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\newcommand{\rh}{{\color{red}h}}

\title{A perspective of GKB's Micro-hydrodynamics}
\author{John Hinch}
\institute{DAMTP, Cambridge}
\date{\today}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}
% \begin{center}
%  \includegraphics[height=8cm]{GKB}
% 
%  G. K. Batchelor \quad 8 March 1920 -- 30 April 2000
% \end{center}
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \titlepage

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Before: his turbulence research 1945--1961}

{\openup1\jot
\begin{itemize}
 \item Understand, explain and exploit Kolmogorov
 \item Townsend's experiments
 \item Troubled cannot solve Navier-Stokes
 \item Marseille 1961 IUTAM/IUGG Congress
  \begin{itemize}
   \item Stewart's 3 decades of $k^{-5/3}$ at $Re=10^8$
   \item Kolmogorov: ERROR (intermittency)
  \end{itemize}
 \item Time off: JFM, Collected papers of G.I.Taylor, textbook
 \item 1967 start of second wave of research 
    {\color{gray}(no more turbulence)}\\ 
  \quad in low-Reynolds-number suspensions of particles \\
  \quad {\color{gray}producing 8 of his top 10 most cited papers }
\end{itemize}
}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
 \frametitle{GKB's Micro-hydrodynamics in perspective}

\begin{itemize}
 \item {\color{purple}Before}
  \begin{itemize}
   \item Stokes\quad $6\pi\mu a V$
   \item Einstein -- viscosity $\mu(1+\frac52c)$, diffusivity $D=kT/6\pi\mu a$
   \item GI  -- drops and viscosity of emulsion
   \item Brenner, Cox, Mason, Giesekus, Saffman, Bretherton
   \item textbook and GI collected papers
        {$\to$ opportunities}
  \end{itemize}
 \item {\color{purple}Batchelor's research}
 \begin{itemize}
   \item Hydrodynamic corrections to Stokes \& Einstein
   \item To start: \\
     \quad (i) bulk stress \\
     \quad (ii) $\mu$ at $c^2$
  \end{itemize}
 \item {\color{purple}What followed}
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
 \frametitle{A start}
 \framesubtitle{{\sl Stress system in a suspension of force-free particles\/} JFM 1970}

Suspension of small particles in a viscous fluid
\begin{itemize}
 \item low Reynolds number flow about particles
 \item each particle sees a linear flow (local rheology)
\end{itemize}

\bigskip

Everywhere
\[
 \sigma_{ij} = {\color{blue}- p\delta_{ij} + 2\mu e_{ij}} 
   {\color{red}+ \sigma^+_{ij}}
\]
with {\color{blue}viscous solvent stress} and 
{\color{red}extra non-zero only inside particles}

\bigskip

Ensemble average (from turbulence research)
\[
 \langle{\sigma_{ij}}\rangle = -  \langle{p}\rangle\delta_{ij} + 2\mu \langle{e_{ij}}\rangle 
  + \langle{\sigma^+_{ij}}\rangle
\]

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{\ldots a start}
 \framesubtitle{{\sl Stress system in a suspension of force-free particles\/} JFM 1970}

 Switch to volume average. Use divergence theorem for force-free particles
\[
 \langle{\sigma^+_{ij}}\rangle = {\color{red}n} 
  \int_{{\color{red}A}} \sigma_{ik}n_kx_j - \mu(u_in_j + u_jn_i)\, dA
\]
with {\color{red}n} number of particles per unit volume and {\color{red}A} surface of typical particle.

\bigskip

%Much cited result - still 30 pa 

\bigskip
{\color{gray}L\&L 1959}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{\ldots a start}
 \framesubtitle{{\sl Stress system in a suspension of force-free particles\/} JFM 1970}

{\openup 2\jot
{\color{purple}
 What followed?}

\begin{itemize}
 \item 880+ citations. 
 \item Evaluation of bulk stress in many rheologies, \\ 
   \qquad {\color{red}once} solved micro-structure evolution
 \item Ensemble average needed for calculating non-local rheology
 \item Homogenisation (1980s) -- a step back, \\ \qquad%
  {except compute in periodic boxes}
\end{itemize}
}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Second result}
 \framesubtitle{{\sl Stress generated in a non-dilute suspension of elongated particles by pure straining motion\/} JFM 1971}


Rods of length $\ell$, radius $b$, number per unit volume $n$.\\
Hence average {\color{red}lateral spacing} $\rh=(2n\ell)^{-1/2}$.
\medskip

Regimes (separation $\rh$ decreasing)
\begin{itemize}
 \item (Very) dilute: $\ell \ll \rh$ \vspace{0.1cm}
 \item {\color{blue} Semi-dilute: $b \ll \rh \ll \ell$} \vspace{0.1cm}
 \item (Nematic phase transition to aligned rods: $\rh = \sqrt{b\ell}$)
 \item Concentrated: $\rh \sim b$
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{\ldots second result}
 \framesubtitle{{\sl Stress generated in a non-dilute suspension of elongated particles by pure straining motion\/} JFM 1971}

Use bulk-stress paper for formula for stress


Use slender-body paper, with {\color{blue}outer} boundary condition at typical separation $h$ in place of at length $\ell$

\bigskip

Result: {\color{red} extensional viscosity}
\[ 
 {\color{red} \mu_{\text{ext}}} = \mu \frac{4\pi n\ell^3}{9\log h/b}
\]


\medskip
{\color{blue}One data point}

\medskip

Much larger than the viscosity of the solvent $\mu$, \\  which is shear viscosity of suspension

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{\ldots second result}
 \framesubtitle{{\sl Stress generated in a non-dilute suspension of elongated particles by pure straining motion\/} JFM 1971}

{\color{purple}
 What followed?}
\medskip

{\openup2\jot
\begin{itemize}
 \item Large effect from small concentration $\to$ TDR
 \item Correct outer with a Brinkman approach by Shaqfeh (1990) 
 \item Anisotropic viscosity ($\mu_\text{ext}\gg\mu_\text{shear}$) produces anisotropic flow \\
\begin{center}
 \includegraphics[height=4cm, angle=90]{../nnf/PiauV.eps}
\end{center}
 \item Anistropic flow $\to$ TDR
\end{itemize}
}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
  \frametitle{Renormalization of hydrodynamic interactions}
  \framesubtitle{{\sl Sedimentation in a dilute suspension of spheres\/} JFM 1972}

Settling velocity of test sphere due to 2nd at distance $r$
\[
 V(r) = V_0 + \Delta V(r) 
\]
with Stokes velocity for isolated sphere $V_0 = 2\Delta\rho g a^2/9\mu$

\bigskip 

Far field form from reflections
\begin{alignat*}{2}
  \Delta V(r)/V_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*}
ignoring directional dependency

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{\ldots renormalization of hydrodynamic interactions}
  \framesubtitle{{\sl Sedimentation in a dilute suspension of spheres\/} JFM 1972}

{\color{red}Naive pairwise addition} of disturbances within large domain $r\le R$, with $n$ spheres per unit volume
\[
 \langle \Delta V \rangle = 
  \int_{r=2a}^{R} V_0\left(\frac{a}{r} + \ldots\right)n\,dV 
  = O\left(V_0 c {\color{red}\frac{R^2}{a^2}}\right)
\]
\rightline{$c=\tfrac{4\pi}{3}n a^3$ the volume fraction.}

\bigskip

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

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{\ldots renormalization of hydrodynamic interactions}
  \framesubtitle{{\sl Sedimentation in a dilute suspension of spheres\/} JFM 1972}

Batchelor's renormalization:

\[
 \Delta V 
   = \left(1 + \tfrac{a^2}{6}{\color{brown}\nabla^2}\right) {\color{orange}u(x)} \big\vert_\text{test sphere}
    + {\color{teal}\text{higher reflections}}
\]

\medskip
Pairwise sum of $O(V_0a^4/r^4)$ {\color{teal}higher reflections} is convergent

\bigskip

Now\quad $\langle {\color{orange}u} \rangle_\text{everywhere} = 0$, \quad so \vspace{-0.2cm}
\begin{align*}
 \langle {\color{orange}u} \rangle_\text{test sphere} 
   &= - \tfrac{11}{2}V_0c \quad\text{ \em{back flow}}
 \\
 \langle  \tfrac{a^2}{6}{\color{brown}\nabla^2u} \rangle_\text{test sphere}
   &= \tfrac12V_0c
 \\
 \langle{\color{teal}\text{higher reflections}} \rangle 
   &= -1.55V_0c
\end{align*}

Hence
\[
  {\color{red}\langle V \rangle = V_0(1 -6.55c)}
\]

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
  \frametitle{\ldots renormalization of hydrodynamic interactions}
  \framesubtitle{{\sl Sedimentation in a dilute suspension of spheres\/} JFM 1972}

{\color{purple}
 What followed?}

\bigskip
\begin{itemize}
 \item  910+ citations
 \item Erroneous applications subtracting wrong infinity
 \item Alternative {\em averaged equation\/} approach\\
   \quad recognising divergences as change of $\rho$ and $\mu$ \\
    \qquad from solvent to suspension values
 \item Much more from Batchelor, e.g.
\end{itemize}
\medskip

{\sl Brownian diffusion of particle with hydrodynamic interactions\/} JFM 1976
\[
 D = \frac{1-6.55c}{6\pi\mu a}\left[\frac{c}{1-c}\frac{\partial\mu}{\partial c}
   = kT(1 + 8c +30c^2)\right]
\]

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{$O(c^2)$ correction to Einstein viscosity} 
  \framesubtitle{{\sl The determination of the bulk stress in a suspension of spherical particles to order $c^2$\/} JFM 1972}


Another renormalization of hydrodynamic interactions,\\ with J.T Green (PhD 1970, without answer)

\medskip

For pure straining, by trajectory calculation of nonuniform probability distribution of separation of pairs
\[
 \mu^* = \mu\left(1 + \tfrac52c + {\color{red}7.6}c^2\right)
\]

For simple shear, problem of closed trajectories ($k={\color{red}5.2}$?)
\[
 \mu^* = \mu\left(1 + \tfrac52c + kc^2\right)
\]

Case of strong Brownian motion (JFM 1977)
\[
 \mu^* = \mu\left(1 + \tfrac52c + {\color{red}6.2}c^2\right)
\]

Note strain-hardening and shear-thinning rheology

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
  \frametitle{A refinement - polydispersity}

{\sl Sedimentation in a dilute polydisperse system of interacting spheres \/} JFM 1982, Parts I, II, Corrigendum
\[
 \langle V_i \rangle = V_{i\,0}\left(1 + S_{ij}c_j\right)
\]


If equal density and nearly equal sizes, then increase of near pairs and
\[
  \langle V \rangle = V_0\left(1 - 5.6c\right)
\]

as in experiments and dilute limit of Richardson-Zaki.

\bigskip

Also

{\sl Diffusion in a dilute polydisperse system of interacting spheres \/} JFM 1983

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
  \frametitle{{\color{purple} What followed} beyond Batchelor's $c^2$}

Brady's Stokesian Dynamics (1988) numerical simulations at {\color{blue}moderate concentrations} of c1000 hard spheres
\bigskip

\centerline{
\includegraphics[height=4cm]{Brady3b.eps}\hspace{1cm}\includegraphics[height=4cm]{Brady1b.eps}
}

\bigskip

Boundary integral methods for emulsions (1996) c12 drops


Ladd's Lattice Boltzmann simulations (1996) c32000 particles

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{{\color{purple} What followed} in sedimentation}

\begin{itemize}
   \item Inclined settling (Boycott effect) 
     \quad{\color{gray} Acrivos 1979}
   \item Structural instability for fibres 
     \quad{\color{gray} Koch \& Shaqfeh 1989}
   \item Fluctuations depend on the size of box 
     \quad{\color{gray} Guazzelli 2001}\\
     \qquad$\langle V'^2 \rangle = O\left(V_0^2c{\color{red}\tfrac{R}{a}} \right)$
\end{itemize}

\bigskip

\centerline{
{\includegraphics[height=4cm]{MontageFib.eps}}\hspace{1.5cm}%
 \includegraphics[height=4cm]{25cm-3-11.eps}
}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
  \frametitle{Suspensions: {\color{purple} what happened in parallel}}

{\openup2\jot
\begin{itemize}
 \item Orientation of non-spheres, with Brownian motion \\
            \rightline{\color{gray} Leal \& Hinch 1972}
 \item Deformation and breakup of drops, emulsions 
     \quad {\color{gray} Acrivos 1970} 
 \item Electrical Double Layers and VdW forces %(k=1000)
     \quad {\color{gray} Russel 1978}
 \item Rough spheres 
   \quad {\color{gray} Leighton 1989}
\end{itemize}
}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{{\color{purple} What followed}}

{\openup2\jot
\begin{itemize}
 \item Rheology, with experiments
 \item Fluid dynamics of non-Newtonian fluids \\
  \quad{Normal stresses, stress relaxation, stress saturation, \\
  \quad elastic boundary layers, anisotropic flow}
 \item Electro- and Magneto- rheological fluids
 \item Microfluidic devices -- exploiting/ignoring techniques\\
  \quad{drop production, mixing, slip at wall} 
 \item Micro-rheology
\end{itemize}
}

\end{frame}

\end{document}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
  \frametitle{Batchelor's fluidized beds}
 \section{Fluidized bed}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
  \frametitle{Batchelor's fluidized beds (not low $Re$)}

{\openup1.5\jot
\begin{itemize}
 \item<1->{\color{gray}{\sl Low Reynolds-number bubbles in fluidized beds\/} Arch Mech 1974}
 \item<1->{\color{gray}{\sl Expulsion of particles from a buoyant blob in a fluidized bed\/} JFM 1994  }
 \item<1->{\sl A new theory of the instability of a uniform fluidized-bed\/} JFM 1988

 \item<1->{\color{gray}{\sl Instability of stationary unbounded stratified fluid\/} JFM 1991 with Nitsche}

 \item<1-> {\sl Secondary instability of a gas-fluidized bed\/} JFM 1993
\end{itemize}
}

\pause
\medskip
\onslide<1->{{\color{red}So} bubbles in gas-luidized, not liquid-fluidized}
   \quad (Jackson 2000)

\pause
\onslide<1->{{\color{red}But} dense regions have higher drag so rise!}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
  \frametitle{Batchelor's last paper}
  \framesubtitle{{\sl Break-up of a falling drop containing disperse particles\/}  JFM 1997 with Nitsche}
 \section{Cloud}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
  \frametitle{Batchelor's last paper}
  \framesubtitle{{\sl Break-up of a falling drop containing disperse particles\/}  JFM 1997 with Nitsche}

\centerline{
\includegraphics[height=6cm]{cloud.eps}
}

\pause
\medskip
{\color{red} But a little later}
 \qquad  -- video by Metzger, Nicolas \& Guazzelli 2007

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
  \frametitle{Batchelor's micro-hydrodynamics}

  \tableofcontents

\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{\ldots a start}
 \framesubtitle{{\sl Stress system in a suspension of force-free particles\/} JFM 1970}

Long discussion in paper, including:

{\openup 2\jot
\begin{itemize}
 \item Antisymmetric stress tensor when couple per unit volume
 \item Move integral from surface of particle to far field: \\
  \qquad Stresslet and Couplet
 \item Energy budget \\
    \quad -- connection to Einstein's dissipation approach
 \item Surface tension/energy
\end{itemize}
}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{A preparation}
 \framesubtitle{{\sl Slender-body theory for particles of arbitrary cross-section in Stokes flow\/} JFM 1970}

Previous studies: Burgers 1938, Tuck 1964, Cox 1970, Tillet 1970

\bigskip
Not quite matched asymptotics, but \ldots\
\begin{itemize}
 \item {\color{blue}Outer} represented by line-distribution of Stokeslets
 \item {\color{blue}Inner} a two-dimensional potential problem gives 
\[
 \text{{\color{blue}Effective radius}} = 
 \begin{cases}
  a & \text{circle} \\
  \tfrac12(b+c) & \text{ellipse} \\
  2^{-n}a & \text{$n$-star}
 \end{cases} 
\]
 by conformal transformation
\end{itemize}

\bigskip

Evaluated force and couple

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{\ldots renormalization of hydrodynamic interactions}
  \framesubtitle{{\sl Sedimentation in a dilute suspension of spheres\/} JFM 1972}

 Assumes {\color{blue}uniform separation} of pairs for result
\[
  \langle V \rangle = V_0(1 -6.55{\color{red}c})
\]
Coefficient smaller if more close pairs.

\bigskip

If {\color{blue}crystal} then
 \[
 V = V_0\left(1 - k{\color{red}c^{1/3}}\right)
\]

\medskip

For sedimentation: given force, find average velocity

For {\color{blue}porous media}: given velocity, find average force
\[
 \langle F \rangle = F_0\left(1 + \tfrac{3}{\sqrt{2}}{\color{red}c^{1/2}} + \tfrac32c\right) 
\]

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
  \frametitle{More sedimentation}

{\sl Structure formation in bidisperse sedimentation \/} JFM 1986
\rightline{{\color{gray}with van Rensburg}}
\bigskip

\begin{minipage}{4cm}
\centerline{
\includegraphics[height=5.5cm]{RWJvR.eps}
}
\end{minipage}
\begin{minipage}{6cm}
 Light and heavy particles of same size, both at $c=0.2$

 \vspace{2cm}
  \rightline{{\color{blue}Fast separation}}
\end{minipage}

\end{frame}

