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.
 06.27.16
 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: $$\frac{\partial \rho v}{\partial t} + \frac{\partial \rho u v}{\partial x} + \frac{\partial \rho v^2}{\partial y} + \frac{\partial \rho w v}{\partial z} = -\frac{\partial P}{\partial y} + \frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \tau_{yy}}{\partial y} + \frac{\partial \tau_{zy}}{\partial z}$$ with $P$ the pressure 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
Starting from the energy equation obtained above $$\frac{\partial \rho E}{\partial t}+\frac{\partial \rho v H}{\partial y}-v\frac{\partial\tau_{yy}}{\partial y}=\frac{\rho \delta Q}{m \Delta t}− \frac{\rho\delta W}{m \Delta t}$$ 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 $k$ the thermal conductivity, and $T$ the temperature.
 09.13.17
 Due on September 20th at 13:30. Do Questions #1, #2, and #4 only.
 $\pi$