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fluidparticles
MigFlow
Commits
c62e7f3b
Commit
c62e7f3b
authored
Dec 15, 2017
by
Jonathan Lambrechts
Browse files
patankar eq
parent
f01f2717
Changes
2
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Equations.tex
→
doc/
Equations.tex
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c62e7f3b
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doc/Patankar.tex
0 → 100644
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c62e7f3b
\documentclass
{
paper
}
\usepackage
[utf8]
{
inputenc
}
\usepackage
{
amsmath
}
\usepackage
{
bm
}
\usepackage
[a4paper]
{
geometry
}
\newcommand
{
\ub
}
{
\mathbf
u
}
\newcommand
{
\Fb
}
{
\mathbf
F
}
\newcommand
{
\gb
}
{
\tilde
{
\mathbf
{
g
}}}
\newcommand
{
\nablab
}
{
\bm
{
\nabla
}}
\begin{document}
\begin{align*}
\ub
^{
n+1
}_
p
&
=
\ub
_
p
^
n+
\frac
{
\Delta
\Fb
_
p
}{
m
}
\\
\Fb
_
p
&
= -
\gamma
^
n(
\ub
^{
n+1
}_
p-
\ub
_
f
^{
n+1
}
) +
\gb
m - V
\nablab
p
\end{align*}
\begin{align*}
\ub
^{
n+1
}_
p
\left
(
\frac
{
m
}{
\Delta
t
}
+
\gamma
^
n
\right
)
&
=
\frac
{
m
}{
\Delta
t
}
\ub
^
n
_
p+
\gamma
^
n
\ub
^{
n+1
}_
f+
\gb
m-V
\nablab
p
\\
%\Fb_p
%&= -\gamma^n\left(-\ub_f^{n+1}+\frac{\Delta t}{m+\gamma^n\Delta t}\left(\frac{m}{\Delta t}\ub^n_p+\gamma^n\ub^{n+1}_f+\gb m-V\nablab p\right)\right)+\gb m -V\nablab p\\
\end{align*}
\begin{align*}
\Fb
_
p
&
=
\left
(
\frac
1
{
\gamma
^
n
}
+
\frac
{
\Delta
t
}{
m
}
\right
)
^{
-1
}
\left
(
\ub
_
f
^{
n+1
}
-
\ub
_
p
^{
n
}
+
\frac
{
1
}{
\gamma
^
n
}
\left
(
\gb
m-V
\nablab
p
\right
)
\right
)
\\
\Fb
_
f
&
= -
\Fb
_
p +
\gb
m -V
\nablab
p
\\
&
=
\left
(
\frac
1
{
\gamma
^
n
}
+
\frac
{
\Delta
t
}{
m
}
\right
)
^{
-1
}
\left
(
\ub
_
p
^
n-
\ub
_
f
^{
n+1
}
+
\frac
{
\Delta
t
}{
m
}
\left
(
\gb
m - V
\nablab
p
\right
)
\right
)
\end{align*}
\end{document}
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