The velocity field may be written in the rectangular Cartesian coordinate system as
[tex]\vec{v} = u(x,y,z) \hat{i} + v(x,y,z) \hat{j} + w(x,y,z) \hat{k}[/tex]
where
u = x3 + 2z²
w = y³ - 2yz
For incompressible flow,
[tex]\nabla . \vec{v} = 0[/tex]
Therefore
[tex] \frac{\partial u}{\partial x} =0 \,\, \Rightarrow \, 3x^{2}=0 \, \Rightarrow \, x=0[/tex]
[tex] \frac{\partial w}{\partial z} =0 \, \Rightarrow -2y=0 \, \Rightarrow \, y=0[/tex]
Also,
[tex] \frac{\partial v}{\partial y} =0 \\ v = f(x,y)[/tex]
Because x = y = 0 in the velocity field, therefore
v = f(x)
Answer: v = v(x) only.
