Viscous friction acting on a solid disk falling in confined fluid: lessons for the scaling analysis

We fill a viscous liquid in a vertically stood cell of millimeter thickness, called the Hele-Shaw cell, and insert a disk in the liquid whose thickness is smaller than the cell thickness. The disk starts falling in the liquid due to gravity opposed by viscous friction. We focus on the case in which lubricating films formed in the gap between the cell surface and the disk surface are thinner than the disk thickness. As a result, we find an apparent scaling regime for the falling velocity of a disk, in which the thickness of the lubricating film characterizes the dynamics. We further show that the apparent scaling regime is explained simply as a result of competition of two scaling regimes, elucidating the physics of the viscous friction. The present study is thus relevant to fundamental issues and applications in various fields in which small-scale physics in the flow at low Reynolds numbers is essential, such as microfluidics, bioconvection, and active matter. The simple scenario for explaining an apparent scaling law demonstrated in the present study would be useful in diverse fields, considering that the generality and strength of scaling analysis in science and that simple arguments usually lead to a few different scaling laws for a given problem.

Scaling analysis has been a powerful tool in various fields in science beyond physics [1,2] and applied mathematics [3], which include biology [4,5], engineering [6], and astronomy [7].A lot of scaling laws have been established with a well defined scaling exponent, with successful explanations.However, it also happens frequently that, although an exponent is clearly observed, a simple scaling argument (or an appropriate reorganization group analysis [1,2]) predicts an exponent slightly deviating from the observed one, as in, for example, the metabolic-rate (R) vs body-mass (M ) relation in biology [8] and the luminosity (L) vs mass (M ) relation in astronomy [9]: in the former (latter) experimentally R ≃ M α (L ≃ M β ) with α ≈ 0.75 (β ≈ 3.7) while a simple dimensional analysis gives α = 5/8 (β = 3).Such apparent scaling laws have been interpreted in case-specific and complex theories, leaving room for debate.Here, we focus on a simple viscous dynamics and demonstrate an apparent scaling law, which is simply explained as a result of competition of two scaling regimes.This simple scenario would be useful for understanding apparent scaling laws in diverse fields in science.
In addition to the significance in the general context of the scaling analysis, the present study could be fundamentally important for various current topics such as microfluidics [10], promising for applications in chemistry, biology, medicine and pharmaceutical industry, and bioconvection [11][12][13], which is created via swimming of a large number of small objects and has received attention in active matter and biology.The central issue is the flow at low Reynolds numbers governed by viscous friction due to small scales at which most liquids including water are highly viscous.
Relevant classic studies on viscous friction are motion of a bubble in a capillary tube [14] and in a Hele-Shaw cell [15].Our focus is on the latter.A number of studies have been performed in the Hele-Shaw geometry, with a main focus on a forced flow and/or with nearly horizontal geometries [16][17][18][19].Recently, we have highlighted the existence of lubricating film between the cell surface and bubble surface, using a vertically stood Hele-Shaw cell of millimeter thickness, filled with viscous liquid to confirm a number of scaling regimes for drag friction acting on fluids surrounded by another immiscible fluid [20][21][22][23].
Other groups have explored closely related issues, using cells with smaller scales [24,25] and comparing numerical results with experiment [26].
In this study, we explore a seemingly simpler case of drag friction acting on a solid disk in a Hele-Shaw cell of millimeter thickness, where similar lubricating films exist between the cell wall and disk surface.Our principal interest is in the case in which the thickness of the lubrication film is smaller than the disk thickness when the disk falling in the direction perpendicular to its axis.As far as we know, this case has not been explored in the literature, while it is closely related a study of transport of strongly confined disks under flow in microfluidic devices [27].As a result, we identified an apparent scaling regime, in which the observed scaling exponent for collapsed data deviates from a predicted exponent.We reveal that the deviation is simply the result of competition of two scaling regimes.In experiments, we filled a Hele-Shaw cell, stood vertically, with a silicone oil (polydimethylsiloxane, PDMS) as shown in Fig. 1 (a).We inserted a metal disk of radius R = 10 mm at the top of the cell with a zero initial speed.We recorded the ensuing falling motion of the metal with a camera after the disk began going down in the cell at a constant speed.The width and height of the cell was 90 and 160 mm, respectively.We checked the thickness D of the cell using a laser sensor (ZS-HLDS5, Omron) and its controller (ZS-HLDC11, Omron) to find D was in the range of 2.2 to 7 mm.The kinematic viscosity ν of PDMS was in the range of 1000 to 10000 cS, which corresponds to the range of the viscosity η = ρ 0 ν, 0.970 to 9.75 Pa•s (the density ρ 0 is in the range 970-975 kg/m 3 ).The metal disk is a stainless-steel SUS403 of density ρ = 7.70 g/cm 3 , which can be manipulated by a magnet placed on the cell surface.The thickness D 0 (< D) of the disk was either 1.88, 2.87, or 3.90 mm (these three thicknesses will be labeled as D 0 = 2, 3, and 4 mm, for convenience, in Fig. 1 to 3).
We used a digital camera (EX-F1, Casio) setting the time interval in the range of 1 to 1/60 second.The digital images were analyzed with a software, Image J.
As demonstrated in Fig. 1 (b), the velocity of falling motion of the disk reached a constant speed, which was reproducible at a given experimental parameter set, η, D 0 , and h (an error in velocity is typically less than 10 per cent).Here, h is the thickness of two viscous liquid films, each of which is sandwiched by an inner surface of cell plate and a surface of the disk.
Obtaining reproducible data as in Fig. 1 (b) is highly non-trivial.The main difficulty is to drop the disk in the cell with precisely maintaining the relation 2h = D − D 0 .The falling speed is expected to be maximized when this relation is satisfied with no tilting.
Thus, we created a gate by setting two plates of thickness comparable to h at the places indicated by the two dashed rectangles at the top of the cell in Fig. 1 (a) to help guarantee the condition at the entry.Even with this gate, we need to carefully insert the disk into the gate and collect the data in fast falling cases.In this way, we could obtain reproducible data as specified above.and we showed all the results in the plots.As a result, in some case data points are closely overlapped, which suggests a small size of errors in the measurements.
The viscosity dependence of the velocity in Fig. 2 indicates that the dynamics is governed by gravity opposed by viscosity.In fact, the inertia can be neglected in most (practically all, as revealed below) of the data.This is because, writing Reynolds number Re as ρ 0 V L/η with introducing a length scale L, the condition Re < 1 is equivalent to L < η/(ρV ), where the average and minimum of η/(ρV ) are 1.5 and 0.02 m, while L in the present case is either h, D 0 , D, or R, which are all smaller than even the minimum of η/(ρV ).
Theoretically, the change in gravitational energy per time is trivially given as As a first trial for comparison between theory and experiment, we assume the largest h-dissipation dominates over the other dissipations.Then, the balance of the change in gravitational energy with dissipation (the h-dissipation only) gives which can be expressed as with In Fig. 3 (a), we plot the renormalized velocity ηV /(∆ρgD 2 0 ) as a function of the renormalized thickness h/D 0 , in the light of Eq. ( 2), using all the data in Fig. 2, where the data described by Eq. (1) should collapse onto a master curve on this plot.This is what we observe in Fig. 3 (a): all the data in Fig. 2 (except the dataset with h ≳ D 0 ) collapse well  4) with α = 0.716.The slope of the fit (solid line) deviates from the slope of the dotted line, which corresponds to α = 1.(b) Successful fit by Eq. ( 6) with the same renormalized axes with (a).(c) The same successful fit by Eq. ( 6) but with a differently renormalized horizontal axis.
on a master curve.
However, the agreement is not perfect.As a result of numerical fitting, we find that the collapsed data are well described by the following expression with k 1 = 0.127 ± 0.001 and α = 0.716 ± 0.008 (in this fitting, the dataset with h ≳ D 0 is excluded).The exponent α deviates from the theoretical prediction α = 1.
This deviation cannot be explained by the slip length which has been discussed for polymer liquids [1,28].This is because a simple linear extrapolation of the data of each color in Fig. 2 (a) may intersect the horizontal axis at a value around h = −0.1 to −0.3 mm and this order of magnitude is too large for the slip length (in the present case, it is at most 10µm).
As a second trial, we include the second largest D-dissipation in addition to the hdissipation, since the first trial based on the h-dissipation only raised problems.Then, the new energy balance can be cast in the following form: where c 1 and c 2 are numerical coefficients (The two energy dissipations are added here, considering a simplest and physically plausible case).This leads to Eq. (2) with where d 1 = 1/c 1 and d 2 = c 2 /c 1 .
Equation ( 6) still predicts a data collapse when ηV /(∆ρgD 2 0 ) is plotted as a function of h/D 0 , which is truly observed in Fig. 3 (a).In Fig. 3 (b), the same plot is presented with the result of an excellent fitting by Eq. ( 2) with (6), where d 1 = 0.34±0.01 and d 2 = 4.90±0.26.Note that even the dataset with h ≳ D 0 is well fit by Eq. ( 6) although this set is excluded in the fitting.Since d 2 = c 2 /c 1 corresponds to the relative importance of the D-dissipation to the h-dissipation as seen in Eq. ( 5), the present fitting predicts that the two dissipations are comparable and the D-dissipation is larger than the h-dissipation.
However, the D-dissipation does not play a dominant role.This is clearly shown in Fig. 3 (c).To understand this, note that Eq. ( 2) with Eq. ( 6) can be expressed as when the D-dissipation dominates over the h-dissipation, i.e., if c 1 << c 2 .This means if the D-dissipation is dominant we should observe most of the collapsed data are on a line with slope one in Fig. 3 (c), which is not the case.We emphasize here that the dataset with h ≳ D 0 (square) in Fig. 3 (c) significantly strengthens our arguments for the D-dissipation, supporting the approach to a line with slope one as D/D 0 increases.Note that thanks to the relation D/D 0 = 1 + 2h/D 0 when the plot ηV /(∆ρgD 2 0 ) vs h/D 0 collapses on to a master curve, so does the plot ηV /(∆ρgD 2 0 ) vs D/D 0 .Further insight can be obtained if we note the following limiting forms for Eq. ( 6): These limiting behaviors can be observed in Fig. 3 (b) with the aide of the dotted line with slope one.In between there exists a quasi-straight-line region with data collapsed.Such a straight-line region, if exists, should possesses a slope smaller one, since the first coefficient 2) in Eq. ( 8) and, thus, the second slopeone region (x >> 1) is shifted below the first (x << 1), as can be confirmed in Fig. 3 (b).This is the reason we obtained α less than one in the above.Note that the dataset with h ≳ D 0 is consistent with this argument: all the data in Fig. 3 (b) are well on the fit in the region between the apparent scaling with slope α and the linear scaling with slope one for x >> 1 in Eq. ( 8), where the set with h ≳ D 0 is in the region rather close to the starting of the linear scaling.Further note that while the linear scaling for x << 1 corresponds to the real scaling for the h-dissipation, that for x >> 1 does not correspond to the real scaling for the D-dissipation.
For completeness, we explain the reason we can neglect the R-dissipation, although this dissipation is expected to be the smallest.We rewrite the balance in Eq. ( 5) with including the R-dissipation: , to obtain Eq. ( 2) with This form predicts that, if d 3 term cannot be neglected, the collapse onto a master curve cannot be observed in Fig. 3 (because the data includes those obtained for different D 0 although R is fixed), which is not true.Thus, we can judge that the remaining R-dissipation can be safely neglected from our argument to explain the collapsed data.
We may learn two lessons from the present analysis.(1) When a log-log plot is seen in a limited range, in principle, it tends to seem a straight line, and when the slope is fairly close to the one predicted from a simple dimensional analysis, one tends to overlook other important physical origin of the problem.(2) In such a case, or even when such an apparent scaling regime spans over a few or several decades, when the exponent deviates from the one predicted from a simple argument, it is worth considering competition of a few scaling regimes, as simply demonstrated in the above.
The present apparent scaling regime demonstrated in Fig. 3 (a) spans at least over one order of magnitude, but a limited range.According to the theory, our data happen to correspond to the ones in the crossover region in between the two asymptotic regimes defined in Eq. ( 8).Judging from the behavior of Eq. ( 8) shown in Fig. 3 (b), we need to further explore regions h/D 0 < 0.01 and > 1 to completely confirm our argument by explicitly establishing real scaling regimes originating from h-and D-dissipations (in addition to the apparent scaling regime).This implies nearly over four orders of magnitude for h/D 0 in total, which is an important but experimentally challenging future problem.Note here that the real scaling for D-dissipation is already reasonably well confirmed in Fig. 3 (c) as mentioned above with the help of the dataset with h ≳ D 0 .
We discuss here a semi-quantitative estimate for c 1 = 1/d 1 , which can be obtained in the following way.The velocity gradient could be precisely ηV /h (with assuming a simple Couette shear flow as in the film between two infinite plates separated by h) almost everywhere on the disk surface except near the edge in the small h limit.Then, in this limit, the total drag force might be obtained simply by multiplying this gradient by the area of both disk surfaces 2πR 2 .Balancing the force thus obtained with the gravitational force πR 2 D 0 ∆ρg, we could obtain a plausible value d 1 = 1/2 (this leads to a one-parameter fitting, resulting in d 2 = 9.25 ± 0.08).It is interesting this plausible value is not far from the previous value (d 1 = 0.34).However, the data for small h/D 0 in the present study slightly better support the previous fitting.This issue could be another motivation for a separate study for h/D 0 < 0.01 and > 1.
As for the coefficient c 2 , when not a solid disk but a fluid drop (viscosity η ′ ) moving in a Hele-Shaw cell filled with another viscous fluid (viscosity η), the formula V = (1/12)∆ρgD 2 /η is derived in the limit η ′ = 0 and D = D 0 in [17] from a result in [15], which has been confirmed different methods [29,30].While the present case corresponds η ′ >> η and D = D 0 + 2h, if the formula could be applied, it would predict c 2 = 12.This estimate is also not far from the value of c 2 obtained from the two-parameter fitting as well as the one-parameter fitting: in the former case, c 2 = d 2 /d 1 = 4.9/0.34 and in the latter case c 2 = 9.25/(1/2).
The present result can be appreciated in the context of a Stokes drag friction F for a solid disk moving in a direction perpendicular to the axis, i.e., in a direction of the disk plane, in a confined geometry specified by two parameters: a measure of confinement C = R/D and the aspect ratio of the disk A = D 0 /R.Under no confinement (C = 0) for a disk of zero thickness (A = 0), the drag friction was given by Oberbeck [31] as F O = 32ηV R/3.After the case of A = 0 and small C was studied [32,33], the case of C = 0 and A ̸ = 0 was considered [34].Then, the case of C ̸ = 0 and A ̸ = 0 was reported [35], where the case with h < D 0 was not studied, which is the main focus of the present study.In terms of the drag friction force The present falling disk problem is inherently different from the case in which the disk is replaced with a fluid "disk."Such a case is studied under no existence of surfactants [21], in which, the thickness of lubricating film h and the disk thickness (or, the shape of fluid drop) are dynamically determined and thus dependent on V , with the former governed by the law of Landau, Levich, and Derjaguin (LLD) [36,37].The drag force F thus found scaled with V 1/3 , in contrast to the present case, in which h is fixed by the (fixed) disk thickness and F scales simply with V .
More complex cases with rigid and mobile surfactants are relevant if fluid drops of various kind, such as armored bubbles or drops, gelified drops, bubbles and drops with surfactants, are utilized.Such cases have been explored mainly on a smaller length scale appropriate for microfluidics [38,39].While direct comparison of such studies with the present case may be irrelevant because of the inherent difference in how the thickness of the lubricating film is determined, the study of cases with surfactants on a larger cm scale as in the present study will be an important future problem.
In the present study, we investigated the falling velocity of a solid disk in viscous liquid in a confined space and the drag friction acting on the disk.We identified an apparent scaling regime, in which two scaling regimes cooperating.The present study provides a simple possibility of explaining an apparent scaling law, which would be useful to any physical scientists, considering the generality and strength of scaling analysis in science.In addition, fundamental understandings on fluid flow at low Reynolds numbers provided in the present study could be valid for small objects in less viscous fluid such as water.Accordingly, the present study are relevant to various fundamental issues and applications, for example, in microfluidics, bioconvection and active matter, in which viscous friction acting on small objects are highly important.

FIG. 1 :
FIG. 1: (a) Illustration of experiment.A metal disk of thickness D 0 is dropped in the cell of width D filled with a viscous oil of kinematic viscosity ν.The side view suggests the existence of two lubricating films of thickness h.The two dashed rectangles at the top of the cell on the left indicate the places at which plates (of thickness ≃ h) are set (see the text for the details).(b) Vertical position x vs time t.Results of three falling experiments are shown for each parameter set specified by (ν, D 0 , h). ν is given in cSt, while D 0 and h are given in mm.(See the text for the precise value D 0 )

FIG. 2 :
FIG. 2: V vs h for h < D 0 .(a) ν = 1000 cSt.(b) ν = 3000 cSt.(c) ν = 5000 and 10000 cSt.Colors and marks differentiate D 0 and ν, respectively.The labels for D 0 and ν are given in mm and cSt.The number of the data points shown here are 240 in total and they are obtained for 46 different parameter set (D 0 , η, h).(The precise values of D 0 are given in the text)

π∆ρgR 2 D
0 V with ∆ρ = ρ − ρ 0 .As for the viscous dissipation, we can think of three possibilities, corresponding to three viscous length scales, h, D, and R. The first one η(V /h) 2 R 2 h, which we call "the h-dissipation," describes dissipation inside the lubrication film of thickness h with its volume scaling as R 2 h.The second "D-dissipation" η(V /D) 2 R 2 D is associated with the viscous flow developed between the cell plates separated by the distance D in a volume scaling as R 2 D. The third "R-dissipation" η(V /R) 2 R 2 D corresponds to the dissipation with the viscous length scale R developed in a volume scaling as R 2 D. If we compare the three dissipations with their coefficients set to unity, the ratios of the D-and R-dissipations to the h-dissipation are h/D and hD/R 2 , respectively, both of which are less than unity in the present experiment.More precisely, the average and median of the ratio first h/D are 0.16 and 0.14, while those of the second ratio hD/R 2 are 0.039 and 0.018.

FIG. 3 :
FIG.3: Renormalized velocity V vs thickness h for the 240 data in Fig.2.(a) Fit by Eq. (4) with α = 0.716.The slope of the fit (solid line) deviates from the slope of the dotted line, which corresponds to α = 1.(b) Successful fit by Eq. (6) with the same renormalized axes with (a).(c) The same successful fit by Eq. (6) but with a differently renormalized horizontal axis.
the present study (0.05 ≲ h/D ≲ 0.3) predicts the following viscous drag and the drag coefficient: F = πηV R 2 (c 1 /h + c 2 /D) and K = (3πR/32)(c 1 /h + c 2 /D) with c 1 and c 2 given through the above values of d 1 and d 2 .Note that K represents the enhancement of the friction due to confinement and thus becomes significantly larger than one under a strong confinement in which R ≫ h and D. This implies that the falling velocity in the present confined case is (3π/4)(D 0 /R)/K times slower than the non-confined case.