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Diffstat (limited to 'content/know/concept/two-fluid-equations/index.pdc')
| -rw-r--r-- | content/know/concept/two-fluid-equations/index.pdc | 18 |
1 files changed, 17 insertions, 1 deletions
diff --git a/content/know/concept/two-fluid-equations/index.pdc b/content/know/concept/two-fluid-equations/index.pdc index df45e73..9ae9dbf 100644 --- a/content/know/concept/two-fluid-equations/index.pdc +++ b/content/know/concept/two-fluid-equations/index.pdc @@ -129,7 +129,7 @@ $$\begin{aligned} Now we have 14 equations, so we need 2 more, for the pressures $p_i$ and $p_e$. This turns out to be the thermodynamic **equation of state**: for quasistatic, reversible, adiabatic compression -of a gas with constant heat capacities (i.e. a *calorically perfect* gas), +of a gas with constant heat capacity (i.e. a *calorically perfect* gas), it turns out that: $$\begin{aligned} @@ -168,6 +168,22 @@ $$\begin{aligned} } \end{aligned}$$ +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: + +$$\begin{aligned} + p = C n^\gamma + \quad \implies \quad + \nabla p + = \gamma \frac{C n^{\gamma}}{n} \nabla n + = \gamma p \frac{\nabla n}{n} + = \gamma k_B T \nabla n +\end{aligned}$$ + +Note that the ideal gas law was not used immediately, +to allow for $\gamma \neq 1$. + ## Fluid drifts |