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---
title: "Euler equations"
sort_title: "Euler equations"
date: 2021-03-31
categories:
- Physics
- Fluid mechanics
- Fluid dynamics
layout: "concept"
---
The **Euler equations** are a system of partial differential equations
that govern the movement of **ideal fluids**,
i.e. fluids without viscosity.
There exist several forms, depending on
the surrounding assumptions about the fluid.
## Incompressible fluid
In a fluid moving according to the velocity vield $$\va{v}(\va{r}, t)$$,
the acceleration felt by a particle is given by
the **material acceleration field** $$\va{w}(\va{r}, t)$$,
which is the [material derivative](/know/concept/material-derivative/) of $$\va{v}$$:
$$\begin{aligned}
\va{w}
\equiv \frac{\mathrm{D} \va{v}}{\mathrm{D} t}
= \pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v}
\end{aligned}$$
This infinitesimal particle obeys Newton's second law,
which can be written as follows:
$$\begin{aligned}
\va{w} \dd{m}
= \va{w} \rho \dd{V}
= \va{f^*} \dd{V}
\end{aligned}$$
Where $$\dd{m}$$ and $$\dd{V}$$ are the particle's mass volume,
and $$\rho$$ is the fluid density, which we assume, in this case, to be constant in space and time.
Then the **effective force density** $$\va{f^*}$$ represents the net force-per-particle.
By dividing the law by $$\dd{V}$$, we find:
$$\begin{aligned}
\rho \va{w}
= \va{f^*}
\end{aligned}$$
Next, we want to find another expression for $$\va{f^*}$$.
We know that the overall force $$\va{F}$$ on an arbitrary volume $$V$$ of the fluid
is the sum of the gravity body force $$\va{F}_g$$,
and the pressure contact force $$\va{F}_p$$ on the enclosing surface $$S$$.
Using the divergence theorem, we then find:
$$\begin{aligned}
\va{F}
= \va{F}_g + \va{F}_p
= \int_V \rho \va{g} \dd{V} - \oint_S p \dd{\va{S}}
= \int_V (\rho \va{g} - \nabla p) \dd{V}
= \int_V \va{f^*} \dd{V}
\end{aligned}$$
Where $$p(\va{r}, t)$$ is the pressure field,
and $$\va{g}(\va{r}, t)$$ is the gravitational acceleration field.
Combining this with Newton's law, we find the following equation for the force density:
$$\begin{aligned}
\va{f^*}
= \rho \va{w}
= \rho \va{g} - \nabla p
\end{aligned}$$
Dividing this by $$\rho$$,
we get the first of the system of Euler equations:
$$\begin{aligned}
\boxed{
\va{w}
= \frac{\mathrm{D} \va{v}}{\mathrm{D} t}
= \va{g} - \frac{\nabla p}{\rho}
}
\end{aligned}$$
The last ingredient is **incompressibility**:
the same volume must simultaneously
be flowing in and out of an arbitrary enclosure $$S$$.
Then, by the divergence theorem:
$$\begin{aligned}
0
= \oint_S \va{v} \cdot \dd{\va{S}}
= \int_V \nabla \cdot \va{v} \dd{V}
\end{aligned}$$
Since $$S$$ and $$V$$ are arbitrary,
the integrand must vanish by itself everywhere:
$$\begin{aligned}
\boxed{
\nabla \cdot \va{v} = 0
}
\end{aligned}$$
Combining this with the equation for $$\va{w}$$,
we get a system of two coupled differential equations:
these are the Euler equations for an incompressible fluid
with spatially uniform density $$\rho$$:
$$\begin{aligned}
\boxed{
\frac{\mathrm{D} \va{v}}{\mathrm{D} t}
= \va{g} - \frac{\nabla p}{\rho}
\qquad \quad
\nabla \cdot \va{v}
= 0
}
\end{aligned}$$
The above form is straightforward to generalize to incompressible fluids
with non-uniform spatial densities $$\rho(\va{r}, t)$$.
In other words, these fluids are "lumpy" (variable density),
but the size of their lumps does not change (incompressibility).
To update the equations, we demand conservation of mass:
the mass evolution of a volume $$V$$
is equal to the mass flow through its boundary $$S$$.
Applying the divergence theorem again:
$$\begin{aligned}
0
= \dv{}{t}\int_V \rho \dd{V} + \oint_S \rho \va{v} \cdot \dd{\va{S}}
= \int_V \dv{\rho}{t} + \nabla \cdot (\rho \va{v}) \dd{V}
\end{aligned}$$
Since $$V$$ is arbitrary, the integrand must be zero.
This leads to the following **continuity equation**,
to which we apply a vector identity:
$$\begin{aligned}
0
= \dv{\rho}{t} + \nabla \cdot (\rho \va{v})
= \dv{\rho}{t} + (\va{v} \cdot \nabla) \rho + \rho (\nabla \cdot \va{v})
\end{aligned}$$
Thanks to incompressibility, the last term disappears,
leaving us with a material derivative:
$$\begin{aligned}
\boxed{
0
= \frac{\mathrm{D} \rho}{\mathrm{D} t}
= \dv{\rho}{t} + (\va{v} \cdot \nabla) \rho
}
\end{aligned}$$
Putting everything together, Euler's system of equations
now takes the following form:
$$\begin{aligned}
\boxed{
\frac{\mathrm{D} \va{v}}{\mathrm{D} t}
= \va{g} - \frac{\nabla p}{\rho}
\qquad
\nabla \cdot \va{v}
= 0
\qquad
\frac{\mathrm{D} \rho}{\mathrm{D} t}
= 0
}
\end{aligned}$$
Usually, however, when discussing incompressible fluids,
$$\rho$$ is assumed to be spatially uniform,
in which case the latter equation is trivially satisfied.
## References
1. B. Lautrup,
*Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
CRC Press.
|
