General Continuity Equation:

The equation is based on the principle of conservation of mass which means mass can neither be created nor be destroyed.

\(\frac{{\partial \rho }}{{\partial t}}\: + \:\frac{\partial }{{\partial x}}\left( {\rho u} \right)\: + \:\frac{\partial }{{\partial y}}\left( {\rho v} \right)\: + \:\frac{\partial }{{\partial z}}\left( {\rho w} \right)\: = 0\)

\(\frac{{\partial \rho }}{{\partial t}}\: + \:\rho \left( {\nabla .\vec V} \right) = 0\)

where

ρ = density of fluid

u = velocity in x-direction

v = velocity in y-direction

w = velocity in z-direction

Special Case:

If the flow is steady then

\(\left( {\frac{{\partial \rho }}{{\partial t}}\: = 0} \right)\)

Thus continuity equation will be

\(\frac{\partial }{{\partial x}}\left( {\rho u} \right)\: + \:\frac{\partial }{{\partial y}}\left( {\rho v} \right)\: + \:\frac{\partial }{{\partial z}}\left( {\rho w} \right)\: = 0\)

For steady, compressible, and two-dimensional flow

\(\frac{{\partial \left( {\rho u} \right)}}{{\partial x}}\: + \:\frac{{\partial \left( {\rho v} \right)}}{{\partial y}}\: = 0\)