Continuity equation in Cartesian coordinates

Statement

The continuity equation expresses conservation of mass: the net rate of mass flowing out of a control volume equals the rate of decrease of mass inside it.

Derivation

Consider a fixed elemental control volume $dx\,dy\,dz$ in a flow with velocity $\vec V = u\hat i + v\hat j + w\hat k$ and density $\rho$.

          y
          ^      _______
          |    /|      /|
          |   /_|_____/ |   dz
          |  |  |____|__|
          |  | /     | /  dy
          |  |/______|/
          +--------------> x
               dx

Mass inflow through the face perpendicular to $x$ (area $dy\,dz$) is $\rho u\,dy\,dz$. Mass outflow through the opposite face is

$$\left[\rho u + \frac{\partial(\rho u)}{\partial x}dx\right]dy\,dz$$

Net outflow in the $x$-direction is $\dfrac{\partial(\rho u)}{\partial x}dx\,dy\,dz$. Similarly for $y$ and $z$. Total net outflow:

$$\left[\frac{\partial(\rho u)}{\partial x} + \frac{\partial(\rho v)}{\partial y} + \frac{\partial(\rho w)}{\partial z}\right]dx\,dy\,dz$$

Rate of decrease of mass in the element: $-\dfrac{\partial\rho}{\partial t}dx\,dy\,dz$.

Equating:

$$\frac{\partial\rho}{\partial t} + \frac{\partial(\rho u)}{\partial x} + \frac{\partial(\rho v)}{\partial y} + \frac{\partial(\rho w)}{\partial z} = 0$$

or in vector form, $\dfrac{\partial\rho}{\partial t} + \nabla\cdot(\rho\vec V) = 0$.

Special cases

• Steady flow ($\partial\rho/\partial t = 0$):

$$\frac{\partial(\rho u)}{\partial x} + \frac{\partial(\rho v)}{\partial y} + \frac{\partial(\rho w)}{\partial z} = 0$$

• Incompressible flow ($\rho$ constant, steady or unsteady):

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \quad \text{i.e. } \nabla\cdot\vec V = 0$$

• For 2D incompressible flow: $\partial u/\partial x + \partial v/\partial y = 0$.