In an incompressible three-dimensional flow field, the velocity components are given by u = ax + byz; υ = cy + dxz. Determine the form of the z component of velocity. If the z component were not a function of x or y what would be the form be?

An incompressible flow field F in a 3D cartesian grid with components u, v, w:

F = u + v + w

where u, v, w are functions of x, y, z

Must satisfy:

∇·F = du/dx + dv/dy + dw/dz = 0

We have a field F defined:

F = u+v+w, u = ax+byz, v = cy+dxz

du/dx = a, dv/dy = c

Recall ∇·F = 0:

∇·F = du/dx + dv/dy + dw/dz = 0

a + c + dw/dz = 0

dw/dz = - a-c

Solve for w by separation of variables:

w = ∫ (-a-c) dz

w = - az - cz + f (x, y)

f (x, y) is some undetermined function of x and y

The question states that w is not a function of x and y, therefore f (x, y) = 0 ...

w = - az - cz