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Diffstat (limited to 'source/know/concept/two-fluid-equations/index.md')
| -rw-r--r-- | source/know/concept/two-fluid-equations/index.md | 47 |
1 files changed, 28 insertions, 19 deletions
diff --git a/source/know/concept/two-fluid-equations/index.md b/source/know/concept/two-fluid-equations/index.md index e224e3e..a00a2f9 100644 --- a/source/know/concept/two-fluid-equations/index.md +++ b/source/know/concept/two-fluid-equations/index.md @@ -98,15 +98,17 @@ leading to the following **continuity equations**: $$\begin{aligned} \boxed{ - \pdv{n_i}{t} + \nabla \cdot (n_i \vb{u}_i) - = 0 - \qquad \quad - \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e) - = 0 + \begin{aligned} + 0 + &= \pdv{n_i}{t} + \nabla \cdot (n_i \vb{u}_i) + \\ + 0 + &= \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e) + \end{aligned} } \end{aligned}$$ -These are 8 equations (2 scalar continuity, 2 vector momentum), +These are 8 equations (2 scalars for continuity, 2 vectors for momentum), 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/), @@ -115,9 +117,13 @@ in particular Faraday's and Ampère's law $$\begin{aligned} \boxed{ - \nabla \cross \vb{E} = - \pdv{\vb{B}}{t} - \qquad \quad - \nabla \cross \vb{B} = \mu_0 \Big( n_i q_i \vb{u}_i + n_e q_e \vb{u}_e + \varepsilon_0 \pdv{\vb{E}}{t} \Big) + \begin{aligned} + \nabla \cross \vb{E} + &= - \pdv{\vb{B}}{t} + \\ + \nabla \cross \vb{B} + &= \mu_0 \Big( n_i q_i \vb{u}_i + n_e q_e \vb{u}_e + \varepsilon_0 \pdv{\vb{E}}{t} \Big) + \end{aligned} } \end{aligned}$$ @@ -129,7 +135,7 @@ it turns out that: $$\begin{aligned} \frac{\mathrm{D}}{\mathrm{D} t} \big( p V^\gamma \big) = 0 - \qquad \quad + \qquad \qquad \gamma \equiv \frac{C_P}{C_V} = \frac{N + 2}{N} @@ -146,7 +152,7 @@ for some constant $$C$$: $$\begin{aligned} \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p}{n^\gamma} \Big) = 0 - \quad \implies \quad + \qquad \implies \qquad p = C n^\gamma \end{aligned}$$ @@ -155,11 +161,13 @@ giving us a set of 16 equations for 16 unknowns: $$\begin{aligned} \boxed{ - \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_i}{n_i^\gamma} \Big) - = 0 - \qquad \quad - \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_e}{n_e^\gamma} \Big) - = 0 + \begin{aligned} + 0 + &= \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_i}{n_i^\gamma} \Big) + \\ + 0 + &= \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_e}{n_e^\gamma} \Big) + \end{aligned} } \end{aligned}$$ @@ -169,15 +177,16 @@ using simple differentiation and the ideal gas law: $$\begin{aligned} p = C n^\gamma - \quad \implies \quad + \qquad \implies \qquad \nabla p = \gamma \frac{C n^{\gamma}}{n} \nabla n = \gamma p \frac{\nabla n}{n} = \gamma k_B T \nabla n \end{aligned}$$ -Note that the ideal gas law was not used immediately, -to allow for $$\gamma \neq 1$$. +Note that we waited until now to use the ideal gas law, +in order to include the case $$\gamma \neq 1$$. + ## Fluid drifts |