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Fluid Mechanics

Continuity Equation

Steady incompressible flow carries the same volume per second through each section of a single stream.

A_{1} v ˉ_{1} = A_{2} v ˉ_{2}

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Source & review

Difficulty

Beginner

Read time

5 min

Prerequisites

Basic algebra and consistent SI units

Review statusNeeds review

Source

OpenStax, College Physics 2e, §12.1 (Paul Peter Urone and Roger Hinrichs)

Textbook reference retained from the OpenFormulaGraph dataset; adaptation credits and license are provided below.

Adapted from OpenFormulaGraph, under CC BY 4.0.

FormuLab adapted the notation, explanations, assumptions, and topic placement, and expanded the numerical check into a worked example.

Definition

Understanding the core concept

Steady incompressible flow carries the same volume per second through each section of a single stream. A smaller cross-sectional area therefore requires a larger mean speed. This is a mass-balance relationship.

Variables & Units

Understanding each component

Symbol	Meaning	Units
$A_{1}$	Inlet cross-sectional area	$m^{2}$
$v ˉ_{1}$	Inlet mean speed	$m/s$
$A_{2}$	Outlet cross-sectional area	$m^{2}$
$v ˉ_{2}$	Outlet mean speed	$m/s$

Real-World Applications

Where this formula is used in practice

Pipe reductions

Find the mean speed through a narrower section.

Flow balances

Compare inlet and outlet volume flow.

Worked Example

Step-by-step calculation with real numbers

Problem

Steady incompressible flow enters an area of 2.00 m² at a mean speed of 3.00 m/s and exits through 1.00 m². Find the outlet mean speed.

Given

A_{1} = 2.00 m^{2}

v ˉ_{1} = 3.00 m/s

A_{2} = 1.00 m^{2}

Solution

A_{1} v ˉ_{1} = A_{2} v ˉ_{2}

v ˉ_{2} = \frac{A _{1} v ˉ _{1}}{A _{2}}

v ˉ_{2} = \frac{( 2.00 ) ( 3.00 )}{1.00} m/s

v ˉ_{2} = 6.00 m/s

Final Answer

v ˉ_{2} = 6.00 m/s

Formula Guide

Info

DifficultyBeginner

Read Time5 min

PrerequisitesBasic algebra and consistent SI units

Tools

Source & review

StatusNeeds review

Last reviewedPending

Source

OpenStax, College Physics 2e, §12.1 (Paul Peter Urone and Roger Hinrichs)

Textbook reference retained from the OpenFormulaGraph dataset; adaptation credits and license are provided below.

Adapted from OpenFormulaGraph, under CC BY 4.0.

FormuLab adapted the notation, explanations, assumptions, and topic placement, and expanded the numerical check into a worked example.

Assumptions

• Use steady incompressible flow with one inlet, one outlet, and no leakage or accumulation.
• Each velocity is the cross-sectional mean normal velocity, not an arbitrary local speed.

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Continuity Equation

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Flow balances

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