Question

The following cases represents the two velocity components, determine ghe third components of velocity such that they satisfy the continuity equation :
(i) u = 4x^2 , v = 4xyz
(ii) u = 4x^2 + 3xy , w = z^3 - 4xy - 2yz.

Answer 1

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0.$

1)

$8x+4xz+\frac{\partial w}{\partial z}=0,$

$w=-\int(8x+4xz)dx=-8xz-2xz^2+C.$

2)

$8x+3y+\frac{\partial v}{\partial y}+3z^2-2y=0,$

$v=-\int(8x+y+3z^2)dy=-8xy-0.5y^2-3yz^2+C.$