1. The manufacture of pipe and the generation of pansons used in blow molding involve flow through an annular die. Our goal is to design an annular die for extruding a pipe at 180C with an outer diameter of 0.0762m (3. in.) and an inner diameter of 0.0635m(2.5in.) for high-density polyethylene (HDPE) at the highest rate possible (here with the rate we mean m/min of the pipe). Assume the limiting factor that sets the upper limit of this rate is the shear stress value w which leads to a melt fracture and occurs at wall shear stress of w=1.0105Pa (the melt fracture is an instability that causes imperfections to the final products and reduces the end-use properties significantly). The rheological analysis for this melt at this temperature suggests that the viscosity of the melt is constant and is equal to 6.19103 Pa.s. (A) (60 points) Obtain a relationship between the wall shear stress and the pressure drop P, as well as the volumetric flow rate, Q. Assume: - The flow is steady, laminar, and isothermal. - There are no entry or exit effects. - Intertia is insignificant (i.e., the Reynolds number is negligible). - The fluid is inelastic (treat it as viscous liquid). - The fluid does not slip on the die surface. - The flow is pressure-driven. - The length of the die is ten times of die gap. To solve this problem: - Define an appropriate coordinate system. - Determine the main component of the velocity vector and its dependence on the coordinates (you can use the continuity equation). - Use a proper set of equations of motion (i.e., momentum balance) and simplify them. - Specify suitable boundary conditions and determine the shear stress and velocity profile. (B) (15) Determine the values of P,Q, and average velocity. Schematic of melt flow in the tubular die