Question

The velocity components of a two dimensional plane motion of a fluid are: $u=\frac{y^3}{3}+2x-x^{2}yandv=x{y}^2-2y-\frac{x^3}{3}$
The correct statement is:

• A. Fluid is incompressible and flow is irrotational
• B. Fluid is incompressible and flow is rotational
• C. Fluid is compressible and flow is irrotational
• D. Fluid is compressible and flow is rotational

Answer 1

The correct option is A Fluid is incompressible and flow is irrotational

Continuity equation
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$
$\Rightarrow (2-2xy)+(2xy-2)=0$
$0=0$
Hence, incompressible fluid.
${\omega }_z=\frac{1}{2}[\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}]$
$=\frac{1}{2}[y^2-\frac{3x^2}{3}-(\frac{3y^2}{3}-x^2)]$
$=\frac{1}{2}[y^2-x^2-y^2+x^2]$
$=0$
Hence, flow is irrotational.