7/31/2019 jcis_lohi01

1/5

ournal of Colloid and Interface Science 242, 15 (2001)

oi:10.1006/jcis.2001.7894, available online at http://www.idealibrary.com on

PRIORITY COMMUNICATION

Oscillatory Driven Cavity with an Air/ Water Interface and an InsolubleMonolayer: Surface Viscosity Effects

Juan M. Lopez and Amir H. Hirsa

Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804; andDepartment of Mechanical Engineering, Aeronautical Engineerin

and Mechanics, Rensselaer Polytechnic Institute, Troy, New York 12180-3590

Received April 5, 2001; accepted July 30, 2001

Flow in a planar cavity bounded by stationary side walls, a flat

as/liquid interface covered by an insoluble monolayer, and driven

y sinusoidal motion of the floor is examined numerically. N avier

tokes computations with the BoussinesqScriven surface model

re presented utilizing the equation-of-state measured for a vita-

min K1 monolayeron the air/water interface. The results identify a

ange of initial surfactant concentration for which the surface ve-

ocity is sensitive to the surface viscosity B (sum of surface shear

nd dilatational viscosities) down to 102 surface Poise. Thus, the

udy suggests a practical method for determining surface viscosi-

es consisting of the measurement of the motion of a tracerparticle

n the interface and comparisons with numerical computations at

arious values ofB. C 2001 Academic Press

Key Words: insoluble surfactants; nonlinear equation-of-state;

urface dilatational viscosity.

1. INTRODUCTION

There is much interest in gas/liquid interfaces, e.g., air/water,

ue in part to the recent interest in microfluidic systems, which

re gaining technological importance. When there is a free sur-

ace in the system (which in some cases is unavoidable, espe-ially when channel walls are not fully wetting, and in many

ases essential, e.g., for gas analysis using microchannels), then

he coupling between the interface and the bulk flow needs to

e considered. When the length scales are small, then interfa-

ial effects can dominate effects of gravity and other forces. For

ery small length scales, the effect of intrinsic surface viscosities

surface shear, s, and surface dilatational, s) can dominate the

ffect of surface elasticity (surface tension gradients). Of the two

urface viscosities, surface shear viscosity has been consistently

measured for a variety of surfactant systems using differenttech-

iques (1, 2). However, the same is not true of the surface dilata-onal viscosity. Determination ofs is difficult since its effects

are coupled to surface tension, both in the normal and tangetial stress balances. General agreement between measureme

of s with different techniques have yet to be demonstrat

(3); results varying by several orders of magnitude have be

reported (2), as well as negative values ofs (4). Most indi

tions are that s may be many orders of magnitude larger th

s (58), so it is important to develop techniques to measur

reliably.

Either the tangential or the normal stress balance may be u

to determine s. One of the techniques that utilizes the norm

stress is the maximum bubble pressure method (9). This tec

nique is not applicable to the measurement of s for insolu

(Langmuir) monolayers, which are of interest to this study. Althe complexities resulting from gradients in surfactant distrib

tion on thesurface of thebubble,e.g.,development of shear str

along the surface, have not been addressed and may account

some of the discrepancies seen between different measureme

of s. Other methods that try to determine s are drop def

mation techniques, but they are not easy to apply to insolub

systems. Although in principle it is possible to spread an ins

uble monolayer on an order 1-mm drop, it is difficult to kno

how much material is present on the surface. Thus, these tec

niques have been primarily applied to soluble systems (101

For larger drops, distortions due to gravity become increasinimportant and the technique requires a microgravity enviro

ment (13). These techniques all assume spherical symmetry,

in practice there is always a lip (e.g., in pendant drop, maxim

bubble pressure method, etc.), so the role of tangential str

may remain unaccounted for.

The classic example that utilizes the tangential stress balan

to determine s is the longitudinal wave method (14). This tec

nique is based on theory that assumes an inertialess limit, hen

the restriction to small barrier speed/frequency. Small barr

speed/frequency is also necessary to avoid making transve

waves as the theory is based on an essentially flat interfaWhen small frequency is used, then the effect of elasticity c

1 0021-9797/01 $35

7/31/2019 jcis_lohi01

2/5

PRIORITY COMMUNICATION

ominate the effect of surface viscosities (the product of Boussi-

esq number and capillary number being small), so errors in

etermining elasticity (e.g., deviations in the equation-of-state)

may lead to large errors in s. Another method using the tangen-

al stress balance employs a transient vortex flow and simulta-

eously measures the surfactant concentration on the interface,

(x , t), and the surface velocity, us(x, t), at a given location x

8). The problem with the transient vortex method is that theow is not time-periodic, so one cannot do phase averaging. As

result, there are large inaccuracies (at least 50% noise level)

n the determination of surface viscosities. Further, the simulta-

eous measurement of both c and us is technically challenging,

specially since nonlinear optics had to be used to measure c.

We propose a cavity-driven flow; by changing from a barrier-

riven flow, we can drive the system at higher Reynolds numbers

based on frequency and amplitude) and still avoid complicated

urface deformation problems.

. EQUATIONS GOVERNING THE DRIVEN CAVITY FLOW

The flow consists of fluid of density , molecular viscosity

and kinematic viscosity = /), contained in a rectangular

egion of width 2L and depth H, and driven by the horizontal

armonic oscillation of the bottom boundary. The top surface

f the fluid is exposed to air and has an insoluble monolayer on

he interface. Initially, everything is at rest, and the surfactant

monolayer is uniformly distributed. At time t= 0, the bottom

late is set to oscillate with horizontal velocity U sin(2 t).

n this study, we consider the two-dimensional problem that is

nvariant in the transverse direction and neglect viscous coupling

n the air side.The governing equations are the two-dimensional Navier

tokes equations, together with the continuity equation and ap-

ropriate boundary and initial conditions. It is convenient to use

streamfunctionvorticity formulation, where the nondimen-

onal velocity vector and the corresponding(scalar) vorticityare

(u, v) = (y ,x ), = x x yy .

We use H as the length scale and the viscous time H2/ as the

me scale. The two-dimensional NavierStokes equations, with

hese scalings, reduce to the evolution equation for the vorticity:

t +y x x y = x x + yy . [1]

nitially, everything is at rest and the monolayer is uni-

ormly spread; (x, y, 0) = (x, y, 0) = 0 and c(x, 0) = c0.

he boundary conditions on the solid boundaries are no-

ip; for the two stationary vertical walls at x = L/H,

(L/H, y, t) = x (L/H, y, t) = 0 and hence (L/H,

, t) = x x (L/H, y, t). For the oscillating bottom,

(x, 0, t) = 0and y (x, 0, t) = Re sin(2 Ret); the two gov-

rning parameters are Re = UH/, the scaled velocity ampli-

ude of the floor, and Re = H2

/, the scaled frequency of theoor oscillation. The vorticity is (x, 0, t) = yy (x, 0, t). On

the air/water interface, being a material surface, (x, 1, t) =

by continuity with its value on the sidewalls, which is set

zero without loss of generality. We assume that the interface

flat, and hence the contact angle at the air/water/solid cont

line is 90 (in a physical laboratory experiment, the location

the contact line can be fixed by depositing a nonwetting paraf

film above the interface on the vertical walls (15); also, t

Froude number, Fr= 2

/g H

3

, for water at room temperatuand a depth H 1 cm, is only about 103). This leaves t

condition for the vorticity on the interface to be specified.

We model the interface using the BoussinesqScriven co

stitutive relation (16). In planar two-dimensional systems w

a flat interface, only the tangential stress balance plays a d

namic role (2). The tangential stress balance is, noting that

the interface v = 0:

(x, 1, t) = uy (x, 1, t) = C1x

Busx

x

,

where C = /H0 is the capillary number, is the s

face tension which varies with surfactant concentration c, 0(c = 0),B = (s + s)/H is the Boussinesq number cor

sponding to the sum of the surface shear and dilatational v

cosities (in general, also functions of c), scaled by H, a

us(x, t) = u(x , 1, t). In (15), we determined that an insolub

monolayer comprised of vitamin K1, for concentrations up

1 mg/m2, had negligible surface shear viscosity. In this stud

we shall treat B as constant since its functional dependence o

is not known a priori. This linearization is self-consistent as t

computational results indicate that variations in c in both tim

and space are small for the parameter ranges considered. T

reduces the surface viscosity term in Eq. [2] to Bus

x x . For we utilize the equation-of-state measured for water/vitamin

in (15). This equation-of-state is plotted in Fig. 1, along with

FIG. 1. Measured equation-of-state (open symbols) for a vitamin K1 mo

layer on air/water interface at 23C (15), together with the curve fit givenEq. [3].

7/31/2019 jcis_lohi01

3/5

PRIORITY COMMUNICATION

tted curve given by

= 66.1+ 6.2 tanh(7.5(1 c/1.38)). [3]

Since the vorticity at the interface depends on surface tension

radients, which in turn depend on the surfactant concentration,

we also need to solve an advectiondiffusion equation for the

urfactant concentration,

ct + (c us)x = Pe

1cx x , [4]

where Pe= /Ds is the surface Peclet number and Ds is the dif-

usivity of the surfactant on the interface. Note that [4] is linear in

so there is no need to nondimensionalize c. Conservation of in-

oluble surfactant on the interface is enforced with the zero-flux

onditions cx (L/H, t) = 0. The zero-flux boundary condition

ssumed for insoluble monolayers is realizable in experiment

or low Fr flow by pinning the contact line via the application of

nonwetting film on the sidewall above the waterline (15, 17).The numerical solution of [1] and [4] together with the bound-

ry and interface conditions follows that used in (15, 18).

pecifically, a second-order centered finite-difference spatial

iscretization is used, with nx = 101 and ny = 51 grid points

n the horizontal and vertical directions, respectively, together

with a second-order predictorcorrector scheme for the time

volution. The time step, t, is governed predominantly by the

dvectiondiffusion Eq. [4] for the range of parameters chosen,

ndneedsto be reduced when the surface velocitybecomes small

which is the case when Marangoni stress is large and/or surface

iscosities are large). The smallest value used was t= 107.

3. RESULTS

Equation[2] shows that the stress on the fluid at the interface is

ue to contributions from elasticity and surface viscosity. These

ontributions are qualitatively different in the way they relate to

he thermodynamic state of the interface (via surfactant concen-

ation) and its kinematics (fluid velocity at the interface), and

heir effects are complementary. To leading order, c is essentially

onstant (this is verified numerically for the Re and Re ranges

onsidered). The elastic term depends on the equation-of-state,

s x = ccx ; for small cx , its contribution is greatest for c valueswhere the equation-of-state is steep, i.e., c large, regardless of

ny velocity gradients. On the other hand, the surface viscosity

erm,Busx x , has B essentially constant (for small cx ), but its con-

ibution to the surface stress depends to leading order on surface

elocity variations. These can be made large by appropriate os-

illatory driving. Also, the elastic contribution dominates when

here is large bulk inertia (flow) that drives a concentration gra-

ient and results in a Marangoni stress that brings the surface

elocity to zero. This occurs for adequately large Re and suit-

bly small Re. Whereas, the viscous contribution dominates

when there are large surface velocity gradients, and is relativelynsensitive to concentration gradients.

The objective of the computations presented here is to demo

strate that the flow under consideration is sensitive to surfa

viscosity effects in some range of parameter space. For this fl

to provide a practical experimental technique to measure surf

viscosities,B, we need to show that it is sensitive toB variatio

in a parameter regime accessible to laboratory measureme

with a reasonable signal-to-noise ratio. We begin by noting t

B only appears in the tangential stress balance (2), and its effon this balance may be completely masked by surface tensi

gradient effects. For a given surfactant system (and bulk liqu

e.g., water at room temperature), the only variable available

adjust the relative contributions of Marangoni stress and surfa

viscosities to the stress balance is the length scale H; C

scales with H and B scales with H1, so as H is reduced,

ratio of surface viscosity to elasticity is increased by H2

practical lower limit is H 1 cm, which would limit the Frou

number to 103 and minimize surface deformations. Fo

vitamin K1 monolayer onwater ofdepth 1 cm at 23C, this giv

C = 1.2 106. In the present calculations, we have conside

FIG. 2. Variation of max |us| and c/c0 with c0 (mg/m2) for Re = 1Re = 16, H/L = 1, and B as indicated.

7/31/2019 jcis_lohi01

4/5

PRIORITY COMMUNICATION

FIG. 3. Profiles of surface velocity, us, and concentration, c, at 10 phases over one oscillation period for Re = 100, Re = 16, H/L = 1, c0 = 2.0 mg/

nd B as indicated.

arious Re and Re with c0 up to 2.25 mg/m2, covering a largeange of the equation-of-state. For each of these parameters,

e have set B = 0, 1, 10, and 100; using our scalings, B = 100

orresponds to (s + s) = 0.9325 g/s (surface Poise). In the

alculations, we have set Pe= 10, which is probably about

wo orders of magnitude too small, but since the surfactant

oncentration gradients are everywhere small for the cases

onsidered, we do not expect the results to be sensitive to Pe.

The flow geometry was selected to accentuate the contribu-

on from surface viscosity. The parameters used to illustrate the

esults are selected with an eventual laboratory experiment in

mind to verify the model and ultimately to measure the surfaceiscosities, B. The depth of the cavity, H= 1 cm, was cho-

en to be as small as possible (in line with the above scaling

rgument), but not so small that Fr becomes large and surface

atness suffers, and surface velocity measurements are diffi-

ult. The frequency of oscillation, Re = 16, was selected to

e large enough to avoid approaching a quasi-static monolayer

18) showed that for steady flow, the effect of surface viscosi-

es is very small at a steady monolayer front), yet small enough

o give adequately large surface velocity and surface velocity

radients and avoid instabilities in the bulk flow. The amplitude

f oscillation, Re = 100, was selected to be large enough to en-

ure a strong surface flow (to keep viscosity in the bulk fluid from

diminishing the surface velocity) and small enough to avoid stabilities in the bulk flow (19). In Fig. 2, the maximum surfa

velocity over a complete period, max |us|, andthe relative chan

in concentration over a complete period, c/c0, are plotted

functions of initial concentration, c0, for B = 0, 1, 10, and 1

in the driven cavity with Re = 100, Re = 16, and H/L =

using a vitamin K1 monolayer. The plot shows that over som

ranges of c0, the effect of surface viscosity is to decrease t

magnitude of the surface velocity, as expected. For c0 < 1

c0 > 1.8 mg/m2, the change in surface velocity is well with

the range that can be sensed in a laboratory measurement. T

change in c is comparatively small, thus use of a constant surfa

viscosity in each calculation is justified.

Profiles ofus and c over one period for a typical set of para

eters (Re = 100, Re = 16, H/L = 1, c0 = 2) are present

in Fig. 3 for B = 0 and B = 100, illustrating the factor of tw

difference in us, which is detectable using now standard flo

measuring techniques.

4. CONCLUSION

The theoretical foundations for a method for determing t

surface dilatational viscosity of insoluble monolayers are p

sented (the planar geometry gives the sum of the surface she

7/31/2019 jcis_lohi01

5/5

PRIORITY COMMUNICATION

nd dilatational viscosities and measurements of surface shear

iscosityvia independent established methods canbe used to iso-

ated the surface dilatational viscosity (20)). The time-periodic

ow was chosen to minimize surface deformations and max-

mize gradients of surface velocity. By considering the nondi-

mensionalized tangential stressbalance, we show that theratio of

he surface viscosity term to the elastic term scales as H2, thus

he cavity depth, H, needs to be minimized. NavierStokes com-utations are presented for a physical monolayer on the air/water

nterface; the equation-of-state for vitamin K1 was used in the

alculations since we have already demonstrated that it forms

well behaved monolayer and quantitative comparisons in a

ifferent flow between NavierStokes computations and mea-

urements have already been made for this monolayer (15). For

range of surfactant concentration where surface elasticity is

mall, the method is sensitive down to order 102 surface Poise.

With increasing surface viscosity, the range of concentration for

which theviscosity can be determined increases. Theflow geom-

try and operating conditions for the case presented correspond

o conditions that are presently realizable in the laboratory, and

roduce a stable bulk flow. For these operating conditions, the

omputed amplitudeof themaximum surface velocityis 40 m/s

n the absence of surface shear viscosity and 20 m/s with sur-

ace viscosity of about 1 surface Poise (for a 1-cm-deep channel

with water at room temperature). This difference is readily mea-

urable by video microscopy of a tracer particle on the surface

5). Thus, this study suggests a practical method for determin-

ng surface viscosities, B, consisting of the measurement of the

motion of a tracer particle on the interface and comparisons with

NavierStokes predictions at various values ofB.

ACKNOWLEDGMENT

This work was supported by NSF Grants CTS-9803478 and CTS-989625

REFERENCES

1. Jiang, T.-S., Chen, J.-D., and Slattery, J. C., J. Colloid Interface Sci. 9

(1983).

2. Edwards, D. A., Brenner, H., and Wasan, D. T., Interfacial Transport Pcesses and Rheology. Butterworth-Heinemann, London, 1991.

3. Lopez, J. M., and Hirsa, A., J. Colloid Interface Sci. 206, 231 (1998).

4. Giermanska-Kahn, J., Monroy, F., and Langevin, D., Phys. Rev. E60, 7

(1999).

5. Maru, H. C., and Wasan, D. T., Chem. Eng. Sci. 34, 1295 (1979).

6. Li, D., and Slattery, J. C., AIChE J. 34, 862 (1988).

7. Avramidis, K. S., and Jiang, T. S., J. Colloid Interface Sci. 147, 262 (19

8. Hirsa, A., Korenowski, G. M., Logory, L. M., and Judd, C. D., Langm

13, 3813 (1997).

9. Kao, R. L., Edwards, D. A., Wasan, D. T.,and Chen, E.,J. Colloid Interf

Sci. 148, 247 (1992).

10. Johnson, D. O., and Stebe, K. J., J. Colloid Interface Sci. 168, 21 (199

11. Tian, Y., Holt, R. G., and Apfel, R. E., Phys. Fluids 7, 2938 (1995).

12. Wantke, K.-D., and Fruhner, H., J. Colloid Interface Sci. 237, 185 (200

13. Holt, R. G., Tian, Y., Jankovsky, J., and Apfel, R. E., J. Acoust. Soc. A

102, 3802 (1997).

14. Maru,H. C.,Mohan,V., andWasan,D. T., Chem. Eng. Sci. 34, 1283 (19

15. Hirsa, A. H., Lopez, J. M., and Miraghaie, R., J. Fluid Mech. 443,

(2001).

16. Scriven, L. E., Chem. Eng. Sci. 12, 98 (1960).

17. Vogel, M. J., Hirsa, A. H., Kelley, J. S., and Korenowski, G. M., Rev.

Inst. 72, 1502 (2001).

18. Lopez, J. M., and Hirsa, A., J. Colloid Interface Sci. 229, 575 (2000).

19. Shankar,P. N.,and Deshpande, M. D.,Annv. Rev. Fluid Mech. 32, 93 (20

20. Ting, L., Wasan, D. T., Miyano, K., and Xu, S.-Q., J. Colloid Interface

102, 248 (1984).