Fundamentals of Fluid Mechanics B Assignment 6 — Stokes and Oseen Flow  
Instructions
Write your solutions in single column format, with one statement following another vertically. Write your solutions neatly so that they are easy to read and verify. Don't write one line with two equal signs. Highlight your answers using a box. Failure to do this will result in a lower score and fewer comments on my part.
04.02.20
Problem #1
(a)  For a constant density flow in spherical coordinates, show that a streamfunction $\psi$ exist and that the velocity components become: $$ v_r = \frac{1}{r^2\sin \theta} \frac{\partial \psi}{\partial \theta} $$ $$ v_\theta = -\frac{1}{r \sin \theta} \frac{\partial \psi}{\partial r} $$ List all assumptions.
(b)  Assuming Stokes flow, the streamfunction for flow over a sphere can be shown to be equal to: $$ \psi=\sin^2\theta \left(\frac{A}{r} + B r + C r^2 + D r^4 \right) $$ Impose proper boundary conditions for flow over a sphere and find $A$, $B$, $C$, and $D$.
(c)  Starting from the $\psi$ function determined in (b) and the velocity components determined in (a), derive an expression for the drag force that the sphere experiences due to viscous effects.
Problem #2
Consider flow past a sphere:
(a)  Using order of magnitude analysis, determine the conditions for which the convection terms can be assumed negligible compared to the other terms in the vicinity of the sphere surface.
(b)  Using order of magnitude analysis, explain why under the conditions found in (a) the convection terms can not be assumed negligible compared to the diffusion terms far away from the sphere surface.
Problem #3
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:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2843952/
Let's determine how long such droplets remain in the air after being exhaled. Specifically, do the following:
(a)  First determine analytically an expression for the velocity of the droplets assuming that the inertia terms (within Newton's law) are negligible. Do this using Stokes solution and Oseen's solution.
(b)  Starting from Newton's law and neglecting aerodynamic drag (due to pressure or shear stresses), determine the time needed for the droplet to hit the ground.
(c)  Determine as accurately as possible (including inertia effects and aerodynamic drag) the time needed for such droplets (both 75 and 5 micrometer diameter) to reach the ground when exhaled from an individual mouth located 1.6 m above the ground. Do this for Stokes and Oseen's solutions. If an exact solution can not be found, write a C code that integrates the equations numerically. The code must be submitted in C but can be originally written in another language and translated with an automatic translater.
(d)  Validate your code or analytical expression found in (c) using what was found in (a) and (b). Explain your logic.
(e)  Verify that the low Reynolds number assumption made in (c) is valid for this problem.
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.
Due on Thursday April 9th at 11:00. Do all problems.
I'm not happy with your solutions for problems #2 and #3. Redo them fully. For problem #3, if you can not succeed in solving the problem analytically, then you have to write a C code and integrate the equations numerically. Only codes written in C will be accepted.
Due on Thursday April 16th at 11:00.
04.12.20
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