Continuity Equation

Equation

\[\boxed{\;\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v}) = 0\;}\]

Incompressible form: \(\;\nabla\cdot\mathbf{v} = 0\). Streamtube (1-D steady) form: \(\;\dot m = \rho A v = \text{const}\), and for constant density \(A_1v_1 = A_2v_2\).

Physical meaning

Mass is neither created nor destroyed. Whatever mass leaves a region must have flowed out through its boundary. The incompressible version says fluid elements keep their volume: squeeze a flow into a narrower channel and it must speed up.

Variables

Symbol	Meaning	SI unit
\(\rho\)	density	kg m⁻³
\(\mathbf{v}\)	velocity field	m s⁻¹
\(\dot m\)	mass flow rate	kg s⁻¹
\(A\)	streamtube cross-section	m²

Assumptions

None beyond the continuum hypothesis and no mass sources/sinks. The incompressible form additionally requires \(D\rho/Dt = 0\) (good for liquids generally, and for gases at Mach ≲ 0.3).

Derivation

Fix a control volume \(V\) with boundary \(S\). Conservation of mass:

\[\frac{d}{dt}\int_V \rho\, dV + \oint_S \rho\mathbf{v}\cdot d\mathbf{S} = 0\]

Divergence theorem on the flux term:

\[\int_V \left[\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})\right] dV = 0\]

Arbitrary \(V\) ⇒ the integrand vanishes pointwise. (Equivalently: set \(f = \rho\) in the Reynolds transport theorem.)

Applications

Nozzles and pipe networks (\(A_1v_1 = A_2v_2\)) — the first tool in every Bernoulli problem
The necking of a falling water stream
Defining the stream function in 2-D
Deriving every other conservation law's differential form

Limitations

Breaks only where the continuum itself breaks (free-molecular flow) — or if you apply the incompressible form to genuinely compressible situations (shocks, high-Mach flight, acoustics).

Euler's equation — momentum sibling
Navier–Stokes — viscous momentum sibling
Laplace's equation — continuity + irrotationality
Rankine–Hugoniot — its jump-condition avatar

Quiz

Q1 (computational). Water enters a nozzle at 2 m/s through a 4 cm² section and exits through 1 cm². Exit speed?

Answer

\(v_2 = v_1 A_1/A_2 = 2\times 4 = 8\) m/s.

Q2 (conceptual). Show that steady flow (\(\partial\rho/\partial t = 0\)) does not imply incompressible flow.