The continuity equation \(\rm \frac{\partial u}{\partial x}+ \frac{\partial v}{\partial y}=0\) is valid for a _____.

Explanation:

The general form of the continuity equation in cartesian coordinates:

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

For steady flow:

\(\frac{{\partial \rho }}{{\partial t}} = 0\)

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

If the fluid is incompressible, then ρ is constant

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

\(\nabla .\vec V = 0\)

if the flow is steady, 2D, incompressible flow 

w → 0 (flow is 2D)

ρ → const (flow is incompressible)

\(\frac{{{\rm{\delta \rho }}}}{{{\rm{\delta t}}}}\) → 0 (flow is steady)

So, final equation becomes

\(\frac{{{\rm{\delta u}}}}{{{\rm{\delta x}}}} + \frac{{{\rm{\delta v}}}}{{{\rm{\delta y}}}}\) = 0