Langmuir 1996, 12, 5963-5968
5963
Hydrodynamic Interaction of Curved Bodies Allowing Slip on Their Surfaces Olga I. Vinogradova Laboratory of Physical Chemistry of Modified Surfaces, Institute of Physical Chemistry, 31 Leninsky Prospect, 117915 Moscow, Russia Received May 31, 1996. In Final Form: August 26, 1996X The hydrodynamics of liquid confined between two approaching curved solids is revisited. Attention is focused on investigating the role of the geometry of curved bodies allowing slip on their surfaces. We derive the equations for pressure and hydrodynamic resistance force to approach of the bodies. We show that they can both be presented as a product of the Reynolds expressions (for pressure and resistance to motion) and corrections for slippage that depend only on the relationships between the slip lengths and the gap. These corrections are shown to be the same for any configuration geometry. The Reynolds parts of the expressions present, in turn, the products of two factors. One of them reflects external conditions, while another one depends only on the curvature of the two surfaces, the relative orientation of their principal radii of curvature, and on the separation. The crucial observation is that this dependence is expressed only through the invariants of the second-order surface. As an application some cases of special colloidal interest are considered. Our analysis provides a method for simple recalculation of pressure or hydrodynamic force from one configuration geometry, or state of the liquid/solid interface, to another.
I. Introduction Hydrodynamics of a liquid confined between curved surfaces moving toward each other is a subject that has attracted much experimental and theoretical research effort. It is usually assumed that if the gap between the surfaces is small compared to their radii of curvature, then the hydrodynamic behavior conforms to the Reynolds theory. The latter is based on a simplification of the Navier-Stokes equations of continuum hydrodynamics by exploiting the special geometry of the thin film (the celebrated lubrication approximation) and no-slip boundary conditions. The Reynolds theory predicts that the resistance to approach is inversely proportional to the minimum separation between the surfaces.1 Instead of imposing zero velocity on the liquid at the solid surface, it is natural to allow for an amount of slippage, described by a slip length b2,3
vs ) b
∂vs ∂n
(1.1)
where vs is the slip (tangential) velocity on the wall, and the axis n is normal to the surface. In principle, eq 1.1 allows one to change from b f 0 (no-slip) to b f ∞ (no tangential stress). Such a slippage may occur in various (smooth) systems. The best known examples are the following: (1) A liquid flowing over a solid will display a length b comparable to the molecular size (for this case, slippage is usually negligible). However, if this surface is lyophobed, the value of b may attain 10-5-10-4 cm.4-8 X Abstract published in Advance ACS Abstracts, November 1, 1996.
(1) Actually, Reynolds considered only parallel surfaces (see Reynolds, O. Philos. Trans. R. Soc. (London) 1886, 157, 177) and did not state this dependence, although it follows from his theory (and is usually attributed to him, especially in the colloid literature). (2) de Gennes, P.-G. C. R. Acad. Sci. 1979, 288B, 219. (3) Vinogradova, O. I. Langmuir 1995, 11, 2213. (4) Schnell, E. J. Appl. Phys. 1956, 27, 1149. (5) Tolstoi, D. M. Dokl. Acad. Nauk SSSR 1952, 85, 1329. (6) Somov, A. N. Kolloidn. Zh. 1982, 44, 160. (7) Churaev, N. V.; Sobolev, V. D.; Somov, A. N. J. Colloid Interface Sci. 1984, 97, 574. (8) Blake, T. D. Colloids Surf. 1990, 47, 135.
S0743-7463(96)00531-8 CCC: $12.00
(2) A polymer melt flowing over a solid (without any chemical attachment between polymer chains and the wall9), for any shear rate is expected to allow an anomalously large value of b, i.e., ∼10-2 cm.2 Slippage studies are usually performed at fixed film thickness, namely, in plane11,12,15 or cylindrical7,16 capillaries, or some shear flows, as for example two parallel oscillating plates. The capillary experiments often suffer from some defects (see for example refs 12 and 17). However, in spite of these, it is clear that effect of slippage has to be taken into account and that in practice the value of b can be varied over a very wide range, depending on the state of the liquid/solid interface. It is apparent that in the case of slippage the hydrodynamic behavior of a liquid confined between two approaching surfaces should be richer than that in the conventional (no-slip) case because the slip lengths b1 and b2 of the two bodies define two additional characteristic length scales of the problem (besides the minimum separation h). In other words, one can expect the possibility of more complex hydrodynamic behavior of a confined liquid (as compared with the predictions of the Reynolds theory) when h becomes comparable with the slip lengths. The first attempts to explore the influence of slippage on drainage of a thin film between the two spheres have been made in refs 18-21 in connection to the coagulation problem. In previous work3 we extended earlier consid(9) In the case of weakly grafted chains, i.e., when different grafted chains do not overlap, slippage is also possible, but only at high stress. However, in this case b is shear-dependent.10-14 (10) Brochard, F.; de Gennes, P.-G. Langmuir 1992, 8, 3033. (11) Migler, K. B.; Hervet, H.; Leger, L. Phys. Rev. Lett. 1993, 70, 287. (12) Migler, K. B.; Massey, G.; Hervet, H.; Leger, L. J. Phys.: Condens. Matter 1994, 6, A301. (13) Ajdari, A.; Brochard-Wyart, F.; de Gennes, P. G., Leibler, L.; Viovy, J. L.; Rubinstein, M. Physica A 1994, 204, 17. (14) Sung, W. Phys. Rev. E 1995, 51, 5862. (15) Henson, D. J.; Mackay, M. E. J. Reol. 1995, 39, 359. (16) Kissi, N. E.; Piau, J. M. C. R. Acad. Sci. 1989, 309, 7. (17) Alexeyev, A. A.; Vinogradova O. I. Colloids Surf. A: Physicochem. Eng. Aspects 1996, 108, 173. (18) Hocking, L. M. J. Eng. Math. 1973, 7, 207. (19) Potanin, A. A.; Ur’ev, N. B.; Muller, V. M. Kolloidn. Zh. 1988, 50, 493. (20) Vinogradova, O. I. Colloids Surf. A: Physicochem. Eng. Aspects 1994, 82, 247. (21) Vinogradova, O. I. J. Colloid Interface Sci. 1995, 169, 306.
© 1996 American Chemical Society
5964 Langmuir, Vol. 12, No. 24, 1996
erations to the case of arbitrary slip lengths, as well as arbitrary radii of curvature of the spheres. Our results suggested that for some separations there may be discernible deviations from the Reynolds theory due to slippage. Thus, for example, if both spheres allow slip, at some finite thicknesses the hydrodynamic force is no longer inversely dependent on gap, but is inversely proportional to the slip lengths, and only logarithmically dependent on the separation.3,18 Beside the configuration of interacting spheres there are other geometries of colloidal interest. Thus, possible deviations from the Reynolds theory due to slippage can play a dramatic role when surface forces are investigated with the drainage technique,22 as well as in the coagulation processes. For experimental reasons, during the dynamic measurements of forces between bodies it is advantageous to have the configuration of two crossed cylinders.24,26-28 In the process of coagulation of particles the latter can be of any form, size, and orientation. However, this (geometric) aspect of the hydrodynamics of thin films confined between slippery surfaces has been given insufficient attention. The aim of the present paper is to extend and generalize our previous analysis to the case of interaction of two curved (not necessarily spherical) bodies allowing slip on their surfaces. Our paper is arranged as follows. In section 2 we define our two curved surfaces, transform them to the geometry of a curved surface interacting with a plane, and calculate the invariants of this (effective) surface. The solution of the equations of motion is described in section 3. The expressions for pressure and hydrodynamic force are derived. We conclude in section 4 with some discussion of the formulas for pressure and hydrodynamic force. The main result is that the contributions from the geometry of the system and from slippage are separated. The geometry factor is shown to be expressed only through the invariants of a curved surface, while the correction for slippage is shown to be the same for any geometry of configuration. II. Analysis of Curved Surfaces We consider two curved bodies W1 and W2 with orthogonal principal radii of curvature R1-, R1+ and R2-, R2+ (R1- < R2-), respectively, immersed in a Newtonian liquid.29 The bodies are sufficiently rigid so that any deformation of their surfaces due to hydrodynamic stresses is negligible. Surface roughness effects are neglected. Contact can be realized only at a single point. We denote O1 and O2 as the points on bodies W1 and W2 for which the separation distance is least and let h be the distance of closest approach. This distance is supposed to be small compared with the minimum radius of curvature30 of their surfaces (i.e. h , R1-). (22) The drainage technique is also used to measure the viscosity of a thin liquid film.23,24 We do not comment here on the viscosity investigations because according to current notions, in many cases of particular interest slippage is apparent and due to changes in viscosity in a very thin layer adjacent to the wall.2,3,25 If so, the model (1.1) is applicable only for the separations which are greater than the thickness of the layer with modified viscosity. (23) Israelachvili, J. N. J. Colloid Interface Sci. 1986, 110, 263. (24) Chan, D. Y. C.; Horn, R. G. J. Chem. Phys. 1985, 83, 5311. (25) Derjaguin, B. V.; Churaev, N. V. Langmuir 1987, 3, 607. (26) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (27) Claesson, P. M.; Christenson, H. K. J. Phys. Chem. 1988, 92, 1650. (28) The configuration of the two spheres is also applicable, see Parker, J. L. Langmuir 1992, 8, 551. (29) Superscripts - and + refer to the minimum and maximum radii, respectively. (30) The curvature is defined as the inverse of the radius of curvature.
Vinogradova
Rectangular (Cartesian) coordinates are defined as follows: the origin of coordinates is at O2; the axis Z coincides with the normal to W2 and is oriented toward O1; the plane Z ) 0 is the plane tangent to W2; the plane Z ) h is the plane tangent to W1; the axes X2 and Y2 are taken in the planes of the principal radii of curvature W2, i.e., they are the principal axes of curvature of W2; the axes X1 and Y1 are the principal axes of curvature of W1. In the region which is close to the origin of coordinates, the surface W2 may be described locally as
Z)-
X22
Y22
+
2R2
2R2-
+ O(X24, Y24)
(2.1)
In a similar way, one can write for the surface W1 of the other body
Z)h+
X12
Y12
+ +
2R1
2R1-
+ O(X14, Y14)
(2.2)
where the ratio h/R1- is a small parameter of our problem. The new system of coordinates is defined with the relation
z)Z+
X22
Y22
+ +
2R2
2R2-
In this system it is possible to write
z ) O(X24, Y24, X14, Y14) for W2 z)h+
X22
Y22 +
2R2+
X12 +
2R2-
Y12 +
2R1+
+ 2R1-
O(X24, Y24, X14, Y14) for W1 Therefore, the problem of interaction of two smooth and quadratically curved (in the neighborhood of the origin of coordinates) bodies is mathematically equivalent to that of drainage of a liquid film confined between a plane and a curved surface of the second order (of the effective elliptic paraboloid). In what follows we are dealing with this effective surface.31 Let us now express X1 and Y1 through X2 and Y2. Assume that X2O2Y2 can be obtained by right rotation of X1O1Y1 through an angle φ. Therefore
X1 ) X2 cos φ - Y2 sin φ Y1 ) X2 sin φ + Y2 cos φ Then, the equation for the effective curved surface in coordinates X2O2Y2 takes the form
A11X22 + 2A12X2Y2 + A22Y22 + 2A34z + A44 ) 0
(2.3)
where
A11 )
1 cos2 φ sin2 φ + + 2R2+ 2R1+ 2R1A44 ) h
(31) To exclude possible confusion we will later specify were necessary whether we are talking about original (two) or the effective (one) curved surfaces.
Hydrodynamic Interaction of Curved Bodies
A12 ) cos φ sin φ
A22 )
Langmuir, Vol. 12, No. 24, 1996 5965
[
1 1 2R1- 2R1+
]
1 cos2 φ sin2 φ + + 2R2 2R12R1+ A34 ) -
(2.4)
1 2
some permutation of rows and columns that does not change the sign of determinant I4 can transform it to the form I4 ) hI3 - I2/4. This expression certainly presents no practical interest. However, it shows that in our case the minimum separation between the surfaces can be expressed through the invariants of the effective surface
h)
4I4 + I2 4I3
(2.6)
Let us now calculate the invariants of the surface (2.3) relative to transformation of Cartesian coordinates, i.e., those functions of the coefficients of the equation for the surface whose values do not change on transformation to a new coordinate system. For the surface (2.3) the known expressions for four invariants of a surface of the second order can be written in the form
Now assume A12 * 0 and make a right rotation of the coordinates X2O2Y2 through an angle φ2 (correspondingly, the coordinates X1O1Y1 are rotated by an angle φ1 ) φ + φ2), such that in the new system of coordinates xO2y the cross-terms of the equation of the surface vanish, i.e., a12 ) 0. The equation of surface (2.3) in these coordinates takes the form
I1 ) A11 + A22
z ) h + a11x2 + a22y2 + O(x2, y2)
|
A A I2 ) A11 A12 12 22
|
|
A11 A12 0 I3 ) A12 A22 0 0 0 0
|
A11 A I4 ) 12 0 0
A12 A22 0 0
0 0 0 A34
| 0 0 A34 A44
|
Substituting here the values of the coefficients given by formulas 2.4, we obtain
I1 )
I2 )
[
[
]
1 1 1 1 1 + + + 2 R- R+ R- R+ 1 1 2 2
1 1 1 + + 4 R -R + R -R + 1 1 2 2 1 1 + - - + sin2 φ + + R2 R1 R2 R1 1 1 2 + + cos φ + R2 R1 R2 R1
(
)
(
I2 ) a11a22
)]
I3 ) 0 (2.5)
I2 4
Let us comment on the expressions derived. We remark first that I1 does not depend on relative orientation of the original surfaces. Second, in the expression for I1 the radii R1-, R1+, R2-, and R2+ are not necessarily the principal radii of curvature of W1 and W2, because the sum of the mean curvatures of the original surfaces is independent of how their axes are oriented (although, of course, Y1 and Y2 are always at right angles to X1 and X2, respectively). Note that I2 depends on φ, and the radii that determine I2 are the principal radii of curvature. Also, it is necessary to say that here we have presented I2 in a form which is analogous to that given by Derjaguin in his derivation of the Derjaguin approximation.32 In our opinion this form is the most compact and successful. We also note that (32) Derjaguin, B. V. Kolloid Z. 1934, 69, 155.
The reason for this rotation is to bring into coincidence the axes of the new coordinates with the planes of principal radii of curvature of the elliptic paraboloid (2.3). The relationships between a11, a22 and the coefficients of the previous system of coordinates can be found in handbooks on analytic geometry and linear algebra. Some useful relations between angles are given in ref 33 (see also ref 34). However, we will not consider that in detail, because our aim here is not to find the values of a11, a22, and φ2 themselves, but the functional dependence of pressure and hydrodynamic force on a11 and a22. If we can demonstrate that this dependence is such that both pressure and force are expressed only through the invariants, our problem would be solved. (This is because the invariants are linked with the parameters of the original surfaces through eqs 2.5.) In the new coordinates, the invariants are connected with the coefficients of the equation for the surface in the following way
I1 ) a11 + a22
I3 ) 0 I4 ) -
(2.7)
I4 ) -
a11a22 4
(2.8)
Thus, I1 and I2 are, respectively, the mean and the Gaussian curvatures of the effective surface. III. Solution of the Equations of Motion The bodies move along the common normal to their surfaces (i.e., the line joining O1 and O2) at velocities v1 and v2, respectively. Assume, that only translational motion is possible, i.e., the bodies have no rotational freedom. This means no rotation about the z-axis in particular (but also no “rolling” about any other axes). This condition is certainly valid during surface force measurements and often fulfilled in the coagulation processes. The flow of a liquid in the gap between bodies should satisfy the system of motion equations. In our case they can be substantially simplified, because the lateral component of the velocity field is large compared with the (33) White, L. R. J. Colloid Interface Sci. 1983, 95, 286. (34) Stewart, A. M. J. Colloid Interface Sci. 1995, 170, 287.
5966 Langmuir, Vol. 12, No. 24, 1996
Vinogradova
normal component. Under this condition the NavierStokes equations of the quantity of motion are reduced to the well-known form35
µ
∂2vt ∂z2
∼ ∇tp
(3.1)
∂ ∂ i+ j ∂x ∂y
Here vx and vy are the projections of the liquid flow rate on the axes x and y with the basis vectors i and j, respectively. We remark that the pressure is a function only of x and y in the first-order approximation, i.e. p ) p(x,y). The continuity equation in Cartesian coordinates takes the form
∂vz + ∇tvt ) 0 ∂z
(3.2)
where vz is the projection of the liquid flow rate on the axis z. The boundary conditions express the slippage of liquid over the hydrophobic surface (1.1). Assume that for the surface W2 the value of slip length is equal to b2 ) b, while for the surface W1 it is b1 ) b(k + 1), where k can be varied from -1 to ∞. Therefore, the boundary conditions will have the form36
z ) O(x4, y4), vz ) 0, vt ) b
At
∂vt ∂z
∂vt At z ) H, vz - vt∇tH ) -v, vt ) -b(k + 1) ∂z
(3.3)
( [ ]
H ) H(x, y) ) h + a11x2 + a22y2 + O(x4, y4)
γ ) γ(H) )
[ ])
(3.8)
(3.5)
v)-
([
] [
(3.6)
]) (
∂ ∂p ∂H 1 ∂ ∂p ∂H γ + γ 2µ ∂x ∂H ∂x ∂y ∂H ∂y -
)
)
γ dp ∂2H ∂2H + 2 (3.9) 2µ dH ∂x2 ∂y
Here, we used the fact that the partial derivatives are only taken by independent variables that explicitly enter into the differential equation. In other words, despite the fact that the variables dp/dH and γ both depend on x and y, they remain constants during partial differentiation. Now, taking second partial derivatives of the function (3.5), we obtain the ordinary first-order differential equation for pressure
µv µv dp ))dH γI1 γ(a11 + a22)
(3.10)
The integration of this expression is not straightforward because of the rather cumbersome form of the inverse function of γ. We refer the reader to ref 3, where the integration of this function has been performed. As a result, we derive that to within an arbitrary additive constant, the expression for pressure is given by
p)
3µv p* I1H2
(3.11)
where the dimensionless function p* is
p* )
The solution of eq 3.1 taking into account the boundary conditions (3.3) and (3.4) leads to
1 ‚∇ p(x, y)[z2 - zβ - bβ] 2µ t
H3 H2 β - bHβ 3 2
Now, we will assume that the solution of eq 3.8 can be found in the form p(x, y) ) p(H(x, y)), i.e., that p is a function only of H. If so, the equation can be rewritten as
(3.4)
where v ) |v1 - v2| is the velocity of body W1 relative to W2, while
vt )
1 ∂ ∂p ∂ ∂p γ + γ 2µ ∂x ∂x ∂y ∂y
where
where µ is the dynamic viscosity, p is the pressure, vt ) vxi + vyj,
∇t )
v)-
[
(
)
2AH B 2H2 B - A ln 1 + + BC C - B B2 H C-A C ln 1 + H C2
(
)]
(3.12)
with
A ) b(2 + k)
where
B ) 2b(2 + k + x1 + k + k2)
H(H + 2b(1 + k)) β ) β(H) ) H + b(2 + k)
C ) 2b(2 + k - x1 + k + k2)
By integrating the continuity eq 3.2
v)
∫0 ∇tvt dz ) ∇t∫0 vt dz - ∇tH[vt]z)H H
H
(3.7)
we obtain the expression for the relative velocity of the bodies (35) Here we use the film thickness as a reference length scale, so that the simplified problem may in principle be not solvable in the sense that the boundary conditions involving other length scales cannot be applied (see Introduction and ref 3). We will see however that in our case the simplified problem appears to be consistent and quite regular. (36) It is perhaps worthwhile to comment briefly on the boundary conditions. Here we used that vt ∼ vs, and that at a curved surface vz is not equal to the relative velocity of solids v due to nonzero projection of vs on the axis z.
Since the expression 3.11 for p obeys the differential equation 3.1 and all the boundary conditions, it is the unique solution for liquid flow between a plane and an elliptic paraboloid with mean curvature I1. The hydrodynamic resistance force acting on the body W1 is opposite to the force exerted by the body W2. In Cartesian coordinates, the expression for this force is given by
Fh ) -
∫-∞∞∫-∞∞(-p + 2µ dzz) dx dy dv
(3.13)
Taking into account the orders of magnitude, we conclude that the term for pressure in eq 3.13 is predominant. It
Hydrodynamic Interaction of Curved Bodies
Langmuir, Vol. 12, No. 24, 1996 5967
is now convenient to transform to cylindrical coordinates. Suppose
x)
r cos θ , xa11
y)
so that the Jacobian is
J)
|
r sin θ
cos θ
xa11
-
sin θ
r cos θ
xa22
xa22
xa11
|
r sin θ
xa22
r )
r
xa11a22
)
xI2
and
and on the separation. The crucial observation is that this dependence can be expressed solely in terms of the invariants of the surface of the second order.37 A remarkable feature of the expression for pressure is that the orientation of surfaces does not play any role. Moreover, pressure depends on geometry of the system only through the mean curvature, so that this dependence is relatively weak. In contrast to this, the hydrodynamic force, being a function of both mean and Gaussian curvatures, depends strongly on the angles between principal axes of the two bodies, as well as on their form and size. Analysis of eq 3.15 shows that the maximum Fh corresponds to zero angle between the principal axes of curvature of the two approaching bodies, while the minimum Fh is observed when this angle is π/2. To illustrate the above conclusions we consider some cases of special colloidal interest. Two spheres of radii R1 and R2, respectively. If
H ) h + r2 Re )
Therefore
Fh )
2π ∞ dθ ∫0 p(H)r dr ) ∫ 0 I
1
x
2
∞ p(H) dH ∫ h I
π
x
(3.14)
we have for invariants (calculated from eqs 2.5).
2
I1 )
As a result, for the hydrodynamic drag we derive
Fh )
3πµv hI1xI2
f*
(3.15)
[
( ) ( )]
B Ah 2h (B + h)(B - A) ln 1 + 2 BC C - B h B (C + h)(C - A) C ln 1 + (3.16) 2 h C IV. Concluding Remarks
Let us now analyze the results obtained. The first conclusion is that a result originally found for two spheres,3 namely that the expressions for both the pressure and the drag force can be represented as a product of the formulas obtained for the no-slip case (the Reynolds part) and a dimensionless function p* or f* giving the correction for slippage, can be generalized to any two bodies of finite curvature. The corrections for slippage depend only on the ratio of separation to the slip lengths b and b(k + 1) of the approaching bodies. We emphasize once again that there is also an important new result that concerns the corrections for slippage. Namely, we derived that these corrections for slippage are the same for curved bodies of any geometry. Therefore, we do not comment here on the behavior of the functions p* and f* because it has been thouroughly analyzed in ref 3 where we considered the interaction of two spheres. The other conclusions are connected with the Reynolds parts of the expressions for p and Fh. In turn, the Reynolds part of the expressions is a product of two factors that reflect different aspects of hydrodynamic interaction of the two bodies. These factors deserve comment. The first factor depends only on the viscosity of the bulk liquid, and the relative approach velocity. Thus it reflects external conditions, and does not depend either on characteristics of bodies, or on the properties of the solid/ liquid interface. The second factor depends only on the configuration geometry, i.e. on the curvature of the two surfaces, the relative orientation of their principal radii of curvatures,
1 1 , I2 ) Re 4R
2
e
so that
where
f* ) -
R1R2 R1 + R2
p)
3µvRe H2
p*,
Fh )
6πµvRe2 f* h
The same expressions were derived in ref 3. Earlier results obtained for some special forms of correction for slippage can be found in refs 18-21. Two cylinders (R2+ ) ∞, R1+ ) ∞) may be crossed (φ ) π/2). Hence, the invariants are
I1 )
(
)
1 1 1 1 + ) , 2 R- RRh 2 1
I2 )
1 1 1 ) 4 R -R - 4R 2 2 1 g
where Rh and Rg are the harmonic and geometric means of the cylinder radii. Correspondingly
p)
3µvRh H
2
p*,
Fh )
6πµvRhRg f* h
Now, if one cylinder is rotated through π/2 so that they become parallel (φ ) 0), then
I1 )
(
)
1 1 1 1 + ) , 2 R- RRh 2 1
I2 ) 0
and it follows that
p)
3µvRh H2
p*,
Fh ) ∞
Once again, we stress that pressure did not change, while the force becomes divergent as a result of rotation (physically, this result is obvious, because the cylinders are considered to be infinitely long). We also note that (37) In actual fact in our problem h is the length scale as well as a characteristics of the configuration geometry. These two physical aspects of a single parameter should not be confused. As a length scale h appears only in the correction for slippage (and has no relation to the invariants), while in the geometrical factor h can be expressed through the invariants of the surface (2.3).
5968 Langmuir, Vol. 12, No. 24, 1996
our expressions for two crossed cylinders can easily be reduced to earlier results.24,38 Thus, the derived formulas allow us, having determined pressure or drag force for some particular case, to find them by simple recalculation for bodies of other forms and sizes or for different conditions at the liquid/solid interface. We believe this to be a very useful result for solving hydrodynamic problems in colloid science. (38) Vinogradova, O. I. Kolloidn. Zh. 1996, 58, 590.
Vinogradova
Acknowledgment. I have benefited from discussions with F. Feuillebois. I am grateful to P. Stein for sending refs 13-16. N. V. Churaev and R. G. Horn are thanked for helpful remarks on the manuscript. This research was supported by Grant No. 96-03-32147 from the Russian Foundation for Basic Research.
LA960531U
