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diff --git a/source/know/concept/euler-equations/index.md b/source/know/concept/euler-equations/index.md index 3730ea3..2654d2b 100644 --- a/source/know/concept/euler-equations/index.md +++ b/source/know/concept/euler-equations/index.md @@ -11,14 +11,13 @@ 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. +i.e. fluids without [viscosity](/know/concept/viscosity/). -## Incompressible fluid -In a fluid moving according to the velocity vield $$\va{v}(\va{r}, t)$$, +## Incompressible fluids + +In a fluid moving according to the velocity field $$\va{v}(\va{r}, t)$$, 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 $$\va{v}$$: @@ -33,14 +32,15 @@ This infinitesimal particle obeys Newton's second law, which can be written as follows: $$\begin{aligned} - \va{w} \dd{m} + \va{w} m = \va{w} \rho \dd{V} = \va{f^*} \dd{V} \end{aligned}$$ -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. +Where $$m$$ and $$\dd{V}$$ are the particle's mass and volume, +and $$\rho$$ is the fluid density, which we assume +to be constant in space and time in this case. +Now, the **effective force density** $$\va{f^*}$$ represents the net force-per-particle. By dividing the law by $$\dd{V}$$, we find: $$\begin{aligned} @@ -51,13 +51,13 @@ $$\begin{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 $$S$$. +and the pressure contact force $$\va{F}_p$$ on the enclosing surface $$\partial V$$. Using the divergence theorem, we then find: $$\begin{aligned} \va{F} = \va{F}_g + \va{F}_p - = \int_V \rho \va{g} \dd{V} - \oint_S p \dd{\va{S}} + = \int_V \rho \va{g} \dd{V} - \oint_{\partial V} p \dd{\va{S}} = \int_V (\rho \va{g} - \nabla p) \dd{V} = \int_V \va{f^*} \dd{V} \end{aligned}$$ @@ -76,31 +76,28 @@ Dividing this by $$\rho$$, we get the first of the system of Euler equations: $$\begin{aligned} - \boxed{ - \va{w} - = \frac{\mathrm{D} \va{v}}{\mathrm{D} t} - = \va{g} - \frac{\nabla p}{\rho} - } + \va{w} + = \frac{\mathrm{D} \va{v}}{\mathrm{D} t} + = \va{g} - \frac{\nabla p}{\rho} \end{aligned}$$ -The last ingredient is **incompressibility**: +The last ingredient is incompressibility: the same volume must simultaneously -be flowing in and out of an arbitrary enclosure $$S$$. +be flowing in and out of an arbitrary enclosure $$\partial V$$. Then, by the divergence theorem: $$\begin{aligned} 0 - = \oint_S \va{v} \cdot \dd{\va{S}} + = \oint_{\partial V} \va{v} \cdot \dd{\va{S}} = \int_V \nabla \cdot \va{v} \dd{V} \end{aligned}$$ -Since $$S$$ and $$V$$ are arbitrary, -the integrand must vanish by itself everywhere: +Since $$V$$ is arbitrary, +the integrand must vanish by itself, +leading to the **continuity relation**: $$\begin{aligned} - \boxed{ - \nabla \cdot \va{v} = 0 - } + \nabla \cdot \va{v} = 0 \end{aligned}$$ Combining this with the equation for $$\va{w}$$, @@ -118,62 +115,104 @@ $$\begin{aligned} } \end{aligned}$$ -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: + +## Compressible fluids + +If the fluid is compressible, +the condition $$\nabla \cdot \va{v} = 0$$ no longer holds, +so to update the equations we demand that mass is conserved: the mass evolution of a volume $$V$$ -is equal to the mass flow through its boundary $$S$$. +is equal to the mass flow through its boundary $$\partial V$$. Applying the divergence theorem again: $$\begin{aligned} 0 - = \dv{}{t}\int_V \rho \dd{V} + \oint_S \rho \va{v} \cdot \dd{\va{S}} + = \dv{}{t}\int_V \rho \dd{V} + \oint_{\partial V} \rho \va{v} \cdot \dd{\va{S}} = \int_V \dv{\rho}{t} + \nabla \cdot (\rho \va{v}) \dd{V} \end{aligned}$$ Since $$V$$ is arbitrary, the integrand must be zero. -This leads to the following **continuity equation**, -to which we apply a vector identity: +The new **continuity equation** is therefore: $$\begin{aligned} 0 = \dv{\rho}{t} + \nabla \cdot (\rho \va{v}) - = \dv{\rho}{t} + (\va{v} \cdot \nabla) \rho + \rho (\nabla \cdot \va{v}) + = \dv{\rho}{t} + \va{v} \cdot \nabla \rho + \rho \nabla \cdot \va{v} + = \frac{\mathrm{D} \rho}{\mathrm{D} t} + \rho \nabla \cdot \va{v} \end{aligned}$$ -Thanks to incompressibility, the last term disappears, -leaving us with a material derivative: +When the fluid gets compressed in a certain location, thermodynamics +states that the pressure, temperature and/or entropy must increase there. +For simplicity, let us assume an *isothermal* and *isentropic* fluid, +such that only $$p$$ is affected by compression, and the +[fundamental thermodynamic relation](/know/concept/fundamental-thermodynamic-relation/) +reduces to $$\dd{E} = - p \dd{V}$$. + +Then the pressure is given by a thermodynamic equation of state $$p(\rho, T)$$, +which depends on the system being studied +(e.g. the ideal gas law $$p = \rho R T$$). +However, the quantity in control of the dynamics +is not $$p$$, but the internal energy $$E$$. +Dividing the fundamental thermodynamic relation by $$m \: \mathrm{D}t$$, +where $$m$$ is the mass of $$\dd{V}$$: $$\begin{aligned} - \boxed{ - 0 - = \frac{\mathrm{D} \rho}{\mathrm{D} t} - = \dv{\rho}{t} + (\va{v} \cdot \nabla) \rho - } + \frac{\mathrm{D} e}{\mathrm{D} t} + = - p \frac{\mathrm{D} v}{\mathrm{D} t} \end{aligned}$$ -Putting everything together, Euler's system of equations -now takes the following form: +With $$e$$ and $$v$$ the specific (i.e. per unit mass) +internal energy and volume. +Using that $$\rho = 1 / v$$, +and substituting the above continuity relation: + +$$\begin{aligned} + \frac{\mathrm{D} e}{\mathrm{D} t} + = - p \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{1}{\rho} \Big) + = \frac{p}{\rho^2} \frac{\mathrm{D} \rho}{\mathrm{D} t} + = - \frac{p}{\rho} \nabla \cdot \va{v} +\end{aligned}$$ + +It makes sense to see a factor $$-\nabla \cdot \va{v}$$ here: +an incoming flow increases $$e$$. +This gives us the time-evolution of $$e$$ due to compression, +but its initial value is another equation of state $$e(\rho, T)$$. + +Putting it all together, +Euler's system of equations now takes the following form: $$\begin{aligned} \boxed{ \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \va{g} - \frac{\nabla p}{\rho} - \qquad - \nabla \cdot \va{v} - = 0 - \qquad + \qquad \quad \frac{\mathrm{D} \rho}{\mathrm{D} t} - = 0 + = - \rho \nabla \cdot \va{v} + \qquad \quad + \frac{\mathrm{D} e}{\mathrm{D} t} + = - \frac{p}{\rho} \nabla \cdot \va{v} } \end{aligned}$$ -Usually, however, when discussing incompressible fluids, -$$\rho$$ is assumed to be spatially uniform, -in which case the latter equation is trivially satisfied. +What happens if the fluid is actually incompressible, +so $$\nabla \cdot \va{v} = 0$$ holds again? Clearly: + +$$\begin{aligned} + \frac{\mathrm{D} \va{v}}{\mathrm{D} t} + = \va{g} - \frac{\nabla p}{\rho} + \qquad \quad + \frac{\mathrm{D} \rho}{\mathrm{D} t} + = 0 + \qquad \quad + \frac{\mathrm{D} e}{\mathrm{D} t} + = 0 +\end{aligned}$$ + +So $$e$$ is constant, which is in fact equivalent to saying that $$\nabla \cdot \va{v} = 0$$. +The equation for $$\rho$$ enforces conservation of mass +for inhomogeneous fluids, i.e. fluids that are "lumpy", +but where the size of the lumps is conserved by incompressibility. |