Navier-Stokes Energy Equation
I've been assigned (for homework in a mathematical modelling course) the task of deriving the Navier-Stokes energy equation in one space dimension:
Consider a fluid flowing through a cylindrical pipe (from left to right; also assume the pipe to be horizontal) with constant cross section $A$, velocity $v(x,t)$, density $\rho(x,t)$, specific internal energy $e(x,t)$, temperature $T(x,t)$, under pressure $p(x,t)$, and subject to viscous stress. (By specific internal energy, we mean internal energy per unit mass. Internal energy arises from intermolecular collisions in the fluid, and should be distinguished from the kinetic energy associated with the macroscopic fluid motion(i.e., $\tfrac{1}{2}mv^2$). By conserving energy, show that \begin{equation} \left[\rho\left(\frac{v^2}{2}+e\right)\right]_{t}+\left[\rho ve+\frac{1}{2}\rho v^3-\kappa(x)T_x+pv-\mu vv_x\right]_x=0, \end{equation} where $\kappa$ is thermal diffusivity and $\mu$ is the viscosity coefficient.
Attached with the problem is this "hint":
Note. 453
The energy density should be easy to identify. For the flux, consider each of the following, which correspond respectively with terms in the equation above: internal energy, kinetic energy, energy lost to heat, energy lost to work against pressure, energy lost to work against viscous stress.
I suppose it is intended for us to use the conservation equation $u_t+f_x=0$ (where $u(x,t)$ is some conserved quantity and $f$ is the flux), but I'm having a lot of trouble interpreting the physical phenomena as equations (I'm a mathematics graduate student who knows embarrassingly little physics).
Looking at the equation above I assume that the energy density at a point is simply the internal energy plus kinetic energy, or $mv^2/2+me$, (since specific internal energy is internal energy is internal energy per unit mass), and then somehow we should be able to argue that $m=\rho A$, and eventually the $A$ will be divided out. The trouble I have with this however is that I believe the equation is mass $=$ volume $\times$ density--not area--and I'm having a little trouble conceptually allowing myself the state $m=A\rho$ with confidence (since any object without three spatial dimensions should have zero volume right?).
For the flux, I'm having real trouble interpreting what the "energy flux" would be, even with his hint.
Any help would be greatly appreciated. Hopefully this question hasn't been asked already (I did a good bit of searching) and that it's been tagged correctly. Thanks in advance.
Also, any recommendations for a physics book geared towards people with a math background would be greatly appreciated as well.
Please let me know if anything is unclear in the statement of the question and I will try my best to clear things up. 453