Problem $4(15$ points$)$

flow

The figure above shows a conduit whose cross section has the shape of an equilaterai urangle.

For the coordinate system shown in the figure, the equations of the three sides are

$z+\frac{b}{2\sqrt[\phantom{2}]{3}}=0$

$z+\sqrt[\phantom{2}]{3}y-\frac{b}{\sqrt[\phantom{2}]{3}}=0$

$z-\sqrt[\phantom{2}]{3}y-\frac{b}{\sqrt[\phantom{2}]{3}}=0$

Following the procedure used in class, look for a solution for the velocity distribution in such a conduit of the following form

$u(y,z)=(z+\frac{b}{2\sqrt[\phantom{2}]{3}})(z+\sqrt[\phantom{2}]{3}y-\frac{b}{\sqrt[\phantom{2}]{3}})(z-\sqrt[\phantom{2}]{3}y-\frac{b}{\sqrt[\phantom{2}]{3}})$

$2$