Correct Answer - Option 1 : u = x + y ; v = x - y
Concept:
For incompressible flow, the Continuity equation should follow-
For 3-D, \(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\)
For 2-D, \(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0\)
Calculation:
Given:
u = x + y ; v = x - y
\(\frac{{\partial u}}{{\partial x}} = 1\)
\(\frac{{\partial v}}{{\partial y}} = -1\),
\(\)\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 1 - 1\)
\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0\)
Hence, u = x + y ; v = x - y represent possible 2-D incompressible flows.
