# Derivation of continuity equation in cartesian coordinates

In this article, we are going to derive the continuity equation in three-dimensional cartesian coordinates. Due to some constraints, I have uploaded images of every equation. Fluid elements in the three-dimensional flow diagram are shown below:

## Derivation of continuity equation: Consider a fluid element control volume with sides dx, dy, and dz as shown in the above figure of a fluid element in three-dimensional flow.

Now let ρ = Mass density of fluid at a particular instant.

u, v, w = Components of the velocity of flow entering three faces of a parallelopiped.

Rate of the mass of fluid entering the face ABCD which is a fluid influx,

= ρ x Velocity in X- direction x (Area of ABCD)

= ρ u dy dz    ………………..eq (1)

Now, Rate of the mass of fluid leaving the face EFGH (fluid efflux),

The gain in mass per unit time due to flow in X- direction is given by the difference between the fluid influx and fluid efflux.

Therefore, mas accumulated per unit time due to flow in X- direction,

Similarly the gain in fluid mass per unit time in the parallelopiped due to flow in Y and Z directions,

The net gain in fluid mass per unit for fluid along three co-ordinate axes,

The rate of change of mass of the parallelopiped (control volume) is,

From equating eq(6) and eq(7), we get, ## Final Continuity Equation:

Now simplify the above equation and rearrange the terms to get continuity equation in cartesian coordinates,

therefore,