Viscous Flow Assignment 1 — Mass, Momentum, and Energy Equations  
The first assignment consists of deriving from basic principles the mass, momentum, and energy transport equations commonly used to solve viscous fluid flow.
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 viscous fluid corresponds to: $$ \frac{\partial \rho v}{\partial t} + \frac{\partial \rho u v}{\partial x} + \frac{\partial \rho v^2}{\partial y} + \frac{\partial \rho w v}{\partial z} = -\frac{\partial P}{\partial y} + \frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \tau_{yy}}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} + B_y $$ with $P$ the pressure, $B_y$ the $y$-component of the body force per unit volume, and $\tau_{ij}$ the shear stress vector along $j$ acting on the faces perpendicular to $i$.
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 $$ \rho \frac{\partial v}{\partial t} + \rho v \frac{\partial v}{\partial y}=-\frac{\partial P}{\partial y}+\frac{\partial \tau_{yy}}{\partial y} $$ show that the total energy transport equation for a viscous fluid corresponds to: $$ \frac{\partial \rho E}{\partial t}+\frac{\partial \rho v H}{\partial y}-v\frac{\partial\tau_{yy}}{\partial y}=\frac{\rho \delta Q}{m \Delta t}− \frac{\rho\delta W}{m \Delta t} $$ with the total energy $E\equiv e+\frac{1}{2}q^2$, the total enthalpy $H\equiv h +\frac{1}{2}q^2$, and $q$ the speed of the flow.
Question #4
Show that, for a 1D problem along $y$ in which the forces are limited to viscous and pressure forces, the following is correct: $$ − \frac{\rho\delta W}{m \Delta t} = \tau_{yy} \frac{\partial v }{\partial y} $$ with $\delta W/\Delta t$ the rate of work done by the fluid element on the environment by forces other than pressure and $m$ the mass of the fluid element.
Due on Wednesday September 19th at 9:00. Do Questions #2 and #4 only.
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