Particle Laden Flows

Matt Hin, Kaiwen Huang, Shreyas Kumar, Gilberto Urdaneta

August 9, 2013

Abstract

We investigate the dynamics of gravity-driven mono- and bidisperse suspensionsconsisting of silicone oil and negatively buoyant particles of different densities exper-imentally. The well-mixed slurry mixtures spread down an inclined plane, exhibitingdistinct flow patterns arising from competition of gravitational sedimentation and vary-ing shear forces. We confirm the results of previous studies where, an initially well-mixed flow evolves towards either a settled regime in which the particles settle to thesubstrate, or a ridged regime in which the particles aggregate at the front of the flow.Our results show that the addition of a second particle species induces or prevents thesetting of particles due to the mismatch in particle densities. We show that the latterdepends strongly on the relative amount of heavy to light particles used. We compareour experimental results to spreading relations as applied to particle-free, thin-fluidfilms. Further, we investigate the evolution of particle concentrations in each of thetwo regimes by using fluorescent particle beads and compare our results to existingtheoretical models.

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Contents

1 Introduction and Motivation 3

2 Bidisperse Bifurcation 4

3 Prefactor Characterization 73.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3.1 Prefactor vs Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3.2 Comparison to theory . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3.3 Prefactor vs Lambda . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Shocks and Flow Fronts 134.1 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Fluorescent Concentration Imaging 17

6 Conclusion and Future Work 20

List of Figures

1 Setup Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Flow Progression, Settled . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Flow Progression, Ridged . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Regime Diagram with Screenshots . . . . . . . . . . . . . . . . . . . . . . . 65 Average Front Position vs Time1/3 . . . . . . . . . . . . . . . . . . . . . . . 86 Prefactor vs Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Prefactor vs Angle - Comparison to Theory . . . . . . . . . . . . . . . . . . 108 Prefactor vs λ - Fluid Front . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Prefactor vs λ - Particle Front . . . . . . . . . . . . . . . . . . . . . . . . . 1210 Numerical Solutions to the PDE Model . . . . . . . . . . . . . . . . . . . . 1311 Ridged Flow Front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1512 Settled Flow Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613 Ridged Intensity Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1814 Settled Intensity Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915 Intensity Correlation Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

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1 Introduction and Motivation

Particle-laden flows are common in nature; from oil spills to mudslides, these kinds ofslurries can be found by only taking a look around us. Given these examples, we can seewhy we would like to better understand the dynamics of these flows: if their behavior iswell understood, then we could contain oil spills, or prevent the damage mudslides causewhen these disasters happen. Slurry flows do not only show up in ecological disasters, thetheory behind particle-laden flows is also used in the mining industry to separate particlesof different species based on their densities, and in the food industry when it is desiredto deposit a layer of some particulate on top of foodstuff. The dynamics of particle-ladenfluids will find applications in a wide range of operations.

Much research has been done on clear fluid flows, as well as on single-species particle-laden flows. In this project, we expand on the results on bidisperse flows by furthercharacterizing the regimes which each flow falls into depending on parameters such as angle,total particle concentration, and light-to-heavy bead ratio. We compare the experimentalresults to new developments in theory, and we try to determine the value and behavior ofthe proportionality constant for x = ct1/3 flows. Furthermore, we redesign our experimentalsetup to track the concentration of particles down the incline, by dying the particles withfluorescent paint and using backlights to make them fluoresce and thus making it possibleto find regions of high concentration of particles.

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2 Bidisperse Bifurcation

In this portion of the project, we are interested in experimentally characterizing the α−φ−λphase space of the slurry flow system. We do this by performing runs varying one parameterat a time and characterizing the regime of the run. Throughout this project however, wefocus on a total particle concentration of λ = 0.4. Fig. 1 shows a schematic representationof our setup. We adjust the angle α between 15◦ and 50◦ at 5◦ intervals for this project.For every run, we prepare a mixture which we pour in the reservoir and then open thegate, recording at least the first 65cm of displacement of the flow down the track, which,depending on the parameters for the specific run, usually takes between 8 to 12 minutes.

The materials we use to prepare the slurry mixtures for the run are PDMS oil, glassbeads, and ceramic beads. The PDMS oil we use has a density of 0.97 g/mL and a viscosityof 1000 cSt, while the glass beads have a density of 2.475 g/mL, and the ceramic beads havea density of 3.8 g/mL, thus both particles being negatively-buoyant in the oil. The beadsare roughly of the same diameter as size difference effects is not the focus of this project,with the glass beads ranging from 0.250 cm to 0.425 cm in diameter, and the ceramicbeads raging from 0.2 cm to 0.4 cm in diameter. In order to more easily differentiateamong the bead species, we use food coloring dyes to dye the glass beads red, and theceramic beads blue. For the cases which we know will fall in the settled regime (basedon previous experiments), we also dye the oil using concentrated cadmium sulfide (PY37)yellow dye at a concentration of 0.9158 grams of dye per 100 mL of oil. Fig. 2 and Fig.3show sample run progressions for settled and ridged cases, respectively.

Figure 1: A sketch of our setup. Our setup consists of a 90cm long by 14cm wide trackwith an adjustable angle α from 5◦ to 70◦. At the top of the track we have a 14.5cm ×7.5cm × 11.5cm reservoir in which we pour the prepared mixture and then open the gateto let the fluid fall down the incline. We record the runs using a camera that captures atleast the first 65cm of the flow down the track.

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Figure 2: An example run for the settled case. Notice the run starts in the well-mixedregime. As the run progresses, fingering instabilities appear, and in the last frame we cansee that the particles settle out of the fluid flow and onto the bottom of the track. Noticealso the lighter red particles separating from the heavier blue particles towards the end ofthe flow.

Figure 3: An example run for the ridged case. Again we see the run starts in the well-mixed case, but it now falls into the ridged regime. We do not see fluid fingers, but ratherparticles concentrating towards the front of the flow and forming ridged fingers.

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We performed more experiments which allowed us to fill out the table in Fig.4, wherewe can more easily see the effect that varying one of the parameters (α or λ) has on theflow. Notice that lower angles and lower light beads concentrations tend to induce thesettling regime, whereas higher angles and higher light beads concentration tend to inducethe ridged regime in the flow. We then expanded the experiments to include angles of 25◦,35◦, and 45◦ for α for λ = 0, 0.25, 0.5, 0.75, 1 and φ = 0.40. These additional runs willbe compared with the previous runs as well as the theoretical results obtained by JeffreyWong in his disperse ODE model.

Figure 4: A screenshot was taken for every run with an angle from 20◦ to 50◦ in 10◦

intervals, at a volume fraction of φ = 0.4, and for λ from 0 to 100, in 25 unit increments. Forthe settled cases the PDMS oil was dyed yellow to better visualize the fingering instabilitiesand make image processing easier. Each image was taken when the front of the flow wasat a displacement of 65 cm down the track.

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3 Prefactor Characterization

3.1 Motivation

In this portion of the project, we are interested in experimentally characterizing the pref-actor of the slurry flows, presuming the following self-similarity solution with prefactorK:

x = Kt1/3.

Note that for settled flows we expect the fluid front of the slurry to behave according toHuppert’s law. In addition, for ridged monodisperse flows, we expect the fluid to have thisone-third proportionality. As for ridged bidisperse flows and the particle fronts of settledflows, we will be assuming that the flows evolve according to the one-third proportionalityand will try to characterize the prefactor in that matter. Note that we will be saving thefront data in addition to any trend analysis, so as to allow for future analyses to have adataset to work with.

3.2 Methodology

We initially hoped to use a log-log plot to extract both the slope and the prefactor. Weran into issues though, since in reality, the data collection process (with the camera) wasnot perfectly synchronized with the start of the run. The camera, for the most part, wasfocused about 10 cm down from the gate - this allowed us to capture the important partof the flow - once it got to steady state. Given that the camera can only capture a portionof the track, we did not want to waste some of the frame with transient behavior that wasa direct result of the initial condition. What this meant though, was that there was aninitial constant offset that differed from run to run. So, our relationship would be of theform

x = Kt1/3 + x0

Taking the logarithm on both sides

log x = logKt1/3 + x0

If we were working in the appropriate coordinate system, x0 = 0, and we would be able toextract the relevant data. However, we have no way of identifying the relevant coordinatesystem. Hence, we adopted an approach use by Ward et al. and in the JFM paper - weassumed that the flow (both the fluid and particle fronts) obeyed a t1/3 proportionality.Furthermore, we defined a uniform x0 and t0 as the point where the flow reached the 15cm mark on the incline. While not ideal, it gave us a replicable method. The best methodwould probably be to fit the front to a function of the form

x = Ktα + x0

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Figure 5: A plot of the average particle front position (y-axis) versus time1/3 (x -axis)for 15 degrees and 100 gsb. A linear fit to determine the prefactor is overlaid in black.Note that the initial transience is ignored. The slope of the black line is taken to be theprefactor.

where K, α and x0 are left as free parameters. This would help confirm the exponent aswell - this is however complicated to implement in MATLAB and is left for future work.

Instead we perform a linear fit on of the average front position against the time raisedto the 1/3. The slope of this fit is used as the prefactor. Note that only the last 200 framesare used in the fit. This ensures that the initial transience is not taken into account in thelinear fit. Figure 5 shows an example of the average front position vs t1/3 with the linearfit overlayed.

The slope of this linear fit is then regarded as the prefactor K.

3.3 Results

The system we are dealing with, atleast experimentally, has three key adjustable parameters- α, the incline angle, λ, the ratio of the lighter particle to the total particle volume andφ, the initial particle concentration. Note that for a given run, these three parameters areconstants. We do not yet have enough data to investigate the relationship between theprefactor and φ (we have data only at φ = 0.4 and φ = 0.2). However, this is not the casewith α and λ. Our preliminary observations are detailed below.

3.3.1 Prefactor vs Angle

Figure 6 shows the data for the prefactor vs angle with λ = 0 and φ = 0.4. We see amonotonic trend, where the prefactor increases with increasing α. We only focus on thefluid front obtained in settled runs, as the fluid front has the richest theory to compare to.

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Figure 6: A plot of the experimentally determined prefactor (y-axis) versus the inclineangle (x-axis). The trend is monotonic and appears to be linear. These prefactors are thatof the fluid front in the settled runs ranging from 15 to 30 degrees for 0 GSB and a volumefraction of 0.4.

This points on this graph are the average of two runs. To present accurate error bars, wewould need more repetitions of these runs. This trend is not as ’nice’ for other λ values(especially bidisperse runs). However, we believe that repeated runs should produce amonotonic trend in the prefactor for other λ values as well.

3.3.2 Comparison to theory

There are two existing theories that predict the dynamics of fronts. The first was developedby Huppert in the 1980s. Huppert used force arguments to derive a self-similarity solutionfor a fluid (no particles) flowing down an incline. This solution assumes an infinite widefluid body. Huppert’s prediction is as follows

xfront =

(9A2 sinα

4ν

)1/3

t1/3 (1)

where A is the axial cross-sectional area,ν is the viscosity. We calculated the prefactorsusing Huppert’s law to see if they agree with the experimental prefacors we got. From theequation, we can see the α and ν are given for each run and we still need to figure out thecross section area A. We want to find out the two parameters for A which are the values

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Figure 7: A plot of the experimentally and theoretically determined prefactor (y-axis)versus the incline angle (x-axis). The experimental prefactors are in red, while the blackrepresents Huppert’s prediction and the green represents Murisic et al.’s prediction. Theseprefactors are that of the fluid front in the settled runs ranging from 15 to 30 degrees for0 GSB and a volume fraction of 0.4.

of x and z. We find x by taking the point at 20 centimeters so that we can ge rid of theintial transience as we do in data analysis; also we add to 20 centimeters an additionallength in the reservoir which varies slightly in each run. The measurement for z is notso feasible as for x. What we do is when each flow progress to 20 centimeters, we insertthe stick perpendicularly to the track plane to get the height of the flow, namely. Thez values we get for 0 GSB at angles from 15 to 30 degree presents very small deviationfrom 0.2centimeters. However, the x values have bigger difference among every run thanz. This indicates the total volume for each run is not ideally constant. While we couldexpect the the settled fluid to obey Huppert’s law at long times, our track is not longenough to observe this behavior. Hence, we also compare the experimental prefactors tothe a second, more recent theory that predicts the position of slurry fronts. Developed byMurisic et al. in 2013 [4], this is a generalization of Huppert’s result. The key differencesare that the different densities of the particles and fluid as well as a new term µ thataccounts for changes in viscosity. Murisic et al’s theoretical form is presented below

xfront =

((ρl + ρp(1 − φ)

9A2 sinα

4µ(φ)

)1/3

t1/3 (2)

Figure 7 presents the experimental results from Figure 6 (in blue) along with bothMurisic et al’s (in green) and Huppert’s(in red) predictions. Unfortunately these do notline up very well, and we are not sure why. One reason is definitely the difficulty inestimating the cross sectional area A. We do not have an effective way of obtaining a

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Figure 8: A plot of the experimentally determined prefactor (y-axis) versus the percentageof GSB in the slurry particles (x-axis). The prefactors at 20 degrees in red, while the blackrepresents 30 degrees. These prefactors are that of the fluid front in the settled runs avolume fraction of 0.4.

height profile, (right now we measure the height at an arbitrary point before fingering setsin). Hence, we are assuming a uniform height profile and uniform front position, both ofwhich are incorrect. An improvement on this would be to account for a non-uniform frontposition and non-uniform height profile (perhaps with a laser line).

3.3.3 Prefactor vs Lambda

The second relationship we investigated was the prefactor versus the particle ratio, λ.Again, we only focussed on the settled runs here - both the fluid front (presented in Figure8 ) and the particle fronts (presented in Figure 9). We do not yet have results for theridged runs as our code does not work very well on the ridged runs. As is seen in Figure8 and Figure 9 initial data suggests that the prefactor is unaffected by the ratio of theparticles. To say this conclusively, we would need error bars, which would require multiplerepetitions of each run.

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Figure 9: A plot of the experimentally determined prefactor (y-axis) versus the percentageof GSB in the slurry particles (x-axis). The prefactors at 20 degrees in red, while the blackrepresents 30 degrees. These prefactors are that of the particle front in the settled runs avolume fraction of 0.4.

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4 Shocks and Flow Fronts

(a) Settled; α = 30, λ0 = 0.25, φ0 = 0.40

(b) Ridged; α = 40, λ0 = 0.25, φ0 = 0.40

Figure 10: Numerical solutions of the PDE model for late time flows in the two regimes:settled and ridged. Note that for the settled flow there are three shocks that correspondto the prediction of three fronts in the flow, corresponding to each particle front and fluidfront. For the ridged flow, we find that there are two shocks, corresponding to two particlefronts.

4.1 Shocks

The dynamics of particle-laden flows have been modeled before for monodisperse cases,however we seek an analogous model for the bidisperse case. We begin with the ODE forthe normal direction and PDE for the axial direction obtained for the monodisperse case[3]. The solutions for these differential equations together form a complete characterizationof the monodisperse flow. By modifying the 2-dimensional ODE to account for the secondparticle species, we can construct an analogous 3-dimensional system in terms of the shearstress σ, total particle concentration φ, and relative concentration of the heavier particle

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species λ = φs1/φ:

σ′ = −(1 − φρ(λ)), (3)

φ′ =1

σ

(φ+ ρ(λ)

(φ2 − 2

9Kc tanα(1 − φ)

))(1 + 2

Kv −Kc

Kc

φ

φm − φ

)−1, (4)

λ′ =2

9Kc(ρs1 − ρs2) cotα

1

σDtr(φ)

φmφm − φ

λ(1 − λ), (5)

where Kv and Kc are viscosity and collision constants (obtained experimentally), φm is themaximal particle concentration, ρs1 and ρs2 are nondimensionalized particle densities, ρ(λ)is the effective slurry density, and Dtr = 0.1φ is the tracer diffusivity. By transformingthe solutions to this ODE, we are then able to construct fluid velocity u(z), and particleconcentrations φ1(z) and φ2(z). We can model the dynamics of the bidisperse case assumingconstant volume. As such we are able to construct a PDE in terms of the conserved massesof the fluid and particles:

yt + F(y)x = 0,

where y = [h, hφ1, hφ2], h is the height of the flow, φ1 is the concentration of particlespecies 1, and φ2 is the concentration of particle species 2. The advection flux F is definedas:

F =

h3∫ 10 u ds

h3∫ 10 φ1u ds

h3∫ 10 φ2u ds

=

[h3f(φ1 + φ2)h3g(φ1) h

3j(φ2)

]Since this PDE constitutes an advection conservation law, we can solve this PDE with anupwind solver. We find that the theory predicts shocks forming in the flow, as seen inFig. 10.

4.2 Fronts

We found that these shocks evident in the model manifest in the experimental runs asfronts. By tracking the blue levels of a narrow region of interest, we were able to identifythe fronts in the experiment that the theory predicted for both the settled and ridgedregimes. For a flow in the settled regime as seen in Fig. 12, we confirm the theory thatthree fronts form. Using the blue levels in our images, we find that the heavier blue particlesform a front further upstream and much later than the lighter red particles. Note that bothparticle fronts were upstream from the front of the fluid. As for a flow in the ridged regimeas seen in Fig. 11, we find that a single front forms at the front of the flow, where theparticles seem to be suspended in the fluid.

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Figure 11: The front of a ridged flow as captured by the blue levels of the specified region ofinterest. For a ridged flow, a single inflection point is visible in the intensity plot, indicatingthat a single visible particle front is formed. Note that in the intensity plot, the blue levelrises to a high amplitude noise region due to the white color of the track.

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Figure 12: The fronts of a settled flow as captured by the blue levels of the specified regionof interest. For the settled flow, two inflection points are visible in the intensity plot,corresponding to the two particle fronts that form. Note that in the intensity plot, theblue level rises to a high amplitude noise region due to the white color of the track.

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5 Fluorescent Concentration Imaging

To image the particle concentration and characterize its evolution in space and time, weturn to using fluorescent beads in an attempt to correlate observed brightness and particleconcentration. We altered the existing experimental apparatus to include a UV blacklightand prepared fluorescent beads. To prepare the fluorescent beads, we dyed transparentGSB-5 glass beads with fluorescent red acrylic paint. Our current batch requires a mixtureof 900g of GSB-5 glass beads with 10mL of red paint and 30mL of water. To more aptlyimage the fluorescence, we also adjusted the camera settings to have 60 frames-per-secondshutter speed, aperture size of f5.6, and light sensitivity of ISO800. We also made sure toauto focus the camera with the room lights on, before switching to manual focus for runimages with only the blacklights.By creating a histogram over a narrow ribbon across an image at a particular instance, weare able to capture the behavior of the particle concentration as a function of position. Asseen in Fig. 13, for a flow in the ridged regime we find that the intensity upstream is quitelow compared to the intensity downstream. In addition, as we follow the flow downstream,the intensity grows to a relatively large value before jumping to a large peak right at thefront of the flow.By performing a similar set of analyses, we find that for a settled run, intensity differencebetween upstream and downstream is not as dramatic. In fact as seen in Fig. 14, the profileof the intensity as a function of position appears to be similar to that of the ridged case,minus the leading edge peak. The intensity grows towards the leading edge and plateaus,before dropping to zero. Note that for the settled regime, the position of this drop inintensity is not at the front of the flow, as fluid does extend past this position.In order to correlate the intensity obtained from the images to a single particle concentra-tion, we performed a series of calibration experiments. These consisted of pouring a knownvolume of fluorescent beads suspended in PDMS oil into a Petri dish and then taking asingle image of the mixture under a UV blacklight. The observed intensities from theseimages were then normalized by the area of the Petri dish and plotted against its corre-sponding particle concentration, as set by the volume fraction of the beads in the mixture.Unfortunately, we were not able to viably correlate these intensities to actual particle con-centrations. We were able to determine a correlation curve between the two quantities,however the curve quickly becomes multivalued as seen in Fig. 15. The range for whichthe curve is single-valued and thus useful for the experiment is between intensities of 0 to1,000,000, which on our calibration curve corresponds to concentrations of φ = 0 ∼ 0.15.Further work must be done to determine a viable experimental set-up so as to correlatefluorescent intensity with physical particle concentration.

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Figure 13: A plot of observed intensity vs. position in the x-direction for a flow in theridged regime with corresponding image captured. Note that we find that the intensitysteadily grows to a region of a modestly high particle concentration before peaking at thefront of the flow and dropping off to zero once more.

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Figure 14: A plot of observed intensity vs. position in the x-direction for a flow in thesettled regime with corresponding image captured. Note that we find that the intensitysteadily grows to a region of high particle concentration before dropping off to zero, wherethe particles settle. Also note that the fluid does extend beyond this region of high con-centration as evident in the bright image.

Figure 15: A plot of Particle concentration versus observed intensity. We see that thecurve that correlates the two variables becomes multivalued beyond φ = 0.1 (Dashed linenot a best-fit curve, drawn for visible auxiliary purposes).

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6 Conclusion and Future Work

We explored bidisperse case with several parameters, particle concentration φ, angle α andGSB to Ceramic ratio λ. We also compare our experimental prefactors for average frontposition with two theoretical statements, Huppert’s law for clear fluid front and Murisic’sstatement for average front position. We found that the our experimental prefactors deviatefrom the two theoretical predictions with two different scalings and display the same trendas the experiments. We also imaged the second particle front of the heavier species andconfirmed particle tend towards the downstream front. Lastly, we observed two distinctparticle behaviors in fluorescent experiments. We will try fixing the scaling between ex-perimental and theoretical pre factors and continue characterizing fluid and particle frontsin different regimes and dispersity. We also seek to determine a viable single-valued corre-lation curve. As for the fluorescent experiments, the particle flows are not uniform enoughso an improvement in experimental set-up for that is also desired.

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References

[1] T. Ward, C. Wey, R. Glidden, A. E. Hosoi, A. L. Bertozzi. Experimental study ofgravitation effects in the flow of a particle-laden thin film on an inclined plane. Physicsof Fluids 083305 (2009).

[2] H. E. Huppert. Flow and instability of a viscous current down a slope. Nature vol. 300,(1982).

[3] N. Murisic, J. Ho, V. Hu, P. Latterman, T. Koch, K. Lin, M. Mata, A. L. Bertozzi.Particle-laden viscous thin-film flows on an incline: Experiments compared with a theorybased on shear-induced migration and particle settling. Physica D (2011).

[4] N. Murisic, B. Pausader, D. Peschka, A. L. Bertozzi. Dynamics of particle settling andresuspension in viscous liquid films. J. Fl. Mech., vol. 717, pp. 203-231 (2013).

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