A viscous molten polymer is pumped through a thin slit between two flat surfaces. The slit has a depth $H$, width $W$, and length $L$, and is inclined upward at an angle $\theta$ to the horizontal $(H \ll W)$. The flow is laminar, and the polymer is non-Newtonian, with properties that can be represented by the power law model.

(a) Derive an equation relating the volume flow rate of the polymer $(Q)$ to the applied pressure difference along the slit, the slit dimensions, and the fluid properties.

(b) Using the definition of the Fanning friction factor $(f)$, solve your equation for $f$ in terms of the remaining quantities. The corresponding solution for a Newtonian fluid can be written as $f=24 / N_{R e}$. Use your solution to obtain an equivalent expression for the power law Reynolds number (i.e., $N_{R e, p l}=24 / f$ ). Use the hydraulic diameter as the length scale in the Reynolds number.

(Note: It is easiest to take the origin of your coordinates at the center of the slit, then calculate the flow rate for one-half the slit and double this to get the answer. Why is this the easiest way?)