2.2.3 Conservation of Mass or Continuity Equation (Differenntial Form)

Using Gauss divergence theorem, we can express the right hand side term of Equation (2.6) as

(2.9)

Substituting Equation (2.9) into (2.6), we obtain

or,

(2.10)

For infinetly small elemental volumes we can always write equation (2.10) as,

(2.11)

This is the continuity equation in the form of a partial differential equation. This is the conservation form of equation. For unsteady, compressible and three dimensional flow the Equation (2.11) can be expressed as

(2.12)

Equation (2.12) can be re-written as

or,

(2.13)

However, the sum of the first two terms of the Equation (2.13) is the substantial derivative of ρ. Thus, from Equation (2.13),

(2.14)

This is the form of continuity equation written in terms of the substantial derivative. This is also called as the non-conservative form of mass conservation or continuity equation.