I'm trying to understand the basics of fluid dynamics and the Navier-Stokes equations by following the short book A Mathematical Introduction to Fluid Dynamics by Chorin and Marsden (I'm an applied mathematician, not a physicist, so please be patient with me. Also I wasn't sure if this question was better suited for MSE or PSE. I chose here.)
As I understand it, to uniquely determine the three important quantities in a fluid mechanics problem which are $\mathrm u$, $P$ and $\rho$, one needs five equations - one for each component of $\mathrm u$ (which can be written as a single vector equation, reducing our number of equations to 3), and two more for the scalars $P,\rho$. To get these three equations, we need three physical conditions. They are
- Conservation of mass
- Conservation of momentum
- Conservation of energy
I want to derive these three equations in the most general form possible. What I mean by that is our fluid may be
- compressible ($\nabla\cdot\mathrm u\neq0$)
- viscous ($\mu,\lambda\neq0$)
- vortical ($\nabla\times\mathrm u\neq0$)
- Subject to thermodynamic effects
The easiest part is conservation of mass. A short derivation will show that mass conservation may be described mathematically via the continuity equation,
$$\partial_t\rho+\nabla\cdot(\rho~\mathrm u)=0\tag{1}$$
Conservation of momentum is considerably more difficult, but after some work I was able to eventually arrive at
$$\rho\frac{D\mathrm u}{Dt}=-\nabla P+\nabla\cdot \boldsymbol{\tau}+\mathrm{F}\tag{2A}$$
Where,
- $D/Dt=\partial_t +\mathrm u\cdot\nabla$ is the material derivative,
- $\mathrm{F}$ denotes any external forces ($\mathrm{F}=0$ in an isolated system),
- $\boldsymbol{\tau}$ is the deviatoric stress tensor, a $(1,1)$ tensor
defined as $$\boldsymbol{\tau}=\lambda\mathbf{I}~\nabla\cdot\mathrm u+\mu\big(\nabla\mathrm u+(\nabla\mathrm u)^{\mathrm T}\big)$$ Where $\mathbf I$ is the identity, $\mu$ is the dynamic viscosity, and $\lambda$ is the bulk viscosity. Using the identity $\nabla\cdot(\nabla\mathrm u)^{\mathrm T}=\nabla(\nabla\cdot\mathrm u)$ we can expand out this equation to reach
$$\rho(\partial_t\mathrm u+\mathrm u\cdot\nabla\mathrm u)=-\nabla P+(\lambda+\mu)\nabla(\nabla\cdot\mathrm u)+\mu\boldsymbol{\triangle}\mathrm u+\mathrm F\tag{2B}$$
This equation is given on page 33 of the linked text.
However, conservation of energy still eludes me. NASA appears to have a published formula, but it is only given in the special case of Cartesian coordinates, and I would much rather a coordinate-free representation. This page gives a coordinate-free description:
$$\partial_t\left[\rho~\left(e+\frac{|\mathrm u|^2}{2}\right)\right]+\nabla\cdot\left[\rho\mathrm u~\left(e+\frac{|\mathrm u|^2}{2}\right)\right]=\nabla\cdot(k\nabla T-P\mathrm u+\boldsymbol{\tau}\cdot\mathrm u)+\mathrm u\cdot\mathrm F+\mathcal{Q}\tag{3}$$
But they don't offer a derivation, nor do they even explain what the symbols $e,k,\mathcal Q$ mean. My guess(?) is that $k$ is some kind of thermal diffusivity, $e$ is a function of position and time and tells you something about the internal or potential energy, and $\mathcal{Q}$ is some external heat source. Am I right? And in most cases we would know $e(\mathrm r,t)$ and $T(\mathrm r,t)$ as given, and not have to solve for them, correct?
If we assume that the fluid has no thermal or potential energy, the energy can be written as purely kinetic: $$E_{\text{tot}}=\frac{1}{2}\int_{\Omega(t)}\rho(\mathrm r,t)~|\mathrm u(\mathrm r,t)|^2~\mathrm d\mu(\mathrm r)$$ Here $\Omega(t)$ is the region that our fluid occupies at time $t$. Using the transport theorem, one can arrive at $$\frac{\mathrm d E_{\text{tot}}}{\mathrm dt}=\frac{\mathrm d}{\mathrm dt}\int_{\Omega(t)}\rho |\mathrm u|^2\mathrm d\mu=\int_{\Omega(t)}\rho\mathrm u\cdot\frac{D\mathrm u}{Dt}\mathrm d\mu$$ So it would appear that a sufficient constraint for conservation of energy in this case is
$$\rho\mathrm u\cdot\frac{D\mathrm u}{Dt}=0\implies \mathrm u\cdot\big(\partial_t\mathrm u+\mathrm u\cdot\nabla\mathrm u\big)=0\tag{4}$$
Which is a slightly modified form of the well known Burgers' equation.
But what if there is thermal and kinetic energy? What can I do then? Can anyone please help me derive and make sense of equation $(3)$, and reduce it to $(4)$ in a special case?
Thanks so much.
EDIT: Definitions for various uses of the symbol $\nabla$.
If $\mathbf{T}$ is an $(r,s)$ tensor, then I define the components of $\nabla \mathbf{T}$ in a coordinate system with Christoffel symbols $\Gamma^i_{jk}$ as (Using Einstein summation)
$$( \nabla \mathbf{T})^{i_{1} \dotsc i_{r}}{}_{j_{1} \dotsc j_{s} \ k} =\begin{matrix} \partial _{k} T^{i_{1} \dotsc i_{r}}{}_{j_{1} \dotsc j_{s}}\\ +\Gamma _{lk}^{i_{1}} T^{l\ i_{2} \dotsc i_{r}}{}_{j_{1} \dotsc j_{s}} +\cdots +\Gamma _{lk}^{i_{r}} T^{i_{1} \dotsc i_{r-1} \ l}{}_{j_{1} \dotsc j_{s}}\\ -\Gamma _{j_{1} k}^{l} T^{i_{1} \dotsc i_{r}}{}_{l\ j_{2} \dotsc j_{s}} -\cdots -\Gamma _{j_{s} k}^{l} T^{i_{1} \dotsc i_{r}}{}_{j_{1} \dotsc j_{s-1} \ l} \end{matrix}$$
I define the components the divergence of a $(1,s)$ tensor $\mathbf{T}$ as
$$(\nabla\cdot\mathbf{T})_{j_1\dots j_s}= (\nabla \mathbf{T})^i{}_{j_1\dots j_s~i}$$
And lastly I define the Laplacian of an $(r,s)$ tensor as
$$(\boldsymbol{\triangle}\mathbf{T})^{i_1\dots i_r}{}_{j_1\dots j_s}=g^{kl}(\nabla(\nabla\mathbf{T}))^{i_1\dots i_r}{}_{j_1\dots j_s~kl}$$
Where $g^{ij}$ is the $i,j$th component of the inverse metric.
