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Exact and approximate boundary conditions of a fluid interface
Exact and Approximate Boundary Conditions of a Fluid Interface A. N. GARAZO, I V. A. K U Z / AND M. A. VILA 2 Instituto de F[sica de Liquidos y Sistemas Bioldgicos, IFLYSIB, 1900 La Plata, Argentina Received March 3, 1986; accepted September 30, 1986 A numerical study of the field velocity for an inhomogeneous fluid is performed. The mentioned fluid is in a stationary flow regime within a canal. When hydrodynamic equations for inhomogeneous systems are used and a special shear viscosity profile is provided, a boundary condition is obtained to integrate once the momentum balance equation. When hypotheses about the tangential and normal velocity gradients' behavior are accepted, the boundary condition takes a simple form and it is possible for the surface shear viscosity coefficient to be defined as an integral along the inhomogeneous region. A discrepancy of 4% is found between the exact and the approximate boundary conditions proposed by F. C. Goodrich (Proc. R. Soc. London Set. A 374, 341 (1981)) when a special viscosity shape profile of adjustable intensity is used. © 1987Academicpress,Inc. I. INTRODUCTION
The important role that boundary conditions play in the solutions to a wide variety of hydrodynamic problems is well known. Among them must be mentioned those concerned with hydrodynamic instabilities, such as Rayleigh-Benard-Marangoni (1) and the theory of capillary wave propagation (2). It is in our interests to elucidate the meaning of a boundary condition for a specific hydrodynamic problem. The interface as a finite phase confined between two liquids or one liquid and its vapor is considered here. Also, the application of ordinary phenomenological hydrodynamic equations in this region is supposed valid, as was shown by Alder and Alley (3). Considering the interface as a finite region goes back to van der Waals ideas. Historically the interface was simply thought of as a mathematical surface having a constant tension as its most notable property. Most recently, the relevant role played by viscosity in the inter1 Fellowship and Member of the Consejo Nacional de Investigaciones Cientificas y Trcnicas (CONICET), respectively. 2 Fellowship of the Comisi6n de Investigaciones Cientlficas Pcia. Bs. As, (CIC).
facial rheology, even in pure fluids, was shown (4). In the context of the thermodynamics of irreversible processes, Bedeaux et aL (5), considering some of the interfacial properties as singular ones, found a boundary condition where thermal and viscoelastic aspects of the interface are taken into account. In these works mentioned above a common characteristic is the two-dimensional view of the interface: therefore it follows that interfacial parameters are effective values. An interface with thickness is characterized at each point by continuous functions with continuous derivatives. These functions, coefficients in the plane hydrodynamics, are z dependent, z being the normal direction toward the interface. The system has a cylindrical symmetry. Because the characteristic interfacial properties change at each point, it is convenient to define surface properties as integrals along z, as was done by Goodrich (6) within the phenomenological approach and by Baus and Tejero (7) within the framework of generalized hydrodynamics. In these works just mentioned a central hypothesis is assumed: the smoothness of the field velocity results in the indetermination of the tangential second derivative of the velocity
49 Journalof ColloidandInterfaceScience, Vol. 119,No. 1, September1987
0021-9797/87 $3.00 Copyright© 1987by AcademicPress,Inc. Allfightsof reproductionin any form reserved,
50
GARAZO, KUZ, A N D VILA
(see Ref. (6), p. 345). This hypothesis is a sine qua non condition for defining integral properties, the rheological coefficients, along the inhomogeneous region. Here we study the validity of these hypotheses in the light of the numerical solution to the hydrodynamic equations for stationary flow inside a canal. If the hypotheses referred to are valid, then the boundary condition obtained through the two- and three-dimensional hydrodynamic formulations will be equivalent, giving the last one explicit information about the structure of the surface coefficient. In Section II the problem is established and the differential equations are given. Section III is devoted to the numerical analysis of the exact and the approximate boundary conditions, when different surface viscosity profiles are employed. Also, comparing the two- and three-dimensional hydrodynamical models, an effective surface shear viscosity value is obtained. Finally some concluding remarks are added. II. S T A T I O N A R Y F L O W OF A N I N H O M O G E N E O U S F L U I D IN A C A N A L
Let there be two fluid phases limiting an inhomogeneous interface in the z direction. The liquids are placed in a canal with fixed lateral walls. The bottom moves with a given horizontal velocity profile and the flow is considered to be in a stationary regime. The fluids are stuck at the motionless lateral walls, while only the central bottom's fluid is supposed to move at velocity Vb. The free upper side is at rest. The equation of motion considering the shear viscosity z dependence (8) for an incompressible fluid is
Ov o(z) -~ (x, y, z; t) = -grad p + div{2~/(z)(Grad v)s},
[ 1]
where (grad v)~,~ = ½[(OvJOx~) + (OvjOx~)] - ½6~ ~3=1 (Ov,dOx.O(a,/~ = 1, 2, 3) v is the velocity of the fluid particle, and p is the pressure. Journal of Colloid and Interface Science, Vol. 119,No. l, September 1987
,Since the canal is unlimited in the x direction, the fluid motion will be only (y, z) dependent. Then Eq. [1] in the stationary regime becomes
02Vx
O/
OVx
z)) = O,
n(z)T(y, z) + Tzz[.(z)-£z (y, _
[21
where the system is not supposed to be acted upon by any external gradient of pressure, Vp --0.
If Eq. [2] is integrated once along z, it becomes
OVx
o--/[,-n,OVx]
Cn
O2V~ n(z)Taz.
[31
This equation represents the exact boundary condition and will be used later. Assuming that the velocity field can be written in a separable form, V(y, z) = F(y). G(z), Eq. [2] is transformed into a pair of differential equations, one for F(y) and another for G(z), with the following boundary conditions: (i) G(0). F ( 0 ) = Vb,
the velocity in the middle point of the canal is Vb.
(ii) F(+-yo/2) = 0,
the liquid is stuck at the lateral walls.
(iii) G(oo)= 0,
far from the floor the fluid velocity is null.
The solution for F(y) fulfilling conditions (i) and (ii) is
F(y) = Vb/a(o). cos(V~ly),
[41
where only the first mode (n = 0) of the general solution ( f ~ = (2n + 1)r/y0) is considered. This mode selection imposes a reasonable trigonometric profile for each constant z plane. On the other hand the differential equation in the z direction is
d2f (z) dz 2 -}-d ( z ) ~ -
Xlf(z) = 0,
[51
where Xl = (~r/yo),z d(z) is the logarithmic derivative of n(z), and f(z) = G(z)/G(O) is an
BOUNDARY
CONDITIONS OF FLUID INTERFACE
51
[6]
sponds to the "velocity macroscopically observed." In an interface with thickness this is a vague affirmation. In our calculation we show that it is possible by trial and error to make a suitable choice for point Zo in such a way that the approximate boundary condition, Eq. [3], is self-satisfied, at least for one of the shear viscosity profiles proposed. Numerical evaluations of Eq. [6] are gathered in Tables 1 and II, for different shear viscosity profiles, at the middle point y = 0 of the inhomogeneous region. Here we examine the results obtained from Tables I and II. From the first, if a hyperbolic profile is used for modeling the shear viscosity for any value of z = z0, within the interface, the approximation concerned is not a good one. In particular, when z0 takes values as zl or zn, limit points of the interfacial region, a poor correspondence is observed. Table II reflects a different situation. When z0 = &i, Zn being the most distant point from the moving floor, the fight- and the left-hand sides of Eq. [6] agree within a mean error of 4%, for each of the 3' values considered; 3' is the amplitude of the maximum viscosity value.
=E 1 where n~ r;(z)dz is the surface shear viscosity and z~, zn show the position where the viscosity becomes constant. In Goodrich's work it is not indicated at which point of the interface the evaluation of 02 Vx(y, Zo)/ay 2 is performed; he asserts that the point Zo corre-
In this simple problem we compare the exact and the approximate boundary conditions for a fluid interface. The exact condition was obtained by integrating once the corresponding hydrodynamic
auxiliary function. The boundary conditions (i) and (iii) are simply f ( 0 ) = 1 andf(oo) -- 0. Equation [5] was solved numerically (9) for each of the different shear viscosity profiles n(z) (see details of viscosity shear profiles in the appendix). A mesh of 3000 points was used for a canal 100 A deep, 200 A wide, and with an interfacial thickness of 10 A. The field velocity found will be used in the next section for discussing one dissipative aspect of the fluid-fluid boundary condition. Ill. B O U N D A R Y C O N D I T I O N S A N D T H E I R NUMERICAL EVALUATIONS
Here we consider the difference between the exact and the approximate boundary conditions proposed by Goodrich, in light of the numerical solution of the field velocity, for an inhomogeneous fluid-fluid interface in stationary motion. When the tangential velocity gradient OV~/Oy is assumed to remain a smooth function of z while the normal 0 Vx/ Oz is assumed to change rapidly, Eq. [3] then becomes the approximate boundary condition nii
n
-
fix OVx Oz i -~-ns
02V~(zo) Oy 2
'
IV. C O N C L U D I N G R E M A R K S
TABLE I Numerical Results of the Exact (Eq. [3]) and Approximate Boundary Conditions (Eq. [6]) for a S m o o t h Shear Viscosity Profile Ao = 0.0031392 ~
A1 (zl) = 0.0017676 b
A1 (Zil) = 0.0012606
e~,(z0 = 43.7% C
e~,(~,o = 59.8%
Note. F o r any z value within the interphase, a p o o r correspondence is observed. Here we show this for z = z~ and z = ZlI. a AO = nix (0/)/92)111 _ ~/[ (ou/oz)h" b
~,(zo) = - . . (0~v/oy')(0, zo);., = Jzi'I.(z)dz.
Ce ~ = ((2xI - 2~o)/Ao). I00. Journal of Colloid and InterfaceScience, Vol. 119, No. 1, September 1987
52 ¸
G A R A Z O , KUZ, A N D VILA TABLE II The Same Boundary Conditions as in Table I for an Abrupt Viscosity Profile 3' Dependent
20 50 100 500 1000
0.004914 0.008595 0:012639 0.02302 0.026149
0.005872 0.01046 0.016 0.03784 0.054695
0.00465 0.008198 0.01211 0.02207 0.02506
19.5 21.7 26.6 64.4 109,2
5.4 4.6 4.2 4.1 4.2
Note. The amplitude of the m a x i m u m value, 3,, ranging from 20 to 1000. W h e n z0 = zn, the m e a n percentage error is 4%, while when z0 = z~, the error systematically increases. ~.b,~See footnotes to Table I.
as is shown in Fig. 1. This agreement is lost when the monotonous profile (e.g., hyperbolic tangent) is used. Also, it must be mentioned that the abrupt profile represents a restricted form of the viscosity profile, and it could be possible, at first, to propose a profile such that both boundary conditions agree exactly. It must be borne in mind that there is neit h e r a microscopic theory, as there is for the elastic coefficients (10) nor experimental measurements that could be invoked in favor of an abrupt profile for viscosity. We outline the following arguments as a reasonable physical viewpoint. I t is k n o w n by experiments (1 l) that the 0.9 viscosity in dense homogeneous fluids agrees monotonously with density. Also it has been shown (12) that some density profiles--oilwater-surfactant interface--functionally be0.6 have like our abrupt viscosity profile. So if visN cosity also follows density in the inhomogeneous region, then it is acceptable to represent the viscosity profile as a bumped function. O, 3 r/(1) \\\\ eff (i) ^ Another remark must be made about the variation of surface viscosity ~s--the integral of the viscosity profile--with the concentration of surfactants c s. At low concentrations 0 30 60 90 Os increase with c s (13) and the interfacial thickness does not change significantly. Thus FIG. 1. f ( z ) = G(z)/G(O) (adimensional velocity), (a) our bumped profile is capable to account suitwhen the interfacehas a finitewidth and the shearviscosity ably for this behavior. is represented by an abrupt profile with maximum amIn conclusion we can say that it is reasonable plitudes, 3,°~ = 20 and 3`(2)= 1000, and (b) when the into identify a real interface with a two-dimenterface is represented by a mathematical surface sional surface endowed with effective propwith an effectiveshear viscosity, ~ce ('~ = Jzl rz. ~(z, 3,(~))dz,for erties, such as surface viscosity. i= 1,2. equations for an incompressible fluid. Expression [3], an integro-differential equation results. Generally speaking, this kind of equation is hard to manage and of scanty use in the solutions to hydrodynamic problems. In order to make its solution feasible we propose two alternative viscosity profiles, one smooth and one abrupt, These solutions are then compared with those approximated and proposed by Goodrich. We find a mean error of 4%, when the sharp -viscosity profile is considered. Also, the resultant velocity fields are in accordance,
zci
Journal of Colloid and Interface Science. Vol. 119,No. 1, September1987
BOUNDARY APPENDIX:
VISCOSITY
CONDITIONS
OF FLUID
53
INTERFACE
PROFILES
Clearly knowledge of V(Z) is necessary for solving the differential equation for z. Provided that this information is in general not accessible, we propose different profiles for considering the behavior of the shear viscosity at the inhomogeneous region. Two types of analytical functions are used. First we employ a hyperbolic tangent in order to make a smooth interpolation between the two bulk values of viscosity qI and qI1. Later we consider an abrupt profile with a peak value the location and intensity of which may be modified freely.
-52Y
50
Z(i) a. Hyperbolic ProJile The interfacial gap among the shear viscosities rlI and vII was interpolated through the function q(z) = &h(0.3(z-
FIG. 3. An abrupt viscosity profile given by the following function:
17(z)=
(z - 30)+(b1’7H=-3°) + 1, 30 G z < 37 i 5[(40 - Z)%-(b~3X40-=)+ I],
37 c z< 40
and the logarithmic derivative
35)) + 21
[b,(l-(z-30)/7)/(z-30)+e’b1’7X’-30’/
of which the logarithmic
derivative d(z) is d(z) =
(z - 30y,
30 G z < 37
b2((40 - z)/3 - 1)/(40 - z) + e(b2’3K40-Z)/ (40 - zp-‘,
37 < z< 40.
d(z) = 0.3 sech’ X (0.3(2 - 35))/[2 + th(0.3(z- 35))]. The function r](z) and its derivative are represented in Fig. 2. b. Sharp Projile
-a,51 * * . 27
30,2
.
, * s * * I
33,4
36,6
39.8
43
26) FIG. 2. The function that represents a smooth shear viscosity profile:
q(z) = th[0.3(z - 35)] + 2. The logarithmic derivative also appears: d(z) = 0.3 sech2[0.3(z - 35)J 2 + th[0.3(z - 35)] .
In the present case an abrupt profile is considered. The maximum viscosity value ~(z,,,) rz qmax= yqI depends on the arbitrary election of the y parameter. This parameter ranged from 20 to 1000, looking at the possibility of surface viscosity modifications due to surfactants. The shear viscosity of region II, qII, was chosen to be five times VI= 1. A viscosity profile fulfilling these requirements for z,, = 37 A is Journal ofcolloid
and Interface Science, Vol. 119, No. 1, September 1987
54
GARAZO, KUZ, AND VILA ~I[(Z --
n(z)={
30)bi8 -bl(z-30)/7 "q- l ], 30~
5~/I[(40 -- z)b2e -b2(40-z)/3 q- 1 ],
37~z~<40. The coefficients bl = In(3' - 1)/ln(7/e) and b2 = ln((~/5) - 1)/ln(3/e) c o m e from the continuity o f ~(z). The function ~(z) and its derivative, is shown in Fig. 3. ACKNOWLEDGMENT The authors thank Dr. Antonio E. Rodriguez for several related discussions. REFERENCES 1. Sorensen, T. S., in "Lectures Notes in Physics 105, Dynamics and Instability of Fluid Interfaces" (T. S. Sorensen, Ed.), pp. 1-74. Springer-Verlag, New York/Heidelberg/Berlin, 1978; Scriven, L. E., and Sternling, C. V., J. FluidMech. 19, 321 (1964).
Journal of Colloid and Interface Science, Vol. 119, No. 1, September 1987
2. Levich, V. G., "Physicochemical Hydrodynamics," p. 591. Prentice-Hall, Englewood Cliffs, NJ, 1962. 3. Alder, B. J., and Alley, W. E., Phys. Today January, 56 (1984). 4. Vila, M. A., Kuz, V. A., and Rodriguez, A. E., J. Colloid Interface Sci. 107, 314 (1985). 5. Bedeaux, D., Albano, A. M., and Mazur, P., Physica 82A, 438 (1976); Bedeaux, D., and Oppenheim, I. Physica 90A, 39 (1978). 6. Goodrich, F. C., Proc. R. Soc. London Ser. A 374, 341 (1981). 7. Baus, M., and Tejero, C. F., Chem. Phys, Lett. 84, 222 (1981). 8. de Groot, S. R., and Mazur, P., "Non Equilibrium Thermodynamics," p. 309. Nortb-Holland, Amsterdam, 1969. 9. Pereyra, V., Lect. Notes Comput. Sci. 76, 67 (1978). 10. Baus, M., 9". Chem. Phys, 76, 2003 (1982). 11. Egelstaff, P. A., and Ring, J. W., "Physics of Simple Liquids," p. 289. North-Holland, Amsterdam, 1968; Van Den Berg, H. R., and Trappeniers, N. J., Chem. Phys. Lett. 58, 12 (1978); Iwasaki, H., and Takahashi, M., J. Chem. Phys. 74, 1930 (1980). 12. Borzi, C., Lipowsky, R., and Widom, B., Faraday Symp. Chem. Soc. 20, 1 (1985). 13. See Ref. (1), p. 205.
The important role that boundary conditions play in the solutions to a wide variety of hydrodynamic problems is well known. Among them must be mentioned those concerned with hydrodynamic instabilities, such as Rayleigh-Benard-Marangoni (1) and the theory of capillary wave propagation (2). It is in our interests to elucidate the meaning of a boundary condition for a specific hydrodynamic problem. The interface as a finite phase confined between two liquids or one liquid and its vapor is considered here. Also, the application of ordinary phenomenological hydrodynamic equations in this region is supposed valid, as was shown by Alder and Alley (3). Considering the interface as a finite region goes back to van der Waals ideas. Historically the interface was simply thought of as a mathematical surface having a constant tension as its most notable property. Most recently, the relevant role played by viscosity in the inter1 Fellowship and Member of the Consejo Nacional de Investigaciones Cientificas y Trcnicas (CONICET), respectively. 2 Fellowship of the Comisi6n de Investigaciones Cientlficas Pcia. Bs. As, (CIC).
facial rheology, even in pure fluids, was shown (4). In the context of the thermodynamics of irreversible processes, Bedeaux et aL (5), considering some of the interfacial properties as singular ones, found a boundary condition where thermal and viscoelastic aspects of the interface are taken into account. In these works mentioned above a common characteristic is the two-dimensional view of the interface: therefore it follows that interfacial parameters are effective values. An interface with thickness is characterized at each point by continuous functions with continuous derivatives. These functions, coefficients in the plane hydrodynamics, are z dependent, z being the normal direction toward the interface. The system has a cylindrical symmetry. Because the characteristic interfacial properties change at each point, it is convenient to define surface properties as integrals along z, as was done by Goodrich (6) within the phenomenological approach and by Baus and Tejero (7) within the framework of generalized hydrodynamics. In these works just mentioned a central hypothesis is assumed: the smoothness of the field velocity results in the indetermination of the tangential second derivative of the velocity
49 Journalof ColloidandInterfaceScience, Vol. 119,No. 1, September1987
0021-9797/87 $3.00 Copyright© 1987by AcademicPress,Inc. Allfightsof reproductionin any form reserved,
50
GARAZO, KUZ, A N D VILA
(see Ref. (6), p. 345). This hypothesis is a sine qua non condition for defining integral properties, the rheological coefficients, along the inhomogeneous region. Here we study the validity of these hypotheses in the light of the numerical solution to the hydrodynamic equations for stationary flow inside a canal. If the hypotheses referred to are valid, then the boundary condition obtained through the two- and three-dimensional hydrodynamic formulations will be equivalent, giving the last one explicit information about the structure of the surface coefficient. In Section II the problem is established and the differential equations are given. Section III is devoted to the numerical analysis of the exact and the approximate boundary conditions, when different surface viscosity profiles are employed. Also, comparing the two- and three-dimensional hydrodynamical models, an effective surface shear viscosity value is obtained. Finally some concluding remarks are added. II. S T A T I O N A R Y F L O W OF A N I N H O M O G E N E O U S F L U I D IN A C A N A L
Let there be two fluid phases limiting an inhomogeneous interface in the z direction. The liquids are placed in a canal with fixed lateral walls. The bottom moves with a given horizontal velocity profile and the flow is considered to be in a stationary regime. The fluids are stuck at the motionless lateral walls, while only the central bottom's fluid is supposed to move at velocity Vb. The free upper side is at rest. The equation of motion considering the shear viscosity z dependence (8) for an incompressible fluid is
Ov o(z) -~ (x, y, z; t) = -grad p + div{2~/(z)(Grad v)s},
[ 1]
where (grad v)~,~ = ½[(OvJOx~) + (OvjOx~)] - ½6~ ~3=1 (Ov,dOx.O(a,/~ = 1, 2, 3) v is the velocity of the fluid particle, and p is the pressure. Journal of Colloid and Interface Science, Vol. 119,No. l, September 1987
,Since the canal is unlimited in the x direction, the fluid motion will be only (y, z) dependent. Then Eq. [1] in the stationary regime becomes
02Vx
O/
OVx
z)) = O,
n(z)T(y, z) + Tzz[.(z)-£z (y, _
[21
where the system is not supposed to be acted upon by any external gradient of pressure, Vp --0.
If Eq. [2] is integrated once along z, it becomes
OVx
o--/[,-n,OVx]
Cn
O2V~ n(z)Taz.
[31
This equation represents the exact boundary condition and will be used later. Assuming that the velocity field can be written in a separable form, V(y, z) = F(y). G(z), Eq. [2] is transformed into a pair of differential equations, one for F(y) and another for G(z), with the following boundary conditions: (i) G(0). F ( 0 ) = Vb,
the velocity in the middle point of the canal is Vb.
(ii) F(+-yo/2) = 0,
the liquid is stuck at the lateral walls.
(iii) G(oo)= 0,
far from the floor the fluid velocity is null.
The solution for F(y) fulfilling conditions (i) and (ii) is
F(y) = Vb/a(o). cos(V~ly),
[41
where only the first mode (n = 0) of the general solution ( f ~ = (2n + 1)r/y0) is considered. This mode selection imposes a reasonable trigonometric profile for each constant z plane. On the other hand the differential equation in the z direction is
d2f (z) dz 2 -}-d ( z ) ~ -
Xlf(z) = 0,
[51
where Xl = (~r/yo),z d(z) is the logarithmic derivative of n(z), and f(z) = G(z)/G(O) is an
BOUNDARY
CONDITIONS OF FLUID INTERFACE
51
[6]
sponds to the "velocity macroscopically observed." In an interface with thickness this is a vague affirmation. In our calculation we show that it is possible by trial and error to make a suitable choice for point Zo in such a way that the approximate boundary condition, Eq. [3], is self-satisfied, at least for one of the shear viscosity profiles proposed. Numerical evaluations of Eq. [6] are gathered in Tables 1 and II, for different shear viscosity profiles, at the middle point y = 0 of the inhomogeneous region. Here we examine the results obtained from Tables I and II. From the first, if a hyperbolic profile is used for modeling the shear viscosity for any value of z = z0, within the interface, the approximation concerned is not a good one. In particular, when z0 takes values as zl or zn, limit points of the interfacial region, a poor correspondence is observed. Table II reflects a different situation. When z0 = &i, Zn being the most distant point from the moving floor, the fight- and the left-hand sides of Eq. [6] agree within a mean error of 4%, for each of the 3' values considered; 3' is the amplitude of the maximum viscosity value.
=E 1 where n~ r;(z)dz is the surface shear viscosity and z~, zn show the position where the viscosity becomes constant. In Goodrich's work it is not indicated at which point of the interface the evaluation of 02 Vx(y, Zo)/ay 2 is performed; he asserts that the point Zo corre-
In this simple problem we compare the exact and the approximate boundary conditions for a fluid interface. The exact condition was obtained by integrating once the corresponding hydrodynamic
auxiliary function. The boundary conditions (i) and (iii) are simply f ( 0 ) = 1 andf(oo) -- 0. Equation [5] was solved numerically (9) for each of the different shear viscosity profiles n(z) (see details of viscosity shear profiles in the appendix). A mesh of 3000 points was used for a canal 100 A deep, 200 A wide, and with an interfacial thickness of 10 A. The field velocity found will be used in the next section for discussing one dissipative aspect of the fluid-fluid boundary condition. Ill. B O U N D A R Y C O N D I T I O N S A N D T H E I R NUMERICAL EVALUATIONS
Here we consider the difference between the exact and the approximate boundary conditions proposed by Goodrich, in light of the numerical solution of the field velocity, for an inhomogeneous fluid-fluid interface in stationary motion. When the tangential velocity gradient OV~/Oy is assumed to remain a smooth function of z while the normal 0 Vx/ Oz is assumed to change rapidly, Eq. [3] then becomes the approximate boundary condition nii
n
-
fix OVx Oz i -~-ns
02V~(zo) Oy 2
'
IV. C O N C L U D I N G R E M A R K S
TABLE I Numerical Results of the Exact (Eq. [3]) and Approximate Boundary Conditions (Eq. [6]) for a S m o o t h Shear Viscosity Profile Ao = 0.0031392 ~
A1 (zl) = 0.0017676 b
A1 (Zil) = 0.0012606
e~,(z0 = 43.7% C
e~,(~,o = 59.8%
Note. F o r any z value within the interphase, a p o o r correspondence is observed. Here we show this for z = z~ and z = ZlI. a AO = nix (0/)/92)111 _ ~/[ (ou/oz)h" b
~,(zo) = - . . (0~v/oy')(0, zo);., = Jzi'I.(z)dz.
Ce ~ = ((2xI - 2~o)/Ao). I00. Journal of Colloid and InterfaceScience, Vol. 119, No. 1, September 1987
52 ¸
G A R A Z O , KUZ, A N D VILA TABLE II The Same Boundary Conditions as in Table I for an Abrupt Viscosity Profile 3' Dependent
20 50 100 500 1000
0.004914 0.008595 0:012639 0.02302 0.026149
0.005872 0.01046 0.016 0.03784 0.054695
0.00465 0.008198 0.01211 0.02207 0.02506
19.5 21.7 26.6 64.4 109,2
5.4 4.6 4.2 4.1 4.2
Note. The amplitude of the m a x i m u m value, 3,, ranging from 20 to 1000. W h e n z0 = zn, the m e a n percentage error is 4%, while when z0 = z~, the error systematically increases. ~.b,~See footnotes to Table I.
as is shown in Fig. 1. This agreement is lost when the monotonous profile (e.g., hyperbolic tangent) is used. Also, it must be mentioned that the abrupt profile represents a restricted form of the viscosity profile, and it could be possible, at first, to propose a profile such that both boundary conditions agree exactly. It must be borne in mind that there is neit h e r a microscopic theory, as there is for the elastic coefficients (10) nor experimental measurements that could be invoked in favor of an abrupt profile for viscosity. We outline the following arguments as a reasonable physical viewpoint. I t is k n o w n by experiments (1 l) that the 0.9 viscosity in dense homogeneous fluids agrees monotonously with density. Also it has been shown (12) that some density profiles--oilwater-surfactant interface--functionally be0.6 have like our abrupt viscosity profile. So if visN cosity also follows density in the inhomogeneous region, then it is acceptable to represent the viscosity profile as a bumped function. O, 3 r/(1) \\\\ eff (i) ^ Another remark must be made about the variation of surface viscosity ~s--the integral of the viscosity profile--with the concentration of surfactants c s. At low concentrations 0 30 60 90 Os increase with c s (13) and the interfacial thickness does not change significantly. Thus FIG. 1. f ( z ) = G(z)/G(O) (adimensional velocity), (a) our bumped profile is capable to account suitwhen the interfacehas a finitewidth and the shearviscosity ably for this behavior. is represented by an abrupt profile with maximum amIn conclusion we can say that it is reasonable plitudes, 3,°~ = 20 and 3`(2)= 1000, and (b) when the into identify a real interface with a two-dimenterface is represented by a mathematical surface sional surface endowed with effective propwith an effectiveshear viscosity, ~ce ('~ = Jzl rz. ~(z, 3,(~))dz,for erties, such as surface viscosity. i= 1,2. equations for an incompressible fluid. Expression [3], an integro-differential equation results. Generally speaking, this kind of equation is hard to manage and of scanty use in the solutions to hydrodynamic problems. In order to make its solution feasible we propose two alternative viscosity profiles, one smooth and one abrupt, These solutions are then compared with those approximated and proposed by Goodrich. We find a mean error of 4%, when the sharp -viscosity profile is considered. Also, the resultant velocity fields are in accordance,
zci
Journal of Colloid and Interface Science. Vol. 119,No. 1, September1987
BOUNDARY APPENDIX:
VISCOSITY
CONDITIONS
OF FLUID
53
INTERFACE
PROFILES
Clearly knowledge of V(Z) is necessary for solving the differential equation for z. Provided that this information is in general not accessible, we propose different profiles for considering the behavior of the shear viscosity at the inhomogeneous region. Two types of analytical functions are used. First we employ a hyperbolic tangent in order to make a smooth interpolation between the two bulk values of viscosity qI and qI1. Later we consider an abrupt profile with a peak value the location and intensity of which may be modified freely.
-52Y
50
Z(i) a. Hyperbolic ProJile The interfacial gap among the shear viscosities rlI and vII was interpolated through the function q(z) = &h(0.3(z-
FIG. 3. An abrupt viscosity profile given by the following function:
17(z)=
(z - 30)+(b1’7H=-3°) + 1, 30 G z < 37 i 5[(40 - Z)%-(b~3X40-=)+ I],
37 c z< 40
and the logarithmic derivative
35)) + 21
[b,(l-(z-30)/7)/(z-30)+e’b1’7X’-30’/
of which the logarithmic
derivative d(z) is d(z) =
(z - 30y,
30 G z < 37
b2((40 - z)/3 - 1)/(40 - z) + e(b2’3K40-Z)/ (40 - zp-‘,
37 < z< 40.
d(z) = 0.3 sech’ X (0.3(2 - 35))/[2 + th(0.3(z- 35))]. The function r](z) and its derivative are represented in Fig. 2. b. Sharp Projile
-a,51 * * . 27
30,2
.
, * s * * I
33,4
36,6
39.8
43
26) FIG. 2. The function that represents a smooth shear viscosity profile:
q(z) = th[0.3(z - 35)] + 2. The logarithmic derivative also appears: d(z) = 0.3 sech2[0.3(z - 35)J 2 + th[0.3(z - 35)] .
In the present case an abrupt profile is considered. The maximum viscosity value ~(z,,,) rz qmax= yqI depends on the arbitrary election of the y parameter. This parameter ranged from 20 to 1000, looking at the possibility of surface viscosity modifications due to surfactants. The shear viscosity of region II, qII, was chosen to be five times VI= 1. A viscosity profile fulfilling these requirements for z,, = 37 A is Journal ofcolloid
and Interface Science, Vol. 119, No. 1, September 1987
54
GARAZO, KUZ, AND VILA ~I[(Z --
n(z)={
30)bi8 -bl(z-30)/7 "q- l ], 30~
5~/I[(40 -- z)b2e -b2(40-z)/3 q- 1 ],
37~z~<40. The coefficients bl = In(3' - 1)/ln(7/e) and b2 = ln((~/5) - 1)/ln(3/e) c o m e from the continuity o f ~(z). The function ~(z) and its derivative, is shown in Fig. 3. ACKNOWLEDGMENT The authors thank Dr. Antonio E. Rodriguez for several related discussions. REFERENCES 1. Sorensen, T. S., in "Lectures Notes in Physics 105, Dynamics and Instability of Fluid Interfaces" (T. S. Sorensen, Ed.), pp. 1-74. Springer-Verlag, New York/Heidelberg/Berlin, 1978; Scriven, L. E., and Sternling, C. V., J. FluidMech. 19, 321 (1964).
Journal of Colloid and Interface Science, Vol. 119, No. 1, September 1987
2. Levich, V. G., "Physicochemical Hydrodynamics," p. 591. Prentice-Hall, Englewood Cliffs, NJ, 1962. 3. Alder, B. J., and Alley, W. E., Phys. Today January, 56 (1984). 4. Vila, M. A., Kuz, V. A., and Rodriguez, A. E., J. Colloid Interface Sci. 107, 314 (1985). 5. Bedeaux, D., Albano, A. M., and Mazur, P., Physica 82A, 438 (1976); Bedeaux, D., and Oppenheim, I. Physica 90A, 39 (1978). 6. Goodrich, F. C., Proc. R. Soc. London Ser. A 374, 341 (1981). 7. Baus, M., and Tejero, C. F., Chem. Phys, Lett. 84, 222 (1981). 8. de Groot, S. R., and Mazur, P., "Non Equilibrium Thermodynamics," p. 309. Nortb-Holland, Amsterdam, 1969. 9. Pereyra, V., Lect. Notes Comput. Sci. 76, 67 (1978). 10. Baus, M., 9". Chem. Phys, 76, 2003 (1982). 11. Egelstaff, P. A., and Ring, J. W., "Physics of Simple Liquids," p. 289. North-Holland, Amsterdam, 1968; Van Den Berg, H. R., and Trappeniers, N. J., Chem. Phys. Lett. 58, 12 (1978); Iwasaki, H., and Takahashi, M., J. Chem. Phys. 74, 1930 (1980). 12. Borzi, C., Lipowsky, R., and Widom, B., Faraday Symp. Chem. Soc. 20, 1 (1985). 13. See Ref. (1), p. 205.