The velocity components of a two dimensional plane motion

The velocity components of a two dimensional plane motion of a fluid are:
u = y³ + 2x − x²y and v = xy² − 2y² − x³
3 3

The correct statement is:

1. Fluid is incompressible and flow is irrotational
2. Fluid is incompressible and flow is rotational
3. Fluid compressible and flow is irrotational
4. Fluid is compressible and flow is rotational

For incompressible flow,

∂u	+	∂v	= 0
∂x		∂y

u =	y³	+ 2x − x²y
	3

v = xy² − 2y −	x³
	3

⇒	∂		y³	+ 2x − x²y		+	∂		xy² − 2y −	x³
	∂x		3				∂x			3

= 2 − 2xy + 2xy – 2 = 0
∴ Flow is incompressible
For irrotational flow,

1		∂v	−	∂u		= 0
2		∂x		∂y

∂

xy² − 2y −

x³

−

∂

y³

+ 2x − x²y

∂x

∂y

=	1	[y² − x² − y² + x²] = 0
	2

∴ Flow is irrotational