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Fluid Mechanics

Bernoulli's Equation

Energy conservation in fluid flow systems

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

Study tools

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Source & review

Difficulty

Intermediate

Read time

8 min

Prerequisites

Basic Physics

Review statusNeeds review

Source

FormuLab initial formula library

Initial content draft pending verification against authoritative course or textbook sources.

Definition

Understanding the core concept

Bernoulli's equation states that in a flowing fluid, an increase in velocity occurs simultaneously with a decrease in pressure or potential energy. Think of it like a trade-off: if fluid speeds up, it 'uses up' some of its pressure energy to gain kinetic energy. This principle explains why airplane wings generate lift and how venturi meters work.

Variables & Units

Understanding each component

Symbol	Meaning	Units
$P$	Static Pressure	$Pa$
$ρ$	Fluid Density	$k g / m^{3}$
$v$	Flow Velocity	$m / s$
$g$	Gravitational Acceleration	$m / s^{2}$
$h$	Height above reference	$m$

Real-World Applications

Where this formula is used in practice

Aircraft Wing Design

Explains lift generation by showing pressure difference above and below wings

Venturi Meters

Measures fluid flow rate by detecting pressure changes in a constricted pipe

Carburetors

Uses pressure drop in constricted airflow to draw fuel into the air stream

Blood Flow Analysis

Models circulation in arteries and veins for medical diagnostics

Worked Example

Step-by-step calculation with real numbers

Problem

Water flows through a horizontal pipe. At point 1, the pressure is 200 kPa and velocity is 2 m/s. At point 2, the velocity increases to 5 m/s. Find the pressure at point 2.

Given

P_{1} = 200, 000 Pa

v_{1} = 2 m/s

v_{2} = 5 m/s

ρ = 1000 kg/m^{3}

h_{1} = h_{2} (horizontal pipe)

Solution

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

P_{2} = P_{1} + \frac{1}{2} ρ (v_{1}^{2} - v_{2}^{2})

P_{2} = 200, 000 + \frac{1}{2} (1000) (2^{2} - 5^{2})

P_{2} = 200, 000 + 500 (- 21)

P_{2} = 200, 000 - 10, 500 = 189, 500 Pa

Final Answer

P_{2} = 189.5 kPa

Formula Guide

Info

DifficultyIntermediate

Read Time8 min

PrerequisitesBasic Physics

Tools

Source & review

StatusNeeds review

Last reviewedPending

Source

FormuLab initial formula library

Initial content draft pending verification against authoritative course or textbook sources.

Assumptions

• Use SI units unless the formula or course context states otherwise.
• Check notation, sign conventions, and applicability against your course material.
• Confirm variables and assumptions before using the formula in graded or professional work.