Continuity Equation Equation, Derivation & Rearrangements

Core idea

Overview

The continuity equation represents the principle of conservation of mass for a moving fluid. For an incompressible fluid flowing through a conduit, the product of the cross-sectional area and the velocity of the fluid remains constant at any point along the flow path.

When to use: Apply this equation when dealing with steady-state flow of incompressible fluids, such as liquids, moving through pipes or ducts. It is applicable when the fluid's density does not change significantly between two points of interest.

Why it matters: This principle is critical in engineering for designing nozzles, venturi meters, and piping systems where speed and pressure control are necessary. It explains why water speeds up when a hose is partially covered, which is a fundamental concept in fluid dynamics and aerodynamics.

Symbols

Variables

v_2 = Velocity 2, A_1 = Area 1, v_1 = Velocity 1, A_2 = Area 2

v_{2}

Velocity 2

m / s

A_{1}

Area 1

m^{2}

v_{1}

Velocity 1

m / s

A_{2}

Area 2

m^{2}

Walkthrough

Derivation

Derivation of the Continuity Equation

The continuity equation expresses conservation of mass for steady flow: what enters a pipe must leave it.

Flow is steady (no time variation in mass flow rate).
Fluid is incompressible (constant density).
Flow fills the pipe cross-section.

1

State Conservation of Mass Flow:

Mass flow rate at point 1 equals mass flow rate at point 2.

\frac{d m _{1}}{d t} = \frac{d m _{2}}{d t}

2

Write Mass Flow Rate in Terms of $ρ, A, v$:

Mass flow rate equals density times area times average velocity.

ρ_{1} A_{1} v_{1} = ρ_{2} A_{2} v_{2}

3

Use Incompressibility ($\rho_1=\rho_2$):

With constant density, $ρ$ cancels, giving the standard continuity equation.

A_{1} v_{1} = A_{2} v_{2}

Result

A_{1} v_{1} = A_{2} v_{2}

Source: Standard curriculum — A-Level Fluid Mechanics

Free formulas

Rearrangements

Solve for $v_{2}$

Make v2 the subject

v_{2} = \frac{A _{1} v _{1}}{A _{2}}

Exact symbolic rearrangement generated deterministically for v2.

Difficulty: 3/5

Solve for $A_{1}$

Make A1 the subject

A_{1} = \frac{v _{2} A _{2}}{v _{1}}

Exact symbolic rearrangement generated deterministically for A1.

Difficulty: 3/5

Solve for $v_{1}$

Make v1 the subject

v_{1} = \frac{v _{2} A _{2}}{A _{1}}

Exact symbolic rearrangement generated deterministically for v1.

Difficulty: 3/5

Solve for $A_{2}$

Make A2 the subject

A_{2} = \frac{A _{1} v _{1}}{v _{2}}

Exact symbolic rearrangement generated deterministically for A2.

Difficulty: 3/5

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

Visual intuition

Graph

The graph of Velocity 2 (v2) against the independent variable A2 is a hyperbola, as the variables are inversely proportional when A1 and v1 remain constant. The curve approaches the axes as asymptotes but never touches them, reflecting that velocity cannot be zero if the cross-sectional area is finite.

Graph type: hyperbolic

Why it behaves this way

Intuition

Imagine water flowing through a garden hose: if you squeeze the nozzle (reducing the area), the water shoots out faster (increasing velocity) to maintain the same volume of water flowing through.

A_{1}

Cross-sectional area of the conduit at the first point of interest.

Represents the 'width' or 'size' of the path available for the fluid to flow through at that specific location.

v_{1}

Average speed of the fluid flowing through the conduit at the first point of interest.

Indicates how quickly the fluid is moving along the flow path at that location.

A_{2}

Cross-sectional area of the conduit at the second point of interest.

Represents the 'width' or 'size' of the path available for the fluid to flow through at a different location downstream or upstream.

v_{2}

Average speed of the fluid flowing through the conduit at the second point of interest.

Indicates how quickly the fluid is moving along the flow path at the second location.

A_{1} v_{1}

Volumetric flow rate (volume of fluid passing a point per unit time).

This product must remain constant for an incompressible fluid in steady flow, meaning if the area shrinks, the velocity must increase proportionally to maintain the same amount of fluid passing through.

Free study cues

Insight

Canonical usage

This equation is used to ensure dimensional consistency, where the product of cross-sectional area and fluid velocity (representing volumetric flow rate) must have the same units on both sides of the equality.

Common confusion

A common mistake is mixing unit systems or using inconsistent units for area and velocity (e.g., area in $c m^{2}$ and velocity in m/s) without proper conversion, leading to incorrect volumetric flow rate values.

Unit systems

$A$ m^2 · Cross-sectional area. Must be consistent with the chosen unit system for velocity (e.g., m^2 in SI, ft^2 in Imperial).

$v$ m/s · Average fluid velocity. Must be consistent with the chosen unit system for area (e.g., m/s in SI, ft/s in Imperial).

$A * v$ m^3/s · Represents the volumetric flow rate, which must be dimensionally consistent on both sides of the equation. Common units include m^3/s (SI) or ft^3/s (Imperial).

One free problem

Practice Problem

A water pipe with a cross-sectional area of 0.5 m² carries water at a velocity of 4 m/s. If the pipe narrows to an area of 0.2 m², what is the new velocity of the water?

Area 10.5 m^2

Velocity 14 m/s

Area 20.2 m^2

Solve for: $v 2$

Hint: Divide the initial flow rate (A1 × v1) by the final area to find the new velocity.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Estimating jet speed from a narrowed pipe.

Study smarter

Tips

Ensure area units are consistent, such as both being in m² or both in cm².
Verify that the fluid is incompressible; for gases, this usually implies speeds well below the speed of sound.
Recall that as the cross-sectional area decreases, the velocity must increase proportionally.

Avoid these traps

Common Mistakes

Mixing diameter and area.
Using compressible flow cases.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The continuity equation expresses conservation of mass for steady flow: what enters a pipe must leave it.

Apply this equation when dealing with steady-state flow of incompressible fluids, such as liquids, moving through pipes or ducts. It is applicable when the fluid's density does not change significantly between two points of interest.

This principle is critical in engineering for designing nozzles, venturi meters, and piping systems where speed and pressure control are necessary. It explains why water speeds up when a hose is partially covered, which is a fundamental concept in fluid dynamics and aerodynamics.

Mixing diameter and area. Using compressible flow cases.

Estimating jet speed from a narrowed pipe.

Ensure area units are consistent, such as both being in m² or both in cm². Verify that the fluid is incompressible; for gases, this usually implies speeds well below the speed of sound. Recall that as the cross-sectional area decreases, the velocity must increase proportionally.

References

Sources

Bird, Stewart, Lightfoot, Transport Phenomena
Wikipedia: Continuity equation
Bird, R. Byron, Stewart, Warren E., Lightfoot, Edwin N. Transport Phenomena. John Wiley & Sons.
Incropera, Frank P., DeWitt, David P., Bergman, Theodore L., Lavine, Adrienne S. Fundamentals of Heat and Mass Transfer. John Wiley & Sons.
Britannica, The Editors of Encyclopaedia. 'Continuity equation'. Encyclopedia Britannica, 22 Sep. 2023.
Wikipedia, 'Continuity equation'.
Bird, Stewart, Lightfoot Transport Phenomena
Incropera, DeWitt, Bergman, Lavine Fundamentals of Heat and Mass Transfer

Continuity Equation

Overview

Variables

Derivation

State Conservation of Mass Flow:

Write Mass Flow Rate in Terms of \(ρ, A, v\):

Use Incompressibility (\(\rho_1=\rho_2\)):

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Frequently Asked Questions

Sources

Continuity Equation

Overview

Variables

Derivation

State Conservation of Mass Flow:

Write Mass Flow Rate in Terms of \(ρ, A, v\):

Use Incompressibility (\(\rho_1=\rho_2\)):

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Volumetric Flow Rate

Frequently Asked Questions

Sources