posted 11 months ago

(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).

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