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Deposition of a particle-laden film on the inner wall of a tube
Deok-Hoon Jeong,1Anezka Kvasnickova,1Jean-Baptiste
Boutin,1David C´ebron,2and Alban Sauret1, ∗
1Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
2Universit´e Grenoble Alpes, CNRS, ISTerre, Grenoble, France
The withdrawal of a liquid or the translation of a liquid slug in a capillary tube leads to the
deposition of a thin film on the inner wall. When particles or contaminants are present in the liquid,
they deposit and contaminate the tube if the liquid film is sufficiently thick. In this article, we
experimentally investigate the condition under which particles are deposited during the air invasion
in a capillary tube initially filled with a dilute suspension. We show that the entrainment of particles
in the film is controlled by the ratio of the particle and the tube radii and the capillary number
associated with the front velocity. We also develop a model which suggests optimal operating
conditions to avoid contamination during withdrawal of a suspension. This deposition mechanism
can also be leveraged in coating processes by controlling the deposition of particles on the inner
walls of channels.
∗asauret@ucsb.edu
arXiv:2010.15992v1 [cond-mat.soft] 29 Oct 2020
2
I. INTRODUCTION
Multiphase flows involving liquid phases, gas, and solid particles are ubiquitous in the dispensing
of liquid, coating of tubings, and flows in porous media [1–4]. The presence of particles in porous
media is especially important since it can lead to clogging events [5]. The liquid initially filling the
tube or the pores is commonly displaced by a non-miscible fluid, e.g., air [6, 7]. As an invading fluid
pushes the other fluid forward in a tube, this latter leaves behind a thin liquid layer or a series of
drops depending on fluid properties and velocity [8–11]. The situation of a slug of wetting liquid
pushed by an immiscible fluid has been considered in various studies, because of its relevance in
liquid dispensing and coating of tubing [8, 12–14]. When the gravitational and inertial forces are
negligible, the thickness of the liquid film h0deposited on the inner wall of a capillary of radius Rby
a Newtonian fluid of dynamic viscosity µand surface tension γfor a front velocity Uis governed by
the competition between viscous and surface tension forces, captured through the capillary number
Ca = µ U/γ [see, e.g., 8, 15]. The film thickness h0in a capillary tube has initially been predicted
by Bretherton in the limit Ca 1: h0/R = 1.34Ca2/3[16]. The evolution of the film thickness for
a broader range of parameters has been obtained through experiments and simulations [see, e.g.,
15]. In particular, the thickness of the film resulting from air invasion in a capillary filled with a
Newtonian liquid is captured by the empirical relation [8]:
h0
R=1.34Ca2/3
1+3.35 Ca2/3.(1)
Different extensions of this model have considered curved [17] and non-cylindrical channels [18]. For
a partially wetting fluid, the contact line motion also plays a role in determining the deposition
patterns [10, 19]. Most past studies have considered one fluid pushed by a second immiscible fluid.
The influence of particles on the deposition and the composition of the thin film coating the inner
wall of the capillary remains elusive. This configuration is nevertheless relevant to model the flow of
suspensions in porous media and contamination of tubings. A description of the thin film formation
is required to account for the influence of the particles dispersed in the liquid. However, because the
film thickness and the diameter of the particles can be of the same order of magnitude, the particles
deform the air/liquid interface and may modify the interfacial dynamics.
Previous works with particulate suspensions have shown that fluid-fluid interfaces can reduce the
length scale of the flow to less than a few particle diameters [20]. This situation occurs between
two fluid-fluid interfaces, for instance, during the pinch-off of suspension, which is strongly modified
by the presence of particles [21–23], and during the spreading and fragmentation of suspension
sheets [24]. When extruding a suspension from a nozzle, the ligament of liquid thins out and
eventually pinches off to generate a droplet due to interfacial effects. In tubings and porous media,
the situation becomes even more complex because of the presence of solid surfaces. This type of
particle confinement is at play during the drainage of suspension films [25], impact of a suspension
drop on a substrate [26, 27], the dip coating of suspensions [28–32], viscous fingering [28, 29],
and bubble rise in confined suspensions [33]. The displacement of the particles is controlled by
the capillary force, the drag force exerted by the fluid, and the friction on the solid substrate. The
complexity of such a situation was recently considered by Yu et al. [34], who investigated the coating
of the air/liquid interface of a long gas bubble translating in a mixture of glycerol and particles
exhibiting a finite contact angle. Yu et al. also reported the possible separation of a bidisperse
suspension by size using the motion of a confined bubble [35]. The influence of the coupling between
particles and interfacial dynamics has also recently been considered in the dip coating configuration,
where a substrate is withdrawn from a particulate suspension [36, 37]. In this situation, it has been
shown that particles dispersed in a wetting fluid are entrained in the coating film when the thickness
at the stagnation point is larger than a certain fraction of the particle diameter [38–41]. These
experiments have shown that particles up to six times larger than the liquid film can contaminate
the withdrawn surface [39]. Besides, this criterion was also found to be observed with biological
microorganisms. The flow topology associated with the dip-coating configuration shares common
3
features with the flow of a slug of liquid in a tube and the withdrawal of a liquid from a circular
capillary. In particular, both configurations involve the presence of a stagnation point that governs
the thickness of the coating film. Therefore, a similar condition for particle entrainment based on the
particle size and the film thickness at the stagnation can be expected. To the best of our knowledge,
no accurate determination of this threshold, which controls the possible entrainment of particles in
a coating film on the wall of a capillary tube, has been obtained.
In this article, we report the deposition of particles during the invasion of air in a capillary tube
initially filled with a dilute suspension. We predict the entrainment threshold of particles in the film
and show that this situation shares common features with the dip-coating configuration, suggesting
that particle entrainment is universal in the formation of thin-films governed by a stagnation point.
II. EXPERIMENTAL METHODS
In our experiments, we initially fill a cylindrical glass capillary tube open to the atmosphere on the
other side. The dilute suspension of non-Brownian particles is then withdrawn at a constant velocity
U, leaving a liquid film of thickness h0on the inner wall of the tube [Fig. 1(a)]. The experiments are
performed in capillary tubes (borosilicate glass from Vitrocom) of inner radii R= [500,750,1000] µm
placed vertically. Between experiments, the capillary tubes were thoroughly cleaned with Isopropanol
(IPA), acetone, and DI water and properly dried with an air gun. The suspension consists of spherical
polystyrene (PS) particles (Dynoseeds TS from Microbeads) dispersed in silicone oil (µ= 0.12 Pa.s,
γ'22 ±2 mN.m−1), which perfectly wets the capillary tube and the particles. We used three
different particle sizes and measured the size distribution of each batch through image analysis. The
different particle radii used in this study are a= 40 ±3.5µm, a= 72 ±4µm and a= 125 ±3µm.
The density of the particles was measured by mixing them with a mixture of DI water and Sodium
Chloride (NaCl) until we reach a close density match. For the particles used here, the density is
found to be ρP= 1056 ±2kgm−3. The dilute suspensions are prepared using a precision scale
(Ohaus, PX series), and the particles are dispersed in the silicone oil using a mechanical stirrer
(Badger Air-Brush Paint Mixer). The density matching between the particles and the liquid allows
us to consider that the particles are neutrally buoyant over the timescale of an experiment. Small
volume fractions are considered, 0.32% < φ < 1%, so that collective effects between particles can
be neglected, and the viscosity is not significantly modified [42, 43]. The Reynolds number is small
Re 1, and inertial effects can be neglected.
We connect one end of the capillary tube to a syringe filled with the dilute suspension. The other
end of the capillary tube is open to the atmosphere. The suspension is first injected in the capillary
and then withdrawn at a constant flow rate Qusing a syringe pump. Withdrawing the suspension,
instead of injecting compressible air, allows us to reach a constant velocity of the air-liquid front
quickly. The velocity of the air-liquid interface Uis directly related to the withdrawal flow rate
through the relation U=Q/ π R2and is also measured by image processing using ImageJ. To
visualize the motion of the suspension, the air-liquid meniscus, and the particles deposited on the
wall of the capillary, we use a backlight LED Panel (Phlox). The motion and the final morphology
of the coating film are recorded with a DSLR camera (Nikon D5300) and, when needed, with a high-
speed camera (Phantom VEO 710L) equipped with a macro-lens (Nikkor 200mm) and microscopic
lens (Mitutoyo). The curved surface of the cylindrical glass capillary tube leads to optical distortion,
and does not allow clear visualization of the liquid film and of the particles deposited on the inner
wall. Therefore, we inserted the cylindrical capillary tube in a square capillary, and injected glycerol
between the capillaries as its refractive index matches the index of the borosilicate glass. This method
ensured that the optical distortions due to the curved walls of the inner capillary are avoided [10].
The threshold velocity for particle entrainment in the coating film is determined as the average of
the largest value of the air/liquid front velocity where no isolated particles are visible in the film
and the smaller value of the air/liquid front velocity where individual particles are entrained.
An example of the experiment is shown in Fig. 1(b), where air invades the capillary tube in
the presence of particles. We observe that (i) the particle reaches the meniscus, (ii) deforms the
4
2R
2a
U
h0
Capillary
tube
U
U
Air
Liquid
Particle
Syringe pump
(a) (b)
FIG. 1. (a) Schematic of the experimental setup. (b) Time sequence (top to bottom) of the entrainement
of a particle of radius a= 125 µm in the liquid film for U= 4.18 mm.s−1in a tube of radius R= 0.75 mm
(Movie: Supplemental Material). The dashed lines indicate the inner walls of the tube. Scale bar is 400 µm.
liquid/air interface, and (iii) is entrained in the coating film whereas its diameter is larger than the
film thickness. After being deposited, the particle strongly deforms the film and does not move
anymore, coating the inner wall of the tube. This example shows that particle of diameter larger
than the film thickness can be entrained contrary to the situation of two rigid boundaries where
sieving effects would occur [5, 44, 45].
III. RESULTS
A. Coating of Newtonian fluids
We first performed the experiments without particles for two silicone oils of different viscosities in
a capillary tube of radius R= 750 µm. The experimental data, reported in figure 2, are captured by
the Taylor’s law [Eq. (1)] in the entire range of capillary numbers considered here [8, 15]. Besides, the
prediction given by h0/R = 1.34Ca2/3, valid in the limit of small capillary numbers Ca = µ U/γ 1,
agrees with the Taylor’s law, within around 10%, for Ca .10−2. We shall see later that for the
experimental parameters considered here the entrainment of particles dispersed in the liquid occurs
for capillary numbers Ca .10−2, thus the Bretherton’s law, h0/R = 1.34Ca2/3, is expected to give
a good prediction and will be used in the following.
B. Deposition threshold of particles in the coating film
The no-deposition and deposition regimes are illustrated in Fig. 3(a). At low front velocity U,
no particles are visible in the film. Particles start to be entrained in the film beyond a threshold
velocity U∗. The particles are randomly dispersed, and the number of particles per surface area
increases with the velocity. The threshold velocity U∗depends both on the radius of the particles
aand the radius of the tube R[Fig. 3(b)]. The situation observed here is reminiscent of the results
5
1
0
FIG. 2. Rescaled thickness h0/R of the coated film on the wall of the capillary tube when varying the
capillary number Ca for µ= 0.12 Pa.s (blue symbols) and µ= 0.24 Pa.s (red symbols). The dashed line and
the solid line represent the theoretical expression h0/R = 1.34Ca2/3and the empirical Taylor’s law given by
Eq. (1), respectively.
2a
(b)
a(μm)
72 125
U(mm/s)
3.0
0.75
1.5
(a)
FIG. 3. (a) Examples of film and particles deposited for increasing velocity (bottom to top) and two particle
radii, 72 µm and 125 µm, in a tube of radius R= 750 µm. Scale bar is 500 µm. (b) Threshold front velocity
U∗beyond which particles are entrained in the coating film as a function of the particle radius afor tube of
radii R= 0.5,0.75, and 1mm.
reported for dip coating [39]. Besides, for every configuration considered here, particles are able to
squeeze into films thinner than the particle diameter.
Examples of the time-evolution of the position of a particle are reported in Fig. 4(a) and 4(b)
for the no-deposition and deposition regimes, respectively. When a particle is not entrained in the
film, it first approaches the interface before recirculating in the bulk [inset of Fig. 4(a)]. In this
regime, all particles follow similar dynamics. The entrainment case exhibits different dynamics, as
illustrated in Fig. 4(b). The particle first approaches the interface in a similar way, but the velocity
of the particle remains smaller than the front velocity. The particle is then entrained in the coating
film where it reaches a zero velocity in the frame of reference of the laboratory, i.e., and the particle
6
(a) (b)
FIG. 4. Particle dynamics (R= 0.75 mm, a= 125 µm) in the (a) no deposition regime for U= 2.07 mm.s−1
and (b) deposition regime for U= 3.45 mm.s−1. The main panels show the time evolution of the position of
the front (solid line) and one particle (circles). Top right insets report the evolution of the rescaled absolute
velocity of the front (solid line) and the particle (circles). The bottom right insets show the corresponding
experimental observations. The small arrows indicate the particle and the scale bars are 500 µm (Movies:
Supplemental Material)
is deposited on the inner wall of the capillary [inset of Fig. 4(b)]. Once deposited on the tube inner
wall, the particle is trapped between the wall and the air/liquid interface so that it does not move
anymore and remains deposited.
These two distinct behaviors suggest that the presence of a stagnation point on the air/liquid
front is a key parameter. Indeed, the streamlines ending at this location separate the region where
the fluid flows into the film and a recirculation region within the bulk. Therefore, we expect that
the thickness h∗at the stagnation point is the relevant parameter that controls the entrainment of
particles in the film [38, 39].
IV. DISCUSSION
A. Theoretical thickness at the stagnation point
We consider the air invasion in a cylindrical tube in the limit Ca 1 leading to the deposition of a
film of uniform thickness h0on the wall. The thickness h0also corresponds to the thickness observed
far from the front and rear menisci during the translation of a long bubble [16]. In this limit, the
lengthscale `, associated with the meniscus, and h0, associated with the liquid film thickness far
from the meniscus, satisfy h0/` = Ca1/31 and |dh/dx| 1 [16]. Besides, hR, so that locally
the situation is 2D and we use the cartesian coordinates system (x,y), where xis along the capillary
and yis oriented inward. The interface is given by h(x, t), and the front is translating at the velocity
U=Uex. The Reynolds number remains small and we use the Stokes’ equations in the lubrication
analysis:
ux+vy= 0 (2a)
0 = −px+µ uyy (2b)
0 = −py(2c)
On the wall of the capillary tube, the velocity field satisfies no-slip boundary conditions, so that
in the reference frame moving with the air/liquid front u(x, 0, t) = −Uand v(x, 0, t) = 0. The
7
boundary conditions at the free surface at the leading order in Ca are ∂u/∂y|y=h= 0 and p=−γ κ,
where κ=∂xxh+ 1/R '∂xx h. Equation 2(c) shows that the pressure field is only a function of x.
Integrating equation 2(b) twice with respect to y, and using the boundary conditions leads to:
u(x, y) = −γ
µ∂xxxhy2
2−h y−U, (3)
To simplify this expression, we consider the classical lubrication equation [see e.g., 46]
ht+γ
3µ∂xh3∂xxxh= 0.(4)
In the reference frame moving with the front at the velocity U, the solution is of the form h(x, t) =
h(x−Ut). The previous equation can be integrated with respect to xtogether with the boundary
conditions h(x→ −∞) = h0and ∂xxxh(x→ −∞) = 0 so that
h2∂xxxh=3µ U
γ1−h0
h,(5)
which corresponds to the canonical Landau–Levich–Derjaguin–Bretherton equation. The velocity of
the interface is obtained using Eq. (3) and Eq. (5) and by setting y=h(x):
us(x) = u(x, y =h) = U3
21−h0
h−1.(6)
The surface velocity changes sign along the interface at least once since far from the meniscus, i.e.,
x→ −∞,h(x) = h0so that us=−U < 0, and for x= 0, i.e., at the meniscus, h0/h ∼0 so that
us=U/2>0. The thickness at the stagnation point h∗corresponds to the point where the surface
velocity vanishes, leading to
h∗=C h0,where C= 3.(7)
This result is similar to the dip coating of a plate [39] and a fiber [41]. Note that this analytical
results applies, a priori, only when the Bretherton law is valid, i.e., Ca 1 and hR.
B. Numerical simulation
We performed numerical simulations of the fluid flow to characterize the prefactor Cin the range
of Ca, where particles are experimentally entrained. The model relies on the method developed by
Balestra et al. [15] and uses a laminar two phase-flow approach with moving mesh, where the fluid-
fluid interface is resolved in each step by an Arbitrary Lagrangian-Eulerian (ALE) method. The
initial velocity of the outer liquid is imposed as U∞.Udthen denotes the average velocity on the
surface of the droplet. The capillary number associated with the droplet is given by Ca = Udµ/γ,
where γis the surface tension at the interface of the droplet and µis the viscosity of the surrounding
liquid. The ratio between the inner and outer viscosity, λ=µin/µ, was fixed at 0.01 since it has been
shown that this value of λgives the same flow profile and film thickness as those obtained at smaller
values of λ[15]. The ratio of densities does not affect the results and is thus set to 1. The model
converges to a stationary solution when the droplet reaches its equilibrium shape and moves with a
steady velocity Ud. The moving mesh helps to avoid large deformations of the mesh associated with
the moving droplet. Furthermore, the problem is solved in the moving frame of reference attached
to the droplet, the droplet hence practically does not change its position. The inlet in the model
is imposed as a laminar Poiseuille flow, with mean velocity U∞−Ud. The velocity at the walls is
imposed as −Ud, and the velocity of the bubble is determined and recalculated at each time step.
We benchmarked the code by ensuring that the numerical simulation recovered the experimental
8
Recirculation
region
S*
Entrainment
region
(a) (b)
FIG. 5. (a) Examples of streamlines and flow field (colormap: |u|/U) obtained numerically for Ca = 10−4,
Ca = 10−3and Ca = 2 ×10−2(from left to right). (b) Zoom on the stagnation point S∗. The red dashed
line separate the recirculating region where the fluid flow back to the bulk and the film region where the
liquid is entrained in the coating film.
results of Aussillous & Qu´er´e for C a ≤10−1[8]. More details on the numerical procedure can be
found in Ref. [15].
Examples of streamlines observed numerically are shown in Fig. 5(a) for various capillary numbers.
The numerical results report that the main features of the flow remain similar, but the thickness of
the liquid film on the wall of the capillary increases when increasing the capillary number. Fig. 5(b)
shows a zoom in the region surrounding the stagnation point S∗that delimitates the region where
the fluid flows into the film and a recirculation region within the bulk.
We measure the thickness h0in the uniform film region. The stagnation point corresponds to
the location at the interface where the surface velocity vanishes allowing us to measure h∗. The
dimensionless thicknesses h0/R and h∗/R are reported in Fig. 6. The simulations agree with the
theoretical expression (7). Furthermore, the numerical results demonstrate that the coefficient C= 3
obtained analytically can also be used with the Taylor’s law given by Eq. (1) for a better prediction
of h∗for Ca ∼10−2so that:
h∗
R=4.02 Ca2/3
1+3.35 Ca2/3.(8)
C. Entrainment threshold
Fig. 7(a) reports that the threshold capillary number Ca∗for particle entrainment increases with
(a/R)2. The larger the particle is, the larger the coating film must be, and thus the associated
capillary number, to entrain the particle. A particle is entrained if the thickness at the stagnation
point is larger than a fraction of the particle diameter 2aleading to the condition h∗≥α a, where
1≤α≤2. The particle can be entrained only if it follows the streamlines entering the coating film,
corresponding to the condition αmin = 1. Besides, if the particle does not deform the air/liquid
interface, no capillary force is exerted on the particle, which will always be entrained in the film, so
that an upper bound is αmax = 2. Assuming that an isolated particle does not significantly modify
the flow topology and using Eq. (8), we obtain the capillary threshold value for particle entrainment:
Ca∗=α
4.023/2a
R3/2
.(9)
9
0
0
FIG. 6. Evolution of h0/R and h∗/R with Ca. The squares are the numerical results, the solid red line is
h0/R = 1.34Ca2/3, and the solid blue line corresponds to h∗/R = 3 h0/R. The red dashed line is given by
Eq. (1), and the blue dashed line by Eq. (8).
Present study
0 50 100 150
0
15
30
45
60
75
90
105
120
135
150
a
*
th
(a) (b)
FIG. 7. (a) Threshold capillary number Ca∗as a function of (a/R)2for different radii of the particles and of
the tubes. Inset: Comparison with the entrainment thresholds obtained for the dip coating of a plate [39],
a thin fiber [41] and the present experiments. In both figures the solid line is Eq. (9) with α= 1.15. (b)
Thickness at the stagnation point calculated theoretically (knowing the experimental parameters) allowing
a particle to be entrained in the coating film as a function of the radius of the particle a. The dashed line
shows the linear law: h∗
th =a. Inset: Entrainement of a 125µm particle in the coating film (Ca = 6.1×10−2)
illustrating that an entrained particle can be smaller than the liquid film thickness owing to the deformation
of the air/liquid interface. The scale bar is 200 µm.
Fig. 7(a) shows that all the experiments performed collapse on a master curve given by Eq. (9),
where α= 1.15 ±0.1. The coupling of the liquid/air interface and the particle is complex. However,
we can provide some rationalization of the value of αby considering the force acting on the particle
near the stagnation point that controls its entrainment into the coating film.
10
β
Air
Front
meniscus
Coating
lm h0
Capillary tube
Liquid
FIG. 8. Schematic of the forces exerted on a particle confined in the liquid at the stagnation point.
D. Force acting on the particle near the stagnation point
The entrainment of the particle is initially controlled by the passive advection of the particle
that follows the streamlines. Once the particle approaches the stagnation point, the experimental
observations show that the particle strongly deforms the air/liquid interface. The particle is assumed
to be in contact with the substrate, owing to its surface roughness of order 100 nm [26, 47]. Therefore,
the entrainment of a particle is governed by the competition between the interfacial force, the viscous
drag, and the friction on the substrate. The three phases situation (particle, air, liquid) leads to a
complex interplay, and we try here to estimate the forces acting on the particle. The experimental
results show that the entrainment threshold occurs for a thickness at the stagnation point h∗of the
same order as the particle radius a[Fig. 7(b)]. Therefore, we consider the force acting on a particle
of radius alocated at the stagnation point of thickness h∗∼a. We further assume that far from
the particle, the interface has a shape similar to the situation without particle.
A schematic of a particle fully wetted by a liquid of thickness h∗around the stagnation point
is shown in Fig. 8. Below, we detail the amplitude and orientation of the different forces to
estimate their influence on particle entrainment. We consider the situation in the frame of reference
moving with the front. Besides, close to the entrainment threshold, the velocity of the particle
nearly vanished at the stagnation point before being able to be entrained in the coating film. The
experimental parameters imposed that inertial effects are negligible since Re 1.
An estimate of the viscous drag acting on the particle can be obtained by considering the average
viscous stress acting on the particle: µ U /(2 h∗) [26]. Furthermore, the viscous stress acts over half
of the particle since h∗∼aso that the total drag is of order
FD∼µ U
2h∗
π a2
2∼µ π U a
4.(10)
The drag force acts along the x-axis and acts as a driving force to entrain the particle (see Fig. 8).
Since the particle is totally wetted by the liquid (contact angle: θ= 0), the capillary force is
acting downard, pushing the particle against the wall of the capillary tube. The amplitude of the
capillary force is given by (see, e.g., [48]):
Fγ= 2 π a γ sin φsin(θ−φ),(11)
where ais the radius of the particle, γthe interfacial tension, φthe filling angle and here θ= 0 since
the silicone oil perfectly wets the particle. For a liquid film of thickness h∼a, the absolute value of
the amplitude of the capillary force reduced to Fγ= 2 π a γ sin2φ. The capillary force is maximum
for φ=π/2, corresponding to h∼a, so that an order of magnitude of the capillary force acting on
11
the particle is
Fγ∼2π a γ (12)
As illustrated in Fig. 8 the capillary force can be decomposed in its ycomponent that pushes the
particle toward the surface of the capillary tube
Fγ|y∼ −2π a γ cos β(13)
and an axial force acting to keep the particle in the liquid reservoir (in the direction of the front):
Fγ|x∼2π a γ sin β(14)
The angle βcomes from the inclination of the air/liquid interface between the front and the region
of uniform coating thickness. Note that we have neglected the radial curvature of the capillary since
both h∗and aare very small compared to R. The slope at the interface close to the stagnation
point is obtained by linearizing the Landau-Levich-Derjaguin-Bretherton (LLDB) equation for the
thickness [38, 49]:
h(x) = h0+ (h∗−h0) exp −(x−x∗)
h0/(3 Ca1/3),(15)
As a result, an approximate solution of the slope is tan β= dh(x∗)/dx∼ −3 Ca1/3. In the range of
parameters considered here leading to the entrainment of the particles Ca ∼10−3−10−2, so that
β∼15o−35o.
The resistance of the particle motion when the front is advancing and trapping of the particle
at the stagnation point occurs because the particle becomes in physical contact with the glass
substrate [26]. The particle is thus subject to a dynamic friction FT=µdFN, where µd'0.3 is
an estimate of the dynamic friction coefficient between the immersed polystyrene particle and the
glass substrate [50] and FNdenotes the normal force confining the particle against the substrate,
i.e. the ycomponent of the capillary force given by Eq. (13). Therefore, the dynamic friction force,
oriented along x < 0 and opposed to the motion of the front, is
FT∼2µdπ a γ cos β(16)
We can now estimate the order of the magnitude of the two opposite forces induced by the
deformation of the interface by the particles. The x-component of the capillary force tends to keep
the particle within the bulk liquid, whereas the y-component confine the particle and leads to the
friction force that favors the particle entrainment during the translation of the air/liquid front. Using
Eqs. (14) and (16) together with the expression of β, leads to an order of magnitude of the ratio of
these forces:
Fγ|x
FT
=tan β
µd
∼3 Ca1/3
µd
,(17)
For the order of magnitude of capillary number leading to capillary entrainment in this study,
Ca ∼10−3−10−2, we found that this ratio is of order unity (between 0.8 and 1.85). Therefore, the
effects of the capillary force induced by the deformation of the interface mostly balance each other.
Using the order of magnitude of the experimental parameters used in this study, we can estimate
the drag force and the friction force that also contribute to entrain the particle in the coating film
and the capillary force along xthat prevents the deposition of the particle in the coating film. An
order of magnitude of the ratio of these two opposite effects is given by
FD+FT
Fγ|x
∼Ca
8+µd
3 Ca1/3∼µd
3 Ca1/3.(18)
12
The two components of the capillary forces mainly balance each other owing to the friction force
induced by the contact of the particle on the wall. The main limitation to entrain the particle in
the film is thus the possibility for a particle to follow the streamlines leading to the coating flow,
corresponding to the condition α∼1 as observed experimentally.
The value of αis also in quantitative agreement with previous experiments on dip-coating of
plates and thin fibers [39, 41] as reported in the inset of Fig. 7 (note that the because of the flat
geometry, the relevant parameter for the plate configuration is (2a/`c)2to account for the difference
in geometry). Therefore, the entrainment of particles in a coating film governed by a stagnation
point is controlled by a universal mechanism for dip coating, coating of the inner wall of a tube,
and is expected to hold for a confined air bubble in a perfectly wetting fluid. The translation of a
long bubble beyond the entrainment threshold would lead to similar results, except that particles
will first be trapped in the liquid film and then released in the liquid at the rear of the bubble. In
particular, contrary to the recent experiments of Yu et al. [51], no attachment of particles at the
air/water interface of the bubble would be observed because the liquid fully wets the particles.
We should emphasize that we have considered diluted suspension in this work so that particles
can be considered as isolated [39]. For larger volume fractions of the suspension, the entrainment
of clusters may be observed as well as the accumulation of particles near the meniscus. We should
emphasize that we have never observed clogging of the tube for the perfectly wetting fluid considered
here, but we have observed, from time to time, the entrainment of some clusters near the entrainment
threshold.
V. CONCLUSION
In this article, we have shown that the liquid coating the inner wall of a capillary tube exhibits
different regimes in the presence of particles: liquid only and deposition of particles. We demon-
strated that a particle is entrained in the film even if its diameter is larger than the film thickness.
Considering the thickness at the stagnation point, the experimental results have been quantitatively
rationalized. In particular, we have provided some physical insights to rationalize the order of mag-
nitude of the parameter α. A similar physical mechanism underlies the entrainment of particles as
observed in both this configuration and in dip coating [38, 39]. This result comes from the common
features between this configuration and the dip-coating of a substrate, the most important of which
is the presence of a stagnation point. Therefore, preventing the contamination of substrate by par-
ticles and biological microorganisms using this strategy, as reported for the dip coating system [39],
is also valid in tubings and can provide guidelines for dispensing of suspensions and contaminated
liquid in tubes. These results are also relevant to environmental processes involving the transport
of small particles in confined geometries, such as the dissemination of colloid and microorganisms
in porous media [52] and provide guidelines to improve the coating of tubings by suspensions [53].
This filtering mechanism could also be an ingredient to explain the increase of bubble rise speed
observed recently in confined suspension [33]. The influence of the wettability of the particle can
further refine the value of the prefactor and is the topic of an ongoing study.
ACKNOWLEDGMENTS
This material is based upon work supported by the National Science Foundation under NSF
CAREER Program Award CBET No. 1944844 and by the ACS Petroleum Research Fund 60108-
DNI9. We thank F. Gallaire and L. Keiser for their help with the numerical model and B. Dincau
13
for a careful reading of the manuscript.
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