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diff --git a/source/know/concept/two-fluid-equations/index.md b/source/know/concept/two-fluid-equations/index.md index 425d50d..e224e3e 100644 --- a/source/know/concept/two-fluid-equations/index.md +++ b/source/know/concept/two-fluid-equations/index.md @@ -11,13 +11,13 @@ layout: "concept" The **two-fluid model** describes a plasma as two separate but overlapping fluids, one for ions and one for electrons. Instead of tracking individual particles, -it gives the dynamics of fluid elements $\dd{V}$ (i.e. small "blobs"). +it gives the dynamics of fluid elements $$\dd{V}$$ (i.e. small "blobs"). These blobs are assumed to be much larger than the [Debye length](/know/concept/debye-length/), such that electromagnetic interactions between nearby blobs can be ignored. -From Newton's second law, we know that the velocity $\vb{v}$ -of a particle with mass $m$ and charge $q$ is as follows, +From Newton's second law, we know that the velocity $$\vb{v}$$ +of a particle with mass $$m$$ and charge $$q$$ is as follows, when subjected only to the [Lorentz force](/know/concept/lorentz-force/): $$\begin{aligned} @@ -27,19 +27,19 @@ $$\begin{aligned} From here, the derivation is similar to that of the [Navier-Stokes equations](/know/concept/navier-stokes-equations/). -We replace $\idv{}{t}$ with a -[material derivative](/know/concept/material-derivative/) $\mathrm{D}/\mathrm{D}t$, -and define $\vb{u}$ as the blob's center-of-mass velocity: +We replace $$\idv{}{t}$$ with a +[material derivative](/know/concept/material-derivative/) $$\mathrm{D}/\mathrm{D}t$$, +and define $$\vb{u}$$ as the blob's center-of-mass velocity: $$\begin{aligned} m n \frac{\mathrm{D} \vb{u}}{\mathrm{D} t} = q n (\vb{E} + \vb{u} \cross \vb{B}) \end{aligned}$$ -Where we have multiplied by the number density $n$ of the particles. +Where we have multiplied by the number density $$n$$ of the particles. Due to particle collisions in the fluid, stresses become important. Therefore, we include -the [Cauchy stress tensor](/know/concept/cauchy-stress-tensor/) $\hat{P}$, +the [Cauchy stress tensor](/know/concept/cauchy-stress-tensor/) $$\hat{P}$$, leading to the following two equations: $$\begin{aligned} @@ -50,7 +50,7 @@ $$\begin{aligned} &= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) + \nabla \cdot \hat{P}_e{}^\top \end{aligned}$$ -Where the subscripts $i$ and $e$ refer to ions and electrons, respectively. +Where the subscripts $$i$$ and $$e$$ refer to ions and electrons, respectively. Finally, we also account for momentum transfer between ions and electrons due to [Rutherford scattering](/know/concept/rutherford-scattering/), leading to these **two-fluid momentum equations**: @@ -67,12 +67,12 @@ $$\begin{aligned} } \end{aligned}$$ -Where $f_{ie}$ is the mean frequency at which an ion collides with electrons, -and vice versa for $f_{ei}$. +Where $$f_{ie}$$ is the mean frequency at which an ion collides with electrons, +and vice versa for $$f_{ei}$$. For simplicity, we assume that the plasma is isotropic and that shear stresses are negligible, in which case the stress term can be replaced -by the gradient $- \nabla p$ of a scalar pressure $p$: +by the gradient $$- \nabla p$$ of a scalar pressure $$p$$: $$\begin{aligned} m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t} @@ -83,8 +83,8 @@ $$\begin{aligned} \end{aligned}$$ Next, we demand that matter is conserved. -In other words, the rate at which particles enter/leave a volume $V$ -must be equal to the flux through the enclosing surface $S$: +In other words, the rate at which particles enter/leave a volume $$V$$ +must be equal to the flux through the enclosing surface $$S$$: $$\begin{aligned} 0 @@ -93,7 +93,7 @@ $$\begin{aligned} \end{aligned}$$ Where we have used the divergence theorem. -Since $V$ is arbitrary, we can remove the integrals, +Since $$V$$ is arbitrary, we can remove the integrals, leading to the following **continuity equations**: $$\begin{aligned} @@ -107,7 +107,7 @@ $$\begin{aligned} \end{aligned}$$ These are 8 equations (2 scalar continuity, 2 vector momentum), -but 16 unknowns $\vb{u}_i$, $\vb{u}_e$, $\vb{E}$, $\vb{B}$, $n_i$, $n_e$, $p_i$ and $p_e$. +but 16 unknowns $$\vb{u}_i$$, $$\vb{u}_e$$, $$\vb{E}$$, $$\vb{B}$$, $$n_i$$, $$n_e$$, $$p_i$$ and $$p_e$$. We would like to close this system, so we need 8 more. An obvious choice is [Maxwell's equations](/know/concept/maxwells-equations/), in particular Faraday's and Ampère's law @@ -121,7 +121,7 @@ $$\begin{aligned} } \end{aligned}$$ -Now we have 14 equations, so we need 2 more, for the pressures $p_i$ and $p_e$. +Now we have 14 equations, so we need 2 more, for the pressures $$p_i$$ and $$p_e$$. This turns out to be the thermodynamic **equation of state**: for quasistatic, reversible, adiabatic compression of a gas with constant heat capacity (i.e. a *calorically perfect* gas), @@ -135,14 +135,14 @@ $$\begin{aligned} = \frac{N + 2}{N} \end{aligned}$$ -Where $\gamma$ is the *heat capacity ratio*, -and can be calculated from the number of degrees of freedom $N$ +Where $$\gamma$$ is the *heat capacity ratio*, +and can be calculated from the number of degrees of freedom $$N$$ of each particle in the gas. -In a fully ionized plasma, $N = 3$. +In a fully ionized plasma, $$N = 3$$. -The density $n \propto 1/V$, -so since $p V^\gamma$ is constant in time, -for some constant $C$: +The density $$n \propto 1/V$$, +so since $$p V^\gamma$$ is constant in time, +for some constant $$C$$: $$\begin{aligned} \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p}{n^\gamma} \Big) = 0 @@ -163,8 +163,8 @@ $$\begin{aligned} } \end{aligned}$$ -Note that from the relation $p = C n^\gamma$, -we can calculate the $\nabla p$ term in the momentum equation, +Note that from the relation $$p = C n^\gamma$$, +we can calculate the $$\nabla p$$ term in the momentum equation, using simple differentiation and the ideal gas law: $$\begin{aligned} @@ -177,13 +177,13 @@ $$\begin{aligned} \end{aligned}$$ Note that the ideal gas law was not used immediately, -to allow for $\gamma \neq 1$. +to allow for $$\gamma \neq 1$$. ## Fluid drifts The momentum equations reduce to the following -if we assume the flow is steady $\ipdv{\vb{u}}{t} = 0$, +if we assume the flow is steady $$\ipdv{\vb{u}}{t} = 0$$, and neglect electron-ion momentum transfer on the right: $$\begin{aligned} @@ -194,9 +194,9 @@ $$\begin{aligned} &\approx q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) - \nabla p_e \end{aligned}$$ -We take the cross product with $\vb{B}$, -which leaves only the component $\vb{u}_\perp$ of $\vb{u}$ -perpendicular to $\vb{B}$ in the Lorentz term: +We take the cross product with $$\vb{B}$$, +which leaves only the component $$\vb{u}_\perp$$ of $$\vb{u}$$ +perpendicular to $$\vb{B}$$ in the Lorentz term: $$\begin{aligned} 0 @@ -205,9 +205,9 @@ $$\begin{aligned} &= q n (\vb{E} \cross \vb{B} - \vb{u}_\perp B^2) - \nabla p \cross \vb{B} - m n \big( (\vb{u} \cdot \nabla) \vb{u} \big) \cross \vb{B} \end{aligned}$$ -Isolating for $\vb{u}_\perp$ tells us -that the fluids drifts perpendicularly to $\vb{B}$, -with velocity $\vb{u}_\perp$: +Isolating for $$\vb{u}_\perp$$ tells us +that the fluids drifts perpendicularly to $$\vb{B}$$, +with velocity $$\vb{u}_\perp$$: $$\begin{aligned} \vb{u}_\perp @@ -216,11 +216,11 @@ $$\begin{aligned} \end{aligned}$$ The last term is often neglected, -which turns out to be a valid approximation if $\vb{E} = 0$, -or if $\vb{E}$ is parallel to $\nabla p$. -The first term is the familiar $\vb{E} \cross \vb{B}$ drift $\vb{v}_E$ +which turns out to be a valid approximation if $$\vb{E} = 0$$, +or if $$\vb{E}$$ is parallel to $$\nabla p$$. +The first term is the familiar $$\vb{E} \cross \vb{B}$$ drift $$\vb{v}_E$$ from [guiding center theory](/know/concept/guiding-center-theory/), -and the second term is called the **diamagnetic drift** $\vb{v}_D$: +and the second term is called the **diamagnetic drift** $$\vb{v}_D$$: $$\begin{aligned} \boxed{ @@ -236,9 +236,9 @@ $$\begin{aligned} It is called *diamagnetic* because it creates a current that induces -a magnetic field opposite to the original $\vb{B}$. -In a quasi-neutral plasma $q_e n_e = - q_i n_i$, -the current density $\vb{J}$ is given by: +a magnetic field opposite to the original $$\vb{B}$$. +In a quasi-neutral plasma $$q_e n_e = - q_i n_i$$, +the current density $$\vb{J}$$ is given by: $$\begin{aligned} \vb{J} @@ -247,7 +247,7 @@ $$\begin{aligned} = \frac{\vb{B} \cross \nabla (p_i + p_e)}{B^2} \end{aligned}$$ -Using the ideal gas law $p = k_B T n$, +Using the ideal gas law $$p = k_B T n$$, this can be rewritten as follows: $$\begin{aligned} @@ -255,7 +255,7 @@ $$\begin{aligned} = k_B \frac{\vb{B} \cross \nabla (T_i n_i + T_e n_e)}{B^2} \end{aligned}$$ -Curiously, $\vb{v}_D$ does not involve any net movement of particles, +Curiously, $$\vb{v}_D$$ does not involve any net movement of particles, because a pressure gradient does not necessarily cause particles to move. Instead, there is a higher density of gyration paths in the high-pressure region, |