The Euler equations

Section 3.4 The Euler equations

Subsection 3.4.1 Incompressible fluids and the Euler equations

Recall from Theorem 3.1.2 that the Jacobian relates infinitessimal volumes in Eulerian and Lagrangian frames via

\begin{equation} \dxyz = J \dXYZ\text{.}\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). If we use Euler’s identity in the theorem, 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 already previously encountered an intuitive definition for an incompressible flow as part of Exercise 2.3.1).

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Theorem 3.4.1. Incompressible fluids.

A fluid is said to be incompressible if it preserves infinitessimal 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.

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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}

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Proof.

Note that if $\DD{J}{t} = 0\text{,}$ then indeed the an infinitessimal volume element must be of static 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.

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So in conclusion, for the case of an incompressible fluid, it suffices to solve the equation

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

instead of the more complicated equation in Theorem 3.2.1.

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By the way, what happens to the mass conservation equation in Theorem 3.2.1 if the fluid is incompressible? This leads to the following corollary.

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Corollary 3.4.2. Constant density along streamlines.

For an incompressible fluid, the density is constant along streamlines, i.e.

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

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Proof.

This relies on setting $\nabla \cdot \bu = 0$ in Theorem 3.2.1.

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Let us return to consider the total sum of equations and unknowns. We have introduced the scalar mass conservation equation found in Theorem 3.2.1 (or alternatively the more simplified equation (3.4.3) for incompressible fluids). Also the vector momentum equation found in Theorem 3.3.1. This yields four scalar equations for five unknowns: the pressure $p\text{,}$ density $\rho\text{,}$ and the three velocity components $\bu\text{.}$

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One way of proceeding is to attempt to establish or to inuit a relationship between pressure and density. For instance, the assumption of an ideal gas law can be derived from kinetic theory, which leads to the empirical law that $p = RT\rho\text{,}$ relating pressure in a linear fashion 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.

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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:

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Note 3.4.3. Constant density assumption.

We often assume that in the situation of liquids, the density is taken to be constant, $\rho = \textrm{constant}\text{.}$

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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.

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Remark 3.4.4.

Confusingly, in many references, authors define a fluid to be incompressible if $\rho$ is constant. However, we see this is not necessarily the case. A fluid can be incompressible and therefore $\DD{\rho}{t} = 0$ without $\rho$ being constant.

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We are finally ready to state the Euler equations.

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Definition 3.4.5. Euler equations.

The Euler equations consist of the continuity equation (3.4.3) and momentum equations (3.3.1), considered in the situation of an incompressible fluid:

\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}

Note this is then four scalar equations for the three unknown velocities in $\bu$ and pressure $p\text{.}$ We shall assume in the course that incompressible fluids have constant density, $\rho\text{.}$ The above Euler equations include the (gravitational) body force $\bg\text{.}$

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Simple applications of the Euler equations are given in Exercise 6.5.1, Exercise 3.6.5 and Exercise 3.6.6

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You will consider the derivation of the Euler equations as part of Exercise 3.6.4.

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Example 3.4.6. Computer graphics and fluid simulations.

There has always been an intimate 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 "bastardised version") 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.

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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, a bounding surfaces might include the ocean floor bottom and the body 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.)

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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.

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For the situation of fluid at a solid boundary, this leads us to formulate the following.

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Note 3.4.7. No-flux condition through a boundary.

For a fluid in contact with a fixed rigid boundary, $\partial V\text{,}$ then 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).

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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 the tangential vector at the boundary.

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Later in Chapter 7, you will study the situation of a viscous fluid, where it will be important to consider the tangential fluid velocity.

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