Fluid Mechanics Assignment 8 — Fully-Developed Flow  
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
Question #1
Starting from the mass conservation equation in cylindrical coordinates: $$ \frac{\partial \rho}{\partial t} + \frac{1}{r} \frac{\partial }{\partial r} (\rho r v_r) + \frac{1}{r} \frac{\partial }{\partial \theta}(\rho v_\theta) + \frac{\partial}{\partial x}(\rho v_x)=0 $$ and the $r$ component of the momentum conservation equation in cylindrical coordinates $$ \rho\left(\frac{\partial v_r}{\partial t} + v_r \frac{\partial v_r}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_r}{\partial \theta} - \frac{v_\theta^2}{r} + v_x \frac{\partial v_r}{\partial x} \right) = B_r - \frac{\partial P}{\partial r} + \frac{\mu}{r}\frac{\partial}{\partial r}\left(r \frac{\partial v_r}{\partial r}\right) + \frac{\mu}{r^2} \frac{\partial^2 v_r}{\partial \theta^2} - \frac{2\mu}{r^2}\frac{\partial v_\theta}{\partial \theta} -\mu \frac{v_r}{r^2} + \mu \frac{\partial^2 v_r}{\partial x^2} $$ and the $x$ component of the momentum equation: $$ \rho\left(\frac{\partial v_x}{\partial t} + v_r \frac{\partial v_x}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_x}{\partial \theta} + v_x \frac{\partial v_x}{\partial x} \right) = B_x - \frac{\partial P}{\partial x} + \frac{\mu}{r}\frac{\partial}{\partial r}\left(r \frac{\partial v_x}{\partial r} \right) + \frac{\mu}{r^2} \frac{\partial^2 v_x}{\partial \theta^2} + \mu \frac{\partial^2 v_x}{\partial x^2} $$ Show that the normalized velocity and the Darcy friction factor for laminar flow in a pipe with radius $R$ under no external forces correspond to: $$ \frac{u}{u_{\rm b}}=2\left( 1-\frac{r^2}{R^2} \right) ~~~{\rm and}~~~ f=\frac{64}{{\rm Re}_D} $$ with: $$ u_{\rm b}\equiv \frac{\dot{m}}{\rho A_{\rm cs}} ~~~{\rm and}~~~ f\equiv \frac{4\tau_{\rm w}}{\frac{1}{2} \rho u_{\rm b}^2} $$ Outline all assumptions. Note: this question is worth double the points awarded to the other questions.
Question #2
A pipeline is used to transport oil from Pohang to Gyeongju with an internal diameter of $(1.2-0.03\times\xi_1)$ m and a length of 20 km. Knowing that the oil has a density of 900 kg/m$^3$ and a viscosity of 0.765 kg/ms, and knowing that the pressure exiting the pipeline must be equal to the atmospheric pressure, find the pressure at the pipeline entrance that is high enough to maintain the oil flowing at a rate of 500 kg/s.
Question #3
Water is flowing in a pipe concatenated to a duct with a rectangular cross section, as shown below:
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Knowing that $L_1=2$ m, $L_2=40$ km, $W=1$ m, $H=1/(1+\xi_1)$ m, $D=0.1$ m, $\rho=1000$ kg/m$^3$, $\mu=10^{-3}$ kg/ms, and that the surface roughness $e=2$ mm (for both ducts), and that the pressure difference between the entrance and the exit is $P_1-P_3=(13.38+\xi_1)$ Pa, calculate the mass flow rate, as well as $P_2-P_1$ and $P_3-P_2$.
Question #4
Water with a density of 1000 ${\rm kg/m^3}$ and a viscosity of $10^{-3}$ kg/ms flows in a duct with a width of $0.01$ m, a height of 0.02 m, a length of 1 m, and a wall roughness $e=0.107$ mm. If the pressure at the pipe entrance is $(1.1+0.01\times\xi_1)$ atm and the pipe exits to atmospheric pressure, find the water mass flow rate. Determine the increase in water mass flow rate if the duct surfaces are polished and can be assumed smooth (i.e. $e\rightarrow0$).
Question #5
Water with a density of 1000 kg/m$^3$ and a viscosity of 10$^{-3}$ kg/ms flows in a smooth pipe with a diameter $D=1$ cm and a length $L=(1+\xi_1)$ m. The pressure difference between the pipe entrance and exit is denoted by $P_1-P_2$. Do the following:
(a)  Find $P_1-P_2$ that yields the largest mass flow rate while keeping the flow laminar.
(b)  Find $P_1-P_2$ that yields the smallest mass flow rate while keeping the flow turbulent.
(c)  Find the mass flow rate for the value of $P_1-P_2$ found in (a)
(d)  Find the mass flow rate for the value of $P_1-P_2$ found in (b)
(e)  Find the mass flow rate for a value of $P_1-P_2$ ten times less than the one determined in (a)
(f)  Find the mass flow rate for a value of $P_1-P_2$ ten times higher than the one determined in (b)
(g)  Plot the mass flow rate as a function of $P_1-P_2$
Due on Wednesday December 3rd
Note: this assignment is given 4.5 points instead of the usual 3. I.e. if you don't submit it, you'll get a $-4.5$ point penalty.
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