Consider the flow in a conduit whose cross-section has the shape of an ellipse. 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):

\[\begin{equation*}v=\frac{G}{2 \mu} \frac{a^{2} b^{2}}{a^{2}+b^{2}}\left(1-\frac{y^{2}}{a^{2}}-\frac{z^{2}}{b^{2}}\right) \tag{6.91}\end{equation*}\]

where \(a\) and \(b\) are the semiaxes of the ellipse corresponding to the \(y\)-and \(z\)-axes.

Find the flow rate, \(Q\), for a given pressure drop. Answer:

\[Q=\frac{\pi G}{4 \mu} \frac{a^{3} b^{3}}{a^{2}+b^{2}}\]