Commit f1ba7f03 authored by Jonathan Lambrechts's avatar Jonathan Lambrechts

doc : single document paper

parent c70cc7d4
DOCS= particle-fluid-interaction ale equations stability stability2
all : ${DOCS:=.pdf}
%.pdf : %.tex
rubber --pdf $<
clean :
rubber --clean ${DOCS}
\documentclass[11pt]{article}
\usepackage{amsmath,amstext}
\usepackage{graphicx,color,bm}
\usepackage{amssymb,amscd}
\usepackage{amsmath,amsfonts}
\usepackage{amsthm}
\usepackage[mathletters]{ucs}
\usepackage[utf8x]{inputenc}
%\usepackage{unicode-math}
\usepackage{newunicodechar}
\usepackage{MnSymbol}
\DeclareUnicodeCharacter{183}{{\,\cdot\,}}
\DeclareUnicodeCharacter{187}{{\big\rrangle}}
\DeclareUnicodeCharacter{171}{{\big\llangle}}
\DeclareUnicodeCharacter{8249}{{\big\langle}}
\DeclareUnicodeCharacter{8250}{{\big\rangle}}
\DeclareUnicodeCharacter{7753}{{\mathbf{n}}} % ṉ
\usepackage{hyperref}
\usepackage{xcolor}
\newcommand{\mvu} {\ensuremath{{\bf{u}}}}
\newcommand{\mvv} {\ensuremath{{\bf{v}}}}
\newcommand{\mvw} {\ensuremath{{\bf{w}}}}
\newcommand{\dx}{\,\text{d}x}
\newcommand{\vc}[1]{\underline{#1}}
\newcommand{\bv}{\bar{\vc{v}}}
\newcommand{\mx}[1]{\mathbf{#1}}
\newcommand{\dd}[1]{\,\mbox{d}{#1}}
\makeatletter
\@namedef{u8:\detokenize{}}{\mathbf{u}}
\makeatother
\newunicodechar{}{u}
\begin{document}
\newcommand{\dpartial}[2]{\frac{\partial #1}{\partial #2}}
\section{Averaging of variables and operators}
Distribution function of the $\alpha$ phase:
......@@ -222,4 +186,3 @@ where
∇v^α = \frac{∇v_α}{c_α} - \frac{v_α∇c_α}{_α}
\]
$$
\end{document}
......@@ -49,7 +49,7 @@
\global\let\svgscale\undefined%
\makeatother%
\begin{picture}(1,0.29480557)%
\put(0,0){\includegraphics[width=\unitlength,page=1]{dessin.pdf}}%
\put(0,0){\includegraphics[width=\unitlength,page=1]{figures/stability/dessin.pdf}}%
\put(0.50000611,0.26107532){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$u_s$}}}%
\put(0.50000611,0.02793328){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$u$}}}%
\end{picture}%
......
\documentclass{paper}
\usepackage[mathletters]{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\DeclareUnicodeCharacter{183}{{\,\cdot\,}}
\newcommand{\s}{_{\text{s}}}
\renewcommand{\ss}[1]{_{\text{s}#1}}
\newcommand{\re}{\mathit{R\kern-.15em e}}
\newcommand{\acc}{a}
\newcommand{\tia}{\tilde {\acc}}
\newcommand{\tiv}{\tilde v}
\newcommand{\aim}{\tia_{\text{im}}}
\newcommand{\aex}{\tia_{\text{ex}}}
\newcommand{\aext}{\acc_{\mathrm{ext}}}
\newcommand{\af}{\acc_{\text{f}}}
\newcommand{\haf}[1]{\hat\acc_{\text{f#1}}}
\newcommand{\vf}{v_{\text{f}}}
\newcommand{\vfree}{v_{\text{free}}}
\usepackage{xcolor}
\newcommand{\nb}[1]{{\color{green!30!black}\begin{itemize}#1\end{itemize}}}
\usepackage{a4}
\usepackage{graphicx}
\usepackage[round]{natbib}
\begin{document}
\bibliographystyle{plainnat}
\section{Avancement des particules}
Pour le moment, on résout en déplacement (si j'ai bien compris ce qu'on entendait par là). C'est à dire que:
\begin{align*}
......@@ -179,5 +154,3 @@ Les calculs sont identiques, il faut juste utiliser $\frac{Δt}{2}$ au lieu de $
\subsection{plus haut ordre de précision}
On va déjà essayer les précédents ...
\bibliography{zotero}
\end{document}
\documentclass{paper}
\usepackage[a4paper,margin=3cm]{geometry}
\usepackage[mathletters]{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath, amstext}
\DeclareUnicodeCharacter{183}{{\,\cdot\,}}
\usepackage{xcolor}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{arrows}
\newcommand{\s}{_{\text{s}}}
\renewcommand{\ss}[1]{_{\text{s}#1}}
\newcommand{\re}{\mathit{R\kern-.15em e}}
\newcommand{\acc}{a}
\newcommand{\tia}{\tilde {\acc}}
\newcommand{\tiv}{\tilde v}
\newcommand{\aim}{\tia_{\text{im}}}
\newcommand{\aex}{\tia_{\text{ex}}}
\newcommand{\aext}{\acc_{\mathrm{ext}}}
\newcommand{\af}{\acc_{\text{f}}}
\newcommand{\haf}[1]{\hat\acc_{\text{f#1}}}
\newcommand{\vf}{v_{\text{f}}}
\newcommand{\vfree}{v_{\text{free}}}
\newcommand{\nb}[1]{{\color{green!30!black}\begin{itemize}#1\end{itemize}}}
\usepackage[round]{natbib}
\def\labelitemi{--}
\usepackage{newunicodechar}
\usepackage{MnSymbol}
\DeclareUnicodeCharacter{187}{{\big\rrangle}}
\DeclareUnicodeCharacter{171}{{\big\llangle}}
\DeclareUnicodeCharacter{8249}{{\big\langle}}
\DeclareUnicodeCharacter{8250}{{\big\rangle}}
\DeclareUnicodeCharacter{7753}{{\mathbf{n}}} % ṉ
\usepackage{hyperref}
\newcommand{\mvu} {\ensuremath{{\bf{u}}}}
\newcommand{\mvv} {\ensuremath{{\bf{v}}}}
\newcommand{\mvw} {\ensuremath{{\bf{w}}}}
\newcommand{\dx}{\,\text{d}x}
\newcommand{\vc}[1]{\underline{#1}}
\newcommand{\bv}{\bar{\vc{v}}}
\newcommand{\mx}[1]{\mathbf{#1}}
\newcommand{\dd}[1]{\,\mbox{d}{#1}}
\makeatletter
\@namedef{u8:\detokenize{}}{\mathbf{u}}
\makeatother
\newunicodechar{}{u}
\begin{document}
\bibliographystyle{plainnat}
\input{equations.tex}
\input{scontact.tex}
\input{particle-fluid-interaction.tex}
\input{stability.tex}
\input{stability2.tex}
\input{imex.tex}
\newpage
\bibliography{zotero}
\end{document}
\documentclass[11pt]{article}
\usepackage{amsmath,amstext}
\usepackage{graphicx,color,bm}
\usepackage{amssymb,amscd}
\usepackage{amsmath,amsfonts}
\usepackage{amsthm}
\usepackage[mathletters]{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{newunicodechar}
\usepackage{tikz}
\usepackage[round]{natbib}
\usetikzlibrary{arrows}
\newcommand{\re}{\mathit{R\kern-.15em e}}
\DeclareUnicodeCharacter{183}{{\,\cdot\,}}
\begin{document}
\bibliographystyle{plainnat}
\noindent
NB : all those papers deal with gaz-particle mixture in general and fluidized bed in particular. I'm not sure this can be applied to subsurface water.
\section*{Particle-fluid interaction forces}
......@@ -62,5 +46,3 @@ In our current implementation (Brinkman law), we have
\[
F = -ρ\frac{με²}{K₁}(u_p - u_f) + \dots\text{, with }K₁ = \frac{k₁ε³}{(1-ε)²}
\]
\bibliography{zotero.bib}
\end{document}
This diff is collapsed.
\documentclass{paper}
\usepackage[mathletters]{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\usepackage{a4}
\usepackage{xcolor}
\DeclareUnicodeCharacter{183}{{\,\cdot\,}}
\begin{document}
\section{Setup}
\section{stability1}
\subsection{Setup}
The particle positions and velocities advance in time with :
\begin{align*}
\tilde{v}_p^{n+1} &= v_p^n + Δt \, f_p\\
......@@ -25,7 +18,7 @@ where $m_p$ is the particle mass, $ρ$ the particle density, $Ω_p$ the domain c
where $c$ is the compacity, $u$ the (Darcy) fluid velocity and $v$ the (Darcy) particle velocity.
So that $f_p$ is actually a function of the particles position and velocity and of the fluid velocity $f_p(x_p, v_p, u)$.
\section{Implicit solid velocity}
\subsection{Implicit solid velocity}
If we do the coupling in a naive, fully explicit way, i.e. $f_p = f_p(x_p^n, v_p^n, u)$, a CFL condition appears which is too restrictive to compute anything.
Fortunately, it is easy to overcome this restriction by threating implicitly the impact of the pressure field on the particle velocity (not, as I tried initially on the particle position which is more difficult).
\[f_p = f_p(x_p^n, \tilde{v}_p^{n+1}, u)\]
......@@ -46,4 +39,3 @@ Injecting $\tilde{v}^{n+1}$, the fluid equations reads :
0 &= ∇·u + ∇·\big(v^n-\frac{Δt}{ρ}c^{n}∇p\big)
\end{align*}
Compared to a fully explicit setup, the additional term in the volume conservation equation corresponds to a pressure diffusion with a diffusivity of $\frac{Δt\, c}{ρ}$.
\end{document}
\documentclass{paper}
\usepackage[mathletters]{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\DeclareUnicodeCharacter{183}{{\,\cdot\,}}
\newcommand{\s}{_{\text{s}}}
\renewcommand{\ss}[1]{_{\text{s}#1}}
\newcommand{\re}{\mathit{R\kern-.15em e}}
\usepackage{a4}
\usepackage{color}
\usepackage{graphicx}
\begin{document}
\section{Stability2}
\paragraph{Instable}
\begin{align*}
&\left\{
......@@ -55,11 +44,10 @@ TODO : il faudrait garder les projections de manière à réutiliser $\frac{∂F
\paragraph{Approche Directe}
Je pense que ça ne peut pas marcher sans tenir compte des contacts dans la linéarisation.
\newpage
\paragraph{condition de stabilité dans le cas 1d homogène}\ \\
Dans un cas où la vitesse et la compacité seraient homogènes, on aurait :
{\centering \def\svgwidth{\columnwidth} \input{dessin.pdf_tex}}
{\centering \def\svgwidth{\columnwidth} \input{figures/stability/dessin.pdf_tex}}
par conservation, on a :
\[u = - u_s\]
si on ne considère que le terme de trainée :
......@@ -101,6 +89,5 @@ Pour ne pas changer l'implémentation, on peut écrire
\end{align*}
Ce qui revient à réduire le pas de temps d'un facteur ridicule, il y a un problème qqpart. En plus, il faut garder $u_p - u_f$ et pas jute $u_p$.
Une autre solution (plus précise ?) serait de sous-itérer le schéma de particule en updatant la force.
\end{document}
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