Equation of Continuity

Beginner Level

During steady flow, fluid cannot disappear or collect inside a pipe. The mass entering each second must equal the mass leaving each second. This is the idea behind the equation of continuity.

This explains why water speeds up when a hose end is squeezed and why a freely falling stream becomes thinner as it accelerates.

Equation of Continuity

Overview

The continuity equation is a mathematical statement of conservation of mass in fluid dynamics. It is used for pipes, tubes, ducts, rivers, blood vessels and power-plant systems.

Learning Outcomes

• Derive continuity from conservation of mass.
• Distinguish mass flow rate from volume flow rate.
• Use the general and incompressible forms.
• Explain the inverse area-speed relation.
• Solve single-pipe and branched-flow problems.
• Apply continuity to water jets, hoses, rivers and blood vessels.

Intermediate Level

Derivation from Conservation of Mass

Consider steady flow through a tube whose cross-sectional area changes. During the same short time Δt, fluid at sections 1 and 2 moves distances Δx₁ and Δx₂.

Each moving portion is a short cylinder. Since volume equals cross-sectional area multiplied by length:

Using mass = density × volume, the masses crossing the sections are:

For steady flow with one entrance, one exit and no source or sink, mass cannot accumulate. Therefore Δm₁ = Δm₂. Cancelling the common time interval gives:

Incompressible Form

For an incompressible fluid, density remains constant, so ρ₁ = ρ₂. The densities cancel from the general equation.

Volume Flow Rate

Volume flow rate Q is the volume passing a cross-section per unit time. A fluid column of length vΔt and area A has volume AvΔt, so:

Area-Speed Relation

When Q is constant, speed varies inversely with cross-sectional area. A narrow section has greater speed; a wide section has smaller speed.

Applications

• A freely falling water stream accelerates, so its cross-sectional area decreases.
• Squeezing a rubber hose reduces its opening and increases water speed.
• For the same river discharge, a deep or wide region has larger area and slower average flow.
• Blood from the aorta divides among many arteries; their combined area controls the new speed.
• Pipe and duct systems use continuity at changing cross-sections.

Solved Examples

Branched Flow

At a junction, total inflow equals total outflow. If one inlet divides into N identical outlets:

Common Mistakes

• Using A₁v₁ = A₂v₂ when density changes significantly.
• Forgetting to include every outlet branch.
• Using radius directly instead of A = πr².
• Mixing cm² with m² when calculating Q.
• Saying speed is inversely proportional to radius instead of area.
• Assuming continuity alone determines pressure.

Short-Question Preparation

Long-Question Preparation

Sticky Notes

• Continuity = conservation of mass.
• General: ρ₁A₁v₁ = ρ₂A₂v₂.
• Incompressible: A₁v₁ = A₂v₂.
• Q = Av.
• Mass flow rate = ρAv.
• Circular area = πr².
• Smaller area → greater speed.

60-Second Revision

Summary

Continuity expresses conservation of mass. The mass flow rate ρAv remains constant during steady flow. If density is constant, Av and Q remain constant, producing the inverse relation between area and speed.

Ask-the-Tutor Modes

• Explain continuity simply.
• Show the full derivation.
• Help me choose the correct form.
• Solve an area-speed numerical.
• Solve a branched-flow problem.
• Explain an application.
• Test me with an MCQ.