Section 3.4 The Euler equations

The conservation laws derived so far, for mass (Theorem 3.2.1) and momentum (Theorem 3.3.1), hold for any inviscid fluid, but they do not yet form a closed system: there are more unknowns than equations. In this section, we close the system by introducing the notion of an incompressible fluid and, in particular, assuming that the density is constant. The result is the Euler equations, first written down by Leonhard Euler in 1757. Together with the boundary conditions at the end of this section, they form a complete mathematical model of an ideal liquid.

The Euler equations are the central model for much of this course. In Section 3.5, we integrate them to obtain Bernoulli’s equation, which links pressure to velocity. When the flow is also irrotational, they reduce to Laplace’s equation for a velocity potential, which is the basis of Chapter 4 and the water wave theory of Chapter 5. Finally, in Chapter 6, the Euler equations are extended to the Navier–Stokes equations by adding the effects of viscosity.

Subsection 3.4.1 Incompressible fluids and the Euler equations

Recall from Theorem 3.1.1 and the proof of Reynolds’ Transport Theorem that the Jacobian relates infinitesimal volumes in the Eulerian and Lagrangian frames via

\begin{equation} \dxyz = J \dXYZ.\tag{3.4.1} \end{equation}

So the Jacobian is a measure of the local expansion or contraction of a fluid (relative to its original state). Combining this with Euler’s identity (Theorem 3.1.2), we are led to an important interpretation for what it means for a fluid to satisfy $\nabla \cdot \bu = 0\text{.}$ This leads us to defining the notion of an incompressible fluid. (You have already encountered an intuitive definition for an incompressible flow as part of Exercise 2.3.1).

Theorem 3.4.1 Incompressible fluids.

A fluid is said to be incompressible if it preserves infinitesimal volumes. Since such volumes are related via (3.4.1), then this is equivalent to

\begin{equation*} \DD{J}{t} \equiv 0, \end{equation*}

within the relevant fluid (and temporal) domain.

The following equivalence (iff) then follows:

\begin{equation} \textrm{Fluid is incompressible} \quad \Longleftrightarrow \quad \nabla \cdot \bu = 0.\tag{3.4.2} \end{equation}

Proof.

Note that if $\DD{J}{t} = 0\text{,}$ then indeed an infinitesimal volume element must have constant volume for all time according to (3.4.1). Then by Euler’s identity in Theorem 3.1.2, this implies $\nabla \cdot \bu = 0.$ Conversely, if $\nabla \cdot \bu = 0\text{,}$ then again by the identity the fluid is incompressible.

So in conclusion, for an incompressible fluid, the mass conservation equation of Theorem 3.2.1 reduces to

\begin{equation} \nabla \cdot \bu = 0,\tag{3.4.3} \end{equation}

together with $\DD{\rho}{t} = 0\text{,}$ as shown in the corollary below. If, in addition, the density is constant, then it suffices to solve (3.4.3).

To see this, consider what happens to the mass conservation equation in Theorem 3.2.1 if the fluid is incompressible. This leads to the following corollary.

Corollary 3.4.2 Constant density following the flow.

For an incompressible fluid, the density of each fluid particle is constant in time (i.e. constant along particle paths):

\begin{equation*} \DD{\rho}{t} = \pd{\rho}{t} + (\bu \cdot \nabla)\rho = 0. \end{equation*}

Proof.

This follows from setting $\nabla \cdot \bu = 0$ in (3.2.2).

Let us now count equations and unknowns. We have the scalar mass conservation equation of Theorem 3.2.1 (or alternatively the simpler equation (3.4.3) for incompressible fluids), together with the vector momentum equation of Theorem 3.3.1. This yields four scalar equations for five unknowns: the pressure $p\text{,}$ density $\rho\text{,}$ and the three velocity components $\bu\text{.}$

One way of proceeding is to attempt to establish or to intuit a relationship between pressure and density. For instance, for a gas, one may use the ideal gas law, $p = RT\rho$ (which can be derived from kinetic theory and is well supported empirically), relating pressure linearly to density, and depending on the temperature, $T\text{,}$ and a (gas) constant $R\text{.}$ Other assumptions of the form $p = p(\rho)$ are possible, and this is involved in the study of gases and compressible fluids.

However, empirically, we observe that in most liquids, the density only varies within a few percent under typical variations of temperature and pressure. Therefore, it is common to assume:

Note 3.4.3 Constant density assumption.

For liquids, we often assume that the density is constant, $\rho = \textrm{constant}\text{.}$

Note, then, that in the case of constant-density fluids, if we consider the mass conservation equation (3.2.1), it follows that $\nabla \cdot \bu = 0\text{.}$ Therefore, from the definition Theorem 3.4.1, we conclude that the fluid is indeed incompressible.

Remark 3.4.4.

Confusingly, many references define a fluid to be incompressible if $\rho$ is constant. However, incompressibility does not require constant density: a fluid can be incompressible, so that $\DD{\rho}{t} = 0\text{,}$ without $\rho$ being constant. An example is the ocean, where density varies with depth due to salinity and temperature.

We are finally ready to state the Euler equations.

Definition 3.4.5 Euler equations.

The Euler equations consist of the incompressibility condition (3.4.3) and the momentum equations (3.3.1), considered in the situation of an incompressible fluid of constant density:

\begin{align} \nabla \cdot \bu &= 0, \tag{3.4.4}\\ \left(\pd{\bu}{t} + (\bu \cdot \nabla)\bu\right) &= - \frac{1}{\rho}\nabla p + \bg.\tag{3.4.5} \end{align}

These are four scalar equations for four unknowns: the three components of $\bu$ and the pressure $p\text{.}$ Although incompressibility does not in general require constant density (see the remark above), in this course we shall take incompressible fluids to have constant density, $\rho\text{.}$ The above Euler equations include the (gravitational) body force $\bg\text{.}$

Simple applications of the Euler equations are given in Exercise 3.6.4 and Exercise 3.6.5.

You will consider the derivation of the Euler equations as part of Exercise 3.6.3.

Example 3.4.6 Computer graphics and fluid simulations.

There is a long-standing link between the research field of fluid mechanics and the entertainment field, where cutting-edge applications in animation, movie, and video-game graphics use ideas from fluid mechanics to model fluids. Many fluid simulators will use things like the Euler equations (or simplified versions of them) to simulate fluids. In some cases, such simulations can even happen in real time. Here is an example of a real-time Euler equation solver coded in JavaScript.

Subsection 3.4.2 Boundary conditions

A great deal of the complexity of fluid motion comes from the conditions that we must consider between the fluid and its bounding surfaces. For water in the ocean, bounding surfaces might range from the ocean floor and the hull of a boat, down to the vegetation in the water or the sand on a beach. This can get quite complex! In this module, and when we first learn fluid dynamics, we only consider simple bounded fluid regions (fluid above a plate, fluid in a box, fluid in a channel, etc.).

In the case of a water wave, a bounding surface will also include the free surface of the wave, itself, which interacts with the surrounding atmospheric gas. This leads to the situation of free-boundary or free-surface conditions.

At a solid boundary, we impose the following condition.

Note 3.4.7 No-flux condition through a boundary.

For a fluid in contact with a fixed rigid boundary, $\partial V\text{,}$ we require that the normal velocity of the fluid there must be zero:

\begin{equation} \bu \cdot \bn = 0 \quad \textrm{on $\partial V$}, \tag{3.4.6} \end{equation}

where $\bn$ is the unit normal to $\partial V\text{.}$ This condition states that the fluid cannot flow through $\partial V$ or separate from $\partial V$ (hence leaving a vacuum).

It turns out that the above no-flux condition is sufficient to provide well-posed boundary conditions for most inviscid fluids. Note that in particular, we have not constrained the tangential velocity of the fluid, i.e. $\bu \cdot \bt\text{,}$ where $\bt$ is any tangent vector to the boundary.

Later in Chapter 6, you will study the situation of a viscous fluid, where it will be important to consider the tangential fluid velocity.

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