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\title{Part IIb. A seminar on fluctuations in sedimentation}
\begin{document}

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\begin{frame}
 \titlepage

\end{frame}

\title{Fluctuations in the velocities of sedimenting particles}
\author{John Hinch}
\institute{DAMTP, Cambridge}
\date{\today}

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\begin{frame}
 \titlepage

In collaboration with \'Elisabeth Guazzelli \& Laurence Bergougnoux

\vspace{10pt}
\rightline{and their students} % D.Chehata G\'omez, S.Ghicini, \& H.Nicolai}
\bigskip
\cpaper{Guazzelli \& Hinch (2011) Ann. Rev. Fluid Mech. {\bf 43}, 97--116}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Fluctuating velocities}
 \framesubtitle{Particles do no fall at a constant speed in a suspension}

Trajectories of two spheres at $\phi=0.3$

\bigskip
\centerline{
 \includegraphics[height=4cm]{TrajPoF95.pdf}
}

\bigskip
\rightline{\cpaper{Nicolai, Herzhaft, Hinch, Oger \&
   Guazzelli. (1995) Phys.\ Fluids\ {\bf 7}, 12--23.}}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{The divergence paradox}

\begin{itemize}
 \vspace{0.3cm}
 \item Theory: depend on size ${\red L}$ of box \qquad
    $w' = V_S\sqrt{\phi{{\red L}\over a}}$ \pause
 \item Experiments: no such dependence
\end{itemize}

\vspace{-0.6cm}
\begin{center}\tiny
 \raisebox{3.5cm}{$w'/V_S$}\includegraphics[height=4.5cm]{FluctuJohn}\break
 \hspace*{2.5cm}{$t/t_S$}\break
\end{center}
\vspace{-0.7cm}
\rightline{\cpaper{Nicolai \& Guazzelli. (1995) Phys.\ Fluids\ {\bf 7}, 3--5.}}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Theory of scaling}

\begin{itemize}
 \item Dilute: pair separated by $r$ have $w' \sim V_S\frac{a}{r}$, \break
  \pause \qquad so averaging with $p \sim n$ (const)\pause
   \vspace{-0.2cm}
  \[\int {w'}^2 p\, dV \quad\mbox{\rm diverges like}\quad V_S^2\phi \frac{\red L}{a}\]
  \rightline{\cpaper{Caflisch \& Luke (1985) Phys.\ Fluids\ {\bf 28}, 759-60.}}
 \pause\item Explanation \hfill{\cpaper{Hinch (1988) {\sl Disorder and Mixing} 153--60}}
\end{itemize}

\pause
\vspace{-0.3cm}
\begin{minipage}{5.5cm}
\begin{picture}(150,100)(0,0)
 \thicklines
 \put(10,90){\line(0,-1){80}}
 \put(10,10){\line(1,0){130}}
 \put(140,10){\line(0,1){80}}
 \thinlines{ \put(75,10){\line(0,1){80}}}
 \put(20,70){$\frac{N}{2} + \sqrt{N}$}
 \put(85,70){$\frac{N}{2} - \sqrt{N}$}\pause
  \red{\curve(
   60,55,
   55,40,
   60,25,
   75,20,
   90,25,
   95,40,
   90,55
   )
 \put(93,52){\vector(-1,1){4}}
 \put(110,30){$w'$}}
\end{picture}
\end{minipage}
\begin{minipage}{5.0cm} \pause
 \[ {\red {w'} } = \displaystyle{\frac{\sqrt{N}mg}{6\pi\mu {\red L}}} \pause
       = V_S\sqrt{\phi \displaystyle{\frac{{\red L}}{a}}}
 \]
\pause
\rightline{{\red {`Poisson' value}}}
\end{minipage}


\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Big effect of a little stratification}

{\cpaper{B{\l}awdziewicz c1995, private communication - ignored.}\pause
 \hfill\cpaper{Luke (2000)  Phys.\ Fluids\ {\bf 12}, 1619--21.}} \pause

\medskip
\begin{itemize}
 \item If vertical change in density exceeds statistical fluctuation, \pause
 \rightline{then heavy side sinks only to level of neutral buoyancy.}

\pause
 \item Blobs smaller than $\ellr$ unaffected, with $\ellr$ given by
\[
 \ellr\frac{\partial n\ellr^3}{\partial z} = \sqrt{n \ellr^3}
 \pause\quad\text{so}\quad
 \ellr = n^{1/5}\left(- \frac{\partial n}{\partial z}\right)^{-2/5}
\]
\pause Hence
\[
 {\blue w'} = V_s\sqrt{\phi\frac{\ellr}{a}} \pause
    = V_S\phi^{3/5}\left(-a\frac{\partial \phi}{\partial z}\right)^{-1/5}
\]
\end{itemize}

\rightline{\cpaper{Tee, Mucha, Cipelletti, Manley \& Brenner (2002) PRL {\bf 89}:054501}}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Computer simulations to test effect of stratification}
 \framesubtitle{Initially stratified}

\pause
\begin{minipage}{5.5cm}
 \raisebox{2.0cm}{\tiny $z$}\hspace{-0.1cm}\includegraphics[height=3cm]{c14}

 \vspace{-0.4cm}\hspace{3cm}{\tiny $\phi$}
\end{minipage}
\begin{minipage}{5cm}

Concentration profile 

at different times

\medskip
{\small $\Delta\phi/\phi=0.4$, 2500 particles, 

average over 40 realisations}
\end{minipage}

\pause
\begin{minipage}{5.5cm}
 \raisebox{1.5cm}{\tiny $\frac{{w'}^2}{n}$}\hspace{-0.3cm}\includegraphics[height=3cm]{decayc}

 \vspace{-0.4cm}\hspace{3cm}{\tiny $t$}
\end{minipage}
\begin{minipage}{5cm}
Velocity fluctuations for 

$\Delta\phi/\phi=$ 0,\ldots,0.4

\medskip
\small{$10^4$ particles, $h=10$}

\bigskip
\pause
{\blue Decay to a plateau value $w'_\infty$}

\end{minipage}

\medskip
\rightline{\cpaper{Chehata G\'omez, Bergougnoux, Guazzelli \& Hinch (2009)
     Phys.\ Fluids\ {\bf 21}: 093304}}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Computer simulations to test effect of stratification}
 \framesubtitle{Initially stratified}

{\blue Plateau value  ${w'_\infty}^2$} plotted against stratification

\pause
\begin{minipage}{7.5cm}

\centerline{
 \raisebox{2cm}{$\frac{\blue {{w'_\infty}}^2}{{\rm Poisson}}$}\hspace{-0.5cm}
 \includegraphics[height=4cm]{winhib4c}
}
\vspace{-0.4cm}
\centerline{\hspace{2cm}\small $n^{1/5}(-\partial n/\partial z)^{-2/5}$}

\end{minipage}
\begin{minipage}{3cm}
\medskip
{\small{Different $n$ \& $\delta x$}}

\end{minipage}
\bigskip

\pause
\begin{itemize}
\item Hence
\[
 {\blue w'_\infty} = 0.94 V_S \phi^{3/5}
   \left(-a\frac{\partial\phi}{\partial z} \right)^{-1/5}
\]
\end{itemize}

\bigskip
\rightline{\cpaper{Chehata G\'omez, Bergougnoux, Guazzelli \& Hinch (2009)
     Phys.\ Fluids\ {\bf 21}: 093304}}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Computer simulations to test effect of stratification}
 \framesubtitle{Initially uniform - stratified in descending front}

\pause
Viewed in windows at different heights: {\color{cyan}{top}}, \red{bottom}

\bigskip
\centerline{
\raisebox{3.5cm}{$\frac{{w'}^2}{{\rm Poisson}}$}
\hspace{-1cm}\raisebox{1cm}{$\frac{\partial\phi}{\partial z}$}
\includegraphics[height=5cm]{vresnx5h8c}
}
\vspace{-0.3cm}\centerline{\hspace{3cm}$t$}

\pause
\bigskip
{\blue Velocity fluctuations reduced when front arrives in window}


\bigskip
%\rightline{\cpaper{Chehata G\'omez, Bergougnoux, Guazzelli \& Hinch (2009)
%     Phys.\ Fluids\ {\bf 21}: 093304}}

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Computer simulations to test effect of stratification}
 \framesubtitle{Initially uniform - stratified in descending front}

{\blue ${w'}^2$ in front } plotted against stratification

\pause
\centerline{
\raisebox{3.5cm}{$\frac{\blue {w'}^2}{{\rm Poisson}}$}
\includegraphics[height=5cm]{nx71012b}
}
\centerline{\hspace{2cm}$\frac{a}{L}\phi^{1/5}\left(-a\frac{\partial\phi}{\partial z}\right)^{-2/5}$}

\pause
\bigskip
{\blue Fair agreement only, but recall time delay for initial value to decay}

%\bigskip
%\rightline{\cpaper{Chehata G\'omez, Bergougnoux, Guazzelli \& Hinch (2009)
%     Phys.\ Fluids\ {\bf 21}: 093304}}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Experiments}
 \framesubtitle{Initially uniform - stratified in descending front}

\begin{itemize}
\item Four experiments at $\phi=0.3\%$, with different box size and different particle sizes and densities. View in fixed window.

\pause
\item Open symbols $w'/V_S$. Filled symbols $-a\partial\phi/\partial z$ (difficult). 
\end{itemize}

\centerline{
\includegraphics[height=5cm]{Figure2}
}

\pause
{\blue I -- Decay of initial state, \pause \quad II -- plateau, \pause \quad III -- in front }

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Experiments}
 \framesubtitle{Initially uniform - stratified in descending front}

\begin{itemize}
\item Velocity fluctuations inhibited by stratification
\end{itemize}
\medskip

\begin{minipage}{5.5cm}
\centerline{\includegraphics[height=4cm]{Figure3a}}
\end{minipage}
\begin{minipage}{5cm}
Filled symbols on plateau (II), 

open in front (I).
\end{minipage}

\bigskip
\rightline{\cpaper{Chehata G\'omez, Bergougnoux, Guazzelli \& Hinch (2009)
     Phys.\ Fluids\ {\bf 21}: 093304}}


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Experiments}

And initial values are the old divergent scaling

\vspace{0.7cm}
\begin{minipage}{4.5cm}
\[ w'_0 = V_S\sqrt{\phi{{\red L}\over a}}\]
\end{minipage}
\begin{minipage}{6cm}
\pause
 \includegraphics[height=5cm]{Figure4}
\end{minipage}

\bigskip
\rightline{\cpaper{Chehata G\'omez, Bergougnoux, Guazzelli \& Hinch (2009)
     Phys.\ Fluids\ {\bf 21}: 093304}}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Diffusing front}

\begin{itemize}
 \item Does front between top of suspension and clear fluid {\blue diffuse}?
\bigskip\pause
 \item Self-diffusivity \quad
   ${\blue D = w'\ellr} \pause = {\dgreen 2.75} V_s a \phi^{4/5}
           \left(-a\partial\phi/\partial z\right)^{-3/5}$ \\
\bigskip\pause
 \item Nonlinear diffusion equation
   \[\frac{\partial\phi}{\partial t}
      - \frac{\partial (V_s\phi)}{\partial z} =
      \frac{\partial{}}{\partial z}\left({\dgreen 2.75} V_s a^{2/5} \phi^{4/5}
        \left(-\frac{\partial\phi}{\partial z}\right)^{2/5}\right)
    \]
\rightline{\cpaper{Mucha \& Brenner (2003) Phys.\ Fluids\ {\bf 15}: 1305-13 }}
\medskip\pause
 \item Numerical value {\dgreen 2.75} of diffusivity from similarity solution \ldots
\end{itemize}


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Diffusing front}

\begin{itemize}
 \item Similarity thickness of front
   \[\vspace{0.3cm}{\delta} = {\dgreen 3.07} a \phi^{1/7} 
        (V_s{\red t}/a)^{\red{ 5/7}}
   \]
\end{itemize}

\pause
\centerline{\raisebox{2cm}{{$\delta$}}\includegraphics[height=5cm]{delta4d}}
\centerline{\hspace{2cm}{${\red t}^{\red {5/7}}(N/h)^{1/7}$}}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Diffusing front}

Similarity plot of concentration profile
\begin{center}
 \raisebox{3cm}{\small $c$}\includegraphics[height=4.5cm]{c16}

 \vspace{-0.3cm} 
\hspace{2cm}{\small $[z-z(c=0.5)]/\delta(t)$}
\end{center}

\pause
\begin{itemize}
 \item Nonlinear diffusion equation predicts concentration profile in
 diffusing front at top of suspension
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Open question: effect of small inertia}

The Poisson estimate was for blobs at low Reynolds numbers.

\pause
With inertia, estimate:
\[
 w' = V_s \phi^{1/3} Re_p^{-1/3}
\] 

\pause\medskip
\begin{minipage}{5.5cm}
 \rightline{\includegraphics[height=5cm]{Resed}\qquad}
\end{minipage}
\begin{minipage}{5cm}
 Recent preliminary experiments (Bergougnoux 2011)
\end{minipage}

\end{frame}

\end{document}


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