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diff --git a/source/know/concept/euler-equations/index.md b/source/know/concept/euler-equations/index.md new file mode 100644 index 0000000..f89d577 --- /dev/null +++ b/source/know/concept/euler-equations/index.md @@ -0,0 +1,182 @@ +--- +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. |