An incompressible Newtonian fluid is flowing in the $z$ direction in a square duct bounded by the surfaces $x=+-H$ and $y=+-H.$ It has been suggested that the velocity of the fluid in the duct is given by:

$v_x=v_y=0$

$v_z=\frac{(p_0-p_L)}{4L}[1-(\frac{x}{H}{)}^2][1-(\frac{y}{H}{)}^2]$

where $p_0-p_L$ is the pressure difference between the ends of the duct, $L$ is the length of the duct, and $$ is the viscosity of the fluid.

Determine the components of gradv for this flow.

$($Note: This is not the true velocity profile in such a duct.$)$