Question
Download Solution PDFThe continuity equation \(\rm \frac{\partial u}{\partial x}+ \frac{\partial v}{\partial y}=0\) is valid for a _____.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFExplanation:
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
Last updated on May 28, 2025
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