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
From equation (4.16) we can write down
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
(i) steady and unsteady flow
(ii) uniform and non-uniform flow, and
(iii) compressible and incompressible flow.
Case (a) Steady flow the continuity equation becomes
and equation (4.22) can be written in a vector form also as