You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: |
    The friction of a sliding plate on a thin immersed granular layer obeys Amonton-Coulomb law. We bring to the fore a large set of experimental results which indicate that, over a few decades of values, the effective dynamical friction-coefficient depends neither on the viscosity of the interstitial fluid nor on the size of beads in the sheared layer, which bears out the analogy with the solid-solid friction in a wide range of experimental parameters. We accurately determine the granular-layer dilatancy, which dependance on the grain size and slider velocity can be qualitatively accounted by considering the rheological behaviour of the whole slurry. However, additional results, obtained after modification of the grain surface by a chemical treatment, demonstrate that the theoretical description of the flow properties of granular matter, even immersed, requires the detailed properties of the grain surface to be taken into account.\
    \
    PACS: 47.57.Gc: Granular flow; 83.50.Ax: Steady shear flows, viscometric flow; 83.80.Hj: Suspensions, dispersions, pastes, slurries, colloids; 81.40.Pq: Friction, lubrication, and wear.
author:
- 'T. Divoux and J.-C. Géminard.'
title: 'Friction and dilatancy in immersed granular matter.'
---

Conducting studies on immersed granular flows remains of primary interest. A host of geophysical or industrial issues deals with mixtures of grains and fluid as submarine avalanches [@Cassar], snow flows or clay suspensions [@coussot]. Also fundamental issues are at stake: one would like to extend the empirical friction law proposed for dense and dry granular flows [@GDRmidi] to immersed ones [@Pouliquen2]. Enlightening previous studies of sheared and immersed granular media are numerous, and different devices have been developed to describe mixtures of grains and fluids (meaning air or liquids). Among them we choose to focus on the three following.

Studying immersed granular matter flowing down an inclined plane, C. Cassar *et al.* measured the dynamical friction-coefficient, $\mu$, for different flow configurations [@Cassar]. Their results were analyzed using an approach inspired by recent results obtained for dry and dense granular flows [@GDRmidi; @Pouliquen]: They report the friction coefficient as a function of the dimensionless parameter $I$, first introduced by Da Cruz *et al.* [@DaCruz], defined to be the ratio of an apt microscopic time scale (inertial, viscous, …) to the relevant macroscopic time scale $\dot \gamma ^{-1}$, where $\dot \gamma$ denotes the shear rate. For immersed granular-matter in the viscous regime [@Courrech], $I \equiv (\dot \gamma \,\eta  _f)/(\alpha \,P_g)$, where $\eta _f$ denotes the viscosity of the intersticial fluid, $P_g$ the pressure exerted on the sheared media and $\alpha$ the normalized permeability of the granular packing [@Cassar]. They propose a semi-empirical law for $\mu (I)$ which describes the whole set of data they report for both aerial and immersed granular flows [@Cassar; @Pouliquen2].

Using a Couette geometry Bocquet *et al.* tuned the pressure within the granular material by applying an upward air flow between the rotating and the stationnary cylinder [@Bocquet]. They found out that mean-flow properties and fluctuations in particle motion are coupled. They introduced an hydrodynamic model which quantitatively describes their experiments: The key feature is that the shear force obtained from this model is found to be proportionnal to pressure and approximately independent of the shear velocity. This model does not include any frictional forces between grains, but contains a phenomenological relationship between the viscosity and the dilation of the media.

Using an experimental setup first designed to perform sensitive and fast force-measurements in the dry case [@Nasuno], Géminard *et al.* brought to the fore a dynamic friction-coefficient $\mu$ in the case of an immersed granular layer sheared by means of a sliding plate [@Losert]. At low imposed normal-stress, the friction force is shown to be independent of the plate velocity, which holds true as long as the granular material is allowed to dilate [@Tardos]. The main difference with the dry case lays in the fact that the slider usually exhibits a continuous sliding instead of the stick-slip motion and in the value of the friction coefficient which is roughly cut down by half [@Losert; @Geminard2]. The dependance of the frictional coefficient on the fluid viscosity and of the associated dependance of the dilatancy on the slider velocity were not reported.

Here we report a set of experimental measurements of the friction coefficient and of the dilatancy in a wide range of fluid viscosities and grain sizes at very low $I$ [@Cassar]. Such a study is relevant for several reasons: First, we point out that the quasi-static regime is unaccessible to the free-surface-flow geometry as size effects crop up in this limit [@Pouliquen2]. In addition, there is a strong discrepancy between the limit of $\mu$ for vanishing $I$ reported in [@Cassar] ($\mu\simeq 0,43$ with $I \simeq 4.10^{-3}$) and those reported for the plane-shear geometry by Géminard *et al.* ($\mu\simeq 0.23$, [@Losert; @Geminard2]) and S. Siavoshi *et al.* ($\mu\simeq 0.54$, [@Kudrolli]), both for $I\simeq 2.10^{-4}$. How can be explained such discrepancies between those three results? We also raise the following questions: What does happen when the fluid viscosity or the bead size are changed? How far does the analogy with the Amonton-Coulomb laws remain relevant? What does the effective friction-coefficient depend on? We choose to stick to the canonical plane-shear geometry for which we know that there is a strong analogy between the friction of a slider on an immersed granular layer and the Amonton-Coulomb law [@Losert]. In the chosen geometry, the layer is free to dilate which makes it possible to measure both the friction coefficient and the dilation of the granular layer at imposed normal stress.

![[]{data-label="fig.schema"}](divoux_fig1.eps){height="5cm"}

*Experimental setup.* - The experimental setup (Fig. \[fig.schema\]) is very similar to the one described in [@Losert; @Geminard2]. A thin plate, the slider, is pushed at the free surface of an immersed granular layer by means of a steel leaf-spring ($k=129 \pm 2$ N.m$^{-1}$) connected to a translation stage driven at constant velocity, $V_s$ by a computer-controlled stepping motor ($V_s$ ranging from 0.1 to 100 $\mu{\rm m.s}^{-1}$). The coupling between the spring and the plate is insured by a metal bead, which avoids applying a torque. The frictional force is monitored by measuring the receding of the spring from its rest position with an inductive sensor (ElectroCorp, EMD1053). The dilatancy is obtained from the vertical displacement of the slider: A second inductive sensor, at rest in the laboratory referential, monitors the distance to a metallic target which endows the slider, which consists in a thin (5 mm) PMMA plate ($76 \times 53$, $53 \times 51$, or $53 \times 24$ mm$^2$). The granular material consists in spherical glass beads (Matrasur Corp.) sieved in order to obtain the three following mean diameters $d = (100 \pm 11)$, $(215 \pm 20)$ and $(451 \pm  40)$ $\mu$m, with a relative standard deviation almost independant of the characteristic grain size. The intersticial fluid consists in distilled water, water and sugar mixtures (viscosity $\eta $ ranging from $1$ to $76$ mPa.s), or Rhodorsil silicon oil (Rhodorsil, viscosity $\eta $ ranging from $71$ mPa.s to $500$ mPa.s). All viscosities were measured, in addition, using an Ubbelohde viscosimeter. The thickness of the granular bed ($6.0$ mm) is always larger than ten bead diameters in order to make sure that the sheared zone is not limited by the bottom of the container and, thus, that edge effects are not at stake [@Kudrolli]. Finally, the contact of the slider with the granular layer is insured by gluing a layer of the largest beads (451 $\mu$m) onto the lower surface. We checked, for a layer of 215 $\mu$m-in-diameter beads, that the friction coefficient is independent of the size of the beads in the glued layer as long as it remains larger than that of the beads in the granular bed (table \[tab.1\], Top).

  -------------------- ------------------ ----------------- -----------------
     $\o$ ($\mu $m)           100                215               451
   $\mu _d$ ($\mu $m)   $0.33 \pm  0.02$   $0.38 \pm 0.02$   $0.37 \pm 0.02$
       S (mm$^2$)        $53 \times 24$    $53 \times 51$    $76 \times 53$
   $\mu _d$ ($\mu $m)   $0.38 \pm  0.02$   $0.42 \pm 0.02$   $0.38 \pm 0.02$
  -------------------- ------------------ ----------------- -----------------

  : []{data-label="tab.1"}

![[]{data-label="fig.FM"}](divoux_fig2.eps){height="6.5cm"}

*Friction coefficient.* - In our experimental conditions, the value of the spring constant $k$ is chosen so as to observe the continuous motion of the slider in the whole accessible range of the driving velocity $V_s$. After a transient regime, the frictional force reaches a steady-state value which is observed to scale up with the slider mass $m$ (Fig. \[fig.FM\]), provided that the buoyancy force is taken into account, and to be independant of the slider velocity $V_s$ (inset, Fig. \[fig.FM\]). In addition, we checked that $\mu$ does not significantly depend on the slider surface-area or aspect-ratio (Table \[tab.1\], Bottom), as already known for the dry case [@GeLo]. We repeated the procedure for different bead-diameter ($d$ from $100$ $\mu $m to $450$ $\mu $m) and fluid viscosity ($\eta$ from $1$ mPa.s to $500$ mPa.s). We found out that $\mu$ neither depends on $\eta$ nor on $d$ in the whole experimental range. In order to encompass those two results and the independence on the slider velocity $V_s$, we report $\mu$ as a function of the Reynolds number $Re \equiv \rho\,d\,V_s/\eta $, where $\rho$ stands for the fluid density. We estimate $\mu  = 0.38\pm 0.03$ for $10^{-5} \leq I \leq 5.10^{-3}$, which nicely supplement the data reported for the free-surface-flow configuration in [@Cassar] that limited to $I\geq 4.10^{-3}$.

![ []{data-label="fig.mueff"}](divoux_fig3.eps){height="6.5cm"}

*Layer dilation.* - As we shall see, in contrast to the frictional coefficient, the dilation of the granular layer is sensitive to the grain size. Experimentally, the moot point consists in obtaining a reproducible reference state. The chosen procedure is as follows: In order to obtain a well-defined state of compaction, we initially push the slider over a distance of approximately $10$ bead-diameters in the steady regime at a given velocity, henceforth denoted $V_{ref}$ (usually $8.8\, \mu$m.s$^{-1}$, excepted when specified). We then stop the translation stage and move it backwards until the spring goes back to its rest position without loosing contact with the slider, which remains at rest (contact loss could make the slider surf over the granular layer as we push it forth at large velocity, meaning above $40\,\mu $m.s$^{-1}$). We then immediately push the slider forwards at various driving velocities, $V_s$, over a few millimeters and monitor the vertical position of the plate. We observe that the total variation of the vertical position of the plate $\Delta h$, or total dilation, increases with the bead diameter $d$ and the velocity $V_s$. By constrast, $\Delta h$ does not significantly depend on the interstitial-fluid viscosity $\eta$ (Fig. \[fig.dilat\]). We checked that these latter measurements neither depend on the preparation of the granular layer (by varying the velocity of reference $V_{ref}$), nor on the slider mass $m$ (Fig. \[fig.dilat\], inset).

![ []{data-label="fig.dilat"}](divoux_fig4.eps){height="6.5cm"}

*Discussion.* - The dependence of the total dilation on the velocity and on the bead diameter can be accounted by the two following ingredients: First, we can guess that, due to the steric interaction between the grains (solid contact between grains or hydrodynamical interaction), the local shear-stress $\sigma_s$ induces a local normal-stress $\sigma_n = \alpha(\phi)\,\sigma_s$. In a first approximation, the coefficient $\alpha$, which describes a geometrical property, depends only on the volume fraction of the grains, $\phi$, and not on the shear rate, $\dot\gamma$. The assumption is correct, at least for a dense suspension in the limit of small $\dot\gamma$ [@huang05]. We point out that, from this local relation between $\sigma_n$ and $\sigma_s$, we recover the apparent friction law, $F = \mu \,m g$ where $\mu = 1/\alpha$, independant on the shear rate $\dot\gamma$ or on the surface area $S$ of the slider provided that $\alpha$ does not significantly depend on $\phi$ [@huang05]. Second, we assume, as already proposed by Bocquet [*et al*]{} [@Bocquet], that the rheological behavior of the immersed granular-material can be accounted, in addition, by $\sigma_s = \eta(\phi) \,\dot\gamma$, where the effective viscosity $\eta$ diverges algebraically as a function of $\phi$ near a critical volume fraction $\phi_c$: $\eta  = {\eta _0}/{\left(1-\phi/\phi _c\right)^{\beta}}$. At this point, assuming that the total dilation $\Delta h$ is mainly due to the dilation of a constant number $N$ of layers underneath the slider and linearizing the velocity profile in this region where the dilation is the larger, we write $\dot\gamma = V_s/(Nd)$ and get the following scaling law: $$\Delta h \propto {V_s}^{1/\beta}{d^{(\beta-1)/\beta}}
\label{scaling}$$ The interpolation of the experimental data reported in Fig. \[fig.dilat\] leads to $\beta = 2.9 \pm 0.3$, which compares quantitatively with the values reported in [@Bocquet] and references therein.

Thus, our experimental measurements are in agreement with the conclusions of references [@Cassar] and [@Bocquet]. However, the discrepancy with the values issued in [@Losert; @Geminard2] and [@Kudrolli] remains unexplained and deserves to be discussed. Few physical origins can be taken responsible for such a dispersion related in the literature. Among them, in the case of spherical glass-beads, the polydispersity of the batches and the properties of the beads surface, especially its roughness. On the one hand, using unsieved samples (polydispersity about 25%), we measured significantly higher values $\mu \simeq 0.47\pm 0.02$ of the friction coefficient, which provides us with a rough estimate of the polydispersity effects. For all the data reported above, we use sieved samples, the standard deviation in the diameter being about $10\%$ as in references [@Losert; @Geminard2; @Kudrolli], so that we estimate that the polydispersity is not enough for explaining the observed discrepancy. On the other hand, only a few observations, dealing with the influence of the surface properties on the effective friction coefficient, have been reported: It has been recently proven that the roughness can drastically alter the dynamic angles of repose for dry materials [@Ottino] and even be a motor for shear-induced segregation in immersed granular-materials [@Pouligny]. The beads used in our study and in [@Kudrolli; @Losert; @Geminard2] may not present the same surface properties. In order to test the influence of the bead surface, we altered $451 \,\mu$m diameter beads by immersing them for $30$ min in a $1.0$ mol.L$^{-1}$ sodium-carbonate solution [@Herve]. The beads were then thoroughly washed up using distilled water and $\mu$ immediatly assessed. We obtained $\mu = 0.30 \pm 0.01$ and, thus, a significant decrease of about 20% by changing only the surface properties, even in immersed granular matter. We are currently performing an extensive and careful study of the friction coefficient as a function of surface roughness of the beads.

As a conclusion, our measurements of the friction coefficient nicely supplement the results reported by Cassar [*et al*]{} [@Cassar] and we interpret our measurements of the dilatancy in the framework of the hydrodynamical model proposed by Bocquet *et al.* [@Bocquet]. The friction coefficient $\mu$ remains constant over a large range of $Re$ values because the dilatancy of the layer is a free parameter that adapts as the slurry is sheared at different velocities. In this regime, the effective friction-coefficient depends neither on the fluid viscosity nor on the bead diameter. However, we underscore that the properties of the grain surface play an important role in the rheological properties of immersed granular-matter and could be responsible for the dispersion of $\mu $ values encountered in the literature. Finally, we emphasize that measurements of $\mu $, defined from the mean value of the friction force in the steady-state regime, does not provide any piece of information about the grain- or fluid-characteristics. As an extension of this work, we are currently focusing on the fluctuations of the frictional force in the steady-state regime (inset, Fig. \[fig.mueff\]), from which we hope to recover a signature of the components of the slurry.

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