Fluid Flow and Bernoulli's Principle
3. Bernoulli's Principle
Learning outcomes
- I can state Bernoulli's Principle.
- I can explain the relationship between pressure and fluid speed.
- I can describe how energy is conserved in flowing fluids.
- I can apply Bernoulli's Principle to explain observed phenomena.
- I can solve simple problems involving pressure and fluid flow.
Why does pressure change when a fluid speeds up? Why can a narrow section of pipe have faster-moving fluid but lower static pressure? How can moving air help produce forces on wings, sails, and roofs?
These questions involve one of the most important ideas in fluid dynamics: Bernoulli's Principle.
In a simplified steady-flow situation:
Where a fluid moves faster, its static pressure tends to be lower, provided other relevant conditions such as height are accounted for.
Bernoulli's Principle is fundamentally an application of conservation of energy to a moving fluid.
From Continuity to Bernoulli
In the previous topic, we learned the continuity equation:
A₁v₁ = A₂v₂
For steady flow of an approximately incompressible fluid:
narrower section → greater fluid speed
But this raises another question:
What happens to the pressure when the fluid speeds up?
Bernoulli's equation helps answer this.
For a horizontal pipe under idealized conditions:
greater speed ↔ lower static pressure
and:
lower speed ↔ higher static pressure
This connection between speed and pressure is one of the central ideas of Bernoulli's Principle.
Bernoulli's Principle
A useful statement of Bernoulli's Principle is:
For steady flow of an ideal fluid along a streamline, the total mechanical energy per unit volume remains constant.
That energy can appear in three main forms:
- pressure energy
- kinetic energy
- gravitational potential energy
If one form increases, another must decrease if no external energy is added and losses are negligible.
Bernoulli's Equation
The mathematical form is:
P + ½ρv² + ρgh = constant
Between two positions:
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
where:
- P = static pressure (Pa)
- ρ = fluid density (kg/m³)
- v = fluid speed (m/s)
- g = gravitational field strength (m/s² or N/kg)
- h = height (m)
Each term has units of:
Pa = J/m³
So each term represents energy per unit volume.
The Three Energy Terms
Pressure Term
P
This represents energy associated with the fluid's static pressure.
Kinetic Term
½ρv²
This represents kinetic energy per unit volume due to fluid motion.
Faster fluid has a larger kinetic term.
Gravitational Term
ρgh
This represents gravitational potential energy per unit volume.
Fluid at greater height has a larger gravitational term.
So Bernoulli's equation can be summarized as:
pressure energy + kinetic energy + gravitational potential energy = constant
under the ideal conditions assumed.
A Horizontal Pipe
Suppose a pipe remains at the same height.
Then:
h₁ = h₂
so the gravitational terms cancel.
Bernoulli's equation becomes:
P₁ + ½ρv₁² = P₂ + ½ρv₂²
Now the relationship becomes easier to see.
If:
v₂ > v₁
then:
½ρv₂² > ½ρv₁²
So the kinetic term has increased.
For the total to remain constant:
P₂ < P₁
Therefore:
In this ideal horizontal-flow case, faster-moving fluid has lower static pressure.
Continuity + Bernoulli
These two principles work together.
Continuity tells us:
narrower pipe → greater speed
for the same steady incompressible flow rate.
Bernoulli tells us:
In an ideal horizontal system:
greater speed → lower static pressure
Therefore:
wide section → slower fluid → higher static pressure
narrow section → faster fluid → lower static pressure
WIDE NARROW WIDE
slower flow faster flow slower flow
higher static P lower static P higher static P
┌──────────────┐ ┌───────┐ ┌──────────────┐
│ → → → │────────│→→→→→→→│────────│ → → → │
└──────────────┘ └───────┘ └──────────────┘This combination is extremely useful in fluid dynamics.
The Venturi Effect
A tube that narrows and then widens again can demonstrate Bernoulli's Principle.
This is often called a Venturi tube.
At the narrow region, called the throat:
- cross-sectional area decreases
- fluid velocity increases
- static pressure decreases
This pressure difference can be measured.
Venturi devices can therefore be used to determine fluid flow rate.
Why Doesn't the Narrow Section Have Higher Pressure?
This is a common source of confusion.
It is tempting to think:
"The fluid is squeezed into a smaller space, so its pressure must increase."
But in a flowing system, the fluid must accelerate as it enters the narrow section.
Energy is required to increase its kinetic energy.
Under ideal horizontal conditions, that energy comes partly from the pressure term.
Therefore:
pressure energy decreases → kinetic energy increases
So the narrow section can have lower static pressure, not higher static pressure.
Pressure and Speed
Consider an ideal horizontal flow.
If fluid speed increases:
kinetic energy per unit volume increases
Since:
P + ½ρv² = constant
the static pressure must decrease.
If fluid speed decreases:
kinetic energy per unit volume decreases
and static pressure can increase.
This is an example of energy transformation rather than energy disappearing.
Worked Example 1: Pressure Change
Water flows through a horizontal pipe.
At Point 1:
v₁ = 2 m/s
P₁ = 120 000 Pa
At Point 2:
v₂ = 6 m/s
Use:
ρ = 1000 kg/m³
Find P₂.
For a horizontal pipe:
P₁ + ½ρv₁² = P₂ + ½ρv₂²
Substitute:
120 000 + ½(1000)(2²) = P₂ + ½(1000)(6²)
Calculate the kinetic terms:
½(1000)(4) = 2000 Pa
½(1000)(36) = 18 000 Pa
Therefore:
120 000 + 2000 = P₂ + 18 000
122 000 = P₂ + 18 000
So:
P₂ = 104 000 Pa
Answer
P₂ = 104 000 Pa
The fluid moves faster at Point 2, so its static pressure is lower.
Checking the Answer
Before accepting a calculation, always ask:
Does the answer make physical sense?
The water sped up:
2 m/s → 6 m/s
Therefore, in the ideal horizontal case, static pressure should decrease.
Our answer shows:
120 000 Pa → 104 000 Pa
So the result is reasonable.
Worked Example 2: Finding Fluid Speed
Water flows through a horizontal pipe.
At Point 1:
P₁ = 150 000 Pa
v₁ = 3 m/s
At Point 2:
P₂ = 142 000 Pa
Use:
ρ = 1000 kg/m³
Find v₂.
Start with:
P₁ + ½ρv₁² = P₂ + ½ρv₂²
Substitute:
150 000 + ½(1000)(3²) = 142 000 + ½(1000)v₂²
150 000 + 4500 = 142 000 + 500v₂²
154 500 = 142 000 + 500v₂²
12 500 = 500v₂²
v₂² = 25
Therefore:
v₂ = 5 m/s
Answer
The fluid speed at Point 2 is:
5 m/s
Again, the lower-pressure location has the greater fluid speed.
What Happens When Height Changes?
Bernoulli's Principle also includes gravitational potential energy.
Consider fluid moving upward through a pipe.
The full equation is:
P + ½ρv² + ρgh = constant
If the fluid moves upward:
h increases
so gravitational potential energy increases.
If the speed remains approximately constant, this increase can come from a decrease in pressure.
Therefore:
greater height can mean lower pressure
even when fluid speed does not change.
This is why we must be careful with the simplified statement:
"faster fluid has lower pressure."
Height also matters.
Worked Example 3: Changing Height
Water flows through a pipe of constant diameter.
Point 2 is 5 m higher than Point 1.
Assume the water speed is the same at both points.
At Point 1:
P₁ = 200 000 Pa
Use:
ρ = 1000 kg/m³
g = 10 N/kg
Because the speeds are equal, the kinetic terms cancel.
Bernoulli's equation becomes:
P₁ + ρgh₁ = P₂ + ρgh₂
Take:
h₁ = 0 m
and:
h₂ = 5 m
Then:
200 000 = P₂ + (1000)(10)(5)
200 000 = P₂ + 50 000
Therefore:
P₂ = 150 000 Pa
Answer
The pressure at the higher point is:
150 000 Pa
Some pressure energy has been converted into gravitational potential energy.
Energy Conservation
Bernoulli's equation is really an energy equation.
Imagine a small amount of fluid moving through a system.
Its energy can shift between:
pressure ↔ motion ↔ height
but under ideal conditions the total remains constant.
For example:
pressure energy decreases → kinetic energy increases
or:
pressure energy decreases → gravitational potential energy increases
This is the same conservation principle encountered throughout physics:
Energy can be transferred or transformed, but it is not created or destroyed.
A Useful Energy Picture
Imagine three "accounts" containing the fluid's mechanical energy:
| Energy Form | Bernoulli Term | Increased By |
|---|---|---|
| Pressure | P | Greater static pressure |
| Kinetic | ½ρv² | Greater fluid speed |
| Gravitational | ρgh | Greater height |
If one account gains energy, another must lose energy when the total remains constant.
This is a useful way to think about Bernoulli problems.
Visualizing Pressure and Velocity
Consider an illustrative horizontal Venturi tube where fluid speed increases as the tube narrows.
The values above are illustrative rather than measured data, but they show the key relationship:
speed rises → static pressure falls
and when the pipe widens again:
speed falls → static pressure can recover
Bernoulli and Airflow
Bernoulli's Principle applies to gases as well as liquids when the assumptions are appropriate.
Air moving around objects can therefore show pressure differences associated with differences in flow speed.
Applications include:
- aircraft wings
- sails
- roofs
- spray devices
- Venturi meters
- carburetors
- laboratory aspirators
Aircraft Wings
Aircraft wings provide an important application of fluid dynamics, but they require a careful explanation.
Air moving around a wing creates a pressure distribution over its surfaces.
The pressure is typically lower over much of the upper surface than over the lower surface, producing part of the upward aerodynamic force called lift.
Bernoulli's equation can help relate regions of different airflow speed to pressure differences.
However, it is incomplete to say:
"Planes fly only because air moves faster over the top."
Lift also involves:
- wing shape
- angle of attack
- airflow deflection
- circulation
- pressure distribution
Bernoulli's Principle and Newton's laws are consistent descriptions of the same fluid behaviour, not competing explanations.
An Important Wing Misconception
You may have heard this explanation:
"Air travelling over the top of a wing must meet the air travelling underneath at the back."
This is called the equal-transit-time explanation, and it is incorrect.
Air particles separated at the front of the wing do not have to arrive at the trailing edge at the same time.
A better explanation considers the complete airflow and resulting pressure distribution around the wing.
This distinction is important when using Bernoulli's Principle correctly.
Spray Bottles and Atomizers
Some spray devices use fast-moving air to help draw liquid into an airflow.
Fast air moving past the top of a tube can be associated with lower static pressure.
If the pressure acting on the liquid elsewhere is greater, the pressure difference can help push liquid upward.
The moving air then breaks the liquid into small droplets.
This principle has been used in:
- perfume atomizers
- paint sprayers
- laboratory equipment
- some fuel-delivery systems
Chimneys
Moving air across the top of a chimney can influence the pressure near its opening.
A pressure difference can help encourage gases to move upward through the chimney.
However, chimney draft is also strongly affected by buoyancy because hot gases inside the chimney are usually less dense than cooler outside air.
This is a good example where several fluid principles operate together.
Strong Winds and Roofs
Fast-moving wind around a building can create pressure differences across surfaces.
These pressure differences can contribute to forces on roofs.
However, real wind flow around buildings is turbulent and complex.
Engineers therefore do not rely on a simple Bernoulli statement alone. They consider:
- external pressure
- internal pressure
- turbulence
- building geometry
- wind direction
- structural strength
This is an important example of applying a scientific model while recognizing its limitations.
Sailing
Sails interact with moving air and can create pressure differences across their surfaces.
The resulting aerodynamic force can be resolved into components that help move the boat.
As with aircraft wings, the complete explanation involves:
- pressure distribution
- airflow deflection
- sail shape
- angle to the wind
Bernoulli's equation can help describe the relationship between airflow speed and pressure in suitable parts of the flow.
Pitot Tubes
Aircraft and other systems can measure fluid speed using differences in pressure.
One device is a Pitot tube.
A Pitot-static system compares:
- static pressure
- stagnation pressure
The pressure difference can be related to fluid speed using Bernoulli's equation.
This allows aircraft instruments to determine airspeed.
Worked Example 4: Using Pressure Difference
Air moves through a horizontal system.
At Point 1:
v₁ = 10 m/s
At Point 2:
v₂ = 20 m/s
Use an air density of:
ρ = 1.2 kg/m³
Calculate:
P₁ − P₂
Start with:
P₁ + ½ρv₁² = P₂ + ½ρv₂²
Rearrange:
P₁ − P₂ = ½ρ(v₂² − v₁²)
Substitute:
P₁ − P₂ = ½(1.2)(20² − 10²)
P₁ − P₂ = 0.6(400 − 100)
P₁ − P₂ = 0.6 × 300
P₁ − P₂ = 180 Pa
Answer
P₁ is 180 Pa greater than P₂.
The faster-moving air at Point 2 has the lower static pressure in this idealized horizontal example.
Combining Continuity and Bernoulli
Some problems require both equations.
Suppose water moves through a horizontal pipe:
A₁ = 0.040 m²
v₁ = 2 m/s
The pipe narrows to:
A₂ = 0.020 m²
and:
P₁ = 130 000 Pa
Use:
ρ = 1000 kg/m³
Find v₂ and P₂.
Step 1: Use Continuity
A₁v₁ = A₂v₂
0.040 × 2 = 0.020 × v₂
v₂ = 4 m/s
Step 2: Use Bernoulli
Because the pipe is horizontal:
P₁ + ½ρv₁² = P₂ + ½ρv₂²
Substitute:
130 000 + ½(1000)(2²) = P₂ + ½(1000)(4²)
130 000 + 2000 = P₂ + 8000
Therefore:
P₂ = 124 000 Pa
Answer
v₂ = 4 m/s
P₂ = 124 000 Pa
So:
area decreases → velocity increases → static pressure decreases
A Problem-Solving Strategy
When solving Bernoulli problems:
Step 1: Draw or examine the system
Identify Points 1 and 2.
Step 2: Record the known quantities
Look for:
- P₁ and P₂
- v₁ and v₂
- h₁ and h₂
- ρ
Step 3: Ask whether continuity is needed
If pipe areas are given, you may first need:
A₁v₁ = A₂v₂
Step 4: Write Bernoulli's equation
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Step 5: Cancel equal terms
If the pipe is horizontal:
h₁ = h₂
so the gravitational terms cancel.
Step 6: Solve
Substitute carefully and include units.
Step 7: Check your reasoning
Ask:
- Did the fluid speed increase?
- Should the static pressure therefore be lower in this ideal horizontal case?
- Did the fluid move higher?
- Should gravitational potential energy have increased?
A physical check is often as important as the arithmetic.
When Does Bernoulli's Equation Work Best?
The simple Bernoulli equation assumes an idealized situation.
It works best when the flow is approximately:
- steady
- incompressible
- low in viscosity
- considered along a streamline
- free from significant energy additions or losses between the selected points
Real fluids experience:
- friction
- turbulence
- viscosity
- pumps
- turbines
- heat transfer
These can change the fluid's mechanical energy.
Engineers therefore use extended forms of the energy equation for complex real systems.
Real Pipes Lose Energy
Imagine water travelling through a long pipe.
Friction between the moving fluid and pipe walls causes mechanical energy to be dissipated, mainly into internal energy.
Therefore, real systems often experience pressure loss.
Factors affecting pressure losses include:
- pipe length
- pipe diameter
- fluid viscosity
- surface roughness
- flow speed
- bends
- valves
- turbulence
So real systems may not behave exactly like the ideal Bernoulli model.
Pumps Add Energy
A pump can add mechanical energy to a fluid.
For example, a pump can:
- increase pressure
- increase flow rate
- move water to greater height
In such systems, the simple Bernoulli equation between points on opposite sides of the pump must be modified to include the energy supplied by the pump.
Bernoulli in Engineering
Bernoulli's Principle is important in many fields.
Civil Engineering
Used when analysing:
- water distribution
- pipelines
- dams
- drainage systems
Mechanical Engineering
Used in:
- pumps
- turbines
- nozzles
- Venturi meters
Aerospace Engineering
Used in understanding:
- airflow
- pressure measurements
- wings
- aircraft instruments
Marine Engineering
Used in:
- propellers
- hull flows
- underwater systems
Biomedical Engineering
Fluid-dynamics principles help scientists understand:
- blood flow
- pressure differences
- medical flow devices
Bernoulli's Principle is therefore not just a classroom idea—it is one part of the foundation of fluid engineering.
Common Mistakes
Mistake 1: "Fast fluid always has low pressure."
This is too general.
The simple inverse relationship is most useful when comparing appropriate points in the same flow, especially at similar heights under ideal conditions.
Pressure also depends on height, energy losses, pumps and other factors.
Mistake 2: "Narrow pipes have higher pressure because the fluid is squeezed."
For steady ideal horizontal flow, the fluid accelerates through the narrower section.
Its kinetic energy increases, while its static pressure decreases.
Mistake 3: Confusing pressure with kinetic energy
Pressure and fluid speed represent different forms of mechanical energy in Bernoulli's equation.
Greater speed does not mean greater static pressure.
Mistake 4: Forgetting height
Bernoulli's equation contains:
ρgh
If height changes significantly, it cannot simply be ignored.
Mistake 5: Forgetting to square velocity
The kinetic term is:
½ρv²
not:
½ρv
A small increase in velocity can therefore cause a relatively large change in the kinetic term.
Mistake 6: Assuming Bernoulli creates energy
Bernoulli's Principle does not create extra pressure or speed.
It describes how mechanical energy is transferred between different forms.
Mistake 7: Using the "equal transit time" explanation for wings
Air travelling over and under a wing does not have to reconnect at the trailing edge at the same time.
Aircraft lift requires a more complete explanation involving the entire airflow and pressure distribution.
Mistake 8: Ignoring friction in real systems
The simple Bernoulli equation describes an idealized flow.
Real pipes lose mechanical energy because of friction, turbulence and other effects.
Mistake 9: Thinking Bernoulli and Newton give competing explanations
They are compatible.
Pressure differences and changes in fluid momentum are connected aspects of the same physical process.
Check Your Understanding
1. Recall
State Bernoulli's Principle in your own words.
2. Energy
Name the three energy terms represented in:
P + ½ρv² + ρgh = constant
3. Predict
Water moves through a horizontal pipe from a wide section into a narrow section.
Predict what happens to:
a. fluid velocity
b. static pressure
Explain your reasoning.
4. Explain
Why does static pressure decrease when fluid speeds up in an ideal horizontal flow?
Use the idea of conservation of energy.
5. Calculate
Water flows through a horizontal pipe.
At Point 1:
P₁ = 160 000 Pa
v₁ = 2 m/s
At Point 2:
v₂ = 4 m/s
Use:
ρ = 1000 kg/m³
Calculate P₂.
6. Height
Water moves upward through a constant-diameter pipe.
Its speed remains approximately constant.
Explain what happens to its pressure as its height increases.
7. Apply
Explain how a Venturi tube uses differences in:
- cross-sectional area
- velocity
- pressure
8. Airflow
Explain how Bernoulli's Principle can help describe pressure differences around an aircraft wing. Why is Bernoulli alone not a complete explanation of lift?
9. Combine Principles
Water flows through a horizontal pipe.
At Section 1:
A₁ = 0.060 m²
v₁ = 2 m/s
P₁ = 150 000 Pa
At Section 2:
A₂ = 0.030 m²
Use:
ρ = 1000 kg/m³
First calculate v₂ using continuity.
Then calculate P₂ using Bernoulli's equation.
10. Challenge
A student says:
"Bernoulli's Principle means that whenever something moves faster through a fluid, the pressure everywhere around it must decrease."
Explain why this statement is too simplistic.
Include at least three conditions or factors that must be considered.
Key Terms
- Bernoulli's Principle – conservation of mechanical energy applied to suitable fluid flow
- Bernoulli's equation – relationship between pressure, velocity and height in ideal fluid flow
- Static pressure – pressure associated with the local thermodynamic state of the fluid, distinct from the kinetic contribution of bulk motion
- Dynamic pressure – quantity ½ρv² associated with fluid motion
- Pressure energy – mechanical energy associated with fluid pressure
- Kinetic energy – energy associated with motion
- Gravitational potential energy – energy associated with height
- Venturi effect – pressure change associated with fluid acceleration through a constriction
- Venturi tube – device containing a narrowed section used to produce measurable pressure differences
- Streamline – path representing the direction of fluid flow
- Steady flow – flow whose conditions at a particular point do not change significantly with time
- Incompressible fluid – fluid whose density remains approximately constant
- Pressure loss – reduction in mechanical pressure associated with friction and other losses
- Pitot tube – device that uses pressure measurements to determine fluid speed
Key Takeaways
- Bernoulli's Principle is an application of conservation of mechanical energy to fluid flow.
- Bernoulli's equation is:
P + ½ρv² + ρgh = constant
- The three terms represent pressure, kinetic, and gravitational contributions to mechanical energy per unit volume.
- In an ideal horizontal flow, greater fluid speed is associated with lower static pressure.
- Continuity explains why fluid speeds up when a pipe narrows.
- Bernoulli explains how this change in speed is related to pressure.
- Together:
narrower area → greater velocity → lower static pressure
for an ideal horizontal flow.
- If fluid moves to greater height, some mechanical energy may be transferred into gravitational potential energy.
- Bernoulli's Principle helps explain and analyze Venturi tubes, nozzles, pressure measurements, airflow, spray systems, and many engineering systems.
- Aircraft lift should not be explained using the incorrect equal-transit-time idea; the full airflow and pressure distribution must be considered.
- Real fluids experience viscosity, friction and turbulence, so real systems may lose mechanical energy.
- Pumps can add mechanical energy to flowing fluids.
- Bernoulli's equation is a model with assumptions, so it should be applied only when those assumptions are reasonable.