Problem $69\mathrm{.}22$

Consider a steady, laminar flow through a straight horizontal tube having the constant elliptical cross section given by the equation tube having the comstant elliptical cross section given by the equation

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

The streamlines are all straight and parallel. Investigate the possibility of using an equation for the $z$ component of velocity of the form

$w=A(1-\frac{x^2}{a^2}-\frac{y^2}{b^2})$

as an exact solution to thes problem. With this velocity distribution, what is the relationship between the pressure gradient along the tube and the volume flowrate through the tube?