On the boundary conditions at an oscillating contact line

ABSTRACT

We investigate principally the pinned contact line and the associated flOw field, adjacent surface

ele-vation, and contact angle. A limited discussion and results for partial slip, oscillating contact lines are included also. Previous experimental results are compared to the theories of Hocking and Miles. It is shown that at low frequencies, the pinned contact line experiments agree reasonably with Hocking; however, at higher frequencies, there is a significant difference. A discussion of the present experimental setup is included; the magnification, Brewster angle, and view camera techniques used are presented. New experimental results of the flow fields and surface elevations in

the pinned contact line regime are presented. Finally, the partiallypinned contact-line problem is addressed. Results from standing (Faraday) wave experiments are presented and the need for an improved contact-line model is discussed. Additionally, a comparison is presented of a mod-ified Tanner's law with the data of Ting and Per-lin.

INTRODUCTION

The contact line is the intersection between two distinct fluids and a solid. In our experiments, the fluids are water and air, the solid is glass. The contact angle, O, is defined as the angle between the tangent to the fluid interface at the contact

line and the water-glass interface. "Contact

angle" is used in place of the more proper

description "apparent contact angle." The various phenomena that occur at the contact-line bound-ary are very important in many fluid flows, for example, the spreading process of liquid drops

To be presented at the Fourth MicrograVity Fluid Physics conference, NASA, 1998

ON TILE BOUNDARY CONDITIONS AT AN OSCILLATING CONTACT

LINE

L. Jiangt, Z. Liu2, M. Perlifl2, and W.W. Schul&

tMechanical Engineering & Applied Mechanics and 2Naval Architecture & Marine Engineering University of Michigan, Ann Arbor, MI 48109, perlin@engin.umich.edu

and the generation and dissipation of waves

inter-acting with a solid surface. To determine and

quantify the contact-line behavior, measurements of the contact-line position and dynamic contact angle (i.e. contact angle with a moving contact line) are conducted, and to quantify the flow,

velocity-field measurements are included. As

opposed to their uni-directional counterparts, oscillatory contact-line boundaries have not been studied experimentally in a significant way until very recently (Ting and Perlin, 1995). We mention

the following papers relevant to the work pre-sented: Stokes (1845), Smith (1968), Hocking (1987), Miles (1990), and Cocciaro et al. (1993). The Stokes paper includes the solution to his so-called "second problem," that of an infinite-length, oscillating plate in a semi-infinite fluid. Smith (1968) discussed the surface waves

gener-ated by viscous forces (le. he contrasts the

Stokes solution to the solution with a free surface

present); however, no contact-line effects are

included. Hocking (1987) used an oscillatory con-tact-line boundary condition (by using two

approximations to Dussan V's model) to calculate the waves generated by a vertically oscillating, upright plate and obtained the amplitude of the radiated waves and the energy dissipation due to contact-line hysteresis. Miles (1990), addressing the same problem as Hocking, used a boundary conditiOn along the plate (similar to that proposed by Navier) associated with a boundary condition at the contact line. Viscosity was included and a non-zero initial free-surface meniscús was also considered (in one case). Cocciaro et al. (1993) conducted experiments to examine the effects of the dynamic contact-line behavior on surface

waves bLbodzontallv oscillating a container.

T8C UEØTE

eboratorium 'icor Stheepshydromechanlca Archief Mekeiweg 2, 228 CD Deift 'ieL

D

contact line and the associated flow field, adjacent

surface elevation, and contact angle. A limited discussion and results for partial slip, oscillating contact lines are included also. Previous

experi-mental results are compared to Hocking and

Miles. It is shown that at low frequencies, the

experiments agree reasonably with Hocking; how-ever, at higher frequencies, there is a significant difference. A method to determine whether this is due to inviscid nonlinearity or viscous effects is

discussed. The model of Smith is shown to

require extension to a much larger horizontal region adjacent to the contact line. A discussion of the experimental setup used is offered includ-ing the magnification, Brewster angle, and view camera techniques used. New experimental results of the flow fields and surface elevations in

the pinned contact line regime are presented.

Finally, the partially-pinned contact-line problem is addressed. Results from Faraday wave experi-ments are presented and the need for an improved

contact-line model is discussed. In addition, a comparison of a modified Tanner's law with the data of Ting & Perlin (1995) is presented.

THE PINNED CONTACT LINE AND

ATTENDANT FLOW FIELD

DUR-ING OSCILLATION OF A SURFACE

PIERCING VERTICAL PLATE IN A

LIQUID

TO determine the dissipation and wave-generation effects that the contact line has at a free surface-solid plate ifltersection (a free surface is where the effect of the air layer on the liquid layer is neglected), we begin with the simplest case, an oscillatory pinned contact line. Ting & Perlin has shown that the Hocking model with/without hys-teresis is inadequate for the case of slip-stick motion. The questions we seek to answer are:

Why, how, and to what extent and depth does the contact line modify the flow field so that surface perturbations greatly exceed those due to the assumption of a no-slip boundary coditin along

layer? These are perplexing questions, even for the simplified problem of the pinned contact line because the boundary condition along the plate is unchanged throughout the liquid régime except as modified by slip (not necessary in pinned case). As a first step, figure 1 compares a modified the-ory of Hockiúg, with the Stokes solution (i.e. the "contact angle" of the Stokes solution is the angle

of a material line that, although not at the free

surface, initially has the value of the static menis-cus), and the data of Ting and Perlin. Hocking's theory is based on an inviscid fluid and a contact-line condition with/without hysteresis (we choose to use the one with complete hysteresis for this comparison), i.e. Vr=ali/ar-Vp=Adii/ax where V

is the relative velocityfluid to solid, 1

is the contact-line position, V, is the plate velocity, the contact angle minus 9Ø0 equals the arctangent of the surface slopeand it has been approximated

by the surface slope, and X is a (capillary) coeffi-cient. To obtain a variation in contact angle repre-sentative of that of water on glass that begins with a small static angle, the static contact angle mea-sured experimentally is added to the Hocking free-surface angle prediction to yield a modified Hocking angle. For a pinned-end condition, Vr 5 zero, and so X is set to zero. This figure shows that the modified Hocking prediction is in reason-able agreement with the lower frequency, 2 Hz, oscillation although it is a poorer predictor for the larger contact angle; however, there is significant difference between the modified Hocking

predic-tion and the experiments at 20 Hz. The Stokes

solution is presented to demonstrate the large dif-ference that exists between its predictions and the

data. It is unknown whether the larger discrep-ancy between the modified Hocking theory and the experiments at 20 Hz is due to nonlinearity (Hocking's solutioñ is for linear waves and

obtained through Fourier transform) or due to vis-cous effects. Interestingly, the waves produced during a 2 Hz oscillation at 1 mm are negligible and one can infer that the solution and experiment agree when this is the case. In the future, experi-ments will be conducted with increased stroke at

-t -06 .0.4 -0.2 0 0.2

(a)

Figure 1. Comparison of Ting & Perlin data (D) with a modified Hocking prediction for no contact-angle hysteresis

( . ) and the "contact angle" of the Stokes soluton (). (a) Stroke amplitude of 1 mm and oscillation fre-qûency of 2 Hz. (b) Stroke amplitude of i mm and oscillation frequency of 20 Hz.

lower frequency to increase nonlinearity (ks, the' wavenumber times stroke amplitude, a dimension-\ less measure of noñlinearity) and with decreased stroke at higher fre4úency to presumably reduce nonlinearity. The experimental result presented in

figure i(a) has an associated ks of 0.032 while

that presented in figure 1(b) has a ks of 0.239 and

is therefore more nonlinear. The experimental result in the 20 Hz experiment exhibits much larger contact angle changes than the theory pre-dicts. We have also graphed the Miles theory (not

shown) that includes a viscoúS boundary layer and other assumptions, and its predictions are slightly worse than those of Hockiñg's as modi-fied. The future experiments will demonstrate whether the disagreement with the modified

Hocking theory is due to nonlinearity or viscosity or a combination of the two. To analytically han-dle the nonlinear, inviscid, pinned condition, one cañ resort to a bispectral or even higher-order

spectral treatment.

Smith (1968) treats the problem of a viscous fluid with a free surface adjacent to an oscillating ver-tical plate. The no-slip boundary condition. is applied along the liquid-solid interface. Neither

surface tension nor a meniscus were included in the analysis; however, one can trace the transitiOn from the Stokes' solution that is valid deep in the liquid to the shearfree surface. Missing from this analysis is the capillary length, I,that provides a possible starting point for additional analysis. That is, the distance from the contact line where the free surface "feels" the contact angle under static conditions, I, greatly exceeds, the bound ary-layer thickness or the viscous length scale, i,,.

In Smith's analysis, the transition is shown to have a second-order effect on the surface Wave motion and occurs only in the comer with viscous length scale in the horizontal and verticàl direc-tioñs. Taking surface tension into account in this analysis may modify the results significantly. We have conducted new experiments to measure as a function of time the flow field and surface elevation in the vicinity of a pinned contact Une. The experimental setup is a vertically oscillating upright glass plate. partially immersed in treated water and the longitudinal elevation view is shown iñ figure 2. The oscillation frequencies are i-20 Hz; however, only two phases of the 1-Hz sinusoidal oscillation with stroke amplitude of

-06 -06 -0.4 -02. 0 02 0.4 06 06

Ps (

(b)

toe too

Figure 2. Experimental setup used to obtain the flow fields and surface-elevation profiles.

0.12 mm (so that the contact line remains pinned) are presented. The Reynolds number based On radian frequency and stroke amplitude is 0.09. The primary difficulty associated with these sur-face-elevation and flow field measurements is that a magnification of about ten is desired at a large (about 27cm) Standoffdistancethe standoff dis-tance is required to effect the removal of the re-reflection of light Scattered by particles from the free surface. The technique used is particle-image velocimetry (PIV) in conjunction with Brewster-angle viewing through a transparent Brewster-angled lower wall (Lin & Perlin), although with the vertically oscillated glass plate, other reflection problems arise from the plate itself. A 4 in x 5 in (10.16 cm x 12.70 cm) view camera with a 250 mm focal-length lens and an adjustable bellows (focal-length and angles of the lens and film plane are accommo-dated) is Used. The object distance is 27 cm, the distance to the film plane is about 280 cm, the pulse duration of the acousto-optic modulated (i.e. triggered) argon-ion laser sheet is 4ms, the bias velocity is 0.1058 cm s (compared to a mâximum plate/contact angle velocity of 0.0747 cm sa), and the time between laser pulses is 40 ms. The magnification is 10.5.

To what depth does the contact-line effect extènd or at what depth does the flow field revert to the Stokes solution is the question we would like to answer. We present in figure 3 two phases, - 26.6° and - 44.6° corresponding to plate velocities of

-0.674 mm s and - 0.537 mm s respectively of. the l-Hz experiment. The axes are in cm and the x-aAis coordinates represent the physical distance from the face of the plate; however the velocity vectors have not been shifted according to the location of the contact line. The magnitude of the vectors in the figures represents about 0.22 mm s

It is seen that nearly uniform flows exist in these small regions near the contact line for these phases, although their directions have changed

with phase. Many additional images need to be processed, corrected for contact-line position, and

analyzed beföre definitive conclusions can be

drawn.

THE. PARTIALLY-PINNED

CON-TACT LINE DURING

FREE-SUR-FACE OSCILLATION AND PLATE

OSCILLATION

00 _0.05

_{0 i}

0.15

02

In this section we discuss the frequency and

damping of Faraday waves (waves generated through resonance by vertial container oscilla-tion),. and the

use of an associated

ad-hoc

approach to analyze the data of Ting & Perlin when the contact line is no longer pinned. The

model simplifies to Tanner's law (Tanner, 1979) if the static contact angle is very small.

We describe experimental data on the frequency and damping of Faraday water waves in glass

tanks with treated water. The measured frequency detuning due tO the contact-line effect is shown

in figure 4. In the contact-line regime of the

wave decay where

the wave dissipation is

dominated by contact-line effects (i.e. this is not the physical contact-line region), the average4 data follows a 2/3 power law with respect tO

wave amplitudea. The (viscous and contact-line)

wave damping Obtained from a demodulated signal also demonstrate amplitude dependence. The damping rate first increases with decreasing amplitude following a. -1/3 power law (1.3 mm <

a < 6 mm), then starts to decrease for smaller

wave amplitude (a < 1.3 mm). Miles (1991)

predicted no amplitude dependence in either frequency detuning or damping when the

00 _{0.05 0.1} 0.1 5

Figure 3. Experimental flow fields measured using the PIV technique. Phases of the 1-Hz sinusoidal oscillation are - 26 60 and - 44 60 corresponding to plate velocities of - 0 674 mm s and - 0 537 mm s respectively The axes are in cm however the velocity vectors have not been shifted according to the location of the contact line. The magnitude of the vectors in the figures represents about 0.22 mm

following Hocking condition is applied at the

contact line for standing water waves

ii, = Ai

at X = 0, (1)

where i ₌

_-

with O the dynamic contact

angle and A the capilláiy coefficient. (We use a subscript on A now as the tank walls are fixed in laboratory coordinates and in the analysis, i.e. V=O) For small difference between the actual frequency w and the natural frequency (without contact-line effect) w, Cocciaro et al. (1993)

showed that the results of Miles (1991) can be

simplified to

FA

2' ¡+ (Acol)

Fw

02

w - w,1 1+(Awl)2 2w

where z represents contact-line damping. Here A is assumed to be constant. We note that if we assume A - equations (2) and (3) give the

0.3

Ó.25 0.2 0.15 0.1 0.05 ¿ 4 44 0.3 0.25 0.2 0.15 0.1 0.05 44 ¿ L s 4

44

4 4 4 4 4 .44 4

lo-, 4 s .10' lO' Fb-In..n.* I(n.m i0'

Figure 4. (a) Frequency difference (co-o)/2ir versus wave amplitude a. (b) Damping ratey.

correct frequency and damping as observed in

our experiments 1/3

Fa

'fc ¡ + a2"3(wl)2' Fw,,

(DW-232.

. (5)

The exact

- 2 / 3 power shown in figure 4

between frequency Shift and wave amplitude is shown in (5). And (4) implies a ¡/3 power law

between contact-line damping and wave

amplitude for small a, and a

- 1/3 power for

large a as shown in figure 4.

The above experiments are contrasted with

similar experiments with Photo-Flo treated watet in the same glass tank. This wetting agent usually

forms a thin film of water forms on the glass even for water waves of small amplitude. The measurements suggest this is true and that the frequency of a decaying elevation approaches a

value close to w, (without cofltact line effect) as

amplitude approaches zero. There is no clear amplitude dependence in either the frequency

shift or the damping

More experiments are conducted for Faraday waves in a small circular cylinder (Pyrex glass,

(4)

2

J--n.

12.8 cm diameter) with treated water, and in a

large circular cylinder (plexiglass) with treated water. Both cylinders ensure uniform contact-line behavior (as compared with waves in rectangular tanks discussed above). Preliminary data suggest no amplitude dependence in the frequency and the damping rate as demonstrated earlier. However, the decaying record may be too shOrt to provide valuable information needed in the small ampli-tude (contact-line dominated) range. These mea-surements suggest very complex contact-line influence on water waves and the sensitivity of contact-line behavior to the material of the solid and the surface property of the liquid.

The above ad-hoc approach for wave frequency

and damping estimate does not explain

the fundamental physics of the contact-line dynamics. Both Cocciaro et al. (1993) and Ting & Perlin (1995) find experimentally a nonlinear relation between contact-line velocity and contact angle, see figure 5. In particular, Ting & Perlin measure a time-dependent with its amplitude proportional to a2, not A a1"3 as

assumed in (4) and (5). One could propose the

following model without contact-angle hysteresis

'd(8e9c) = IV,.I

. (6)

lo_ lo' 10'

where 0e is the static contact angle (assumed to be 900 in (1), but is approximately 300 to 50° in

experiments.) The choice of a

cubic fit is

inspired by the 1/3 exponent that appeared in the adhoc approach above. However, matching (6) directly with Ting & Perlin is difficult because of the strong hysteresis loop in the contact angle

versus contact-line velocity diagram (figure

5(b)). Such hysteresis is partially caused by the large hysteresis iñ the static contaçt angle (17°), and maybe partially caused by inertiál effects. However, equation (6) exhibits some qualitative features observed by Ting & Perlin. For example,

both (6) and figure 5(b) show that tbe contact-line velocity is the largest at the. smallest contact angle. (6) also predicts that if the maximum Vr is amplitude dependent (experiments show IV,J,,

-aa, 1 <a < 2), the smallest 9 has a much weaker dependence òp ampljtude. Again, this agrees with figure 9 of Ting & Perlin.

Models for unidirectional, steady contact-li ne motión also exhibit a cubic relation U - A 9 (92

e2), but A is only a functioñ of surface tension and viscosity. For small static contact angle 9e

« I, Tanñer's law U - A92 is recovered. The

mechanism behind Tanner's

law and

(6) is probably the same, i.e. the contact-angle vañation is caused by hydrodynamics very close

to the contact line and can be described by a

balance between capillary' force and viscous force. If we assume that (6) applies to Faraday

waves in the rectangular tank, we can

equivalently assume that the Young's force at the contact line F= cos O -cos O is balanced by the

viscOus force near the contact line. The

dissipation over one wave cycle is then

proportional to F 'a where a is again the wave amplitude. Using (6) for small contact atgle O

we obtain F 2

_-

_{Vr2"3. If we further assume}

thatVr - w a, the damping rite can be estimated

by

2/3

Fa

aa

-1/3

7=

a

Energy _a2

which is the amplitude dependence measured in our Fàraday wave damping räte. The amplitude dependence of

the wave frequency

remains unresolved.

Even with some qualitative agreements with Tiug

&. Perlin, (6) is difficult to apply in the linear eigenvalue analysis for water waves. Thus we will ty to apply a more elaborate aäalysis with the belief that hydrodynamics can describe the wetting condition in oscillating flows, and that

the viscoUs and capillary balañce temaifls essential in determining the relation between

contact-line velocity and (apparent) contact

oc

4

o

l.

el.

/

v/v

Cocciaro, B., Faetti,

S. & Nobili,

M. 1993

"Expenmental investigation of capillary effects on

surface gravity waves in a cylindrical containec non-wetting boundary conditions," J. Fluid Mech. 246, 43. 66.

Hocking L. M. 1987 'Wave produced by a vertically oscillating plate," J. Fluid Mech. 179, 267-281

-rlang, L 1997 "Nonlinear gravity-capillazy water

waves," Ph.D. Dissertation, Univ. Michigan.

Lin, H..!. & Perlin, M. "Improved methods for thin, boundary layer investigations," Exper. in Fluids, to appear.

Miles, J.W. 1990 "Capillary-viscous forcing of Surface Waves,"). Fluid Mech. 219,635-46.

Miles, J.W. 1991 "Capillary boundary layer for standing waves," J. Fluid Mech. 222; 197-205.

Smith, S.H. 1968 "On the creation of surface waves by viscous forces," Q. J. Mech. Appi. Math., XXI, 439-450.

Stokes, G.G. 1845 "On the theories of the internal

friction of fluids in motion," TranL Camb. Philos. Soc.,

8,287. 0.0 -e' 0.0 -0.4 0.2 -.1

.5

_{0.5 1.0} V,l Op

Figure 5. (a) Contact angle versus contact-line vólocity inferred from Cocciaro et aL (1993); (b) Contact angle ver-sus relative contact-line velocity for 2 Hz oscillation (Ting & Perlin) The inserts demonstrate the corner vortex at

the maximum plate position. Stroke amplitude is 3 mm. The solid and dash_{curves are predicted by (1).}

REFERENCES

Tanñer, L. 1979 "The spreading of silicone oil drOpson horizontal surfaces," J. Phys D 12, 1473-1484.

Ting, C.L. & Perlin, M. 1995 "Boundary conditions in the vicinity of the contact line at a vertically oscillating upright plate: an experimental investigation," J. Fluid Mech. 295, 263-300. 60 40 20 o -1.5 -I -0.5 O 0.5 1.5