Continuity Equation

1The continuity equation is simply a mathematical expression of the principle of conservation of mass. For a control volume that has a single inlet and a single outlet, the principle of conservation of mass states that, for steady-state flow, the mass flow rate into the volume must equal the mass flow rate out.

in = ṁout

Mass entering per unit time = Mass leaving per unit time

This equation is called the continuity equation for steady one-dimensional flow. For a steady flow through a control volume with many inlets and outlets, the net mass flow must be zero, where inflows are negative and outflows are positive.

This principle can be applied to a streamtube such as that shown above. No fluid flows across the boundary made by the streamlines so mass only enters and leaves through the two ends of this streamtube section.

When a fluid is in motion, it must move in such a way that mass is conserved. To see how mass conservation places restrictions on the velocity field, consider the steady flow of fluid through a duct (that is, the inlet and outlet flows do not vary with time).

Differential Form of Continuity Equation

A general continuity equation can also be written in a differential form:

∂⍴∂t + ∇ . (⍴ ͞v) = σ

where

• ∇ . is divergence,
• ρ is the density of quantity q,
• ⍴ ͞v is the flux of quantity q,
• σ is the generation of q per unit volume per unit time. Terms that generate (σ > 0) or remove (σ < 0) q are referred to as a “sources” and “sinks” respectively. If q is a conserved quantity (such as energy), σ is equal to 0.