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Diffstat (limited to 'source/know/concept/euler-equations')
| -rw-r--r-- | source/know/concept/euler-equations/index.md | 46 |
1 files changed, 23 insertions, 23 deletions
diff --git a/source/know/concept/euler-equations/index.md b/source/know/concept/euler-equations/index.md index ddaa27a..3730ea3 100644 --- a/source/know/concept/euler-equations/index.md +++ b/source/know/concept/euler-equations/index.md @@ -18,10 +18,10 @@ the surrounding assumptions about the fluid. ## Incompressible fluid -In a fluid moving according to the velocity vield $\va{v}(\va{r}, t)$, +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 $\va{v}$: +the **material acceleration field** $$\va{w}(\va{r}, t)$$, +which is the [material derivative](/know/concept/material-derivative/) of $$\va{v}$$: $$\begin{aligned} \va{w} @@ -38,20 +38,20 @@ $$\begin{aligned} = \va{f^*} \dd{V} \end{aligned}$$ -Where $\dd{m}$ and $\dd{V}$ are the particle's mass volume, -and $\rho$ is the fluid density, which we assume, in this case, to be constant in space and time. -Then the **effective force density** $\va{f^*}$ represents the net force-per-particle. -By dividing the law by $\dd{V}$, we find: +Where $$\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}$$ -Next, we want to find another expression for $\va{f^*}$. -We know that the overall force $\va{F}$ on an arbitrary volume $V$ of the fluid -is the sum of the gravity body force $\va{F}_g$, -and the pressure contact force $\va{F}_p$ on the enclosing surface $S$. +Next, we want to find another expression for $$\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} @@ -62,8 +62,8 @@ $$\begin{aligned} = \int_V \va{f^*} \dd{V} \end{aligned}$$ -Where $p(\va{r}, t)$ is the pressure field, -and $\va{g}(\va{r}, t)$ is the gravitational acceleration field. +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: $$\begin{aligned} @@ -72,7 +72,7 @@ $$\begin{aligned} = \rho \va{g} - \nabla p \end{aligned}$$ -Dividing this by $\rho$, +Dividing this by $$\rho$$, we get the first of the system of Euler equations: $$\begin{aligned} @@ -85,7 +85,7 @@ $$\begin{aligned} The last ingredient is **incompressibility**: the same volume must simultaneously -be flowing in and out of an arbitrary enclosure $S$. +be flowing in and out of an arbitrary enclosure $$S$$. Then, by the divergence theorem: $$\begin{aligned} @@ -94,7 +94,7 @@ $$\begin{aligned} = \int_V \nabla \cdot \va{v} \dd{V} \end{aligned}$$ -Since $S$ and $V$ are arbitrary, +Since $$S$$ and $$V$$ are arbitrary, the integrand must vanish by itself everywhere: $$\begin{aligned} @@ -103,10 +103,10 @@ $$\begin{aligned} } \end{aligned}$$ -Combining this with the equation for $\va{w}$, +Combining this with the equation for $$\va{w}$$, we get a system of two coupled differential equations: these are the Euler equations for an incompressible fluid -with spatially uniform density $\rho$: +with spatially uniform density $$\rho$$: $$\begin{aligned} \boxed{ @@ -119,13 +119,13 @@ $$\begin{aligned} \end{aligned}$$ The above form is straightforward to generalize to incompressible fluids -with non-uniform spatial densities $\rho(\va{r}, t)$. +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 $S$. +the mass evolution of a volume $$V$$ +is equal to the mass flow through its boundary $$S$$. Applying the divergence theorem again: $$\begin{aligned} @@ -134,7 +134,7 @@ $$\begin{aligned} = \int_V \dv{\rho}{t} + \nabla \cdot (\rho \va{v}) \dd{V} \end{aligned}$$ -Since $V$ is arbitrary, the integrand must be zero. +Since $$V$$ is arbitrary, the integrand must be zero. This leads to the following **continuity equation**, to which we apply a vector identity: @@ -172,7 +172,7 @@ $$\begin{aligned} \end{aligned}$$ Usually, however, when discussing incompressible fluids, -$\rho$ is assumed to be spatially uniform, +$$\rho$$ is assumed to be spatially uniform, in which case the latter equation is trivially satisfied. |