# Continuity Equation in Three Dimensions in a Differential Form

Fig. 4.5 explains the concept of control volume and control surface. Fig. 4.6 shows an elemental rectangular parallelepiped of fluid. The volume (control) is dx d y dz.

We have rate at which mass enters the control volume= Rate at which mass leaves the control volume+ rate of accumulation of mass in the control volume … (4.16)

Thus in x direction

Rate of mass entering surface ABCD = p u d y d z

Rate of mass leaving surface

EFGH = (

From equation (4.16) we can write down

Pu dy dz =

dy dz + Rate of accumulation of mass in control volume in x direction.

Rate of accumulation of mass in the control volume in x direction

Similarly, Rate of accumulation of mass in the control volume in y direction.

and rate of accumulation of mass in the control volume in z direction.

Adding equations (4.17), (4.18) and (4.19) we get

Total rate of accumulation in the rectangular parallelopiped

Equation (4.20) is the general equation of continuity. It is applicable to