Viscous Flow Assignment 5 — Fully-Developed Flow  
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 $1.2$ 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, $H=1$ 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 $P_1-P_3=13.38$ 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 $1.1$ 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
You are working for a power plant, and one assignment given to you is to measure the wall roughness in an old rusted pipe. The pipe has a length of 2 m and a diameter of 2 cm. Because the pipe's length is much greater than its diameter, it is difficult to measure directly the height of the bumps on its interior surface. For this reason, you decide to measure the average height of the bumps indirectly through a fluid dynamics experiment: you attach a small pump to one extremity of the pipe and force water (viscosity of $10^{-3}$ kg/ms and density of 1000 kg/m$^3$) to flow through the pipe. You measure a mass flow rate of 1.57 kg/s and a force acting on the pipe due to fluid friction of 11.78 N. Knowing the latter, do the following:
(a)  Find $e$, the average height of the bumps within the rusted pipe
(b)  Find the percent increase in mass flow rate should the rusted pipe be substituted by a pipe with the same diameter and length but with smooth inner walls ($e \rightarrow 0$)
Hints: (i) Because the length is much greater than the diameter, the flow can be assumed fully-developed throughout; (ii) The pressure increase through the pump can be assumed constant; (iii) Part (a) and part (b) can be answered independently of each other.
Question #6
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 $L=1$ 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 $P_1-P_2$
5.  $8\times10^{-3}$ cm, 31.2%.
6.  0.018 kg/s, 73 Pa, 120 Pa.
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