Derive differential continuity equation

The continuity equation expresses conservation of mass: mass is neither created nor destroyed.

Derivation

Take a fixed small control volume (a cube) of sides $dx,\ dy,\ dz$ with density $\rho$ and velocity components $u,v,w$.

        y
        ^   +---------+
        |  /         /|
        | +---------+ |   mass in  : rho*u dy dz
        | |         | |   at x face
        | |         | +   mass out : [rho*u +
        | |         |/      d(rho*u)/dx dx] dy dz
        | +---------+
        +-------------> x

Net mass efflux in the x-direction:

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

Similarly for y and z. Total net outflow of mass per unit time:

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

Rate of decrease of mass inside the element is $-\dfrac{\partial\rho}{\partial t}dx\,dy\,dz$. Equating the two and dividing by the volume:

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

or in vector form

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

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):

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=\nabla\cdot\vec V=0$$

This means the velocity field is divergence-free, and volume is conserved for a fluid element.