**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.

ṁ_{in} = ṁ_{out}

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.