In polymer extrusion processes, a viscous polymer of viscosity u is forced to flowsteadily from left to right (distance L) in the annular area between two fixedconcentric cylinders by applying

a pressure difference Pout - Pin. The inner cylinderis solid, whereas the outer one is hollow; their radii are R1 and R2, respectively.The problem, which could occur in the extrusion of plastic tubes, is to find thevelocity profile in the annular space and the total volumetric flow rate Q. Note thatcylindrical coordinates are now involved. (a) Giving reasons, simplify the continuity equation at steady state using cylindricalcoordinates \frac{\partial \rho}{\partial t}+\frac{1}{r} \frac{\partial}{\partial r}\left(\rho r v_{r}\right)+\frac{1}{r} \frac{\partial}{\partial \theta}\left(\rho v_{\theta}\right)+\frac{\partial}{\partial z}\left(\rho v_{z}\right)=0 (b) Giving reasons, simplify the Navier-Stokes equations for the velocity component which is not zero. \rho\left(\frac{\partial v_{r}}{\partial t}+v_{r} \frac{\partial v_{r}}{\partial r}+\frac{v_{\theta}}{r} \frac{\partial v_{r}}{\partial \theta}+v_{z} \frac{\partial v_{r}}{\partial z}-\frac{v_{\theta}^{2}}{r}\right)=-\frac{\partial p}{\partial r}+\mu\left[\frac{\partial}{\partial r}\left(\frac{1}{r} \frac{\partial}{\partial r}\left(r v_{r}\right)\right)+\frac{1}{r^{2}} \frac{\partial^{2} v_{r}}{\partial \theta^{2}}+\frac{\partial^{2} v_{r}}{\partial z^{2}}-\frac{2}{r^{2}} \frac{\partial v_{\theta}}{\partial \theta}\right]+\rho g_{r} \rho\left(\frac{\partial v_{\theta}}{\partial t}+v_{r} \frac{\partial v_{\theta}}{\partial r}+\frac{v_{\theta}}{r} \frac{\partial v_{\theta}}{\partial \theta}+v_{z} \frac{\partial v_{\theta}}{\partial z}+\frac{v_{r} v_{\theta}}{r}\right)=-\frac{1}{r} \frac{\partial p}{\partial \theta}+\mu\left[\frac{\partial}{\partial r}\left(\frac{1}{r} \frac{\partial}{\partial r}\left(r v_{\theta}\right)\right)+\frac{1}{r^{2}} \frac{\partial^{2} v_{\theta}}{\partial \theta^{2}}+\frac{\partial^{2} v_{\theta}}{\partial z^{2}}+\frac{2}{r^{2}} \frac{\partial v_{r}}{\partial \theta}\right]+\rho g_{\theta} \rho\left(\frac{\partial v_{z}}{\partial t}+v_{r} \frac{\partial v_{z}}{\partial r}+\frac{v_{\theta}}{r} \frac{\partial v_{z}}{\partial \theta}+v_{z} \frac{\partial v_{z}}{\partial z}\right)=-\frac{\partial p}{\partial z}+\mu\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial v_{z}}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} v_{z}}{\partial \theta^{2}}+\frac{\partial^{2} v_{z}}{\partial z^{2}}\right]+\rho g_{z} (c) State the two boundary conditions needed to solve the simplified Navier-Stokes equations from (b). (d) Assuming negligible gravity effects, solve the equation derived in (b) subjected to the boundary conditions from (c) to show that the velocity profile of the viscous polymer flowing horizontally along the annulus is given by: u_{\mathrm{z}}=\frac{1}{4 \mu}\left(\frac{P_{O U T}-P_{I N}}{L}\right)\left[r^{2}-R_{1}^{2}+\frac{R_{1}^{2}-R_{2}^{2}}{\ln \left(\frac{R_{1}}{R_{2}}\right)} \ln \left(\frac{R_{1}}{r}\right)\right] (e) Derive an expression for the shear force (i.e., friction F,) arising from the shearing stress between the fluid flow (z-direction) and the external cylinder radial surface wall (r-direction).