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 to the horizontal (H W). The flow is...

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 to the horizontal (H 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 f ¼ 24=NRe. Use your solution to obtain an equivalent expression for the power law Reynolds number (i.e., NRePL ¼ 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?)