Continuity Equation

The continuity equation expresses the conservation of mass in fluid flow. It states that mass cannot be created or destroyed within a flow field.

Equations

General Form

$\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \vec{v})=0$

Steady Flow

$\nabla \cdot (\rho \vec{v})=0$

Incompressible Flow

$\nabla \cdot \vec{v}=0$

One-Dimensional Steady Flow

${\rho }_1{A}_1{v}_1={\rho }_2{A}_2{v}_2=\dot{m}$

Where:

• ρ = fluid density
• v = velocity vector
• A = cross-sectional area
• ṁ = mass flow rate

Notes

• Mass in = Mass out for any control volume
• If area decreases → velocity increases (for constant density)
• If density changes → velocity adjusts to conserve mass
• Assumes no mass sources or sinks
• Valid for all fluids (compressible and incompressible)
• Steady vs unsteady forms depend on time dependence
• Compressible effects important at high speeds

Related

• Navier-Stokes Equations
• Bernoulli’s Equation