The velocity components in the \(x\) and \(y\) directions are given by \(u=\lambda xy^{3}-x^{2}y; v=xy^{2}-\frac{3}{4}y^{4}\). Find the value of \(\lambda\) for a possible flow field involving incompressible fluid.
For a 2-D incompressible flow the continuity equation is \[\frac{\partial u}{\partial x}+\frac{\partial v }{\partial y}=0\]
\[\frac{\partial u}{\partial x}=\frac{\partial }{\partial x}\left ( \lambda xy^{3}-x^{2}y \right )=\lambda y^{3}-2xy\]
\[\frac{\partial v }{\partial y}=\frac{\partial }{\partial y}\left ( xy^{2}-\frac{3}{4}y^{4} \right )
\\=2xy-3y^{3}\]
Therefore from the above continuity equation \[\lambda y^{3}-2xy+2xy-3y^{3}=0
\\\Rightarrow y^{3}\left ( \lambda -3 \right )=0
\\\Rightarrow \lambda =3\]
