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\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Small particles in a viscous fluid}

\blue{Course in three parts}

\vspace{20pt}

\blue{1. A quick course in micro-hydrodynamics}

\vspace{15pt}

\blue{2. Sedimentation of particles}

\vspace{15pt}

\blue{3. Rheology of suspensions}

\vspace{20pt}
Good textbook for parts 1 \& 2: 

{\em A Physical Introduction to Suspension Dynamics} \break
by Elisabeth Guazzelli, Jeffrey F. Morris and Sylvie Pic \break
(Cambridge Texts in Applied Mathematics 2012).

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Part 2. Sedimentation of particles}
 \tableofcontents[hideallsubsections]
\end{frame}

\section{An isolated particle}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Part 2. Sedimentation of particles}
 \tableofcontents[currentsection,hideothersubsections]
\end{frame}

\subsection{A sphere}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Sedimentation of an isolated sphere}

Recall Stokes drag on a sphere $6\pi\mu a\bfU$

\pause
\bigskip
Force balance, with densities $\rho_s$ of sphere and $\rho_f$ of fluid
\begin{align*}
   0 &\quad =& \rho_s\tfrac{4\pi}3 a^3g &\quad- &\rho_f\tfrac{4\pi}3 a^3g &\quad -& 6\pi\mu a\bfU \\
 \text{no inertia} & & \text{weight} && \text{buoyancy} & & \text{Stokes drag}
\end{align*}

\pause
\bigskip
So Stokes settling velocity
\[
 \bfU_S = \frac{2\Delta\rho a^2g}{9\mu}
\]

\end{frame}
\subsection{Stokes settling velocity}
\subsection{A spherical viscous drop}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Stokes settling velocity}

\[
 \bfU_S = \frac{2\Delta\rho a^2g}{9\mu}
\]

\pause
\bigskip
E.g. $1\,\mu{\rm m}$ sphere, $\Delta\rho=10^3\,{\rm kg\,m^{-3}}$, 
water $\mu = 10^{-3}\,{\rm Pa\,s}$ 
gives $U_S=2\mu{\rm m\,s^{-1}}$, \break i.e. falls through diameter in a second at $Re=10^{-6}$.

\pause
\bigskip
Dependence $\propto a^2$, means $100\,\mu{\rm m}$ sphere in water would fall $10^4$ times faster at $1\,{\rm cm\,s^{-1}}$ at $Re=1$.

\pause
\bigskip
E.g. $1\,\mu{\rm m}$ sphere, $\Delta\rho=10^3\,{\rm kg\,m^{-3}}$, 
air $\mu = 1.8\,10^{-5}\,{\rm Pa\,s}$ 
gives $U_S=100\mu{\rm m\,s^{-1}}$ at $Re=10^{-5}$.

\pause
\bigskip
Drag on a spherical drop $2\pi\mu a \frac{2\mu+3\mu_i}{\mu+\mu_i}$.

\end{frame}

\subsection{Flow past an ellipsoid}    %%%% other shapes
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Stokes flow past an ellipsoid}

For principle semi-diameters $a_1, a_2, a_3$ Oberbeck (1876) found 
\pause
\[
 \text{Force}\quad F_1 = - \frac{16\pi\mu U_1}{L + a_1^2K_2}
 \qword{and Couple}
   G_1 = - \frac{16\pi\mu(a_2^2 + a_3^2)}{3(a_2^2K_2 + a_3^2K_3)}
\]
\pause 
where
\[{
 L = \int_0^\infty \frac{d\lambda}{\Delta(\lambda)} 
  \qword{and}
 K_i =  \int_0^\infty \frac{d\lambda}{(a_i^2+\lambda)\Delta(\lambda)}
}\]
with $\Delta^2 =(a_1^2+\lambda)(a_3^2+\lambda)(a_3^2+\lambda)$.

\pause
\medskip
For \blue{a disk} $a_1 \ll a_2=a_3$
\[
 F_1 \sim 16\pi\mu a_2U_1, \quad F_2 \sim \tfrac{32}{3}\mu a_2 U_2,
 \quad G_i \sim \tfrac83\mu a_2\Omega_i
\]

\pause
\medskip
For \blue{a rod} $a_1 \gg a_2=a_3$, where $\ln = \ln\tfrac{2a_1}{a_2}$
\[
 F_1 \sim \frac{4\pi\mu a_1 U_1}{\ln - \tfrac12}, \quad
 F_2 \sim \frac{8\pi\mu a_1 U_2}{\ln + \tfrac12}, \quad
 G_1 \sim \tfrac{16}{3}\pi\mu a_1a_2\Omega_1, \quad
 G_2 \sim \frac{\tfrac83\pi\mu a_1^3\Omega_2}{\ln - \tfrac12}
\]

\pause
\bigskip
\blue{Important conclusion Drag $\approx 6\pi\mu\times{\rm (largest\ diameter)}$.}

\end{frame}

\section{Two particles}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Part 2. Sedimentation of particles}
 \tableofcontents[currentsection,hideothersubsections]
\end{frame}

\subsection{Two equal spheres}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Fall of two equal spheres}

Fall at constant separation by reversibility, sliding sideways.

\pause
\bigskip
Fall faster, up to 50\% faster if close and vertical.

\medskip
\centerline{\includegraphics[scale=0.5]{twospheres}}

\pause
\bigskip
Approximations for large separations
\[
 U = U_S\left(1 + \frac{3a}{2r} + \frac{a^3}{r^3} \right), 
   \quad U_S\left(1 + \frac{3a}{4r} + \frac{a^3}{r^3} \right)
\]

\end{frame}

\subsection{Two unequal spheres}
\subsection{Three equal spheres}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Fall of two unequal spheres}

Larger fall faster, so will overtake smaller.

\pause
\bigskip
Denser fall faster, so will overtake less dense.

\pause
\bigskip
Interesting interaction between two spheres with same isolated fall speeds, one smaller and denser -- dance round one another.

\pause
\bigskip
Three equal spheres: a fast pair can catch up a slow third, interact, and a pair emerge leaving behind a different third, 
\pause
\break i.e. ever changing configuration.

\end{frame}

\subsection{Method of Reflections}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Method of Reflections}

An iterative method for far-field interactions between particles (use lubrication for close particles).

\pause
\bigskip
\blue{Zeroth approximation}: Treat particles as isolated. Each create disturbances \blue{$\bfu^{0}(\bfx)$}.

\pause
\medskip
\blue{First reflection}: Particles react to \blue{$\bfu^{0}$} as if isolated. Each create disturbances \blue{$\bfu^{1}(\bfx)$}.

\pause
\medskip
\blue{Second reflection}: Particles react to \blue{$\bfu^{1}$} as if isolated. Each create disturbances \blue{$\bfu^{2}(\bfx)$}.

\pause
\medskip
\blue{and so on to more reflections}

\pause
\bigskip
Can use computer algebra to go 100 reflections.

\end{frame}

\subsubsection{Zeroth approximation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Method of Refections: zeroth approximation}

\blue{Zeroth approximation}: each sphere falls as effectively isolated
\begin{gather*}
  \bfu^{0} = \bfu^{01} + \bfu^{02} \\
   \bfu^{01}(\bfx) 
 = \bfU_1\left(\frac{3a_1}{4r_1}+\frac{a_1^3}{4r_1^3} \right) +
   \frac{(\bfU_1\cd{\bf r}_1){\bf r}_1}{r_1^2}\left(\frac{3a_1}{4r_1}-\frac{3a_1^3}{4r_1^3} \right)
\end{gather*}
with $\bfU_1$ isolated fall speed of 1, centred at $\bfx_1$,
${\bf r}_1 = \bfx-\bfx_1$.

\pause
\bigskip
And $\bfu^{02}(\bfx)$ similar.

\end{frame}

\subsubsection{First reflection}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Method of Refections: first reflection}

\blue{First refection}: Sphere 2 sees $\bfu^{01}(\bfx)$ in neighbourhood of 
$\bfx=\bfx_2$
\[
 \bfu^{01}(\bfx) = \underset{\red{(1)}}{\bfu^{01}(\bfx_2)}
     + \underset{\red{(2)}}{{\bf r}_2\cd\nabla\bfu^{01}\vert_{\bfx_2}}
     + \underset{\red{(3)}}{{\bf r}_2{\bf r}_2:\nabla\nabla\bfu^{01}\vert_{\bfx_2}} + \ldots
\]

\pause
\bigskip
\red{(1)} dominates, a uniform flow. Sphere 2 (now force free) moves with this, i.e. additions to fall speed $O(U_1a_1/r_{21}) + O(U_1a_1^3/r_{21}^3)$.

\pause
\bigskip
\red{(2)} smaller, a linear shear flow. Sphere 2 (couple free) rotates with vorticity (antisymmetric part of $\nabla u^{01}$). Cannot deform with straining symmetric part, so reacts with a \dpink{force-dipole $O(a_2\, \mu a_2\, a_2U_1a_1/r^2_{21})$}.

\pause
\bigskip
\red{(3)} even smaller, a quadratic flow. Reacts with \brown{force quadrupole 
$O(a_2^2\, \mu a_2\, a_2^2U_1a_1/r^3_{21})$}. \break
Addition to fall speed by Faxen 
$\frac16a_2^2\nabla^2 u^{01}= O(a^2_2 U_1a_1/r_{21}^3)$.

\end{frame}

\subsubsection{Second reflection}
\subsubsection{Collecting terms}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Method of Reflections: second reflection, collecting terms}

\blue{Second reflection}: Sphere 1 sees $\bfu^{12}(\bfx)$ from \dpink{dipole}, \brown{quadrupole},\ldots from sphere 2 as uniform + linear + quadratic flows 
\pause
Additional fall speed due to uniform flows
\[
 O\left(\dpink{U_1\frac{a_1a_2^3}{r_{21}^2}}\frac{1}{r_{12}^2}\right) +
  O\left(\brown{U_1\frac{a_1a_2^5}{r_{21}^3}}\frac{1}{r_{12}^3}\right)
\]

\pause
\bigskip
\blue{Collecting contributions} to fall speed
\[
 {U_1}
     + {U_2\frac{a_2}{r_{12}} + U_2\frac{a_2^3}{r_{12}^3} + U_2\frac{a_1^2a_2}{r_{12}^3}} +
   {U_1\frac{a_1a_2^2}{r_{21}^2r_{12}^2} + U_1\frac{a_1a_2^5}{r_{21}^3r_{12}^3}}
\]
$+ 1/r^8+ \ldots$ from 2nd reflection $+ 1/r^7 + \ldots$ from 3rd + more. 

\end{frame}

\section{Finite clouds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Part 2. Sedimentation of particles}
 \tableofcontents[currentsection,hideothersubsections]
\end{frame}

\subsection{Spherical cloud}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{A spherical cloud}

A spherical cloud of particles behaves as a drop of a more viscous liquid. 

\pause
Cloud radius $R$.
Particles radius $a$, density difference $\Delta\rho$, volume fraction $\phi$, viscosity of suspension $\mu^*(\phi)$. 

\pause
Fall speed
\[
 \frac{2\Delta\!\rho\,a^2g}{9\mu} +
    \frac{2\Delta\!\rho\,\phi R^2g}{9\mu}\frac{2(\mu+\mu^*)}{2\mu+3\mu^*}
\]
First term from fall of individual particles is much smaller.

\pause
\bigskip
Due to edge being ill defined, there is a small loss of particles into the wake.

\pause
\medskip
This loss produces a toroid. In a very tall tank, breaks up into two drops, process repeating (Metzner, Nicolas \& Guazzelli 2007).

\end{frame}

\subsection{Boycott effect}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Boycott effect}

Boycott observed in 1920 that blood settles faster if the test tubes are inclined to the vertical.

\pause
\bigskip
Much used in large scale industrial settlers.

\pause
\bigskip
Suspension only settles short vertical distance to lower inclined side and then slides slowly as a dense suspension to bottom.

\pause
\bigskip
Clear fluid released rises rapidly on underside of upper side.

\pause
\medskip
Potential shear-flow instabilities remixing.

\pause
\bigskip
http://youtu.be/6I5PNdAHlLI

\end{frame}

\section{Suspensions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Part 2. Sedimentation of particles}
 \tableofcontents[currentsection,hideothersubsections]
\end{frame}

\subsection{Hindered settling}
\subsection{Richardson-Zaki}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Settling of a uniform suspension}

While two spheres settle faster than one, a suspension settles slower, often much slower.

\pause
\medskip
Called `hindered settling'. Due to `back flow' of liquid.

\pause
\bigskip
Richardson-Zaki (1954) empirical correlation, for volume fraction $\phi$
\begin{minipage}{0.5\linewidth}

\centerline{\includegraphics[scale=0.4]{RZ}}

\end{minipage}
\begin{minipage}{0.4\linewidth}
\vspace*{-20pt}
\[
 U(\phi) = U(0)(1-\phi)^{4.5}
\]
\end{minipage}

\pause
\medskip
Note near maximum packing reduced to $\frac1{20}$ -- like porous media.

\end{frame}

\subsection{Batchelor renormalisation}
\subsubsection{Summing interactions in a dilute suspension}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
 \frametitle{Summing interactions with other particles in a dilute suspension}

Settling velocity of test sphere due to 2nd at distance $r$
\[
 U(r) = U_0 + \Delta U(r) 
\]
with Stokes velocity for isolated sphere $U_0 = 2\Delta\rho g a^2/9\mu$


\pause
\bigskip 
Far-field form from reflections
\begin{alignat*}{2}
  \Delta U(r)/U_0 \quad=\quad & \tfrac{a}{r} + \tfrac{a^3}{r^3} & \quad\text{1st reflection} \\
   & + \tfrac{a^4}{r^4} + \tfrac{a^6}{r^6} +\ldots & \quad\text{2nd reflection} \\
   & + \tfrac{a^7}{r^7} + \tfrac{a^9}{r^9} +\ldots & \quad\text{3rd reflection} \\
  &  +\ldots
\end{alignat*}

\end{frame}

\subsubsection{Divergence in naive sum}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Divergence in naive sum}


{\color{red}Naive pairwise addition} of disturbances within large domain $r\le R$, with $n$ spheres per unit volume
\[
 \langle \Delta U \rangle = 
  \int_{r=2a}^{R} U_0\left(\frac{a}{r} + \ldots\right)n\,dV 
 \pause 
= O\left(U_0 \phi {\color{red}\frac{R^2}{a^2}}\right)
\]
$\phi=\tfrac{4\pi}{3}n a^3$ the volume fraction.

\bigskip
\pause
{\color{red} The divergence problem:}
\begin{itemize}
 \item Does mean settling velocity depend on size of domain?\pause
 \item Or is it an intrinsic property independent of domain?\pause \\
   \quad i.e. is pairwise addition naive?
\end{itemize}

\end{frame}

\subsubsection{Batchelor's 1972 renormalisation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Batchelor's 1972 renormalisation}

Batchelor's renormalisation:

\[
 \Delta U 
   = \left(1 + \tfrac{a^2}{6}\nabla^2\right) u(x) \big\vert_\text{test sphere}
    + \text{higher reflections}
\]
\pause
Pairwise sum of $O(U_0a^4/r^4)$ higher reflections is convergent

\medskip
\pause
Now\quad $\langle u \rangle_\text{everywhere} = 0$, 
\pause
\quad so \vspace{-0.2cm}
\[
 \langle u \rangle_\text{test sphere} 
   = - \tfrac{11}{2}U_0\phi
\] 
\pause 
\vspace{-0.2cm}
\[
 \langle  \tfrac{a^2}{6}\nabla^2u \rangle_\text{test sphere}
   = \tfrac12U_0\phi
\] 
\pause 
\vspace{-0.2cm}
\[
 \langle\text{higher reflections} \rangle_\text{test sphere} 
   = -1.55U_0\phi
\]

\pause
Hence
\[
  {\color{red}\langle U \rangle = U_0(1 -6.55\phi)}
\]

\end{frame}

\subsubsection{Polydispersity in sizes}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Polydispersity in sizes}

Batchelor's result assumes uniform distribution of equal spheres
\[
 {\langle U \rangle = U_0(1 -6.55\phi)}
\]

\pause
\bigskip
Real suspension have spheres with only nearly equal radii, although same density.

\pause
\smallskip
Hence spheres move past one another, leading to a non-uniform distribution of pair-separations. 

\pause
\smallskip
Hence practical result (Salin 1986)
\[
 {\langle U \rangle = U_0(1 -5.6\phi)}
\]

\end{frame}

\section{Fluctuations in velocity of particles}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Part 2. Sedimentation of particles}
 \tableofcontents[hideallsubsections]
\end{frame}

\end{document} 

