This is the desired form of mass conservation equation for an infinitesimal control volume in Cartesian coordinate system. It is applicable for major categories of flows such as steady/unsteady, inviscid/viscous, incompressible/compressible. Many a times, it is referred as equation of continuity because it requires no assumptions except the fact that density and velocity are continuous functions. Alternatively, the continuity equation is also expressed in cylindrical coordinate system which is useful in many practical flow problems. In this case, any arbitrary point is defined by the coordinates (r θ z) where, z is the distance along z- axis, r is the radial distance and θ is the rotational angle about the axis as shown in Fig. 2.5.2. Then, the conversion is possible using the transformation as given below,

(2.5.11)

 

Fig. 2.5.2: Definition of cylindrical coordinate system.

Thus, the general continuity equation in cylindrical coordinate system becomes,

(2.5.12)

- If the flow is steady, then all the properties are functions of position only. So the Eqs (2.5.10 & 2.5.12) reduces to,

(2.5.13)

- In a special case, if the flow is incompressible, then density changes are negligible i.e. , regardless of whether the flow is steady or not. So, the Eq. (2.5.13) is still valid without the density term.

(2.5.14)