Convective Heat Transfer Assignment 1 — Fluid Transport Equations  
Question #1
Derive Fourier's law of heat conduction in a gas: $$ q^{\prime \prime}_x=-k \frac{\partial T}{\partial x} $$ with $$ k=\frac{5 k_{\rm B}}{4 \sigma}\sqrt{\frac{3 RT}{2}}$$ with $k$ the thermal conductivity, $\sigma$ the collision cross-section, $k_{\rm B}$ the Boltzmann constant and $R$ the gas constant.
Question #2
Starting from the principle of conservation of mass, show that: $$ \frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x} + \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} =0$$ with $\rho$ the mass density, and $u,v,w$ the $x,y,z$ components of the velocity vector.
Question #3
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 #4
Starting from the 1st law of thermo $$ {\rm d}(mh)-V {\rm d}P=\delta Q-\delta W $$ the $y$ momentum equation in 1D $$ \rho \frac{\partial v}{\partial t} + \rho v \frac{\partial v}{\partial y}=-\frac{\partial P}{\partial y}+\frac{\partial \tau_{yy}}{\partial y} $$ and Fourier's law $\vec{q}^{\prime\prime}=-k \vec{\nabla}T$, show that the total energy transport equation for a viscous fluid corresponds to: $$ \frac{\partial \rho E}{\partial t} + \frac{\partial \rho v H}{\partial y} = \frac{\partial }{\partial y}\left( k \frac{\partial T}{\partial y} \right) + \frac{\partial v \tau_{yy}}{\partial y} $$ with the total energy $E\equiv e+\frac{1}{2}q^2$, the total enthalpy $H\equiv h +\frac{1}{2}q^2$, $q$ the speed of the flow, $k$ the thermal conductivity, and $T$ the temperature.
Due on Wednesday March 22nd at 16:30.
Note: please make an effort to solve question #4 in 2D. Maybe I will ask this question in the midterm or final..
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