2017 Viscous Flow Final Exam  
Saturday December 16th, 2017
16:00 — 19:00

Question #1
A crankshaft journal bearing in an automobile engine is lubricated by oil with a kinematic viscosity of $10^{-4}$ m$^2$/s and a density of 885 kg/m$^3$:
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The bearing inner diameter $D_{\rm i}$ is of 10 cm, the bearing outer diameter $D_{\rm o}$ is of 11 cm, and the bearing rotates at $7200$ rpm. The bearing is under no load so the clearance is symmetric. Determine the torque per unit depth and the power dissipated per unit depth.
Question #2
Starting from Newton's law $\vec{F}_y=m\frac{dv}{dt}$ and the mass conservation equation show that the $y$-component of the momentum transport equation for a viscous fluid corresponds to: $$ \frac{\partial \rho v}{\partial t} + \frac{\partial \rho u v}{\partial x} + \frac{\partial \rho v^2}{\partial y} + \frac{\partial \rho w v}{\partial z} = -\frac{\partial P}{\partial y} + \frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \tau_{yy}}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} $$ with $P$ the pressure and $\tau_{ij}$ the shear stress vector along $j$ acting on the faces perpendicular to $i$.
Question #3
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 #4
Consider an aquarium with a width of $W=2$ m, a depth of $2$ m and a height of $H=2$ m as follows:
aquarium.png  ./download/file.php?id=3949&sid=87e3262c6c6d0786d9a91a5da362490c  ./download/file.php?id=3949&t=1&sid=87e3262c6c6d0786d9a91a5da362490c
Initially, water ($\rho=1000$ kg/m$^3$, $\mu=0.001$ kg/ms) fills fully the aquarium. You wish to drain the aquarium through a tube located at its bottom. Knowing that the length of the tube is of $L=1$ m, that its diameter is of 2 cm, that the surface roughness of the tube is of $e/D=0.05$, and that the tube is standing vertically with no bends, do the following:
(a)  Determine whether the flow in the tube is laminar or turbulent when the aquarium is 100% full.
(b)  Determine whether the flow in the tube is laminar or turbulent when the aquarium is 10% full.
(c)  Determine how many seconds are required to empty 90% of the aquarium.
Question #5
Consider fully-developed turbulent flow in a channel with a viscosity of $\mu=10^{-3}$ kg/ms, a pressure gradient of $\frac{dP}{dx}=-1000$ Pa/m, and a density of 1000 kg/m$^3$. We wish to solve this problem using a finite volume method. For this purpose, we take advantage of the symmetry of the problem and place the nodes as follows:
Given the node positions and values for the turbulence eddy viscosity:
Node$y$, m$\mu_t$, kg/ms
and the governing equation: $$ \frac{\partial}{\partial y}\left((\mu+\mu_t)\frac{\partial u}{\partial y} \right)= \frac{dP}{dx} $$ do the following:
(a)  Using the finite volume method, derive an algebraic equation for the inner node (node 2).
(b)  Using the finite volume method, derive an algebraic equation for the symmetry node (node 3).
(c)  Outline the algebraic equation for the wall node (node 1).
(d)  Solve the algebraic equations obtained in (a), (b), and (c) using Gaussian elimination or any other method of your choice and find the velocities $u_1$, $u_2$, $u_3$.
Question #6
Consider air with uniform properties entering in and flowing on a pipe as follows:
Q6.png  ./download/file.php?id=3957&sid=87e3262c6c6d0786d9a91a5da362490c  ./download/file.php?id=3957&t=1&sid=87e3262c6c6d0786d9a91a5da362490c
The pipe has a length $L=1$ m, an inner diameter $D_i=19$ cm, and an outer diameter $D_o=20$ cm. Knowing that the air viscosity is of $\rm 10^{-5}~kg/ms$, the air velocity is of $\rm 10~m/s$, and that the air density is of $\rm 1~kg/m^3$, do the following:
(a)  Calculate as accurately as possible the boundary layer thickness on the external surface of the pipe at $x=L$.
(b)  Calculate as accurately as possible the drag due to friction in Newton acting on the pipe in the positive $x$ direction.
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