### Question 38667

Verified

Transport Phenomenon

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}

(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:

(c) State the two boundary conditions needed to solve the simplified Navier-Stokes equations from (b).

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).

### Question 36763

Verified

Transport Phenomenon

Use your own words to explain Transit-Oriented Development (TOD).

### Question 36762

Verified

Transport Phenomenon

What is Traffic Calming?

### Question 36761

Verified

Transport Phenomenon

Use your own words to define Eco-driving.

### Question 36760

Verified

Transport Phenomenon

Use your own words to define Alternative Fuel Vehicles (AFVS).

### Question 36759

Verified

Transport Phenomenon

) What is a Sustainable Transportation System?

### Question 36758

Verified

Transport Phenomenon

Use your own words to explain what a Neckdown is.

### Question 36757

Verified

Transport Phenomenon

Use your own words to define Bicycle Boulevard.ist

### Question 36756

Verified

Transport Phenomenon

s) What is Bus Rapid Transit (BRT)? What are typical features of a BRT system?

### Question 36755

Verified

Transport Phenomenon

What are the key differences between Mass Transit and Paratransit?