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Bernoulli's Equation

Bernoulli's equation relates pressure, flow velocity, and elevation for an ideal, incompressible, and steady fluid flow along a streamline.

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

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Core idea

Overview

Derived from the principle of conservation of energy, the equation states that the sum of static pressure, dynamic pressure, and hydrostatic pressure remains constant along a streamline. It is foundational in fluid mechanics for determining how fluid flow characteristics change when piping geometry or elevation varies. This idealization assumes no friction losses and constant fluid density.

When to use: Apply when analyzing steady, incompressible, frictionless (inviscid) flow along a streamline where fluid properties do not change over time.

Why it matters: It is essential for designing piping systems, aircraft wings, and hydraulic devices, allowing engineers to calculate velocity changes based on pressure differentials.

Symbols

Variables

P = Pressure, $ρ$ = Fluid Density, g = Gravity, h = Height

P

Pressure

Variable

ρ

Fluid Density

Variable

g

Gravity

Variable

h

Height

Variable

Free formulas

Rearrangements

Solve for $P$

Make P the subject

P = C - \frac{1}{2} ρ v^{2} - ρ g h

Isolate the pressure term by subtracting the kinetic and potential energy density terms from the constant.

Difficulty: 1/5

Solve for $v$

Make v the subject

v = \frac{2 ( C - P - ρ g h )}{ρ}

Isolate the velocity term by moving other components, multiplying by 2, dividing by density, and taking the square root.

Difficulty: 3/5

Solve for $g$

Make g the subject

g = \frac{C - P - \frac{1}{2} ρ v ^{2}}{ρ h}

Isolate the gravity term by subtracting P and kinetic energy, then dividing by the density and height.

Difficulty: 2/5

Solve for $h$

Make h the subject

h = \frac{C - P - \frac{1}{2} ρ v ^{2}}{ρ g}

Isolate the height term by moving other components and dividing by the density and gravity.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Think of a fluid particle as a budget-conscious traveler moving through a pipe. The total 'energy budget' is fixed; the particle can spend its wealth on static pressure (crowd density), kinetic energy (speed), or potential energy (elevation). If the pipe narrows (speed increases) or moves uphill (elevation increases), the particle must 'spend' its static pressure to pay for the change, illustrating a strict trade-off.

P

Static Pressure

The internal stress or 'stored' energy of the fluid exerted on the surroundings; think of this as the fluid's resting potential.

\frac{1}{2} ρ v^{2}

Dynamic Pressure (Kinetic Energy Density)

The energy cost associated with the motion of the fluid; faster-moving fluid effectively 'uses' more of the total energy budget for its momentum.

ρ g h

Hydrostatic Pressure (Potential Energy Density)

The gravitational 'tax' or 'reward' based on vertical position; being higher up requires more stored energy to maintain that elevation.

Signs and relationships

+: The addition signs represent the additive nature of energy in a closed system; since energy is conserved in an ideal (inviscid) fluid, the sum of these different energy forms must remain invariant along a streamline.

One free problem

Practice Problem

A horizontal pipe with a cross-sectional area of 0.02 m² narrows to 0.01 m². If water flows at 2 m/s in the wider section with a pressure of 200 kPa, what is the pressure in the narrow section (density = 1000 kg/m³)?

P1200000

v12

v24

Solve for: $P + \frac{1}{2} ρ v^{2} + ρ g h$

Hint: Use the continuity equation A1v1 = A2v2 to find the velocity in the second section, then apply Bernoulli's.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In an aircraft wing, the air speed increases over the curved upper surface compared to the lower surface, causing a pressure drop that creates lift according to Bernoulli's principle.

Study smarter

Tips

Always define a reference datum (h=0) before setting up the equation.
Ensure the fluid is treated as incompressible; if Mach number > 0.3, use compressible flow equations instead.
Remember that the equation only strictly applies along a single streamline.

Avoid these traps

Common Mistakes

Neglecting the hydrostatic pressure term (rho*g*h) when there is a significant elevation change.
Attempting to apply the equation to systems with significant viscous losses (e.g., long pipes with friction) without using the Energy Equation extension.
Confusing static pressure with stagnation pressure.

Common questions

Frequently Asked Questions

Apply when analyzing steady, incompressible, frictionless (inviscid) flow along a streamline where fluid properties do not change over time.

It is essential for designing piping systems, aircraft wings, and hydraulic devices, allowing engineers to calculate velocity changes based on pressure differentials.

Neglecting the hydrostatic pressure term (rho*g*h) when there is a significant elevation change. Attempting to apply the equation to systems with significant viscous losses (e.g., long pipes with friction) without using the Energy Equation extension. Confusing static pressure with stagnation pressure.

In an aircraft wing, the air speed increases over the curved upper surface compared to the lower surface, causing a pressure drop that creates lift according to Bernoulli's principle.

Always define a reference datum (h=0) before setting up the equation. Ensure the fluid is treated as incompressible; if Mach number > 0.3, use compressible flow equations instead. Remember that the equation only strictly applies along a single streamline.

References

Sources

White, F. M. (2011). Fluid Mechanics (7th ed.). McGraw-Hill Education.
Batchelor, G. K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press.

Bernoulli's Equation

Overview

Variables

Rearrangements

Intuition

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Continuity Equation

Frequently Asked Questions

Sources