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Surface Rheology III. Stress on a Spherical Fluid SurfaceD. A. Edwards and D. T. Wasan Citation: J. Rheol. 32, 473 (1988); doi: 10.1122/1.549979 View online: http://dx.doi.org/10.1122/1.549979 View Table of Contents: http://www.journalofrheology.org/resource/1/JORHD2/v32/i5 Published by the The Society of Rheology Related ArticlesWall slip and spurt flow of polybutadiene J. Rheol. 52, 1201 (2008) Particle–particle and particle-matrix interactions in calcite filled high-density polyethylene—steady shear J. Rheol. 48, 1167 (2004) Cyclic generation of wall slip at the exit of plane Couette flow J. Rheol. 47, 737 (2003) Dynamics of end-tethered chains at high surface coverage J. Rheol. 46, 427 (2002) Slip at polymer–polymer interfaces: Rheological measurements on coextruded multilayers J. Rheol. 46, 145 (2002) Additional information on J. Rheol.Journal Homepage: http://www.journalofrheology.org/ Journal Information: http://www.journalofrheology.org/about Top downloads: http://www.journalofrheology.org/most_downloaded Information for Authors: http://www.journalofrheology.org/author_information

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Surface Rheology III.Stress on a Spherical Fluid Surface

D. A. EDWARDS* and D. T. WASAN Department ofChemicalEngineering, Illinois Institute of Technology, Chicago,

Illinois 60616

Synopsis

The surface stress boundary condition developed in an invarient form in the preceding contribution to this series for a highly curved fluid interface is expressedin component form for a spherical fluid surface. The stress condition is used toexamine bounded and unbounded viscous flow over a microemulsion droplet.Surface viscous effects are shown to be increasingly significant with increasingsurface curvature, with higher order surface viscosity coefficients gaining a hydrodynamic relevance in the large surface curvature limit.

INTRODUCTION

The surface stress boundary condition at a fluid-fluid interfacemay be considerably more complex than the classical jump condition of the bulk phase normal stresses, particularly when thefluid surface exhibits an intrinsic surface rigidity. Such rigidityis normally attributed to an appreciable surfactant adsorption,and is often quantified by coefficients of surface shear and dilatational viscosity. It was shown in Part II of this series! thatother surface viscosity coefficients arise for highly curved Newtonian interfaces corresponding to the higher moments of theclassical zeroth moment surface viscosity coefficients. In the finalcontribution to the series, we demonstrate that these higher moment coefficients of surface viscosity, while necessarily of asmaller magnitude than the zeroth moment coefficients, arenevertheless hydrodynamically relevant for sufficiently curvedfluid interfaces.

We begin with a presentation of the explicit surface stressboundary conditions for a highly curved spherical fluid surface,

*Current address: Department of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, 32000 Israel.

© 1988 by The Society of Rheology, Inc. Published by John Wiley & Sons, Inc.Journal of Rheology, 32(5),473-484 (1988) CCC 0148-6055/88/050473-12$04.00


474 EDWARDS AND WABAN

derived in an invarient form in Part II. l Dimensional analysisof the boundary conditions reveals that the effect of increasingsurface curvature is to generally increase the significance ofsurface viscous stress. Finally, a hydrodynamical illustration ofthe quantitative relevance of higher order surface viscous effectsis given for bounded and unbounded flow over a micro emulsionfluid droplet.

SPHERICAL SURFACE STRESS CONDITION

The highly curved spherical fluid surface represents the mostpractically significant surface geometry and additionally satisfiesthe condition of parallelism between the tangential coordinatesurfaces within the interfacial region. The highly curved sphere(Fig. 1) may be viewed as a microemulsion droplet, or in a limitingsense, as a swollen micelle.

(The following surface stress conditions proceed directly fromEq. (45) of Part IIY

Normal Stress Condition

The normal component of the surface stress condition is givenby

(1)

~R r

I, 2

• 'l

Fig. 1. The spherical fluid surface.


SURFACE RHEOLOGY III

where

2 ( 1 a. 1 au</»T'; = -p - 3 JL R sin OaO(vosm 0) + R sin 0 or/J .

475

(2)

Tangential Stress Condition

The tangential components of the surface stress condition aregiven by

A 1 au K: + JL: a [ 1 ( d . dUd»]T ro - Tre = R - + R - R' - (Ue sin 0) + -.-ao ae sin e oe ar/J

[2VO 1 a ( 1 laue a . ])]

+ JL: R2 + R sin ear/J R sin e ar/J - de(vq,sm 0) + F~

(3)

and

T _ T _ 1 oa K: + JL: 0r</> rd> - R sin 0 or/J + R sin e ar/J

.[R s~n oCJ8(uesin e) + ~~)] + JL:

• [~: - ~ (Jae(R s~n e[ ~~ - (Jae(v</> sin 8)]) ]

+ F~

where

T = R.i(~)re JL iJr r

and

a (uq,)Trq,=JLR- -.or r

Here we have defined effective viscosities

and

(4)

(5)

(6)

(7)

(8)


476 EDWARDS AND WASAN

(9)

In (1)-(8) a ' is used to designate the inner droplet phase, (J' isthe surface tension, (J'j is the bending stress, (J'2 is the torsionstress, K

S, K1, and K~ are the zeroth, first, and second mOL ents of

surface excess dilatational viscosity, respectively, p:, p,L and p,~

are similarly the zeroth, first, and second moments of the surface excess shear viscosity, F" is the surface excess force, p is thebulk phase pressure, p: is the bulk phase shear viscosity and v isthe bulk phase linear velocity evaluated at the fluid surface.

In the small curvature limit (all higher moment surface parameters become vanishingly small) Eqs. (1), (3), and (4) are identical to the Boussinesq-Scriver. form'"" with the addition of thenormal surface excess force in (1).

But the limit of small surface curvature is not the most significant limit from the surface rheological viewpoint. As we now showthrough dimensional considerations, surface excess rheologicalproperties are far more significant for large surface curvature.

SURFACE CURVATURE AND SURFACE RHEOLOGY

We have all witnessed the stability which even a curved glasssurface can exhibit, particularly when the curvature radius issmall (compare the stability of a small glass bead to an ordinarydinner glass or picture window). This qualitative observation isuseful in understanding the increase in fluid surface rigidity withincreasing surface curvature; a phenomenon which bears a relevance to many hydrodynamic problems, including emulsification,demulsification, flow through porous media, thin film drainageand stability, and the flow of foams and emulsions.

Each of these examples have in common the existence ofsurfacecurvature, and in certain cases, large surface curvature. However,theoretical investigations often neglect the effects of surfacecurvature when characterizing the quantitative effect of surfacerheological properties in such examples'r" thus eliminating animportant feature of the hydrodynamic problem.

Consider the dimensionless form of the spherical azimuthalcomponent of the surface stress condition (3), which for agas-liquid surface may be written as

a(ue)_aa (K:+P,:) a[ 1 (a(_. ) au<p)]--= -=- - - + - -.- - Ve sm 0 +-or r 00 R p, ee sin 0 ee a¢

(P,:)[2- 1 a( 1 [aue a(- . )])]+ - ue+-.-- -.- --- u sin sp,R sin f} a¢ sm f} a¢ af} <P


where

and


vV=

Va

(J"

(J"=-/LVa

(J"Z

(J"z=-Rz/L Va

_ rr = Ii:

477

The dimensionless groups which express the ratio of surfaceviscous forces to bulk viscous forces in (9) may be written as

(10)

(11)

(12)

(13)

(14)

and

(15)

Similar dimensionless groups of course occur for the hybrid moments of surface viscosity /LI., /L~', Kf., and K~'.

The curvature radius R is clearly an important parameter ineach of these dimensionless groups, however the numerical significance of the curvature radius upon the ratios of surface to



bulk viscous forces becomes more apparent by means of a simpleexample.

We compare the significance of surface viscous forces to bulkviscous forces for the flow near a small curvature droplet with a5 mm radius to the flow near a large curvature droplet with a500 Aradius.

Considering the interfacial thickness to be =100 A, we choosethe following surface viscosity values, based upon very small values of fLSand K S

fL' = 10- 7~S

S _ 10- 13 g em!J.I-

s

S _ 10- 19 g cm2

J1.2 -s

S _ 10- 11 g emKI- --

S

2s _ 10- 17 gem

K2 - --.S

(16)

Then with a bulk shear viscosity value of J1. = .01 g/(cm s) wehave for the 5 mm droplet

N", = 2 X 10- 3

Nil.' = 2 X 10-5

N K1 = 4 x 10- 10

Np.l = 4 x 1O-11

NK~ = 8 X 10- 15

Np.~ = 8 X 10- 17

(17)

illustrating that the surface viscous stresses are negligible compared to bulk viscous forces for such small values of the surfacerheological parameters. But for the 500 Adroplet

N" = 200

Nil.' = 2 Nil. I = 0.4 Np.~ = 0.08. (18)

Not only are the surface viscous forces considerably larger forthe small "large curvature" droplet, but the first and second moment surface viscous stresses, which are negligible for the largedroplet approach the order of the zeroth moment surface viscousforces for the small droplet.

It would then appear that the significance of surface rheologicalproperties to microscale hydrodynamic flows near the fluid interface should not be assessed without considering surface curvature.


SURFACE RHEOLOGY III 479

CREEPING FLOW OVER AN EMULSION DROPLET

Consider the creeping flow of an unbounded incompressibleNewtonian fluid over a microemulsion droplet (Fig. 2) as an example of large surface curvature and its effects upon flow nearthe fluid surface through the first and second moments of surface viscosity.

The purely rheological flow problem (i.e., neglecting surfacetension gradients) was first considered by Boussinesq" using theBoussinesq surface model." Agrawal and Wasan" later obtainedthe same result using Scriven's surface model." Both solutionswere applicable to spherical droplets of moderate to large curvature radii.

The equations of motion for flow in the droplet phase and inthe continuous phase yield the following general solutions"

vr = U cos O[C + Dr2] (19)

Va = U sin O[-C - 2Dr 2] (20)

and

u, = U cos o[1 + ~ + ~]

V 8 = U sin o[ -1 +~ - ~]2r 3 2r

where A, B, C, and D are constants.

J!.

Fig. 2. Streaming flow over a microemuleion droplet.

(21)

(22)



The stress boundary condition at the droplet surface followsfrom (3), which together with the condition of velocity continuityprovides

V r = Or = 0 f/ r = a

Vo = 00 V r = a

(23)

(24)

and

A a(00) a (vo) _ (K: + j.t;) a ( 1 a ( , (J))1Jil- - - 1Jil- - - - --,-- Vo smar r ar r a iJ(J a sm (J a(J

,Vo _+ 2JLe 2 f/ r - a.a

Substituting (19)-(22) in (23) and (24) yields

(25)

Vr = U cos 0 [ 1 + ~ ( 1 - (;y) -(;Yl (26)

• Vo = U sin (J [ -1 - :r ( 1 + (;y) - ~ (;Yl (27)

Or = U cos 0 [ - ~ (1- (~y) - ~ (1 - (~y)1 (28)

and

Using (27) and (29) in (25) yields

vr = cos (J [1 - r - ~ f(l - r)G 1 (30)

t, = sin () [ -1 - ~ r + : f(l + r)G 1 (31)

s, = cos 0 [ - ~ (1 - f-2) + (1 - f- 2)G 1 (32)

and


where

and

and


G = 37i + 2 + 21<:3(p. + 1) + 2K:

vv =-u

f=~r

p.p. =

p.

481

(34)

(35a)

(35b)

(35c)

(35d)

The terminal velocity of a rising or falling droplet may be determined from (30)-(33) by balancing the pressure forces on thedroplet surface with the bouyancy forces to obtain

(36)

(37)~ (~~+p.+p.)

U-~ 2~ _3a_-3 a g2 .

p. -;; K s + 2p. + 3p.

The small curvature form of (36) (limit of vanishingly small firstand second moment quantities) is of course identical to theBoussinesq solution (37).

Due to the symmetry of the spherical fluid surface, viz.,

Is = -Rb = R 2c ,



where Is> band c are the first, second, and third fundamentaldyadics of the surface, respectively, the higher moments of surface viscosity do not independently introduce new surface viscousstress, but act to alter the "effective" viscosity of the surface,this being evident in Eqs. (1), (3), and (4) as well as (36). Thissymmetry is of course exceptional.

The dramatic effects of surface curvature indicated by dimensional analysis are not however directly apparent in (36). This isprimarily due to the unbounded nature of the flow.

The classical bounded flow problem of a spherical fluid dropletsettling within a concentric spherical shell (Fig. 3) may be solvedsimilar to (36) (see Happel and Brenner") for large surface curvature to yield

where

(

2 K: ~ )--+ I-t+ I-tU - 2 2P - P 3 a 1--a --g -

3 I-t 2 s 2 3~ K-K e + I-t + I-ta

(38)

K= y+z (39)

Fig. 3. Sedimentation of droplet in solid shell.


with

and

¥=

SURFACE RHEOLOGY III 483

Here

aA ~b'

In Figure 4 we plot the effective surface dilatational viscosityversus sedimentation velocity for various droplet radii, illustrating quantitatively the significance which even very small surfacerheological parameters may exhibit for large surface curvature.

1.2

•::~

1.0

•0~

0.8,:~HU a••0~~

>

· a .• m!!a A::~ 1 100 A 11•" 2 1.0 lUI 7•~ 0.2 3 1.0 •• 1s0

p-~.O.21/cc.~ u-i)'-lcP. :\-0.9~

SURFACE DILATATIOHAL VISCOSITY, K: (Ip)

Fig. 4. Sedimentation VB. effective surface viscosity.



CONCLUSIONS

We have developed explicit surface stress boundary conditionsfor the highly curved spherical fluid surface. The static limit ofthe equations corresponds to the theory of Murphy." The smallcurvature limit of the equations is similar to the BoussinesqScriven form2--4 with the addition of the surface excess normalforce and in the absence of normal surface velocities.

We have, through dimensional considerations, shown largesurface curvature, in general, to have a marked effect upon surface viscous stresses, so that even very small surface rheologicalcoefficients may result in considerable surface stress for highlycurved fluid surfaces.

Finally, we have illustrated quantitatively the influence ofzeroth and higher moments of surface viscosity upon thebounded and unbounded creeping flow over a spherical microemulsion droplet.

This work was supported by the National Science Foundation. One of us (D. E.)

was also supported by an Amoco Foundation Doctoral Fellowship.

References

1. D.A. Edwards and D. T. Wasan, J. Rheol., 32,473 (1988).2. M.J. Boussinesq, Ann. Chim. Phys., 29,349 (1913a).3. L. E. Scriven, Chern, Eng. Sci., 12, 98 (1960).4. J.e. Slattery, Chem. Eng, Sci., 19,379 (1964).5. B, P. Radoev, D. S. Dimitrov, and I. B. Ivanov, Call. Polym. Sci., 252,

50 (1974).6. I. B. Ivanov and D. S. Dimitrov, Call. Po/ym. Sci., 252,982 (1974).7. A. Scheludko, D. Platikanov, and E. Manev, Disc. Faraday Soc., 40,

253 (1965).8. e. A. Miller, Surface Colloid Sci., 10, 227 (1978).9. Z. Zapryanov, A. K. Maholtra, N. Aderangi, and D. T.Wasan, Int.J. Mult.

Flow, 9, 105 (1983).10. M. J. Boussinesq, Ann. Chim. Phys., 29, 357 (1913b).11. S.K. Agrawal and D. T. Wasan, Chem. Eng. J., 18,215 (1979).12, J. C. Slattery, Momentum, Energy and Mass Transfer in Continua, Krieger,

Huntington, New York, 1981, p. 111.13. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics,

Martinus Nijhof Publishers, Boston, 1983, p. 130.14. e.J. Murphy, Ph.D. thesis, University of Minnesota (1966).

Received February 2,1986Accepted December 7,1987
