2016 Viscous Flow Final Exam  
Monday December 19th 2016
10:00 — 13:00

ANSWER ALL 6 PROBLEMS; ALL PROBLEMS HAVE EQUAL VALUE; NO NOTES OR BOOKS; USE VISCOUS FLOW TABLES THAT WERE DISTRIBUTED.
Question #1
Starting from the $x$-component of the momentum equation: $$ \rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \right) = -\frac{\partial P}{\partial x} + \mu \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial^2 u}{\partial y^2} + \mu \frac{\partial^2 u}{\partial z^2} $$ and from the mass conservation equation: $$ \frac{\partial \rho}{\partial t} + \frac{\partial }{\partial x}(\rho u) + \frac{\partial }{\partial y}(\rho v) + \frac{\partial }{\partial z}(\rho w) = 0 $$ Show that the skin friction coefficient and the thickness of a laminar boundary layer correspond to: $$ C_f=0.647 \cdot {\rm Re}_x^{-0.5} {\rm ~~~~and~~~~} \delta/x=4.64 \cdot {\rm Re}_x^{-0.5} $$ Outline all assumptions.
12.14.16
Question #2
Starting from the Navier-Stokes equations and the mass conservation transport equation, show that the constant-density constant-viscosity $y$-momentum equation corresponds to: $$ \rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \right) = -\frac{\partial P}{\partial y} + \mu \frac{\partial^2 v}{\partial x^2} + \mu \frac{\partial^2 v}{\partial y^2} + \mu \frac{\partial^2 v}{\partial z^2} $$
Question #3
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$^3$:
figure4.png  ./download/file.php?id=3176&sid=c637f285fa390f02e2e20a95335a85ec  ./download/file.php?id=3176&t=1&sid=c637f285fa390f02e2e20a95335a85ec
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 #4
Consider a reservoir attached to two rusted pipes as follows:
Q4.png  ./download/file.php?id=3175&sid=c637f285fa390f02e2e20a95335a85ec  ./download/file.php?id=3175&t=1&sid=c637f285fa390f02e2e20a95335a85ec
Water with a viscosity of $\mu=10^{-3}$ kg/ms and a density of $\rho=1000$ kg/m$^3$ flows out of the reservoir due to gravity effects. The two pipes are rusted and both have a $e/D$ ratio of 0.05. Knowing that $H_1=40$ m, $H_2=5$ m, $H_3=20$ m, $L_2=2$ m, $D_1=1~$m, and $D_2=0.1$ m, do the following:
(a)  Find $\dot{m}$, the mass flow rate coming out of the system.
(b)  Determine if the flow is laminar or turbulent within the two pipes.
Question #5
Consider a fluid flowing around an object as follows:
Q5.png  ./download/file.php?id=3173&sid=c637f285fa390f02e2e20a95335a85ec  ./download/file.php?id=3173&t=1&sid=c637f285fa390f02e2e20a95335a85ec
Knowing the fluid has a density of $\rho_\infty=1~{\rm kg/m^3}$, a pressure of $P_\infty=100~$kPa, a viscosity of $2\cdot 10^{-5}$ kg/ms and a freestream velocity $U_\infty$ of 30 m/s, and knowing that the depth of the object is of $D=1$ m, the height of the object is of $H=0.002$ m, and the length of the object is of $L=3$ m, determine as accurately as possible the force that the object feels along the $x$ direction due to the motion of the fluid. Note: the fluid flows around and within the solid.
Question #6
Consider fully-developed laminar flow in a channel with a viscosity of $\mu=10^{-3}$ kg/ms, a pressure gradient of $\frac{dP}{dx}=-100$ 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:
Q6.png  ./download/file.php?id=3174&sid=c637f285fa390f02e2e20a95335a85ec  ./download/file.php?id=3174&t=1&sid=c637f285fa390f02e2e20a95335a85ec
Given the node positions:
Node$y$, m
10.0
20.01
30.03
40.11
and the governing equation: $$ \frac{\partial^2 u}{\partial y^2}= \frac{1}{\mu}\frac{dP}{dx} $$ do the following:
(a)  Using the finite volume method, derive an algebraic equation for the inner nodes (nodes 2 and 3).
(b)  Using the finite volume method, derive an algebraic equation for the symmetry node (node 4).
(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 $u_1$, $u_2$, $u_3$, and $u_4$.
Answers
3.  10.48 N, 7903 W/m.
4.  110.96 kg/s, turbulent.
5.  34 N.
6.  0, 105 m/s, 285 m/s, 605 m/s.
12.26.16
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