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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

(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