biological processes.
• Bernoulli’s principle alone does not fully explain lift on an airfoil, because circulation, viscosity, and Newton’s laws also matter.
• Real fluids experience friction, turbulence, and energy loss, which limit ideal Bernoulli behavior.

History of Bernoulli’s Principle

Bernoulli’s principle is named after the Swiss mathematician and physicist Daniel Bernoulli. He published the principle in his 1738 book Hydrodynamica, one of the foundational works of fluid mechanics.

Daniel Bernoulli came from a famous family of mathematicians. His father, Johann Bernoulli, and uncle, Jacob Bernoulli, were also important contributors to mathematics and physics. Daniel studied medicine, mathematics, and mechanics, eventually focusing on the behavior of fluids.

Bernoulli recognized that a moving fluid contains different forms of energy:

• Pressure energy
• Kinetic energy
• Gravitational potential energy

He showed that these forms can convert into one another while the total energy remains constant in ideal flow. This insight laid the foundation for modern fluid dynamics.

Later scientists expanded Bernoulli’s work:

• Leonhard Euler developed the Euler equations for fluid flow.
• Claude-Louis Navier and George Gabriel Stokes incorporated viscosity into fluid dynamics, producing the Navier–Stokes equations.
• Ludwig Prandtl introduced boundary-layer theory, helping explain real aerodynamic lift.

What Is Bernoulli’s Principle?

Bernoulli’s principle states:

In a flowing fluid, regions where the fluid moves faster have lower pressure than regions where the fluid moves more slowly.

The principle follows from energy conservation. If fluid speed increases, kinetic energy increases. Since total energy remains constant, pressure energy must decrease.

A common demonstration involves blowing across the top of a sheet of paper. The air above the paper moves faster, reducing pressure above the sheet. The higher pressure below pushes the paper upward.

Another example occurs when narrowing a hose nozzle. Water speeds up as it passes through the smaller opening, while pressure decreases.

Bernoulli’s Equation

Bernoulli’s equation mathematically expresses the relationship among pressure, velocity, and height in a fluid.

$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$

Where:

• $P$ = fluid pressure
• $\rho$ = fluid density
• $v$ = fluid speed
• $g$ = acceleration due to gravity
• $h$ = height above a reference point

The three terms represent:

1. Pressure energy per unit volume
2. Kinetic energy per unit volume
3. Gravitational potential energy per unit volume

For two points in a flowing fluid:

$P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2$

This form is especially useful in solving practical fluid-flow problems.

Bernoulli Equation in Different Forms

Bernoulli’s equation appears in several equivalent forms depending on the application. Each form expresses conservation of mechanical energy in fluid flow.

Standard Pressure Form

The most common version relates pressure, velocity, and height:

$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$

This form expresses energy per unit volume.

Energy Per Unit Mass Form

Dividing the equation by density gives energy per unit mass:

$\frac{P}{\rho} + \frac{v^2}{2} + gh = \text{constant}$

This form is common in theoretical fluid mechanics.

Head Form of Bernoulli’s Equation

Dividing by $\rho g$ produces the “head” form used in hydraulics and civil engineering:

$\frac{P}{\rho g} + \frac{v^2}{2g} + h = \text{constant}$

The terms represent:

• Pressure head
• Velocity head
• Elevation head

Each term has units of length, usually meters.

Engineers often interpret these quantities as equivalent heights of fluid columns.

Bernoulli Equation with Energy Losses

Real fluids lose energy because of friction and turbulence. Engineers therefore often include a loss term:

$P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2 + \text{losses}$

This modified form better describes real pipe systems.

Derivation of Bernoulli’s Principle

Derivation from Conservation of Energy

Bernoulli’s equation comes directly from the conservation of mechanical energy.

Consider a small volume of fluid moving through a pipe. As it moves:

• Pressure forces perform work on the fluid.
• The fluid’s speed may change.
• The fluid’s height may change.

The work done on the fluid equals the change in kinetic plus potential energy.

For an incompressible fluid:$$W = \Delta KE + \Delta PE$$

Pressure work is:$$P_1V – P_2V$$

Change in kinetic energy:$$\frac{1}{2}\rho V(v_2^2 – v_1^2)$$

Change in gravitational potential energy:$$\rho gV(h_2 – h_1)$$

Combining these expressions and dividing by volume yields Bernoulli’s equation.

Derivation from Euler’s Equation

Bernoulli’s equation can also be derived from the Euler equations for inviscid flow. This more advanced derivation treats fluid motion continuously rather than as moving packets of fluid.

This approach is common in advanced fluid mechanics and aerodynamics courses.

Bernoulli’s Principle and Continuity

Bernoulli’s principle often works together with the continuity equation.

For incompressible flow:

$A_1v_1 = A_2v_2$

Where:

• $A$ = cross-sectional area
• $v$ = flow speed

If area decreases, velocity increases.

The continuity equation explains why fluid speeds up in narrower regions, while Bernoulli’s equation explains the associated pressure change.

Worked Example Problem

Water Flowing Through a Pipe

Water flows through a horizontal pipe that narrows from a diameter of 6.0 cm to 3.0 cm. The speed in the wider section is 2.0 m/s. Find:

1. The speed in the narrow section
2. The pressure difference between the sections

Assume incompressible flow and neglect viscosity.

Step 1: Use the Continuity Equation

The continuity equation states:

$A_1v_1 = A_2v_2$

Because area is proportional to diameter squared:$$\frac{A_1}{A_2} = \left(\frac{6.0}{3.0}\right)^2 = 4$$

So:$$v_2 = 4v_1$$$$v_2 = 4(2.0) = 8.0 \text{ m/s}$$

Step 2: Apply Bernoulli’s Equation

Since the pipe is horizontal:$$P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$$

Rearranging:$$P_1 – P_2 = \frac{1}{2}\rho(v_2^2 – v_1^2)$$

Using:

• $\rho = 1000 \text{ kg/m}^3$ρ=1000 kg/m3
• $v_1 = 2.0 \text{ m/s}$v1​=2.0 m/s
• $v_2 = 8.0 \text{ m/s}$v2​=8.0 m/s

$$P_1 – P_2 = \frac{1}{2}(1000)(64 – 4)$$ $$P_1 – P_2 = 30000 \text{ Pa}$$

Answer

• Speed in narrow section: 8.0 m/s
• Pressure decrease: 30,000 Pa

The faster-moving water has lower pressure.

Assumptions of Bernoulli’s Principle

Bernoulli’s equation relies on several assumptions.

Incompressible Flow

The fluid density remains constant.

This assumption works well for liquids and for gases moving much slower than the speed of sound.

Nonviscous Flow

The fluid has negligible internal friction.

Real fluids always have viscosity, but the approximation is often reasonable.

Steady Flow

Fluid properties do not change with time at a given point.

Flow Along a Streamline

The equation applies along a streamline unless the flow is irrotational.

No External Energy Input

No pumps, turbines, or heat transfer add or remove energy.

Limitations of Bernoulli’s Principle

Bernoulli’s principle is extremely useful, but it does not apply universally.

Viscosity and Friction

Real fluids lose energy through friction. This energy becomes heat.

Viscous losses are especially important in:

• Long pipes
• Turbulent flow
• Blood circulation
• Boundary layers

Turbulence

Bernoulli’s equation works best for smooth laminar flow. Turbulent flow introduces chaotic motion and energy dissipation.

Compressible Flow

At high speeds, especially above Mach 0.3, gas density changes become significant.

In supersonic flow, compressibility effects dominate.

Rotational Flow

Strong vortices or rotating fluids may violate simple Bernoulli assumptions.

Misapplication to Open Systems

Bernoulli’s equation should not be applied between unrelated fluid regions unless they lie along the same streamline or the flow is irrotational.

Examples of Bernoulli’s Principle

Venturi Tube

A Venturi tube narrows in the middle. Fluid speed increases in the narrow section while pressure decreases.

Venturi tubes measure flow rates in pipes.

Perfume Atomizer

Fast-moving air over a tube lowers pressure and draws liquid upward into a spray.

Chimneys

Wind moving rapidly across the top of a chimney lowers pressure, helping smoke rise.

Shower Curtain Effect

A shower curtain sometimes moves inward because fast-moving air from the water stream lowers pressure inside the shower.

Curveballs and Spinning Balls

A spinning ball drags air around with it. Air moves faster on one side, creating pressure differences that curve the ball’s path.

This effect is called the Magnus effect.

Blood Flow

Bernoulli’s principle helps explain pressure changes in narrowed blood vessels.

Doctors use related principles in medical diagnostics.

Real-World Demonstrations and Experiments

Bernoulli’s principle is easy to demonstrate using inexpensive classroom materials. These demonstrations help students visualize the relationship between fluid speed and pressure.

Paper Strip Lift Demonstration

Hold a strip of paper below your lower lip and blow across the top.

The paper rises instead of falling.

Fast-moving air above the paper lowers pressure above the strip. Higher pressure below pushes the paper upward.

Ping-Pong Ball and Hair Dryer

Place a ping-pong ball in the upward air stream from a hair dryer.

The ball remains suspended in the air stream.

Fast-moving air around the ball creates lower pressure, while surrounding air pressure helps stabilize the ball in the moving stream.

Blowing Between Hanging Sheets of Paper

Hang two sheets of paper a few centimeters apart and blow between them.

Instead of moving apart, the sheets move together.

The faster-moving air between the sheets lowers pressure between them compared with the surrounding air.

Balloon Levitation Demonstration

A lightweight ball or balloon can float in a stream of air from a blower or leaf blower.

The air moving around the object produces pressure differences that stabilize the object in the flow.

Venturi Bottle Sprayer

A straw placed into liquid while air blows across the top causes the liquid to rise and spray outward.

Fast-moving air lowers pressure at the top of the tube, allowing atmospheric pressure to push liquid upward.

Bernoulli Bag Demonstration

A large plastic bag inflates rapidly when air is blown into it from several centimeters away rather than with the mouth sealed to the opening.

The moving air entrains surrounding air into the bag due to pressure differences and fluid momentum transfer.

Applications of Bernoulli’s Principle

Bernoulli’s principle has many practical applications.

Aviation

Aircraft wings generate lift partly through pressure differences associated with airflow.

Engineering

Applications include:

• Flow meters
• Carburetors
• Fuel injectors
• Ventilation systems
• Hydraulic systems

Meteorology

Pressure differences associated with air motion influence weather systems and wind formation.

Medicine

Bernoulli-based calculations help estimate blood flow speed and pressure in arteries and heart valves.

Sports

The principle helps explain:

• Curveballs
• Golf ball flight
• Soccer free kicks
• Tennis spin

Bernoulli’s Principle and Airfoil Lift

Airfoil lift is one of the most famous applications of Bernoulli’s principle.

An airfoil, such as an airplane wing, changes airflow around it. Air generally moves faster over the curved upper surface than beneath the wing.

According to Bernoulli’s principle:

• Faster airflow above the wing creates lower pressure.
• Slower airflow below creates higher pressure.
• The pressure difference produces lift.

However, this explanation alone is incomplete.

The Modern Understanding of Lift

Lift involves several related effects:

• Pressure differences
• Downward deflection of air
• Circulation around the wing
• Viscosity and boundary layers
• Newton’s third law

The wing redirects air downward. The reaction force pushes the wing upward.

Bernoulli’s principle and Newton’s laws are complementary descriptions, not competing explanations.

Common Oversimplification

A popular but incorrect explanation claims that air traveling over the top of the wing must meet the air below at the trailing edge simultaneously.

This “equal transit time” idea is false. Air over the top typically moves much faster and reaches the trailing edge sooner.

Boundary Layers and Flow Separation

Bernoulli’s equation assumes nonviscous flow, but real fluids have viscosity. Near solid surfaces, viscosity becomes extremely important.

Boundary Layers

A boundary layer is a thin region of fluid near a surface where viscosity strongly affects motion.

At the surface itself, the fluid velocity is nearly zero relative to the surface because of the no-slip condition. Velocity gradually increases farther from the surface.

Boundary layers strongly influence:

• Drag
• Lift
• Turbulence
• Heat transfer

Aircraft wings rely heavily on boundary-layer behavior.

Laminar and Turbulent Boundary Layers

Boundary layers may be:

• Laminar, with smooth orderly flow
• Turbulent, with chaotic mixing

Turbulent boundary layers increase drag but often resist flow separation better than laminar layers.

Flow Separation

Flow separation occurs when fluid no longer follows the contour of a surface.

For an airfoil, separation often happens when the angle of attack becomes too large. The airflow detaches from the upper surface, greatly reducing lift.

This condition is called a stall.

Why Flow Separation Matters

Flow separation increases:

• Drag
• Turbulence
• Energy loss

It decreases:

• Lift
• Aerodynamic efficiency

Modern aircraft design carefully manages boundary layers to delay separation and reduce drag.

Compressible Flow and Supersonic Effects

Bernoulli’s equation usually assumes incompressible flow, but gases become compressible at high speeds.

Compressible Flow

When gas speed approaches a significant fraction of the speed of sound, density changes become important.

For air, compressibility effects become significant around Mach 0.3.

Under these conditions:

• Density changes cannot be ignored.
• Pressure and temperature vary more strongly.
• Simple Bernoulli equations become inaccurate.

More advanced compressible-flow equations are required.

Supersonic Flow

When flow exceeds the speed of sound:

• Shock waves form.
• Pressure changes become abrupt.
• Energy losses increase dramatically.

Supersonic aircraft and rockets must account for:

• Shock heating
• Wave drag
• Compressibility effects

The simple incompressible form of Bernoulli’s equation does not adequately describe these flows.

Stagnation Pressure

Compressible-flow analysis often uses stagnation pressure, which is the pressure a moving fluid would have if brought to rest without energy loss.

This concept is important in:

• Jet engines
• Wind tunnels
• Pitot tubes
• Aerospace engineering

Bernoulli’s Principle vs Pascal’s Principle vs Archimedes’ Principle

Fluid mechanics includes several important principles that describe how fluids behave under different conditions. Students often confuse Bernoulli’s principle, Pascal’s principle, and Archimedes’ principle because all three involve fluids and pressure. However, they describe very different phenomena.

How the Principles Relate

All three principles involve fluids, but they focus on different aspects of fluid behavior:

• Bernoulli’s principle describes energy changes in moving fluids.
• Pascal’s principle describes pressure transmission in confined fluids.
• Archimedes’ principle describes buoyancy and displacement.

Real systems often involve more than one principle at the same time. For example, an airplane wing involves Bernoulli’s principle for airflow and Archimedes’ principle in a very small way because the aircraft displaces air.

Common Misconceptions About Bernoulli’s Principle

“Faster Fluids Always Have Lower Pressure”

Not always. The relationship depends on the flow conditions and streamline behavior.

“Bernoulli’s Principle Violates Newton’s Laws”

It does not. Bernoulli’s principle follows from conservation of energy and is fully consistent with Newtonian mechanics.

“Lift Comes Only from Bernoulli’s Principle”

Lift also depends on circulation, viscosity, and downward momentum transfer.

“Pressure Drops Because Molecules Are Farther Apart”

Pressure changes arise from energy conservation and fluid dynamics, not simply molecular spacing.

“Bernoulli’s Principle Works Everywhere”

The principle only applies under specific assumptions.

FAQs

Is Bernoulli’s principle a law?

It is usually called a principle or equation rather than a law because it depends on assumptions about fluid behavior.

Does Bernoulli’s principle apply to gases?

Yes. It applies to gases when compressibility effects are small.

Why does pressure decrease when velocity increases?

Energy conservation requires an increase in kinetic energy to come from somewhere. Pressure energy decreases as velocity increases.

Is Bernoulli’s principle the same as the Venturi effect?

No. The Venturi effect is a specific application of Bernoulli’s principle combined with continuity.

Does Bernoulli’s principle explain tornadoes?

Partly. Pressure differences and fluid motion play roles, but tornadoes involve complex atmospheric dynamics beyond simple Bernoulli flow.

Can Bernoulli’s principle explain why roofs blow off during storms?

Yes. Fast-moving air above a roof can lower external pressure while indoor pressure remains higher, producing an upward force.

Is Bernoulli’s equation valid for turbulent flow?

Not strictly. Turbulence introduces energy losses and chaotic motion that violate ideal assumptions.

Why do airplanes still fly upside down?

Airplanes can fly inverted because lift depends strongly on angle of attack and airflow redirection, not just wing shape.

References and Further Reading

• Batchelor, G.K. (2000). An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press. ISBN 978-0-521-66396-0.
• Darrigol, O.; Frisch, U. (2008). “From Newton’s mechanics to Euler’s equations”. Physica D: Nonlinear Phenomena. 237 (14–17): 1855–1869. doi:10.1016/j.physd.2007.08.003
• Mulley, Raymond (2004). Flow of Industrial Fluids: Theory and Equations. CRC Press. ISBN 978-0-8493-2767-4.
• Oertel, Herbert; Prandtl, Ludwig; Böhle, M.; Mayes, Katherine (2004). Prandtl’s Essentials of Fluid Mechanics. Springer. ISBN 978-0-387-40437-0.
• Tipler, Paul (1991). Physics for Scientists and Engineers: Mechanics (3rd extended ed.). W. H. Freeman. ISBN 978-0-87901-432-2.