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+---
+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.