Question

Which of the following pairs of velocity components u and v satisfy the continuity equation for a two dimensional flow of an incompressible fluid?
1. u = (2x² + 3y³) ; v = -3xy
2. u = (3x - y) ; v = (2x + 3y)
3. u = A sin xy ; v = -A sin xy
4. u = cx ; v = -cy

Answer 1

Correct Answer - Option 4 : u = cx ; v = -cy

Explanation:

Continuity equation for a two – dimensional flow of an in compressible fluid is

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

For option a: \(\frac{\partial }{{\partial x}}\left( {2{x^2} + 3{y^3}} \right) + \frac{\partial }{{\partial y}}\left( { - 3xy} \right) = 4x - 3x = x \ne 0\;\)

For option b: \(\frac{\partial }{{\partial x}}\left( {3x - y\;} \right) + \frac{\partial }{{\partial y}}\left( {2x + 3y} \right) = 3 + 3 \ne o\)

For option c: \(\frac{\partial }{{\partial x}}\left( {Asin\;xy} \right) + \frac{\partial }{{\partial y}}\left( { - A\sin xy} \right) = Aycosxy - Axcosxy \ne 0\)

For option d: \(\frac{\partial }{{\partial x}}\left( {{c_x}} \right) + \frac{\partial }{{\partial y}}\left( { - cy} \right) = c - c = 0\)

⇒ Option d satisfy the continuity equation