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.
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
