Introduction to CFD Assignment 2 — Euler Equations  
The second assignment consists of deriving from basic principles the mass, momentum, and energy transport equations commonly used in CFD.
Question #1
Starting from the principle of conservation of mass, show that the mass conservation equation for a fluid corresponds to: $$ \frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x} + \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} =0$$ with $\rho$ the mass density, and $u,v,w$ the $x,y,z$ components of the velocity vector.
Question #2
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 fluid corresponds to: $$ \frac{\partial \rho v}{\partial t} + \frac{\partial \rho u v}{\partial x} + \frac{\partial \rho v^2}{\partial y} + \frac{\partial \rho w v}{\partial z} = -\frac{\partial P}{\partial y} $$ with $P$ the pressure.
Question #3
Starting from the 1st law of thermo $$ {\rm d}(mh)-V {\rm d}P=\delta Q-\delta W $$ the $y$ momentum equation in 1D $$ \rho \frac{\partial v}{\partial t} + \rho v \frac{\partial v}{\partial y}=-\frac{\partial P}{\partial y} $$ show that the total energy transport equation for a fluid corresponds to: $$ \frac{\partial \rho E}{\partial t} + \frac{\partial \rho v H}{\partial y} = 0 $$ 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, and $T$ the temperature.
Due on Tuesday March 21st at 16:30
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