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,

`\\abla . \vec{v} = 0`
Therefore

`(\partial u)/(\partial x) =0 \,\, \Rightarrow \, 3x^(2)=0 \, \Rightarrow \, x=0`

`(\partial w)/(\partial z) =0 \, \Rightarrow -2y=0 \, \Rightarrow \, y=0`
Also,

`(\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.