Question

An incompressible steady flow pattern is given by u=x^3+2z^2 and w=y^3-2yz. what is the most general form of the third component v(x,y,z) which satisfies continuity?

Answer 1

The velocity field may be written in the rectangular Cartesian coordinate system as
$\vec{v} = u(x,y,z) \hat{i} + v(x,y,z) \hat{j} + w(x,y,z) \hat{k}$
where
u = x3 + 2z²
w = y³ - 2yz

For incompressible flow,
$\nabla . \vec{v} = 0$
Therefore
$\frac{\partial u}{\partial x} =0 \,\, \Rightarrow \, 3x^{2}=0 \, \Rightarrow \, x=0$
$\frac{\partial w}{\partial z} =0 \, \Rightarrow -2y=0 \, \Rightarrow \, y=0$
Also,
$\frac{\partial v}{\partial y} =0 \\ v = f(x,y)$

Because x = y = 0 in the velocity field, therefore
v = f(x)

Answer: v = v(x) only.