Commit 09d36187 authored by Josh Borrow's avatar Josh Borrow
Browse files

Minor fixes to the formatting of equations in the theory for P-U

parent c7960649
......@@ -386,14 +386,14 @@ evolution of the internal energy, as opposed to the entropy.
For P-U, the following choice of parameters in the formalism of \S
\ref{sec:derivation} provides convenient properties:
\begin{align}
x_i =& (\gamma - 1) m_i u_i~, \\
\tilde{x}_i =& 1,
x_i =&~ (\gamma - 1) m_i u_i, \\
\tilde{x}_i =&~ 1,
\label{eq:sph:pu:xichoice}
\end{align}
leading to the following requirements to ensure correct volume elements:
\begin{align}
y_i =& \sum_{j} (\gamma - 1) m_j u_j W_{ij} = \bar{P}_i,\\
\tilde{y}_i =& \sum_{j} W_{ij} = \bar{n}_i, \\
\tilde{y}_i =& \sum_{j} W_{ij} = \bar{n}_i,
\label{eq:sph:pu:yichoice}
\end{align}
with the resulting variables representing a smoothed pressure and particle
......@@ -401,8 +401,8 @@ number density. This choice of variables leads to the following equation of
motion:
\begin{align}
\frac{\mathrm{d} \mathbf{v}_i}{\mathrm{d} t} = -\sum_j (\gamma - 1)^2 m_j u_j u_i
&\left[\frac{f_{ij}}{\bar{P}_i} \nabla_i W_{ij}(h_i) ~+ \right. \\
&\frac{f_{ji}}{\bar{P}_j} \nabla_i W_{ji}(h_j) ~+ \\
&\left[\frac{f_{ij}}{\bar{P}_i} \nabla_i W_{ij}(h_i) ~+ \right. \nonumber \\
&\frac{f_{ji}}{\bar{P}_j} \nabla_i W_{ji}(h_j) ~+ \nonumber \\
& \left.\nu_{ij}\bar{\nabla_i W_{ij}}\right]~.
\label{eq:sph:pu:eom}
\end{align}
......@@ -411,9 +411,10 @@ the final term.
The $h$-terms are given as
\begin{align}
f_{ij} = 1 - & \left[\frac{h_i}{n_d (\gamma - 1) \bar{n}_i \left\{m_j u_j\right\}}
\frac{\partial \bar{P}_i}{\partial h_i} \right] \times \\
& \left( 1 + \frac{h_i}{n_d \bar{n}_i} \frac{\partial \bar{n}_i}{\partial h_i} \right)^{-1}
f_{ij} = 1 - \left[\frac{h_i}{n_d (\gamma - 1) \bar{n}_i \left\{m_j u_j\right\}}
\frac{\partial \bar{P}_i}{\partial h_i} \right]
\left( 1 + \frac{h_i}{n_d \bar{n}_i}
\frac{\partial \bar{n}_i}{\partial h_i} \right)^{-1}
\label{eq:sph:pu:fij}
\end{align}
with $n_d$ the number of dimensions. In practice, the majority of $f_{ij}$ is
......
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