Capillary rivulet rise in real-world corners

Journal Pre-proof Capillary Rivulet Rise in Real-World Corners Felix Gerlach, Jeanette Hussong, Ilia V Roisman, Cameron Tropea

PII:

S0927-7757(20)30123-0

DOI:

https://doi.org/10.1016/j.colsurfa.2020.124530

Reference:

COLSUA 124530

To appear in:

Colloids and Surfaces A: Physicochemical and Engineering Aspects

Received Date:

23 December 2019

Revised Date:

23 January 2020

Accepted Date:

30 January 2020

Please cite this article as: Felix Gerlach, Jeanette Hussong, Ilia V Roisman, Cameron Tropea, Capillary Rivulet Rise in Real-World Corners, (2020), doi: https://doi.org/10.1016/j.colsurfa.2020.124530

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier.

Graphical Abstract Capillary Rivulet Rise in Real-World Corners

Jo

ur n

al

Pr

e-

pr

oo

f

Felix Gerlach,Jeanette Hussong,Ilia V Roisman,Cameron Tropea

Highlights Capillary Rivulet Rise in Real-World Corners Felix Gerlach,Jeanette Hussong,Ilia V Roisman,Cameron Tropea • Modification of existing 𝑡1∕3 theory for practical dynamic capillary rise situations • Excellent agreement between theory and experiment

Jo

ur n

al

Pr

e-

pr

oo

f

• Retrospective explanation of previous experimental results found in literature

Capillary Rivulet Rise in Real-World Corners Felix Gerlach, Jeanette Hussong, Ilia V Roisman and Cameron Tropea∗ Institute of Fluid Mechanics and Aerodynamics, Technische Universität Darmstadt, Darmstadt, Germany

ARTICLE INFO

ABSTRACT

Keywords: capillary rise rivulet wetting

The capillary rise of liquid in a perfect corner wedge has been well studied in the past, yielding a 𝑡1∕3 rise behavior with no upper bound on the height of the rivulet. However, in real-world situations the wedge corner is often not exactly sharp, either because of roughness or fabrication imperfections. In the following study the behavior of capillary rivulet rise in rounded corners is examined experimentally and theoretically, yielding a deviation from the 𝑡1∕3 behavior and also a maximum rivulet height. The agreement between experiment and theory is very good.

Jo

ur n

al

f

oo

pr

Pr

The present study addresses dynamic capillary rise against gravity in inner corners, in which no inherent length scale is imposed, as would be the case in wicking applications or in vertical tubes. For such systems a 𝑡1∕3 time evolution for the rivulet height in the corner is predicted by the theory of [1] or [2] for small wedge angles. A more universal law for capillary rise in corners was presented in [3] in which quadratic and cubic corners were examined and in which the 𝑡1∕3 behavior was shown to be independent of the actual shape of the corner. While past studies have examined ideal corners, the novel feature of the present study is that the corner exhibits some finite curvature, more typical of real-world, rounded corners. Furthermore, the theory derived for this situation is not restricted to small opening angles, but only to the criteria for existence of an infinite rivulet rise, expressed as 𝜃 + 𝛼 < 90◦ [4], where 𝛼 is the half-angle of the corner and 𝜃 is the threephase contact angle. The latter is assumed to be constant everywhere. According to this criterion, for a 90◦ corner opening angle, an infinite rivulet rise should only occur for contact angles below 45◦ , which was numerically confirmed in [5]. Lopez de Ramos et al. [6] analyzed the influence of rounded corners on the rise in rectangular capillary tubes. They derived a maximum rise height of the rivulets in the corners which decreases with increasing curvature of the main meniscus inside the tube. For an open geometry like the one used in the present study, where an infinite radius of the main meniscus can be assumed, the predicted final rise height of Lopez de Ramos et al. becomes identical with the one derived in this study when making the assumption of the rivulet radius becoming equal to the corner radius (Eq. (6)). Lopez de Ramos et al. did not analyze the dynamics of the rivulet rise in a rounded corner, which is the novel contribution of the current study. Such corners have relevance for a large number of applications, such as patterned film coating or wetting of rough surfaces. Any experimental investigation of rivulet rise in verti-

cal corners will invariably exhibit some length scale arising from the finite size of the surfaces used to construct the corner. The conditions under which this length scale can be disregarded with regards to the rivulet dynamics have been recently investigated in [7]. This study has shown that interactions between spontaneous capillary rise in inner corners and outer corner geometries only occur for lengths between the corners one √ order smaller than the capillary length, defined as 𝑙𝜎 (= 𝜎∕𝜌𝑔), where 𝜎 is the surface tension, 𝜌 the density of the fluid and 𝑔 the gravitational acceleration. The actual geometry of a machined or constructed corner is seldom possible to measure exactly and furthermore, the effective inner curvature may not be uniform over the entire length of the corner in which the rivulet rises. Similar geometric uncertainties arise if the surfaces building the corner exhibit roughness. Also in the experimental part of this study these values are not known; however, the fitting of experimental observations to the theoretical rise dynamics allows one to estimate effective values of inner corner curvature. Similarly, data from past studies can be used to estimate to what extent past studies have approximated ideal corners or are more representative of real-world corners. This is achieved by examining their deviation from the ideal 𝑡1∕3 behavior.

e-

1. Introduction

∗ Corresponding

author

[email protected] (C. Tropea) ORCID (s): 0000-0002-1506-9655 (C. Tropea)

F Gerlach et al.: Preprint submitted to Elsevier

2. Materials and methods The current study uses the experimental setup published in [7] and thus, this setup will only be briefly described here. An electro-pneumatic linear drive is used to dip the sample into a liquid pool to start the spontaneous wetting. Front illumination allows tracking of the rivulet from the video recordings. The capillary rise is examined in the inner corner of a milled 1.4404 stainless steel sample as sketched in Fig. 1. Silicone oil (Polydimethylsiloxane, brand: ELBESIL SILIKONÖL B) is used as a wetting liquid, exhibiting a capillary length (𝑙𝜎 ) of approximately 1.48 mm. The sample also has two outer corners, separated from the inner corner by 15 mm and 30 mm respectively; however, the study [7] has shown that the rivulet rise in the inner corner is completely unaffected by the contact line behavior at the outer corners. A specific aim of the present experiments is to observe the capillary Page 1 of 7

Capillary rivulet rise

A-A z

xw

tip of a perfect rivulet

B

B

hperfect

apparent tip

g

w

h

rivulet z,t

R0

B-B

A

pr Rw

x

rivulet

wall

w

A

e-

z

Figure 2: Sketch of the liquid rivulet, definition of the main geometrical parameters and the coordinate system.

al

Pr

rise over very long times; hence, the sample is very long in height (100 mm). Stainless steel has been used to avoid any corrosion effects when cleaning the silicone oil from the surface between experiments. Three different types of silicone oil were used in order to vary the dynamic viscosity 𝜇 in the experiments. Silicone oils with 10, 20 and 50 cSt were used, which all exhibit a very low contact angle (𝜃 < 20◦ ) and a surface tension 𝜎 ranging from 20.2 to 20.8 mN/m. Only a narrow region 43 mm in height was filmed during the experiments, allowing a maximum recording time. This region is indicated with the dotted line in Fig. 1. The overall duration of the recordings was more than 108 minutes for each experiment. The rivulet rise for each of the three silicone oils was observed five times and the results were averaged. The standard deviations provide an indication of the repeatability.

oo

y

f

meniscus

Figure 1: The sample used for experiments, the lengths are not to scale. The dotted frame shows the observed edge of the sample.

3. Rivulet flow on a real surface

Jo

ur n

In the following section theoretical expressions for the rivulet rise in real-world corners will be derived. The flow in a corner is governed by surface tension, gravity and viscous stresses. Since the Reynolds number associated with this flow is much smaller than unity the inertial effects in the rivulet flow are very small and can be neglected. The shape of the rivulet in the corner is three dimensional. An exact analysis of such flow is therefore rather complicated. However, far from the liquid bulk and its main meniscus on the flat surfaces building the corner, the rivulet is thin and thus the long-wave approximation can be applied for the estimation of the thickness distribution and the evolution of the rivulet height. This approximation has been used in several previous studies, including [1, 2, 5, 7, 8]. In the present study the existing approaches to flow modeling in the rivulet have been generalized to account for a curvature of the corner, rather than assuming an exact 90◦ angle.

3.1. Stationary solution Consider a coordinate system with the 𝑧 coordinate running in the corner with the origin of coordinates at the ideal F Gerlach et al.: Preprint submitted to Elsevier

corner vertex and at the level of the bulk liquid, as sketched Fig. 2. The rivulet thickness is denoted 𝛿(𝑧, 𝑡), as defined in the cross-section B - B shown in Fig. 2. In the long-wave approximation, 𝜕𝛿∕𝜕𝑧 ≪ 1, the curvature of the free interface can be roughly estimated as √ 1 − 2 cos 𝜃 𝜕 2 𝛿 𝜅= − . (1) 𝛿(𝑧, 𝑡) 𝜕𝑧2

We consider only the flow at distances from the main meniscus much larger than the rivulet thickness. The second derivative of the rivulet thickness on 𝑧 near the main meniscus is determined by the meniscus thickness 𝜕 2 𝛿∕𝜕𝑧2 ∼ 1∕𝑅0 , where 𝑅0 is the characteristic radius of the meniscus, shown in Fig. 2, A - A. For a single corner, as in the present experiments, the value of the characteristic radius of the meniscus is determined by gravity and surface tension [9, 10]. Since no other length scale is given, the radius √𝑅0 is therefore comparable to the capillary length, 𝑅0 ∼ 𝜎∕𝜌𝑔. The expression for the curvature 𝜅 in this remote asymptotic solution can therefore be further simplified for the region where 𝛿 ≪ 𝑅0 𝜅≈−

𝐴 , 𝛿(𝑧, 𝑡)

𝐴=

√ 2 cos 𝜃 − 1.

(2)

Moreover, since the velocity component in the direction Page 2 of 7

Capillary rivulet rise

normal to the solid wall of the capillary is negligibly small, the pressure is uniform in the cross-section normal to the 𝑧axis and can be estimated using the Young-Laplace equation (3)

At large times the rivulet near the main meniscus approaches a static shape. This shape can be estimated by equating the expression for pressure (3) with the hydrostatic pressure, leading to an equation for the rivulet thickness in the form (4)

𝛿stat =

𝐴𝜎 , 𝜌𝑔𝑧

as

f

An approximate solution for the static meniscus is (5)

𝑡 → ∞.

where 𝐵 is the dimensionless friction coefficient determined by the shape of the rivulet cross-section and the indices denote partial derivatives. The evolution equation (8) is derived from the mass and axial momentum balance equations, accounting for the viscous stresses, gravity and capillary effects in the rivulet. At large times and large 𝑧 the solution for 𝛿 approaches the similarity solution far from the main meniscus/bulk: 𝛿(𝑧, 𝑡)

=

𝐿𝐹 (𝜉)(𝑡 − 𝑡0 )−1∕3 ,

(9a)

𝜉

=

(9b)

𝐿

=

𝐺

=

𝑊 − 𝐺𝑧(𝑡 − 𝑡0 )−1∕3 , ]1∕3 [ 𝐴𝐵𝜇𝜎 , 24𝜌2 𝑔 2 [ ]1∕3 8𝐵𝜌𝑔𝜇 , 3𝐴2 𝜎 2

(9c) (9d)

Pr

The solution for 𝛿stat is positive only if 𝜃 < 𝜋∕4, a result compatible with the criteria given in [4]. In a perfect corner the static rivulet height can be infinite. In the case of a real-world corner the wall curvature limits the height of the rivulet. Denote 𝑅𝑤 as the characteristic curvature radius of the wall in the corner, as shown in Fig. 2, B - B. It is obvious that this radius is the lower bound for the radius of curvature of the rivulet surface. The rivulet thickness 𝛿 cannot be smaller than the thickness 𝛿𝑤 associated with the characteristic curvature of the wall at the corner. The condition 𝑅𝑤 = |𝜅 −1 | yields, with the help of (2), the following expression for the maximum height of a rivulet in a real-world corner exhibiting an inner curvature of 𝑅𝑤 .

oo

𝜎𝐴 + 𝜌𝑔𝑧 = 0. 𝛿stat

The long-wave equation for the evolution of the rivulet thickness 𝛿(𝑧, 𝑡) in a perfect corner is obtained in [5] ( ) 4𝑔𝜌𝛿𝑧 𝛿 2 − 2𝐵𝜇𝛿𝑡 + 𝐴𝜎 𝛿𝑧𝑧 𝛿 + 2𝛿𝑧2 = 0, (8)

pr

−

3.2. Flow in a rivulet at large times and far from the corner

e-

𝑝 = 𝜎𝜅(𝑧).

For 𝜃 = 0 the maximum static rise height 𝑧max according to Eq. (6) is recovered, while it reduces to zero for a contact angle approaching 45◦ , which agrees well with the findings in [4]. The maximum rise height decreases by 5% for a contact angle of 9.8◦ .

𝜎 . 𝜌𝑔𝑅𝑤

(6)

al

𝑧max =

Jo

ur n

As an example, the maximum height 𝑧max for the values of 20 cSt silicone oil and a wall curvature 𝑅𝑤 = 20 𝜇m is approx. 110 mm, which is larger than the present measurement height of 43 mm and larger than the present sample itself. This expression for 𝑧max is in complete agreement with the result presented in [6]. In the derivation of Eq. (6) the radius of the rivulet was directly compared to the radius of the corner 𝑅𝑤 . In reality contact angles higher than zero lead to a bigger rivulet radius for a given rivulet thickness 𝛿. Since Eq. (3) has to be satisfied, rivulets exhibiting higher contact angles show smaller thicknesses 𝛿 at a given height, as theoretically shown by [7]. Due to this reduction of 𝛿, the rivulet thickness becomes zero before the radius of the rivulet matches the radius of the corner. Therefore, for higher contact angles an additional factor emerges which modifies Eq. (6) for a right-angle corner to become: √ ( ) 𝑡𝑎𝑛2 (𝜃) 2 1 − 𝑡𝑎𝑛 −1 2 (𝜃)+1 𝜎 . (7) 𝑧max = √ 𝜌𝑔𝑅𝑤 2−1 F Gerlach et al.: Preprint submitted to Elsevier

where 𝑊 is a constant determined from the boundary conditions far from the rivulet tip, 𝜉 is a similarity variable and 𝐹 (𝜉) is a dimensionless rivulet thickness. 𝐿 and 𝐺 are coefficients to simplify the expression of 𝛿(𝑧, 𝑡) and the dimensionless similarity variable 𝜉. The time shift 𝑡0 is associated with the time required for the creation of the main meniscus. The height ℎperfect of the tip of the rivulet in a perfect corner, corresponding to 𝜉 = 0 in (9b), is obtained in the form ℎperfect =

𝑊 (𝑡 − 𝑡0 )1∕3 . 𝐺

(10)

Function 𝐹 (𝜉) is a solution of the ordinary differential equation −𝐹𝜉 𝐹 2 + 𝐹𝜉𝜉 𝐹 + 𝐹 + 2𝐹𝜉2 − 𝑊 𝐹𝜉 + 𝜉𝐹𝜉 = 0

(11)

and subject to the boundary conditions 𝐹 (0) = 0,

𝐹 (𝑊 ) =

4 , 𝑊 −𝜉

(12)

since 𝜉 = 0 corresponds to the rivulet tip and the region 𝜉 → 𝑊 corresponds to the meniscus, whose shape at large times approaches the static solution (4). The value of the parameter 𝑊 is estimated from the computations of the ordinary differential equation (11). The boundary condition (12) is satisfied only when 𝑊 ≈ 3.1623. This value is obtained by the shooting method. Page 3 of 7

Capillary rivulet rise

In the neighborhood of the rivulet tip, where 𝜉 ≪ 1 and 𝐹 (𝜉) ≪ 1, the approximate solution is obtained by linearization of (11) 𝐹 (𝜉) =

𝑊 𝜉, 2

at

(13)

𝜉 → 0.

Expression (13) can be expressed in dimensional form with the help of (9a) (14)

2𝐴𝑅𝑤 (𝑡 − 𝑡0 )2∕3 . 𝑥𝑤 = 𝐺𝐿𝑊

(15)

ℎ = ℎperfect −𝑥𝑤 =

𝑊 (𝑡 − 𝑡0 )1∕3 2𝐴𝑅𝑤 (𝑡 − 𝑡0 )2∕3 − . (16) 𝐺 𝐺𝐿𝑊

𝑡 − 𝑡0 = 𝜏𝑇 , 3

𝐻

=

(17a)

𝐵𝜇𝜎𝑊 6

ur n

𝜂𝐻,

al

The expression (16) for the apparent rivulet height ℎ(𝑡) in a real-world corner can be written in the dimensionless form using (9) and (16) ℎ =

points. For this fitting 𝑡0 was set to zero because its value is negligibly small compared to the overall measurement time 𝑡max , as seen in Fig. 3. In the fitting routine the data points are weighted by the average time span between the previous and next data points to compensate for the non-uniform distribution of data points in time during the measurements.

4. Experimental results

The fitting procedure mentioned above, applied to the experimental results of each silicone oil, resulted in values of 𝑅𝑤 and 𝐵 as summarized in Table 1. Fitting to data from all experiments simultaneously led to a value for 𝑅𝑤 of approximately 23 𝜇m. Since 𝑅𝑤 is a geometric parameter of the sample, its empirically determined value should be constant, independent of which oil is used. Table 1 indicates that this is not exactly the case. On the other hand, the deviation found among the three oils is not large and moreover, the mean value of 𝑅𝑤 = 23 𝜇m is plausible for a sample manufactured by milling. One explanation for the deviations seen in Table 1 can be that the corner curvature is not exactly constant over the entire sample height and, since the liquids of different viscosity wet a different portion of the sample over the same time, this could lead to a variability of 𝑅𝑤 , determined over different total absolute observation times. Moreover, all three silicone oils have such a low contact angle that it cannot be measured with high accuracy with the available equipment. So it is not possible to detect whether the contact angle of the three silicone oils are slightly different, which would effect 𝐵 and through this the measured 𝑅𝑤 . The rivulet rise for the three silicone oils is shown graphically in Fig. 3, whereby each curve is the average over the five runs conducted for each oil. The vertical bars, placed only on selected values of rivulet height, indicate one standard deviation over the five runs (visible at the 50 cSt plot). The time and height have been made dimensionless using the factors given in Eq. 17b. It can be seen that after a short initial fast rise, the rise height exhibits the expected 𝑡1∕3 power law (dashed/dotted

Pr

In fact, the value of the shift of the rivulet tip 𝑥𝑤 is influenced also by flow disturbances caused by the curvature of the real corner. In particular, the value of 𝐵 should also depend slightly on the ratio 𝛿𝑤 ∕𝛿. This influence is not accounted for in the present model, however. it is negligibly small in the regions where 𝛿𝑤 ∕𝛿 ≪ 1. The position of the tip of the rivulet can be now determined as

Figure 3: The average rivulet rise for the three silicone oils. The height and time are divided by the characteristic length and time scales defined in Eq. 17b. The black dot marks the point corresponding to time and height 3.4 × 106 and 38 in conventional dimensionless scaling.

f

where 𝑥 is the distance from the rivulet tip in a perfect corner. In a real-world corner the rivulet tip is determined by the wall curvature. The rivulet thickness 𝛿 at the rivulet tip should be equal to the thickness 𝛿𝑤 = 𝐴𝑅𝑤 of the real curved surface of the corner, as shown in Fig. 2. Therefore, the corresponding distance 𝑥 = 𝑥𝑤 can be roughly estimated as

oo

𝑥 = ℎperfect − 𝑧,

pr

𝐺𝐿𝑊 𝑥 (𝑡 − 𝑡0 )−2∕3 , 2

e-

𝛿≈

𝜎𝑊 , 8𝜌𝑔𝑅𝑤

𝑇 =

192𝐴2 𝜌2 𝑔 2 𝑅3𝑤

(17b)

These scales allow the dimensionless height 𝜂 = ℎ∕𝐻 to be expressed in the form:

Jo

𝜂 = 𝜏 1∕3 − 𝜏 2∕3 ,

(18)

for all experimental parameters. Note that the value of 𝑅𝑤 should be the same for all experiments performed with the same sample, since it is a property only of the wall morphology. The effect of the roughness on the rivulet height can be evaluated by the ratio 𝑥𝑤 ∕ℎperfect . This ratio can be estimated from (18) as 𝑥𝑤 ∕ℎperfect = 𝜏 1∕3 . Let us say that the effect of the corner curvature is notable if 𝑥𝑤 ∕ℎperfect > 10−1 . This condition is satisfied at times 𝜏 > 10−3 . In this study the values of 𝑅𝑤 and 𝐵 are estimated using least squares fitting of the experimental data for the real rivulet rise height over time to Eq. (16) for all recorded data F Gerlach et al.: Preprint submitted to Elsevier

Page 4 of 7

Capillary rivulet rise Oil type 10 cSt 20 cSt 50 cSt All

𝑅𝑤 [m] 2.129 × 10−5 2.199 × 10−5 2.392 × 10−5 2.294 × 10−5

𝐵 74.09 70.83 80.01 69.01

Table 1 Parameter values determined for the different oil types individually and using all results simultaneously.

Jo

ur n

al

pr

Pr

line) at 𝑡 = 𝑡0 . However, at longer times the rise velocity decreases, as theoretically predicted by Eq. (16). The predicted behavior according to Eq. (16) is also shown in this figure as a dotted line, whereby the parameter values used correspond to those found when fitting the experiments of all silicone oils simultaneously (𝑅𝑤 = 2.294 × 10−5 , 𝐵 = 69.01). The agreement between experiment and theory can be considered excellent. In previous studies of dynamic capillary rise, in which the corner curvature was not considered important, the time and height were made dimensionless using the capillary length √ 𝑙𝜎 and a dimensionless time defined as 𝑡𝜎 = 𝜇∕ 𝜌𝑔𝜎 [3, 7, 11, 12]. A direct comparison of the results shown in Fig. 3 with previous results is therefore not possible. For orientation however, the point marked with a black dot at the dimensionless time 𝜏 = 0.6 × 10−2 and height 𝜂 = 0.15 in Fig. 3 corresponds to a dimensionless time and height of 3.4 × 106 and 38 using the conventional non-dimensionalization. Fig. 4 demonstrates the sensitivity of the rivulet rise to the corner radius. To obtain these results the dimensionless friction factor 𝐵 = 69.01 from the experimental results and the liquid properties of 20 cSt silicone oil are inserted into Eq. (18) to calculate the corresponding dimensional rivulet height over time. Corner radius changes of 10 𝜇m change the slope of the rivulet rise only slightly, which demonstrates the high sensitivity of the estimation of 𝑅𝑤 to errors in the height measurement. For example, even at a physical time of about 100 minutes a corner radius difference of 10 𝜇m leads to a change in rivulet rise height of only approximately 6.46 mm. As stated before the contact angles of the three types of silicone oil on the sample are not known exactly. While the

data in Table 1 was obtained using an assumed contact angle of 0◦ , the fitting was repeated for all possible contact angles ranging from 0◦ to 45◦ , resulting in the same value of 𝑅𝑤 . The contact angle independent fitting of 𝑅𝑤 is not directly obvious from the equations. 𝑅𝑤 determines the shape of the real rivulet rise as shown in Fig. 4. The horizontal scaling, i.e. the speed of the rivulet rise, is determined by the contact angle dependent parameters 𝐴 and 𝐵, see Eq. (8). While the contact angle dependency of 𝐴 is given in Eq. (2), 𝐵 is the only free parameter influencing the speed of the rivulet rise; hence, it has an unique value for a given measured speed of rivulet rise. This means that an arbitrarily chosen contact angle leads to a different 𝐴, resulting in a different value of 𝐵, but which yields the same characteristic time scale 𝑇 (Eq. (17b)) as for the physically correct contact angle. Since 𝐵∕𝐴2 = const., the fitting of 𝑅𝑤 can be performed with an arbitrary contact angle value.

e-

Figure 4: The rivulet height over time for 20 cSt silicone oil and different corner radii, calculated from Eqns. (18) and (17) using 𝐵 = 69.01, as measured in the experiment.

oo

f

Figure 5: Comparison of the 1/3 power law and the present theory with the data of Deng et al. [11] for the H samples. The black cross shows the point at which the present theory predicts the rivulet to reach its final height.

F Gerlach et al.: Preprint submitted to Elsevier

5. Discussion and Conclusions While the rivulet rise in real-world corners was not analyzed explicitly in the vast majority of previous studies, many studies presumably used samples with finite corner radii. To verify the validity of the newly proposed theory, the plotted rivulet rise data from one previous publication was compared to the present theory. Deng et al. [11] produced V-shaped grooves in pure copper using a hard tool. They measured the geometry of the V-shapes. The width and depth of the grooves and the opening angle of the back corner are given in Table 2. They used one set of samples called H, in which they varied the height of the V-grooves, and one set of samples called W, in which they varied the width of the V-grooves. The two sets share one common sample, which is called H5 and W2 in the two sets respectively. Even though Deng et al. used grooves smaller than the capillary length, [7] recently showed that these grooves are still large enough to allow the rivulet to propagate unaffected by the outer corners of the groove. Fig. 5 and Fig. 6 show a comparison of the experimental results with the 1/3 power law and the present theory. Page 5 of 7

Capillary rivulet rise

Figure 7: The quantities of 𝑅𝑤 and 𝐵 obtained for the measurement data of Deng et al. [11] plotted against the corner opening angle. The upright triangles show the values of 𝐵 for acetone with 𝜃 = 24.3−25.9◦ and the downward triangles show the values of 𝐵 for ethanol with 𝜃 = 19.8 − 21.8◦ [11].

e-

pr

mensionless friction factor decreases with increasing corner angle, this behavior reverses in the data of [13] at approximately 10◦ to 20◦ , resulting in an increasing dimensionless friction factor for an increasing corner angle. Deng et al. measured the static contact angle of their liquid/solid systems to be at least 19.8◦ , which agrees well with the observed proportionality between corner angle and dimensionless friction factor 𝐵. It can be seen that ethanol (downward triangles) with a contact angle of 𝜃 = 19.8 − 21.8◦ yields higher values of 𝐵 than acetone (upright triangles) with a contact angle of 𝜃 = 24.3 − 25.9◦ [11]. This increase in the dimensionless friction factor fits to the results of [13] and to physical expectations. Due to the rivulet being thinner and more aligned with the walls for smaller contact angles (compare cross-section B - B shown in Fig. 2) the velocity gradient and the friction forces decrease for higher contact angles when the overall thickness of the rivulet increases. The fact that the difference in 𝐵 between the two liquids is more pronounced for higher corner angles agrees with the results of [13]. For small corner angles the rivulet forms a wedge between the walls, in which the overall shape is only slightly affected by the contact angle. For increasing corner angles the relative thickness of the rivulet decreases and the curvature of the free surface (given by the contact angle) becomes more significant for the overall shape and thickness of the rivulet, exhibiting an increasing effect on the dimensionless friction factor 𝐵. In addition to comparing data from Deng et al. to the present theory, the data from Tani et al. [12] was analysed. Tani et al. fabricated rectangular channels in hard PMMA (Poly(methyl methacrylate)) by micro-milling. The rivulet rise of different kinds of silicone oil was tracked over a maximum time of more than 395 minutes, but still the available plotted data does not allow a clear differentiation between the 1/3 power law and the present theory. The data exhibits slightly different power laws, which leads to large scatter in their dimensionless plots. When automatically fitting the data to the present theory, two samples of Tani et al. show a rise rate higher than the 1/3 power law, resulting in negative corner radii and only for one sample a physical corner ra-

Pr

It can be seen that this data is in excellent agreement with the present theory, clearly deviating from the 1/3 power law. The data of Deng et al. [11] was only measured for a real time of 30 seconds, but they exhibit a very long dimensionless time scale so that the difference between the 1/3 power law and the present theory is clearly evident. The study was performed with ethanol and acetone, which resulted in different asymptotic rise heights, not explained by Deng et al. For Figs. 5 and 6 the data for acetone and ethanol was fitted separately to allow 𝐵 to be different for both liquids showing a contact angle difference of about 5◦ . A black cross in Figs. 5 and 6 marks the point at which Eq. 18 predicts the maximum rivulet height. It can be seen that this prediction is in good agreement with the experimental results.

oo

f

Figure 6: Comparison of the 1/3 power law and the present theory with the data of Deng et al. [11] for the W samples. The black cross shows the point at which the present theory predicts the rivulet to reach its final height.

Jo

ur n

al

Both sample sets of Deng et al. were fitted independent of each other and the fitted quantities are listed in Table 2. Although the samples H5 and W2 are the same, they result in slightly different values. This is likely due to small uncertainties associated with manually extracting data points from the published plots. Fig. 7 shows the quantities 𝑅𝑤 and 𝐵 from Table 2, plotted against the respective corner angles. Except for the smallest corner angle of 24.1◦ from sample W1, 𝑅𝑤 shows a clear tendency to increase with increasing corner angle. Since it is not known whether Deng et al. used one or more machine tools for fabricating the different groove angles it cannot be confirmed whether this observed trend is physically correct. If however the same tool was used in several passes to manufacture the wider grooves, the larger corner radius would be plausible. he increase of 𝐵 with corner angle can be explained when comparing to the study of Ayyaswamy et al. [13]. In this publication the dimensionless friction factor for the flow in triangular grooves was calculated numerically. Although the data cannot be directly compared, because it is obtained with gravity directed perpendicular to the rivulet movement, it can be used to understand the behavior of the dimensionless friction factor. While for small contact angles the diF Gerlach et al.: Preprint submitted to Elsevier

Page 6 of 7

Capillary rivulet rise Sample H1 H2 H3 H4 H5=W2 W1 W2=H5 W3 W4

Width [𝜇𝑚] 450 450 450 450 450 400 450 500 600

Depth [𝜇𝑚] 500 600 650 750 850 850 850 850 850

Corner angle 42.5◦ 36.3◦ 33.9◦ 30.4◦ 26.7◦ 24.1◦ 26.7◦ 30.1◦ 34.5◦

𝑅𝑤 [m] 5.229 × 10−5 4.571 × 10−5 4.033 × 10−5 3.644 × 10−5 3.507 × 10−5 3.722 × 10−5 3.503 × 10−5 3.659 × 10−5 4.006 × 10−5

𝐵acetone 5.937 4.670 3.246 2.975 2.696 2.973 2.647 2.749 3.450

𝐵ethanol 8.480 5.908 4.215 3.501 3.066 3.814 3.019 3.430 4.761

Table 2 The geometry of the different samples used and reported by Deng et al. [11] and the fitted corner radii. The sample codes H5 and W2 refer to the same physical sample.

f

pr

oo

[1] Lei-Han Tang and Yu Tang. Capillary rise in tubes with sharp grooves. Journal de Physique II, 4(5):881–890, 1994. [2] FJ Higuera, A Medina, and A Linan. Capillary rise of a liquid between two vertical plates making a small angle. Physics of Fluids, 20(10):102102, 2008. [3] Alexandre Ponomarenko, David Quéré, and Christophe Clanet. A universal law for capillary rise in corners. Journal of Fluid Mechanics, 666:146–154, 2011. [4] Paul Concus and Robert Finn. On the behavior of a capillary surface in a wedge. Proceedings of the National Academy of Sciences of the United States of America, 63(2):292, 1969. [5] Vignesh Thammanna Gurumurthy, Daniel Rettenmaier, Ilia V Roisman, Cameron Tropea, and Stephen Garoff. Computations of spontaneous rise of a rivulet in a corner of a vertical square capillary. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 544:118–126, 2018. [6] A. Lopez de Ramos and R. L. Cerro. Liquid filament rise in comers of square capillaries: a novel method for the measurement of small contact angles. Chemical Engineering Science, 49(14):2395–2398, 1994. [7] Felix Gerlach, Maximilian Hartmann, and Cameron Tropea. The interaction of inner and outer surface corners during spontaneous wetting. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 583:123977, 2019. [8] Mark M Weislogel. Compound capillary rise. Journal of Fluid Mechanics, 709:622–647, 2012. [9] EA Boucher. Capillary phenomena: properties of systems with fluid/fluid interfaces. Reports on Progress in Physics, 43(4):497, 1980. [10] William Joseph O’Brien, Robert G Craig, and Floyd Avery Peyton. Capillary penetration between dissimilar solids. Journal of Colloid and Interface Science, 26(4):500–508, 1968. [11] Daxiang Deng, Yong Tang, Jian Zeng, Song Yang, and Haoran Shao. Characterization of capillary rise dynamics in parallel micro v-grooves. International Journal of Heat and Mass Transfer, 77:311– 320, 2014. [12] Marie Tani, Ryuji Kawano, Koki Kamiya, and Ko Okumura. Towards combinatorial mixing devices without any pumps by open-capillary channels: fundamentals and applications. Scientific reports, 5:10263, 2015. [13] P. S. Ayyaswamy, I. Catton, and D. K. Edwards. Capillary flow in triangular grooves. Journal of Applied Mechanics, 41:332, 1970.

Funding

ur n

al

Pr

This sensitivity of the rivulet rise rate and height to the value of corner radius 𝑅𝑤 underlines the fact that small local variations of 𝑅𝑤 may have significant influence on the rivulet propagation. Local variations of 𝑅𝑤 can arise due to non-uniformities introduced during the corner fabrication or through roughness elements. While these cannot be captured in a closed analytic form, as given above for idealized corners, the physical origins of their influence on rivulet propagation are now apparent. In summary, the proposed theory describing the dynamic capillary rise in real-world corners fits excellently to the present experimental results and to past literature data from Deng et al. Furthermore, it could be shown how sensitive the fitted value for the corner radius 𝑅𝑤 is to deviations in the height measurement and that the fitting of 𝑅𝑤 can be performed assuming arbitrary contact angles.

References

e-

dius of 24.7 𝜇m is obtained. The results yielding power laws above 1/3 are assumed to be caused by two factors. First, high viscosity liquids up to 500 cSt were used in the study of Tani et al., which prolong the initial phase of higher rise rates before reaching the 1/3 power law. Second, the height measurements have error bars up to ± 2.5 mm, which explain possible deviations in the rise rate and uncertainties in 𝑅𝑤 , as shown in Fig. 4.

Jo

The authors acknowledge the financial support by the German Research Foundation (DFG) within the Collaborative Research Centre 1194 "Interaction of Transport and Wetting Processes".

CRediT authorship contribution statement Felix Gerlach: Experimental part, methodology, data interpretation. Jeanette Hussong: Conceptualization of this study, methodology. Ilia V Roisman: Theoretical modeling. Cameron Tropea: Conceptualization of this study, methodology, data interpretation.

F Gerlach et al.: Preprint submitted to Elsevier

Page 7 of 7