A non-Newtonian fluid, described by the power law model, is flowing through a thin slit between two parallel planes of width $W$, separated by a distance $H$. The slit is inclined upward at an angle $\theta$ to the horizontal.

(a) Derive an equation relating the volumetric flow rate of this fluid to the pressure gradient, slit dimensions, and fluid properties.

(b) For a Newtonian fluid, this solution can be written in a dimensionless form as

\[f=24 / N_{R e, h}\]

where the Reynolds number, $N_{R e, h}$, is based on the hydraulic diameter of the channel. Arrange your solution for the power law fluid in dimensionless form and solve for the friction factor, $f$.

(c) Set your result from (b) equal to $24 / N_{R e, h}$, and determine an equivalent expression for the power law Reynolds number for slit flow.