Fundamentals of Fluid Mechanics B Assignment 1 — Mean Free Path, Viscosity, and Viscous Diffusion  
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.
Question #1
Starting from the conservation of momentum principle, derive Newton's law for a system with mass aggregation as follows: $$ \sum \vec{F} = m \frac{d \vec{v}}{dt} - \sum_k \dot{m}_k \vec{v}_{{\rm REL,}k} $$ Do the proof with 3 small masses $dm_1$, $dm_2$, $dm_3$ being added to one large mass $m$.
Question #2
Starting from the Newton's law for a system with mass aggregation: $$ \sum \vec{F} = m \frac{d \vec{v}}{dt} - \sum_k \dot{m}_k \vec{v}_{{\rm REL,}k} $$ Derive the $x$ momentum equation: $$ \rho \frac{d u}{dt} = -\frac{\partial P}{\partial x} - \frac{\partial f_{ux}}{\partial x} - \frac{\partial f_{uy}}{\partial y} - \frac{\partial f_{uz}}{\partial z} +B_x $$ Define clearly the viscous fluxes $f_{ux}$, $f_{uy}$, etc, and the body force $B_x$.
Question #3
Starting from the definition of the viscous flux $f_{ux}$, prove that $f_{ux}$ is equal to: $$ f_{ux}=-\mu\frac{\partial u}{\partial x} $$ with the viscosity equal to: $$ \mu\equiv \frac{2}{3} \sqrt{\frac{k_{\rm B} T m}{\pi \sigma^2}} $$ Explain assumptions and limitations if any. In deriving the latter, you can use the following relationship for the mean free path in a gas: $$ \lambda=\frac{1}{\sqrt{2} N \sigma} $$
Question #4
Prove that the mean free path in a gas taking into consideration the relative velocity of the molecules corresponds to: $$ \lambda=\frac{1}{\sqrt{2}N\sigma} $$ Note: you need to prove everything in this equation, including the $\sqrt{2}$ on the denominator.
Do Questions #2, #3, #4 only. Due on January 30th at 11:00 am.
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