数理解析研究所講究録
1191 巻 2001 年 51-52

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Front Aggregation and Labyrinthine Patternin the Drying Process of Water-Granule Systems
Yoshihiro Yamazaki
Graduate School of Science, Hiroshima University

1. Introduction
In connection with the statistical approach to the macroscopic collective behavior ofa group of microscopic movable (or motile) units, many studies concerning the behaviorof granular materials have been carried out. Recently we found the interesting patternformation in the drying process of water-granule mixture, and we have done someexperiments by focusing our attention on the change of the pattern as a function of thegranule content of the mixture [Y. Yamazaki and T. Mizuguchi, J. Phys. Soc. Jpn.,
69 (2000), pp.2387-2390.]. In this conference we reported our experimental results andproposed the model to reproduce the dynamics of the pattern formation based on thephase field model.

2. Experiment
We prepared the experimental cell which consisted of a homogeneous mixture ofwater and granules (we use cornstarch), sandwiched between two glass slides. And thecell was set horizontally to dry. As a control parameter for this experiment, wethe weight ratio of granules to water .
Figure 1 shows the typical pattern after the drying process. The black and white
regions represent the compact granular region ( -region) and air region, respectively.
.
show the patterns for
The actual size of these figures is.
different .
$\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}^{t}\mathrm{d}$$\phi_{g}$$\mathrm{C}$$12.7[\mathrm{m}\mathrm{m}]\cross 12.7[\mathrm{m}\mathrm{m}]$$\phi_{g}$$((\mathrm{a})\phi_{g}=0.02, (.\mathrm{b})\phi_{g}=0.2,$$(\mathrm{a})\sim(\mathrm{c})$$(\mathrm{c})\phi_{g}=0.6)$.

$\backslash \cdot..\cdot.\cdot.\sim\prime\prime,\backslash \mathrm{t}\mathrm{t}\backslash ^{\mathrm{s}_{j}}’j.- 7.\backslash \cdot’\vee:’\sim\cdot\wedge’\cdot.‘.‘ f,\wedge.\backslash \sim\wedge r.\wedge\overline{r}_{t}^{A^{\wedge^{-}\backslash }\backslash _{\mathrm{r}}}’.,|.4^{\cdot}.\backslash )_{\star^{-;}\backslash _{t}..,\mathrm{x}-}^{\mathrm{r}.\prime}J\backslash |,‘-\neg’\cdot\wedge\cdot..\bigvee,\grave{t}_{\iota\tau}^{\backslash *}\neg\dot{J}\backslash \backslash P^{\backslash }\backslash \wedge\neg’\downarrow\backslash ^{\backslash \backslash }\backslash \backslash$$.\cdot’.1\aleph^{f}.\cdot\acute{r}-:\sim\iota_{\sim}d^{\Gamma\backslash -*\kappa.\cdot\}_{\{.*}’}\backslash \sim\backslash .j-^{\iota}j_{-}T_{r_{\iota}}...\}\backslash \cdot\backslash \backslash \wedge\backslash \backslash \alpha\sim\backslash _{\tau}\backslash *$.
(a) $\phi_{g}=0.02$(b) $\phi_{g}=0.2$

Fig.1 : The final patterns for different(c) $\phi_{g}=0.6$$\phi_{g}$.

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From the observation, the following features were clarified.
(i) By the water-air front advancing, granules are swept and gathered near the front
in the water region. Therefore, the aggregation of granules occurs with the shiftof the water-air fronts (Front Aggregation).

case, the rearrangement of water-air fronts brings about the disconnec(ii) In lowtion of C-region.
$\phi_{g}$increases, the isolated compact granular regions tend to connect with each(iii) Asother. And -region takes the form of the labyrinthine pattern.
$\phi_{g}$$\mathrm{C}$

3. ModelWe proposed the following model for the front aggregation dynamics of granules:$\tau\dot{\phi}=$$\dot{\psi}$$=$$-\phi(\phi-1)(\phi-A(\psi))+\epsilon\nabla\cdot(K(\psi)\nabla\phi)$,$\nabla\cdot(D(\psi)\nabla\psi)-\nabla\cdot(C(\psi)\nabla\phi)$

Here, is a phase field representing water $(\phi=1)$ and air $(\phi=0)$ regions.
denotesthe density of granules. The coefficients $A(\psi),$ $K(\psi),$ $C(\psi),$ $D(\psi)$ are dependent on
the granular density and should be determined by the physical properties of waterair surface (vapor pressure, surface tension) and the granular material (wettability,friction).
$\psi$$\phi$

4. Res.ults
By the numerical simulation of the above model, the disconnection of -region wasreproduced in low granular density case. We note that -region is assumed to be afluid o.beying Darcy’s law in this model. This assumption produces the diffusion fieldin -region. The diffusion field, in addition to the prevention of the motion of water-air
fronts due to the existence of granules, occurs the instability of a flat water-air front.
Thisis considered to be the origin for the pattern formation.
$\mathrm{C}$$\mathrm{C}$$\mathrm{C}$$\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}\dot{\mathrm{a}}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$Acknowledgment
This study is collaborated with Prof. M. Mimura (Hiroshima-u), Dr. T. Mizuguchi(Kyoto-u), and Mr. T. Watanabe (Hiroshima-u).