Consider the flow in a conduit whose cross-section has the shape of an equilateral triangle. State the differential equations and the associated boundary conditions.

Show that the following expression satisfies the velocity profile in the \(x\)-direction (the flow direction):

\[\begin{equation*}v_{x}=\frac{G}{36 \mu L}(2 \sqrt{3} z+L)(\sqrt{3} z+3 y-L)(\sqrt{3} z-3 y-L) \tag{6.90}\end{equation*}\]

where \(L\) is the length of each side of the equilateral triangle and the origin has been set at the centroid of the tube.
Find the flow rate for a given pressure drop. (Answer: \(\sqrt{3} G L^{4} /(320 \mu)\).)