Consider the flow down a prismatic channel having a rectangular cross section of width \(b\). The channel bottom makes an angle \(\theta\) with the horizontal. Show that

\[ \frac{d y}{d x}=\frac{\tan \theta-\left(n^{2} Q^{2}\right) /\left(A^{2} R_{h}^{4 / 3} \kappa^{2}\right)}{1-Q^{2} /\left(A^{2} g y \cos ^{2} \theta\right)} \]

where \(y\) is the vertical fluid depth, \(A=b y \cos \theta\), and \(Q\) is the volume flow rate. Discuss the form of the equation for small values of \(\theta\left(\theta \sim 1^{\circ}\right)\)