Question

The general equation of continuity for three-dimensional flow an incompressible fluid for steady flow is:
1. \(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\)
2. \(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 1\)
3. \(\frac{{\partial u}}{{\partial x}} = \frac{{\partial v}}{{\partial y}} = \frac{{\partial w}}{{\partial z}} = 0\)
4. None of these

Answer 1

Correct Answer - Option 1 : \(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\)

Explanation:

Continuity equation in three dimensions:

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

The above equation is valid for:

1. Steady and unsteady flow.
2. Uniform and non-uniform flow.
3. Compressible and incompressible flow.

For Steady flow:

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

If the fluid is Incompressible and Steady:

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