2021 Fundamentals of Fluid Mechanics B Final Exam  
Tuesday May 11th 2021
10:30 — 12:30
INSTRUCTIONS
  NO NOTES OR BOOKS EXCEPT FOR THE FUNDAMENTALS OF FLUID MECHANICS TABLES.
  ONLY A BAREBONE CALCULATOR IS ALLOWED WITHOUT SDCARD SLOTS OR A LARGE SCREEN.
  ALL QUESTIONS HAVE EQUAL VALUE; ANSWER ALL 4 QUESTIONS.
  WRITE YOUR SOLUTIONS IN SINGLE COLUMN FORMAT, WITH ONE STATEMENT FOLLOWING ANOTHER VERTICALLY.
  ALWAYS START YOUR SOLUTIONS USING THE EQUATIONS IN THE TABLES.
  OUTLINE ALL ASSUMPTIONS MADE.
  WRITE YOUR SOLUTIONS NEATLY SO THAT THEY ARE EASY TO READ AND VERIFY.
  USE A PENCIL OR A PEN WITH BLACK INK. DO NOT USE BLUE OR RED INK.
  DON'T WRITE ONE LINE WITH TWO EQUAL SIGNS.
  HIGHLIGHT YOUR ANSWERS USING A BOX.
  SCAN YOUR SOLUTIONS USING CAMSCANNER AND CREATE ONE PDF FILE PER QUESTION. UPLOAD THE FOUR PDF FILES ON THE D2L.
05.05.21
Question #1
Starting from the momentum equation in conservative form: $$ \frac{\partial \rho u}{\partial t} + \frac{\partial \rho u^2}{\partial x} + \frac{\partial \rho u v}{\partial y} + \frac{\partial \rho u w}{\partial z} = -\frac{\partial P}{\partial x} + \frac{\partial \tau_{xx}}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} +B_x $$ And using the Reynolds transport theorem: $$ \frac{d}{dt} \int_V \vec{F} dV = \int_V \frac{\partial \vec{F}}{\partial t}dV +\int_S \left(\vec{v}_{\rm cv} \cdot \vec{n} \right) \vec{F} dS $$ Prove that the $i$th component of the viscous momentum equation in integral form corresponds to: $$ \frac{{\rm d}}{{\rm d} t} \int_V \rho \vec{v}_i \,{\rm d} V + \int_S \rho \vec{v}_i (\vec{v}-\vec{v}_{\rm cv}) \cdot \vec{n}\, {\rm d} S = -\int_S P \vec{n}_i \,{\rm d} S + \int_S \sum_{j=1}^3 \tau_{ji} \vec{n}_j \,{\rm d} S + \int_V \vec{B}_i \, {\rm d} V $$ where $\vec{v}$ is the velocity of the flow and $\vec{v}_{\rm cv}$ is the velocity of the control volume surfaces.
Question #2
Consider air flowing in a system made of two pipes and a piston arranged as follows:
F2021Q2.png
Knowing that $L_1=0.1~$m, $L_2=0.2~$m, $R_1=3~$mm, $R_2=5$ cm, that the pressure at the pipe system exit is $P_1=101300~$Pa, that the viscosity of the air is $\mu=2\times10^{-5}~$kg/ms, that the density of the air can be set to $1~$kg/m$^3$, and that the piston travels towards the left with the speed $U_{\rm p}=1~$m/s, do the following:
(a)  Find an analytical expression for the pressure on the right of the piston, $P_4$, that is function of $P_1$, $\rho$, $\mu$, $U_{\rm p}$, $L_1$, $L_2$, $R_1$, and $R_2$.
(b)  Using the analytical expression found in (a), determine the pressure $P_4$.
(c)  Determine the Reynolds numbers in pipe A and in pipe B and check if the assumptions made in finding (a) and (b) are valid.
Question #3
Consider a boundary layer forming on a flat plate and with the freestream velocity $u_\infty$ being parallel with the $x$ axis:
F2021Q3.png  ./download/file.php?id=7856&sid=f04f7c79e8f9bf9147bdece44303f9b7  ./download/file.php?id=7856&t=1&sid=f04f7c79e8f9bf9147bdece44303f9b7
Do the following:
(a)  Find an expression for the normalized $y$-velocity component $v_y/u_\infty$ within the boundary layer. The expression for $v_y/u_\infty$ should be function of the Reynolds number ${\rm Re}_x$ and $y/\delta$ only, with $\delta$ the boundary layer thickness.
(b)  Plot $v_y/u_\infty$ as a function of the non-dimensional wall distance $y/\delta$ (with $\delta$ the boundary layer height). The plot should be detailed. For instance, if there are minimums or maximums of $v_y/u_\infty$, the locations and magnitude of such extrema should be derived and indicated on the plot.
(c)  Find an expression for the ratio $v_y/v_x$ at the location $y=\delta$.
Question #4
The average diameter of droplets originating from expiratory activities (breathing, talking, coughing) can be taken to be 75 micrometers but with a substantial fraction having a diameter as small as 5 micrometers. Let's determine how long such droplets remain in the air after being exhaled at a height of 1.6 m above the ground. Specifically, do the following:
(a)  First determine analytically an expression for the velocity of the droplets and the time needed for the droplets to hit the ground assuming that the inertia terms (within Newton's law) are negligible. Do this using Stokes solution and Oseen's solution. Find the time in seconds and velocity in m/s using the Stokes and Oseen's solutions for a large and small droplet diameter.
(b)  Starting from Newton's law and neglecting aerodynamic drag (due to pressure or shear stresses), determine the time in seconds needed for the droplets to hit the ground. Do this for both Stokes and Oseen's solutions and for a large and small droplet diameter.
(c)  Based on the results obtained in (a) and (b), provide an approximation to the time needed for the droplets (either large or small) to hit the ground taking into consideration aerodynamic drag as well as inertia terms within Newton's law. Do this for both Stokes and Oseen's solution. Note: you should not find an exact answer, but a very good approximate answer. Part (c) should take you less than a half page to answer.
You can assume that the droplet properties are similar to the ones of water. You can take the water and air properties at ambient conditions (1 atm and 300 K) from the tables.
05.10.21
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