Derivation of Euler Equations for Fluid Flow

Overview: the Euler [flow] equations are the equations of motion for an idealized fluid. We just derive them and conclude with a few remarks.

Derivation

The derivations of the Euler equation boil down to the same game plan: derive $F=ma$ for a fluid parcel. (A "fluid parcel" is an infinitesimal volume of fluid, more a heuristic than a rigorous notion.)

The forces acting on a fluid parcel, in a simplified idealized setting, are just the fluid's pressure. We may find it by integrating over the surface of the fluid parcel: \begin{equation} \vec{F} = -\oint_{\partial V}p(\vec{x},t)\vec{n}\,\D A \end{equation} where $\vec{n}$ is the outward-pointing normal vector on the bounding surface $\partial V$ of the fluid parcel. (The negative sign is due to the normal vector being outward-pointing.) The divergence theorem yields \begin{equation} \vec{F} = -\int_{V}(\nabla p(\vec{x},t))\,\D V. \end{equation} This is half the equation of motion.

The other half is the "mass times acceleration" term. We generically describe a fluid by its velocity vector field $\vec{u}(\vec{x}, t)$. Now we can sum over its acceleration vector for each point in the fluid parcel, weighted by its density $\rho$, to give us \begin{equation} m\vec{a} = \int_{V} \rho(\vec{x}, t)\frac{\mathrm{D}\vec{u}(\vec{x},t)}{\mathrm{D}t}\,\D V. \end{equation} Here we use Stokes' notation \begin{equation} \frac{\mathrm{D}\vec{u}(\vec{x},t)}{\mathrm{D}t} = \partial_{t}\vec{u}(\vec{x},t) + (\vec{u}(\vec{x},t)\cdot\nabla)\vec{u}(\vec{x},t) \end{equation} which is the material time derivative.

Now we set equals to equals by Newton's second Law, we get \begin{equation} \int_{V} \rho(\vec{x}, t)\frac{\mathrm{D}\vec{u}(\vec{x},t)}{\mathrm{D}t}\,\D V=-\int_{V}(\nabla p(\vec{x},t))\,\D V \end{equation} for arbitrary regions $V$. This gives us Euler's equations of motion \begin{equation} \rho(\vec{x}, t)\frac{\mathrm{D}\vec{u}(\vec{x},t)}{\mathrm{D}t} =-\nabla p(\vec{x},t). \end{equation} Usually books divide through by density $\rho$, and add an external force term (usually Earth's gravitational field $\vec{g}$) to the right-hand side.

Theorem 1. Euler's flow equations of motion for fluid is \begin{equation} \frac{\mathrm{D}\vec{u}(\vec{x},t)}{\mathrm{D}t} = \frac{-1}{\rho(\vec{x}, t)}\nabla p(\vec{x},t). \end{equation}

A few remarks are in order:

1. This is an idealization insofar as friction between fluid parcels is negligible (which is reflected by a very small viscosity).
2. Exact solutions to the Euler flow may be found in Marchioro and Pulvirenti's Mathematical Theory of Incompressible Nonviscous Fluids. One large class of solutions include potential flow of incompressible fluids, where $\vec{u}=\nabla\varphi$ the velocity is the gradient of a scalar potential. Incompressibility amounts to $\nabla\cdot\vec{u}=0$ in this case. (Note to future me: insert link to incompressibility condition when I get around to writing it)
3. Intuitively, compressible fluids are gases, incompressible fluids are liquids.
4. In $n$ dimensions, Euler equations has $n$ unknowns from the velocity vector, 1 from the pressure, and 1 from the density. But there is one equation per dimension, i.e., there are $n$ equations for $n+2$ unknowns. Consequently, Euler equations are under-determined, and we need 2 more equations to determine all unknowns. We usually add an equation for the conservation of mass (the continuity equation) and either work with incompressible fluid or compressible fluids with additional equations from thermodynamics.

References

The most elegant (and difficult) presentation of Euler's equations I have found is in Landau and Lifshitz. It's the subject of the first chapter of their Fluid Mechanics, which is just amazing (but nearly impenetrable if you don't already have a good familiarity with fluids).

1. Leonhard Euler, "Principes généraux du mouvement des fluides". Mémoires de l'académie des sciences de Berlin 11 (1757) 274–315; English translation arXiv:0802.2383
2. Landau and Lifshitz, Fluid Mechanics.
3. Demetrios Christodoulou, "The Euler Equations of Compressible Fluid Flow". Bulletin of the American Mathematical Society 44, no.4 (2007) 581–602
4. John K. Hunter, An Introduction to the Incompressible Euler Equations. Notes dated Sep 25, 2006.
5. M. Zingale, Notes on the Euler equations. Dated April 16, 2013.
6. Chorin and Marsden, A mathematical introduction to fluid mechanics. Springer, third ed., 1993.
7. C. Marchioro, M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids. Springer, 1994. Discusses exact solutions to Euler flow.