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Squeeze fluid film of spherical hydrophobic surfaces with wall slip
ARTICLE IN PRESS
Tribology International 39 (2006) 863–872 www.elsevier.com/locate/triboint
Squeeze fluid film of spherical hydrophobic surfaces with wall slip C.W. Wu, P. Zhou, G.J. Ma State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China Received 3 June 2005; received in revised form 16 August 2005; accepted 25 August 2005 Available online 11 October 2005
Abstract Isothermal squeeze film flow of Newtonian fluid between spherical hydrophobic surfaces with wall slip is investigated using a limiting shear stress model and complementary algorithm. Wall slip velocity is controlled by the liquid–solid interface limiting shear stress. It is found that the wall slip dramatically decreases the hydrodynamic support force of the squeeze fluid film. In the case of large wall slip the hydrodynamic support force increases only slightly with the decrease in the film thickness. We find that wall slip decreases with increasing film thickness and limiting shear stress, but increases with increasing fluid viscosity and approaching velocity. An empirical equation is given for prediction of the fluid load support capacity. The possible effect of pressure on wall slip is also discussed. It is found that fluid pressure suppresses wall slip after the proportionality coefficient of limiting shear stress reaches a critical threshold. However, almost no effect is found when it is below this critical threshold. Good agreements exist between the present theoretical predictions and some existing experimental observations. r 2005 Elsevier Ltd. All rights reserved. Keywords: Wall slip; Squeeze film; Limiting shear stress
1. Introduction The velocity boundary condition at a liquid–solid interface (either hydrophobic or hydrophilic) has been an old but challenging question for hundreds of years not only in thin film lubrication but also in fluid mechanics. The classical Reynolds theory and fluid mechanics theory assume that no wall slip (boundary slip) occurs at the liquid–solid interfaces, i.e., the so-called no-slip boundary condition [1–3]. The validity of this assumption has been doubted for over a century. An early experiment [4] observed that wall slip occurred for water flowing at a hydrophobic surface. The development of microfluidics and MEMS (micro-electromechanical system) based devices has recently prompted the research interests in this area [5–11]. Wall slips were observed not only on hydrophobic surfaces [5–9] but also on hydrophilic surfaces [5,10,11], not only for a simple fluid but also for a complex Corresponding author. Tel.: +86 411 84706353; fax: +86 411 84708393. E-mail address: [email protected] (C.W. Wu).
0301-679X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2005.08.001
fluid [6]. This indicates that the wetting property of a surface is not the only parameter affecting the wall slip. Theoretical simulations using molecular dynamics (MD) also confirmed that wall slip occurs at a liquid–solid interface [12,13]. During the past decade, wall slip has received much attention in lubrication mechanics [14–20]. There are usually two models used to describe wall slip: the slip length model (SLM) [4–11] and the limiting shear stress model [15–20]. In the slip length model, the slip velocity, V s , is assumed to be proportional to local shear rate qu=qz, i.e., V s ¼ bqu=qz, where b, a constant, is the fictive slip length inside the solid surface at which the no-slip boundary condition holds. u is the fluid velocity, and z is the distance perpendicular to the surface. At a small range of shear rate and small wall slip, the slip length was experimentally found to be nearly constant [10,11] (nanometer scale), and thus the slip length model may adequately describe such a wall slip. At high shear rate, however, experimental observations [7–9] and MD study [13] showed that the slip is strongly nonlinear and can approximately be described by the limiting shear stress model [13]. Experimental observations [21,22] also showed that the wall slip
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Nomenclature F f f b h h0 k p R r t u Vs
hydrodynamic force, N yield function hydrodynamic load support coefficient slip length, m fluid film thickness, m minimum film thickness, m proportionality constant film pressure, Pa sphere radius, m radial coordinate, m time, s fluid velocity, m/s slip velocity, m/s
does not start immediately after a fluid is sheared, but starts after the shear rate reaches a critical value. The concept of the limiting shear stress was first proposed by Smith [23,24] for lubricants, and later was confirmed by Bair and Winer [25–27], where the solid surface is metal (hydrophilic). The limiting shear stress model, as shown in Fig. 1, assumes that wall slip occurs only after the surface shear stress t reaches the limiting shear stress tL , where the shear stress is a constant (shearrate independent) but the apparent shear rate can reach any values (the true shear rate equals the critical threshold). When totL , no slip takes place. What is shown in Fig. 1 is an ideal limiting shear stress model. Experiments usually give a transition region from no-slip to slip flowing the dashed curve. In thin film lubrication, either a high shear stress or a high shear rate is often encountered, and the limiting shear stress model was usually used [15–20]. In the present paper we use the limiting shear stress model to
v t tL t0 Z l T0 TL
Subscripts i L 0 s
denoting surfaces 1 and 2 limiting initial limiting slip
study the isothermal squeeze film flow between spherical hydrophobic surfaces. 2. Theory and numerical method The surface interaction of two approaching spheres is one of the most important topics in many scientific research areas, such as thin film lubrication, colloid interface science, etc. When two spherical surfaces lubricated by a fluid film are approaching each other, the squeezed fluid film will become thinner and thinner, and the shear rates at the lubricated surfaces become larger and larger. Thus a wall slip may occur at the interface of fluid and solid. Based on this principle, the surface force apparatus (SFA) [8,9,21] and the atomic force microscope (AFM) [7,10,11] have recently been used to indirectly measure the wall slip of a squeezed fluid film between a rigid sphere and a flat surface. Very strong shear-rate-dependent slip was found at high shear rates. Now let us use the limiting shear stress model to study the squeeze film flow, as shown in Fig. 2. From the Reynolds lubrication theory [28], one obtains the fluid velocity uðr; zÞ in r direction uðr; zÞ ¼
τL
v ¼ qh=qt, squeeze velocity, m/s shear stress, Pa limiting shear stress, Pa initial shear strength, Pa lubricant viscosity, Pa s slack variable, Pa ¼ t0 R=Zv ¼ tL R=Zv
1 qp 2 z ðz zhÞ þ ðV s2 V s1 Þ þ V s1 2Z qr h
(1)
and the shear stress inside the fluid film
Shear stress τ
qu 1 qp V s2 V s1 ¼ ð2z hÞ þ Z, (2) qz 2 qr h where p is the fluid film pressure, t the shear stress, Z the fluid viscosity, h the gap height, and V s1 and V s2 are the slip velocities at interfaces 1 and 2, respectively. For the first order approximation h is given by [28]
tðr; zÞ ¼ Z
tgα=η
Apparent shear rate
Fig. 1. Schematic of stress–rate curve of the limiting shear stress model. No slip occurs when tptL and slip takes place after t4tL .
hðrÞ ¼ h0 þ r2 =2R,
(3)
where h0 is the minimum thickness of the fluid film, r the radius coordinate, and R the sphere radius. For a squeeze
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where k is a proportionality coefficient, t0 is the initial limiting shear stress at the ambient pressure depending on interface property and temperature, and subscript i denotes surface 1 and 2, respectively. For a pure squeeze film flow between two smooth surfaces as shown in Fig. 2, t1 X0 and t2 p0, one obtains the following slip control equations:
z
ν
R
Surface 2
h0 h o
Surface 1
r
system of two spherical surfaces with radii of R1 and R2 , we can use an equivalent radius R ¼ ð1=R1 þ 1=R2 Þ1 in Eq. (3). Obviously tðr; zÞ is a linear function of z. That is to say, the maximum shear stress in the fluid film occurs at either surface 1 or surface 2 in an isothermal and Newtonian fluid flow. The shear stresses at surfaces 1 and 2 are qu h qp V s2 V s1 þ Z, (4) t1 ¼ Z ¼ qz z¼0 2 qr h qu h qp V s2 V s1 t2 ¼ Z þ Z. ¼ qz z¼h 2 qr h
(5)
The flow continuity condition yields the Reynolds equation for an isothermal squeeze film flow of Newtonian fluid between the approaching spherical surfaces: q rh3 qp 1 q qh rhðV s1 þ V s2 Þ r ¼ 0. (6) qr 12Z qr 2 qr qt One can obtain the classical Reynolds equation if no wall slip occurs, i.e., V s1 ¼ V s2 ¼ 0. For rigid surfaces with an approaching (squeezing) velocity v, qh=qt ¼ v ¼ constant, Eq. (6) reduces to (7)
The wall slip velocities can be written as the following complementary problem with the interface shear stresses: V si ¼ 0
if jti jptLi
ði ¼ 1; 2Þ,
(8a)
V si 40
if jti j ¼ tLi
ði ¼ 1; 2Þ,
(8b)
where tLi is the limiting shear stress at interface i. Usually it is a function of the surface wettability, temperature, fluid pressure, etc. For lubricants it increases linearly with the fluid pressure starting with an initial limiting shear stress and is given as [25–27] tLi ¼ t0i þ ki p,
f 1 ¼ t1 tL1 p0,
(10a)
f 2 ¼ t2 tL2 p0.
(10b)
Using Eqs. (4)–(5), and introducing slack variables li , Eqs. (10a)–(10b) are reduced to h qp iþ1 V s2 V s1 þ ð1Þ Z tLi þ li ¼ 0, (11a) fi ¼ 2 qr h
Fluid
Fig. 2. Schematic of a squeeze film system of an approaching sphere and a smooth surface.
h3 qp hðV s2 þ V s1 Þ þ rv ¼ 0. 6Z qr
865
(9)
li V si ¼ 0;
li X0; V si X0 ði ¼ 1; 2Þ.
(11b)
Numerical solution of Eq. (6) or Eq. (7) under control of the slip control equations (11a)–(11b) yields the film pressure and the slip velocities. Finite element method and complementary algorithm were used in the present work. This numerical technique was described in the analyses of a viscoplastic lubricant flow [16] and wall slip occurring at a journal bearing [18]. The solid surfaces are assumed to be rigid and no elastic deformation was considered since a large fluid pressure cannot be built up between two slippage surfaces. The region of numerical solution was taken to be 0or=Ro0:2 because most of the fluid pressure is concentrated on the contact center. The boundary conditions are: qp=qr ¼ 0 (null volume flow in r direction) at r ¼ 0 and p ¼ 0 at r=R ¼ 0:2. The wall slip is determined by the surface shear strength, gap geometry, fluid viscosity and approaching velocity. 3. Numerical analyses of squeeze film flow with wall slips 3.1. Surfaces with the same slip property If the two solid surfaces have the same slip property, then we have: tL1 ¼ tL2 ¼ tL , V s1 ¼ V s2 ¼ V s and t1 ¼ t2 ¼ t. For the simplicity of analysis, we take the proportionality coefficients of limiting shear stress k1 ¼ k2 ¼ 0 in this section and will discuss their effects in Section 3.3. Fig. 3 shows how the wall slip to be developed during the squeeze process of a fluid film when the dimensionless surface limiting shear stress T L ¼ T 0 ¼ t0 R=Zv ¼ 107 (tL ¼ t0 due to k1 ¼ k2 ¼ 0). Wall slip leads to a reduction in fluid film pressure in the slip zone (Fig. 3a). It can be seen that when the surface separation h0 =R ¼ 2:7 105 , the surface shear stress is below the limiting shear stress in the entire squeezed region (see Fig. 3b), and thus no slip occurs (see Fig. 3c). However, when h0 =R ¼ 2:6 105 , the slip just starts in a small area where the surface shear stress reaches its limiting value. When the fluid film further decreases, the surface shear stress reaches its limiting value in a larger and larger zone.
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C.W. Wu et al. / Tribology International 39 (2006) 863–872 100 τLR/ην=109
10-1
f*
108
107
10-2
2/3
f *=1-e-2.26(LR/)
106
(h0 / R)
10-3 10-8
10-7
10-6
10-5
10-4
102
103
h0/R
(a) 100 h0/R=5×10-6
10-1
f*
10-6
10-2
0.2×10-6
10-3 10-1
100
102 τLR/ην (106)
(b)
Fig. 4. (a) Hydrodynamic load support coefficient, f , the ratio of the hydrodynamic forces with wall slip to that without wall slip, versus film thickness parameter h0 =R for several values of the surface limiting shear stress; (b) hydrodynamic load support coefficient f versus normalized surface limiting shear stress for three values of film thickness parameters h0 =R. The open symbols are numerical solutions and the curves are obtained by Eq. (12). The two surfaces have the same slip property, i.e., tL1 ¼ tL2 ¼ tL ¼ t0 and k1 ¼ k2 ¼ k ¼ 0. The normalized limiting shear stress T L ¼ T 0 ¼ tL R=Zv ¼ t0 R=Zv ¼ 107 .
Fig. 3. Development of wall slip for several values of surface separations: (a) fluid film pressures; (b) surface shear stress distributions and (c) wall slip velocities. The two surfaces have the same slip property with normalized limiting shear stress T L ¼ T 0 ¼ tL R=Zv ¼ t0 R=Zv ¼ 107 and k1 ¼ k2 ¼ k ¼ 0.
Therefore, the slip amplitude is increased and the size of slip zone is enlarged. A hydrodynamic load support coefficient, f , which is defined as the ratio of the hydrodynamic force with wall
slip to that without wall slip [29], is plotted in Fig. 4a for several values of the surface limiting shear stresses. f is approximately proportional to the film thickness parameter h0 =R at small h0 =R. At large h0 =R, however, f approaches to 1 in a nonlinear manner. This is supported by the experimental observations [8,9]. Fig. 4b gives the log–log plot of f versus the dimensionless surface shear stress. When f o0:2 it is approximately proportional to ðtL R=ZvÞ2=3 . This is to say, the film thickness effect on the hydrodynamic force is stronger than the surface limiting shear stress. It is found that the numerical solutions (open symbols in Fig. 4) can be fitted by the following empirical equation: f ¼ 1 e2:26ðtL R=ZvÞ
2=3
ðh0 =RÞ
.
(12)
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The maximum error deviating from the full numerical solution is less than 8%. Eq. (12) gives a theoretical prediction for the hydrodynamic load support coefficient with a reasonable accuracy. For comparison, the slip length model gives an analytical solution for the hydrodynamic load support coefficient [29]: h0 h0 6b f ¼ 1þ 1 . (13) ln 1 þ h0 3b 6b Eq. (13) indicates that the hydrodynamic load support coefficient, f , depends only on the ratio of slip length b to the minimum film thickness h0 , but is independent of the fluid viscosity Z, approaching velocity v of the sphere, and the film thickness parameter h0 =R. If h0 =R is kept unchanged, however, the present work predicted that f increases with increasing R and tL , but decreases with increasing Zv. In other words, wall slip increases with increasing Zv, but decreases with increasing R and tL . The effect of radius R on wall slip is the same as the theoretical prediction of Einzel et al. [30], and the effects of Z and v are supported by the experimental observation [10]. 3.2. Surfaces with different slip properties First, we suppose that the spherical surface (surface 2) is a perfect hydrophobic surface with a null shear strength, i.e., the dimensionless limiting shear stress T L2 ¼ tL2 R=Zv ¼ 0. Such an ideal slippage spherical surface is approaching a smooth plane with different values of surface shear strength. To our surprise, it is found that such a squeeze film system can give only one quarter of the maximum load support coefficient, f , as long as the shear strength on the lower surface (surface 1) is high enough to inhibit any wall slip, as shown in Fig. 5a. f tends to null at small gap (high shear rate) and small surface shear strength at the lower surface (T L1 ¼ tL1 R=Zv ! 0). If the two surfaces are switched, the same situation occurs. In an extreme case that both of the surfaces have null shear strength, no hydrodynamic pressure is generated ( f ¼ 0). Second, we suppose that the spherical surface has a shear strength high enough to suppress any slip, for example T L2 ¼ tL2 R=Zv ¼ 1015 . The load support coefficient ranges from 0.25 to 1 when the normalized shear strength, T L1 , ranges from 0 to 109 , as shown in Fig. 5b. What is shown in Fig. 5 tells us that the match of slippage properties of the two surfaces in a spherical squeeze film system plays an important role in determining the load support coefficient and wall slip. 3.3. Effect of fluid pressure on wall slip The macro-rheological measurements of lubricants in a wetting surface by Bair and Winer [25–27] at high pressures showed a pressure-dependent limiting shear strength, as indicated in Eq. (9). For a wetting mineral-oil/steel interface, the order of k may range from 0.001 to 0.1. For example, for the silicone-oil/steel interface it has an
Fig. 5. Variations of the hydrodynamic load support coefficient for solid surfaces with different slip properties: (a) the upper surface is a perfect slip surface (null shear strength); (b) the upper surface has shear strength high enough to suppress slip. k1 ¼ k2 ¼ k ¼ 0.
order of about 0.007 [31]. For a nonwetting or partially wetting surface, however, no experimental data have been reported so far to determine the proportionality coefficient. But we believe it is a very small value according to the existing experimental observations [8,9]. For simplicity, we first investigate here the situation when the two surfaces have the same slip property, i.e., t01 ¼ t02 ¼ t0 , k1 ¼ k2 ¼ k. Fig. 6 shows how the proportionality coefficient affects wall slip and fluid film pressure. Increasing the proportionality coefficient will help to suppress wall slip (Fig. 6c) and thus gives rise to a high fluid pressure (Fig. 6a). Comparing Fig. 6b with Fig. 6d we can see more clearly how the surface shear stress and the surface limiting shear stress changes with the proportionality coefficient k. The former equals the latter in the slip zone, but is less than the latter in the no-slip zone. In the situation studied here, the surface shear stress is always less than its limiting value when k ¼ 0:005. Fig. 7 shows the effect of proportionality coefficient k on f for several
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Fig. 6. Effect of the proportionality coefficient, k, on the fluid film pressure and wall slip: (a) fluid film pressures; (b) wall shear stress distributions; (c) wall slip velocities; and (d) the surface limiting shear stresses. T 0 ¼ t0 R=Zv ¼ 107 and h0 =R ¼ 106 .
Fig. 7. Effect of the proportionality coefficient, k, on the hydrodynamic load support coefficient when both of the surfaces have the same slip property with a normalized initial shear strength t0 R=Zv ¼ 107 .
values of the surface gap. It can be seen that at a large k, no slip occurs ( f ¼ 1). However, it is very interesting that at a small k, for example ko0:6 103 , the effect of k on the wall slip is hardly found. The proportionality coefficient may affect slip in a complex manner depending on the difference between t0 and p. From Eq. (9) it can be seen that only when the product kp is increased to the same order as the initial shear strength t0 , does k considerably affect the wall slip. Fig. 8 shows effects of the proportionality coefficient of limiting shear stress on the hydrodynamic load support coefficient when the two surfaces have different initial shear stresses. If surface 1 has a higher limiting shear stress than surface 2, numerical solutions show that no slip occurs at surface 1. Now we investigate the effect of the proportionality coefficient at surface 2 (sphere) on the wall slip and hydrodynamic force. The fluid film thickness in Fig. 8a is one order smaller than that in Fig. 8b. The proportionality coefficient, k2 , affects the wall slip in three
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1.2 T02=108
1.0
107 0.8
f*
105 T02=0
0.6
0.4 T02=109 k1=0.01
0.2
h0/R=5×10-7 0.0 10-5
10-4
10-3 k2
(a) 1.2
Fig. 9. Comparison of the predicted hydrodynamic force with those measured by SFA technique [8]. The fluid was deionized water and the solid surfaces were atomically smooth micas coated with a methylterminated close-packed monolayer of condensed octadecyltriethoxysiloxane (OTE). R ¼ 2 cm, Zv ¼ 1:75 1010 N=m (note: we got this value by fitting the no-slip curve given in Fig. 1 of Ref. [8]; no viscosity datum was supplied and the indicated squeeze velocity seems to have problem there), k ¼ 0, tL ¼ t0 ¼ 0:33 Pa (tL R=Zv ¼ 3:77 107 ). For comparison the predictions of slip length model (SLM) are also shown in the inset.
8
T02=10
1.0
f*
0.8 107
0.6
0.4
105
0.2
T02=109 k1=0.01
0
h0/R=5×10-6
0.0 10-5
(b)
10-4
10-3 k2
10-2
10-1
Fig. 8. Effect of the proportionality coefficient at spherical surface, k2 , on the hydrodynamic load support coefficient: (a) h0 =R ¼ 5 107 and (b) h0 =R ¼ 5 106 .
zones: (i) almost no effect exists at small values of k2 (large slip zone); (ii) the hydrodynamic force increases with the proportionality coefficient in medium levels of k2 (small slip zone); and (iii) no effect exists again at large k2 (no slip zone). In the situation under study, the proportionality coefficient gives rise to almost the same effect on wall slip and fluid load support when the dimensionless initial shear stress at surface 2 ranges from 0 to 105 . 4. Comparison with experiments Comparison of the hydrodynamic force predicted by the present work with those measured by SFA [8] for the hydrodynamic force of a squeeze film flow of deionized water is given in Fig. 9. The solid surfaces were atomically smooth micas coated with a methyl-terminated closepacked monolayer of condensed octadecyltriethoxysiloxane (OTE). On such a poorly wetting surface, the advancing contact angle of water was about 110 C. The theoretical prediction agrees reasonably with the experimental data in a large range of shear rate. After the
minimum fluid film thickness h0 is less than about 150 nm (h0 =R 7:5 106 ), wall slips start at both of the surfaces and it is as if the hydrodynamic force reached a saturation level beyond which only a limited amount of further increase was possible. The measured hydrodynamic load support coefficient, f , at a large film thickness is slightly larger than the theoretical prediction. This may be due to the experimental error. At a small film thickness (h0 o5 nm), however, the measured data are slightly smaller than those predicted. The sources of experimental error in this area may be more complex than that in the large film thickness. One of the most probable explanations is that the driving velocity, v, induced from the amplitude of oscillation in SFA system, was reduced by compliance of the apparatus at the lesser values of film thickness. On the other hand, the continuous media assumption used in the theoretical analysis may meet a problem for a very small film thickness. However, our numerical results show that the elastic deformation effect of the solid surfaces is so small that the hydrodynamic force changes no more than 0.1% (h0 45 nm). In the inset of Fig. 9 SLM predictions using Eq. (13) are also shown, best fitting f either in small h0 or in large h0 . Evident is the big deviation of SLM from the experiments. The theoretical prediction of the present paper is also compared with the experimental data measured by Craig et al. [7] using AFM technique (see Fig. 10). A good agreement is obtained again. Since no effect of the fluid pressure on wall slip was reported in the literature [7,8,32], we did not consider the effect of proportionality coefficient on wall slip (k1 ¼ k2 ¼ 0).
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870 150 No-slip
SLM: b=12nm 100 F (nN)
Experiment[7]
50 Theory
0 0.0
0.1
0.2
0.3
0.4
0.5
1/h0 (nm-1)
Fig. 10. Comparison of the predicted hydrodynamic force with those measured by AFM technique [7]. Both of the surfaces of the approaching sphere (silica) and the flat wall (mica) were coated with a 15.1 nm thickness of gold film. The fluid is an aqueous sucrose with viscosity Z ¼ 38:9 mPa s. tL ¼ t0 ¼ 300 Pa, R ¼ 10:4 mm, v ¼ 21:6 mm=s.
5. Discussions In the present paper, we investigate the effect of wall slip on the isothermal squeeze film flow of Newtonian fluid between spherical hydrophobic surfaces based on the concept of interfacial limiting shear stress [see Eq. (9)]. Apart from the work of Smith [24] and Bair and Winer [25–27], several other authors [33–37] have also studied the limiting shear stress of lubricants. The limiting shear stress consists of two important parameters: the initial limiting shear stress, t0 , and the proportionality coefficient, k. In a wetting interface of oil and steel, the reported initial shear strength usually is very large, varying from 0.16 to 8 MPa [27,31,35], but the results should be treated with care because it is a difficult parameter to measure. The proportionality coefficient, k, has been examined in several ways and by a few research groups [31,35–37] with the results that range from about 0.007 to 0.15 and the variation is found to be temperature-dependent. For a nonwetting or partially wetting surface, the limiting shear stress is very small and was reported in only a few papers [9,21,22,32]. It was found to depend on surface wettability, surface roughness, fluid viscosity, etc. A small value of the limiting shear stress of about 0.3 Pa was reported for the interface of deionized water and atomically smooth micas (r.m.s roughness 0:2 nm) coated with a methyl-terminated close-packed monolayer of condensed OTE [32]. However, the limiting shear stress is about 0.6 Pa for the interface of tetradecane and the same solid surface. The limiting shear stress may decrease with surface contact angle [32] and fluid viscosity [10]. The roughness effect on the limiting shear stress is not so clear now. Some researchers [5,9,32] reported that surface roughness inhibits wall slip or increases the limiting shear stress, but others [10] reported that it increases wall slip. In Section 4 for
liquid slippage on a nonwetting or partially wetting surface, the effect of proportionality coefficient was not considered, i.e., the limiting shear stress equals the initial limiting shear stress. However, due to the small value of the limiting shear stress, we believe the proportionality coefficient is very small for a nonwetting or partially wetting surface. Therefore, some researchers [16,19,20,22] neglected its effect. Spikes and Granick [22] also discussed a similar slip problem with a different viewpoint. They developed a new equation for Newtonian fluid flow in the presence of wall slip. Slip is envisaged to occur only when a critical surface shear stress is reached, and once slip starts, it is described by the slip length model. But they think the critical shear stress is pressure-independent. It is to be noted that flow visualization of lubricants at steel gap (highly wetting surface) at high pressure (up to 1 GPa) and shear stress (order of 10 MPa) by Bair et al. [38–40] showed that the slip occurred at the midplane within the film, where the temperature and shear rate were localized, but not at the wall. At such a high pressure the viscosity–pressure relation has to be considered and thus the lubricant viscosity may be so high that it looks like a solid behavior (near the glass transition of the lubricant). However, the fluid pressure studied here is very small. For example, the calculated maximum fluid pressures at h0 ¼ 5 nm and 8 nm in Fig. 9 are about 2.8 kPa and 2.2 kPa, respectively. The match of slippage properties of two surfaces significantly affects the hydrodynamic load support coefficient of a squeeze film between spherical surfaces. Thus in both AFM and SFA measuring systems, the slip should be carefully deduced from measurements of the hydrodynamic load support coefficient. The same group of the load support coefficients may give rise to different explanations for the slip mechanism if the range of shear rate is not very large. For example, if we fit the experimental data in Fig. 10 only after the separation h0 410 nm (1=h0 ¼ 0–0.1), both the constant slip length model and the limiting shear stress model can fit the experimental data pretty well. However, in the case of large range of separation (large range of shear rate) the slip length model cannot agree with the experiment as shown in Fig. 10. We also noted that some researches, for example Charlaix’s group [41,42] using an SFA measuring system, did not report a shearrate-dependent slip behavior. We believe that one of the important reasons may be the small shear rate range under their study. Of course, other mechanisms, such as surface contamination, roughness, nanobubbles or small thickness of gaseous film, match of surface slippage properties, etc., may also affect the slip behavior. The mechanism study of wall slip is out of the scope of the present paper. In this paper, we did not consider the elastic deformation of the solid surfaces. In the case of high squeeze velocity and without wall slip, effect of elastic deformation of the solid surfaces may need studying [43]. In a very small film thickness, the elastic deformation and the surface force may play a role in determining the total interaction force
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[44]. Both experimental measurements (SFA and AFM) and our theoretical prediction may give a little large error for a very small film thickness, i.e., h0 o3–5 nm. However, for the hydrophobic surfaces with small squeeze velocity and large wall slip as we studied here, effect of the elastic deformation on the wall slip and hydrodynamic force should be negligible. This has been supported by the experimental measurements of squeeze film of spherical hydrophobic surfaces with wall slips [7–11].
871
7. Acknowledgments The authors would like to express their high appreciation to V.S.J. Craig for his original experimental data supplied for Fig. 10. This research was supported by NSFC (10421002, 10332010, 10272028) and SRFDP (20030141013).
References 6. Conclusions The wall slip in isothermal squeeze film of a smooth spherical surface and a flat surface is studied in the present paper. Wall slip for a Newtonian fluid flowing at a solid surface is analyzed using the limiting shear stress model and complementary algorithm. The wall slip velocity is controlled by the limiting shear stress at the liquid–solid interface. It is found that good agreements exist between the present theoretical predictions and some existing experimental observations of squeeze film flow for nonwetting or partially wetting surfaces, especially at high shear rates. The numerical results show that after the maximum surface shear stress reaches the limiting shear stress, a wall slip occurs and then spreads out upon increasing the squeezing velocity or decreasing the film gap (increasing shear rate). After the wall slip starts, the hydrodynamic force increases very slowly with further decrease of the film gap. The hydrodynamic load support coefficient decays with a negative exponential function of the surface limiting shear stress and film thickness. For two surfaces with the same slip property and negligible proportionality coefficient of the limiting shear stress, the following empirical equation is given for prediction of the hydrodynamic load support coefficient: f ¼ 1 e2:26ðtL R=ZvÞ
2=3
ðh0 =RÞ
.
If h0 =R is kept unchanged, f is reduced with increasing Zv, but increased with increasing R and tL . In other words, if h0 =R is kept unchanged, we predicted that wall slip increases with increasing Zv, but reduces with increasing R and tL . In the squeeze film flow of spherical surfaces with wall slip, surface shear strength, surface curvature radius, fluid viscosity, the separation and squeezing velocity of two surfaces are coupled together to affect the slip behavior. The possible effect of pressure on wall slip is also discussed using a proportionality coefficient of the limiting shear stress. It is found that pressure starts to suppress wall slip after the proportionality coefficient reaches a critical threshold, but almost no effect is found when the proportionality coefficient is very small. Finally, it is found that the match of slippage properties of the two surfaces in a spherical squeeze film system is another important factor affecting the wall slip behavior.
[1] Reynolds O. On the theory of lubrication and its application to Mr Beauchamp Tomer’s experiments, including an experimental determination of the viscosity of olive oil. Philos Trans R Soc 1886;177:157–67. [2] Lamb H. Hydrodynamics. New York: Dove; 1932. [3] Faber T. Fluid dynamics for physicists. Cambridge: Cambridge University Press; 1995. [4] Schnell E. Slippage of water over nonwettable surfaces. J Appl Phys 1956;27:1149–52. [5] Pit R, Hervet H, Leger L. Direct experimental evidence of slip in hexadecane: solid interfaces. Phys Rev Lett 2000;85(5):980–3. [6] Hervet H, Leger L. Flow with slip at the wall: from simple to complex fluids. C R Phys 2003;4:241–9. [7] Craig VSJ, Neto C, Williams DRM. Shear-dependent boundary slip in an aqueous Newtonian liquid. Phys Rev Lett 2001;87(5):054504. [8] Zhu Y, Granick S. Rate-dependent slip of Newtonian liquid at smooth surfaces. Phys Rev Lett 2001;87(9):096105. [9] Zhu Y, Granick S. Limits of the hydrodynamic no-slip boundary condition. Phys Rev Lett 2002;88(10):106102. [10] Bonaccurso E, Butt HJ, Craig VSJ. Surface roughness and hydrodynamic boundary slip of a Newtonian fluid in a completely wetting system. Phys Rev Lett 2002;90(14):144501. [11] Bonaccurso E, Kappl M, Butt HJ. Hydrodynamic force measurements: boundary slip of hydrophilic surfaces and electrokinetic effects. Phys Rev Lett 2002;88:076103. [12] Sun M, Edner C. Molecular dynamics study of flow at a fluid–wall interface. Phys Rev Lett 1992;69:3491–4. [13] Thompson PA, Troian SM. A general boundary condition for liquid flow at solid surfaces. Nature 1997;389:360–2. [14] Kaneta M, Nishikawa N, Kameishi K. Observation of wall slip in elasto-hydrodynamic lubrication. ASME J Tribol 1990;112:447–584. [15] Wu CW, Hu LC. Wall slippage and oil film collapse. J Dalian Univ Tech 1993;33:172–8. [16] Wu CW, Zhong WX, Qian LX, et al. Parametric variational principle for viscoplastic lubrication model. ASME J Tribol 1992;113:731–5. [17] Stahl J, Jacobson BO. A lubricant model considering wall-slip in EHL line contacts. ASME J Tribol 2003;125:523–32. [18] Wu CW, Ma GJ. Abnormal behavior of a hydrodynamic lubrication journal bearing caused by wall slip. Tribol Int 2005;38:492–9. [19] Spikes HA. The half-wetted bearing. Part 1: Extended Reynolds equation. Proc Inst Mech Eng 2003;217:1–14. [20] Spikes HA. The half-wetted bearing. Part 2: Potential application in low load contacts. Proc Inst Mech Eng 2003;217:15–26. [21] Zhu Y, Granick S. No-slip boundary condition switches to partial slip when fluid contacts surfactant. Langmuir 2002;18:10058–63. [22] Spikes H, Granick S. Equation for slip of simple liquids at smooth solid. Langmuir 2003;19:5065–71. [23] Smith FW. Lubricant behavior in concentrated contact systems—the castor oil–steel system. Wear 1959;2:250–63. [24] Smith FW. Lubricant behavior in concentrated contact—some rheological problems. ASLE Trans 1960;3:18–25. [25] Bair S, Winer WO. Shear strength measurements of lubricants at high pressure. ASME J Lubr Technol 1979;101:251–6. [26] Bair S, Winer WO. The shear stress rheology of liquid lubricants at pressure of 2 to 200 MPa. ASME J Tribol 1990;112:246–52.
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[27] Bair S, Winer WO. The high pressure high shear stress rheology of liquid lubricants. ASME J Tribol 1992;114:1–13. [28] Cameron A. Basic lubrication theory. 3rd ed. Chichester, UK: Ellis Horwood Ltd; 1981. [29] Vinogradova OI. Drainage of a thin liquid film confined between hydrophobic surfaces. Langmuir 1995;11:2213–20. [30] Einzel D, Panzer P, Liu M. Boundary condition for fluid flow: curved or rough surface. Phys Rev Lett 1990;64:2269–72. [31] Wilson WRD, Huang XB. Viscoplastic behavior of a silicone oil in a metalforming inlet zone. ASME J Tribol 1989;111:585–90. [32] Granick S, Zhu Y, Lee H. Slippery questions about complex fluids flowing past solids. Nat Mater 2003;2:221–7. [33] Ho¨glund E, Jacbson BO. Experimental investigation of the shear strength of lubricants subjected to high pressure and temperature. ASME J Tribol 1986;108:571–8. [34] Ho¨glund E. The relation between lubricant shear stress strength analyzer and chemical composition of the base oil. Wear 1989;130:213–24. [35] Kato K, Iwasaki T, Kato M, Inoue K. Evaluation of limiting shear stress of lubricants by roller test. JSME Int J Ser C 1993;36(4): ‘512–22. [36] O¨stensen JO, Wikstrom V, Ho¨glund E. Interaction effects between temperature, pressure and type of base oil on lubricant shear strength coefficient. Tribol Finn J Tribol 1993;11(4):123–32.
[37] Wikstrom V, Ho¨glund E. Investigation of parameter affecting the limiting shear stress-pressure coefficient: a new model incorporating temperature. ASME J Tribol 1994;116:612–20. [38] Bair S, Winer WO. A rheological basis for concentrated contact friction. Proceedings of the 20th Leeds–Lyon symposium on tribology. 1994. p. 37–44. [39] Bair S, Khonsarit M, Winer WO. High-pressure rheology of lubricants and limitations of the Reynolds equation. Tribol Int 1998; 31:573–86. [40] Bair S, McCabe C. A study of mechanical shear bands in liquids at high pressure. Tribol Int 2004;37:783–9. [41] Cottin-Bizonne C, Cross CB, Steinbeger A, Charlaix E. Boundary slip on smooth hydrophobic surfaces: intrinsic effects and possible artifacts. Phys Rev Lett 2005;94:056102. [42] Baudry J, Charlaix E, Tonck A, Mazuyer D. Experimental evidence for a large slip effect of a nonwetting fluid–solid interface. Langmuir 2001;17:5232–6. [43] Yang PR, Wen SZ. Pure squeeze action in an isothermal elastohydrodynamically lubricated spherical conjunction. Part 1. Theory and dynamic load results. Wear 1989;142(1):1–16. [44] Vinogradova OI, Feuillebois F. Interaction between elastic bodies via surface forces. 1. Power-law attraction. Langmuir 2002;18: 5126–32.
Tribology International 39 (2006) 863–872 www.elsevier.com/locate/triboint
Squeeze fluid film of spherical hydrophobic surfaces with wall slip C.W. Wu, P. Zhou, G.J. Ma State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China Received 3 June 2005; received in revised form 16 August 2005; accepted 25 August 2005 Available online 11 October 2005
Abstract Isothermal squeeze film flow of Newtonian fluid between spherical hydrophobic surfaces with wall slip is investigated using a limiting shear stress model and complementary algorithm. Wall slip velocity is controlled by the liquid–solid interface limiting shear stress. It is found that the wall slip dramatically decreases the hydrodynamic support force of the squeeze fluid film. In the case of large wall slip the hydrodynamic support force increases only slightly with the decrease in the film thickness. We find that wall slip decreases with increasing film thickness and limiting shear stress, but increases with increasing fluid viscosity and approaching velocity. An empirical equation is given for prediction of the fluid load support capacity. The possible effect of pressure on wall slip is also discussed. It is found that fluid pressure suppresses wall slip after the proportionality coefficient of limiting shear stress reaches a critical threshold. However, almost no effect is found when it is below this critical threshold. Good agreements exist between the present theoretical predictions and some existing experimental observations. r 2005 Elsevier Ltd. All rights reserved. Keywords: Wall slip; Squeeze film; Limiting shear stress
1. Introduction The velocity boundary condition at a liquid–solid interface (either hydrophobic or hydrophilic) has been an old but challenging question for hundreds of years not only in thin film lubrication but also in fluid mechanics. The classical Reynolds theory and fluid mechanics theory assume that no wall slip (boundary slip) occurs at the liquid–solid interfaces, i.e., the so-called no-slip boundary condition [1–3]. The validity of this assumption has been doubted for over a century. An early experiment [4] observed that wall slip occurred for water flowing at a hydrophobic surface. The development of microfluidics and MEMS (micro-electromechanical system) based devices has recently prompted the research interests in this area [5–11]. Wall slips were observed not only on hydrophobic surfaces [5–9] but also on hydrophilic surfaces [5,10,11], not only for a simple fluid but also for a complex Corresponding author. Tel.: +86 411 84706353; fax: +86 411 84708393. E-mail address: [email protected] (C.W. Wu).
0301-679X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2005.08.001
fluid [6]. This indicates that the wetting property of a surface is not the only parameter affecting the wall slip. Theoretical simulations using molecular dynamics (MD) also confirmed that wall slip occurs at a liquid–solid interface [12,13]. During the past decade, wall slip has received much attention in lubrication mechanics [14–20]. There are usually two models used to describe wall slip: the slip length model (SLM) [4–11] and the limiting shear stress model [15–20]. In the slip length model, the slip velocity, V s , is assumed to be proportional to local shear rate qu=qz, i.e., V s ¼ bqu=qz, where b, a constant, is the fictive slip length inside the solid surface at which the no-slip boundary condition holds. u is the fluid velocity, and z is the distance perpendicular to the surface. At a small range of shear rate and small wall slip, the slip length was experimentally found to be nearly constant [10,11] (nanometer scale), and thus the slip length model may adequately describe such a wall slip. At high shear rate, however, experimental observations [7–9] and MD study [13] showed that the slip is strongly nonlinear and can approximately be described by the limiting shear stress model [13]. Experimental observations [21,22] also showed that the wall slip
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Nomenclature F f f b h h0 k p R r t u Vs
hydrodynamic force, N yield function hydrodynamic load support coefficient slip length, m fluid film thickness, m minimum film thickness, m proportionality constant film pressure, Pa sphere radius, m radial coordinate, m time, s fluid velocity, m/s slip velocity, m/s
does not start immediately after a fluid is sheared, but starts after the shear rate reaches a critical value. The concept of the limiting shear stress was first proposed by Smith [23,24] for lubricants, and later was confirmed by Bair and Winer [25–27], where the solid surface is metal (hydrophilic). The limiting shear stress model, as shown in Fig. 1, assumes that wall slip occurs only after the surface shear stress t reaches the limiting shear stress tL , where the shear stress is a constant (shearrate independent) but the apparent shear rate can reach any values (the true shear rate equals the critical threshold). When totL , no slip takes place. What is shown in Fig. 1 is an ideal limiting shear stress model. Experiments usually give a transition region from no-slip to slip flowing the dashed curve. In thin film lubrication, either a high shear stress or a high shear rate is often encountered, and the limiting shear stress model was usually used [15–20]. In the present paper we use the limiting shear stress model to
v t tL t0 Z l T0 TL
Subscripts i L 0 s
denoting surfaces 1 and 2 limiting initial limiting slip
study the isothermal squeeze film flow between spherical hydrophobic surfaces. 2. Theory and numerical method The surface interaction of two approaching spheres is one of the most important topics in many scientific research areas, such as thin film lubrication, colloid interface science, etc. When two spherical surfaces lubricated by a fluid film are approaching each other, the squeezed fluid film will become thinner and thinner, and the shear rates at the lubricated surfaces become larger and larger. Thus a wall slip may occur at the interface of fluid and solid. Based on this principle, the surface force apparatus (SFA) [8,9,21] and the atomic force microscope (AFM) [7,10,11] have recently been used to indirectly measure the wall slip of a squeezed fluid film between a rigid sphere and a flat surface. Very strong shear-rate-dependent slip was found at high shear rates. Now let us use the limiting shear stress model to study the squeeze film flow, as shown in Fig. 2. From the Reynolds lubrication theory [28], one obtains the fluid velocity uðr; zÞ in r direction uðr; zÞ ¼
τL
v ¼ qh=qt, squeeze velocity, m/s shear stress, Pa limiting shear stress, Pa initial shear strength, Pa lubricant viscosity, Pa s slack variable, Pa ¼ t0 R=Zv ¼ tL R=Zv
1 qp 2 z ðz zhÞ þ ðV s2 V s1 Þ þ V s1 2Z qr h
(1)
and the shear stress inside the fluid film
Shear stress τ
qu 1 qp V s2 V s1 ¼ ð2z hÞ þ Z, (2) qz 2 qr h where p is the fluid film pressure, t the shear stress, Z the fluid viscosity, h the gap height, and V s1 and V s2 are the slip velocities at interfaces 1 and 2, respectively. For the first order approximation h is given by [28]
tðr; zÞ ¼ Z
tgα=η
Apparent shear rate
Fig. 1. Schematic of stress–rate curve of the limiting shear stress model. No slip occurs when tptL and slip takes place after t4tL .
hðrÞ ¼ h0 þ r2 =2R,
(3)
where h0 is the minimum thickness of the fluid film, r the radius coordinate, and R the sphere radius. For a squeeze
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where k is a proportionality coefficient, t0 is the initial limiting shear stress at the ambient pressure depending on interface property and temperature, and subscript i denotes surface 1 and 2, respectively. For a pure squeeze film flow between two smooth surfaces as shown in Fig. 2, t1 X0 and t2 p0, one obtains the following slip control equations:
z
ν
R
Surface 2
h0 h o
Surface 1
r
system of two spherical surfaces with radii of R1 and R2 , we can use an equivalent radius R ¼ ð1=R1 þ 1=R2 Þ1 in Eq. (3). Obviously tðr; zÞ is a linear function of z. That is to say, the maximum shear stress in the fluid film occurs at either surface 1 or surface 2 in an isothermal and Newtonian fluid flow. The shear stresses at surfaces 1 and 2 are qu h qp V s2 V s1 þ Z, (4) t1 ¼ Z ¼ qz z¼0 2 qr h qu h qp V s2 V s1 t2 ¼ Z þ Z. ¼ qz z¼h 2 qr h
(5)
The flow continuity condition yields the Reynolds equation for an isothermal squeeze film flow of Newtonian fluid between the approaching spherical surfaces: q rh3 qp 1 q qh rhðV s1 þ V s2 Þ r ¼ 0. (6) qr 12Z qr 2 qr qt One can obtain the classical Reynolds equation if no wall slip occurs, i.e., V s1 ¼ V s2 ¼ 0. For rigid surfaces with an approaching (squeezing) velocity v, qh=qt ¼ v ¼ constant, Eq. (6) reduces to (7)
The wall slip velocities can be written as the following complementary problem with the interface shear stresses: V si ¼ 0
if jti jptLi
ði ¼ 1; 2Þ,
(8a)
V si 40
if jti j ¼ tLi
ði ¼ 1; 2Þ,
(8b)
where tLi is the limiting shear stress at interface i. Usually it is a function of the surface wettability, temperature, fluid pressure, etc. For lubricants it increases linearly with the fluid pressure starting with an initial limiting shear stress and is given as [25–27] tLi ¼ t0i þ ki p,
f 1 ¼ t1 tL1 p0,
(10a)
f 2 ¼ t2 tL2 p0.
(10b)
Using Eqs. (4)–(5), and introducing slack variables li , Eqs. (10a)–(10b) are reduced to h qp iþ1 V s2 V s1 þ ð1Þ Z tLi þ li ¼ 0, (11a) fi ¼ 2 qr h
Fluid
Fig. 2. Schematic of a squeeze film system of an approaching sphere and a smooth surface.
h3 qp hðV s2 þ V s1 Þ þ rv ¼ 0. 6Z qr
865
(9)
li V si ¼ 0;
li X0; V si X0 ði ¼ 1; 2Þ.
(11b)
Numerical solution of Eq. (6) or Eq. (7) under control of the slip control equations (11a)–(11b) yields the film pressure and the slip velocities. Finite element method and complementary algorithm were used in the present work. This numerical technique was described in the analyses of a viscoplastic lubricant flow [16] and wall slip occurring at a journal bearing [18]. The solid surfaces are assumed to be rigid and no elastic deformation was considered since a large fluid pressure cannot be built up between two slippage surfaces. The region of numerical solution was taken to be 0or=Ro0:2 because most of the fluid pressure is concentrated on the contact center. The boundary conditions are: qp=qr ¼ 0 (null volume flow in r direction) at r ¼ 0 and p ¼ 0 at r=R ¼ 0:2. The wall slip is determined by the surface shear strength, gap geometry, fluid viscosity and approaching velocity. 3. Numerical analyses of squeeze film flow with wall slips 3.1. Surfaces with the same slip property If the two solid surfaces have the same slip property, then we have: tL1 ¼ tL2 ¼ tL , V s1 ¼ V s2 ¼ V s and t1 ¼ t2 ¼ t. For the simplicity of analysis, we take the proportionality coefficients of limiting shear stress k1 ¼ k2 ¼ 0 in this section and will discuss their effects in Section 3.3. Fig. 3 shows how the wall slip to be developed during the squeeze process of a fluid film when the dimensionless surface limiting shear stress T L ¼ T 0 ¼ t0 R=Zv ¼ 107 (tL ¼ t0 due to k1 ¼ k2 ¼ 0). Wall slip leads to a reduction in fluid film pressure in the slip zone (Fig. 3a). It can be seen that when the surface separation h0 =R ¼ 2:7 105 , the surface shear stress is below the limiting shear stress in the entire squeezed region (see Fig. 3b), and thus no slip occurs (see Fig. 3c). However, when h0 =R ¼ 2:6 105 , the slip just starts in a small area where the surface shear stress reaches its limiting value. When the fluid film further decreases, the surface shear stress reaches its limiting value in a larger and larger zone.
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C.W. Wu et al. / Tribology International 39 (2006) 863–872 100 τLR/ην=109
10-1
f*
108
107
10-2
2/3
f *=1-e-2.26(LR/)
106
(h0 / R)
10-3 10-8
10-7
10-6
10-5
10-4
102
103
h0/R
(a) 100 h0/R=5×10-6
10-1
f*
10-6
10-2
0.2×10-6
10-3 10-1
100
102 τLR/ην (106)
(b)
Fig. 4. (a) Hydrodynamic load support coefficient, f , the ratio of the hydrodynamic forces with wall slip to that without wall slip, versus film thickness parameter h0 =R for several values of the surface limiting shear stress; (b) hydrodynamic load support coefficient f versus normalized surface limiting shear stress for three values of film thickness parameters h0 =R. The open symbols are numerical solutions and the curves are obtained by Eq. (12). The two surfaces have the same slip property, i.e., tL1 ¼ tL2 ¼ tL ¼ t0 and k1 ¼ k2 ¼ k ¼ 0. The normalized limiting shear stress T L ¼ T 0 ¼ tL R=Zv ¼ t0 R=Zv ¼ 107 .
Fig. 3. Development of wall slip for several values of surface separations: (a) fluid film pressures; (b) surface shear stress distributions and (c) wall slip velocities. The two surfaces have the same slip property with normalized limiting shear stress T L ¼ T 0 ¼ tL R=Zv ¼ t0 R=Zv ¼ 107 and k1 ¼ k2 ¼ k ¼ 0.
Therefore, the slip amplitude is increased and the size of slip zone is enlarged. A hydrodynamic load support coefficient, f , which is defined as the ratio of the hydrodynamic force with wall
slip to that without wall slip [29], is plotted in Fig. 4a for several values of the surface limiting shear stresses. f is approximately proportional to the film thickness parameter h0 =R at small h0 =R. At large h0 =R, however, f approaches to 1 in a nonlinear manner. This is supported by the experimental observations [8,9]. Fig. 4b gives the log–log plot of f versus the dimensionless surface shear stress. When f o0:2 it is approximately proportional to ðtL R=ZvÞ2=3 . This is to say, the film thickness effect on the hydrodynamic force is stronger than the surface limiting shear stress. It is found that the numerical solutions (open symbols in Fig. 4) can be fitted by the following empirical equation: f ¼ 1 e2:26ðtL R=ZvÞ
2=3
ðh0 =RÞ
.
(12)
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The maximum error deviating from the full numerical solution is less than 8%. Eq. (12) gives a theoretical prediction for the hydrodynamic load support coefficient with a reasonable accuracy. For comparison, the slip length model gives an analytical solution for the hydrodynamic load support coefficient [29]: h0 h0 6b f ¼ 1þ 1 . (13) ln 1 þ h0 3b 6b Eq. (13) indicates that the hydrodynamic load support coefficient, f , depends only on the ratio of slip length b to the minimum film thickness h0 , but is independent of the fluid viscosity Z, approaching velocity v of the sphere, and the film thickness parameter h0 =R. If h0 =R is kept unchanged, however, the present work predicted that f increases with increasing R and tL , but decreases with increasing Zv. In other words, wall slip increases with increasing Zv, but decreases with increasing R and tL . The effect of radius R on wall slip is the same as the theoretical prediction of Einzel et al. [30], and the effects of Z and v are supported by the experimental observation [10]. 3.2. Surfaces with different slip properties First, we suppose that the spherical surface (surface 2) is a perfect hydrophobic surface with a null shear strength, i.e., the dimensionless limiting shear stress T L2 ¼ tL2 R=Zv ¼ 0. Such an ideal slippage spherical surface is approaching a smooth plane with different values of surface shear strength. To our surprise, it is found that such a squeeze film system can give only one quarter of the maximum load support coefficient, f , as long as the shear strength on the lower surface (surface 1) is high enough to inhibit any wall slip, as shown in Fig. 5a. f tends to null at small gap (high shear rate) and small surface shear strength at the lower surface (T L1 ¼ tL1 R=Zv ! 0). If the two surfaces are switched, the same situation occurs. In an extreme case that both of the surfaces have null shear strength, no hydrodynamic pressure is generated ( f ¼ 0). Second, we suppose that the spherical surface has a shear strength high enough to suppress any slip, for example T L2 ¼ tL2 R=Zv ¼ 1015 . The load support coefficient ranges from 0.25 to 1 when the normalized shear strength, T L1 , ranges from 0 to 109 , as shown in Fig. 5b. What is shown in Fig. 5 tells us that the match of slippage properties of the two surfaces in a spherical squeeze film system plays an important role in determining the load support coefficient and wall slip. 3.3. Effect of fluid pressure on wall slip The macro-rheological measurements of lubricants in a wetting surface by Bair and Winer [25–27] at high pressures showed a pressure-dependent limiting shear strength, as indicated in Eq. (9). For a wetting mineral-oil/steel interface, the order of k may range from 0.001 to 0.1. For example, for the silicone-oil/steel interface it has an
Fig. 5. Variations of the hydrodynamic load support coefficient for solid surfaces with different slip properties: (a) the upper surface is a perfect slip surface (null shear strength); (b) the upper surface has shear strength high enough to suppress slip. k1 ¼ k2 ¼ k ¼ 0.
order of about 0.007 [31]. For a nonwetting or partially wetting surface, however, no experimental data have been reported so far to determine the proportionality coefficient. But we believe it is a very small value according to the existing experimental observations [8,9]. For simplicity, we first investigate here the situation when the two surfaces have the same slip property, i.e., t01 ¼ t02 ¼ t0 , k1 ¼ k2 ¼ k. Fig. 6 shows how the proportionality coefficient affects wall slip and fluid film pressure. Increasing the proportionality coefficient will help to suppress wall slip (Fig. 6c) and thus gives rise to a high fluid pressure (Fig. 6a). Comparing Fig. 6b with Fig. 6d we can see more clearly how the surface shear stress and the surface limiting shear stress changes with the proportionality coefficient k. The former equals the latter in the slip zone, but is less than the latter in the no-slip zone. In the situation studied here, the surface shear stress is always less than its limiting value when k ¼ 0:005. Fig. 7 shows the effect of proportionality coefficient k on f for several
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Fig. 6. Effect of the proportionality coefficient, k, on the fluid film pressure and wall slip: (a) fluid film pressures; (b) wall shear stress distributions; (c) wall slip velocities; and (d) the surface limiting shear stresses. T 0 ¼ t0 R=Zv ¼ 107 and h0 =R ¼ 106 .
Fig. 7. Effect of the proportionality coefficient, k, on the hydrodynamic load support coefficient when both of the surfaces have the same slip property with a normalized initial shear strength t0 R=Zv ¼ 107 .
values of the surface gap. It can be seen that at a large k, no slip occurs ( f ¼ 1). However, it is very interesting that at a small k, for example ko0:6 103 , the effect of k on the wall slip is hardly found. The proportionality coefficient may affect slip in a complex manner depending on the difference between t0 and p. From Eq. (9) it can be seen that only when the product kp is increased to the same order as the initial shear strength t0 , does k considerably affect the wall slip. Fig. 8 shows effects of the proportionality coefficient of limiting shear stress on the hydrodynamic load support coefficient when the two surfaces have different initial shear stresses. If surface 1 has a higher limiting shear stress than surface 2, numerical solutions show that no slip occurs at surface 1. Now we investigate the effect of the proportionality coefficient at surface 2 (sphere) on the wall slip and hydrodynamic force. The fluid film thickness in Fig. 8a is one order smaller than that in Fig. 8b. The proportionality coefficient, k2 , affects the wall slip in three
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1.2 T02=108
1.0
107 0.8
f*
105 T02=0
0.6
0.4 T02=109 k1=0.01
0.2
h0/R=5×10-7 0.0 10-5
10-4
10-3 k2
(a) 1.2
Fig. 9. Comparison of the predicted hydrodynamic force with those measured by SFA technique [8]. The fluid was deionized water and the solid surfaces were atomically smooth micas coated with a methylterminated close-packed monolayer of condensed octadecyltriethoxysiloxane (OTE). R ¼ 2 cm, Zv ¼ 1:75 1010 N=m (note: we got this value by fitting the no-slip curve given in Fig. 1 of Ref. [8]; no viscosity datum was supplied and the indicated squeeze velocity seems to have problem there), k ¼ 0, tL ¼ t0 ¼ 0:33 Pa (tL R=Zv ¼ 3:77 107 ). For comparison the predictions of slip length model (SLM) are also shown in the inset.
8
T02=10
1.0
f*
0.8 107
0.6
0.4
105
0.2
T02=109 k1=0.01
0
h0/R=5×10-6
0.0 10-5
(b)
10-4
10-3 k2
10-2
10-1
Fig. 8. Effect of the proportionality coefficient at spherical surface, k2 , on the hydrodynamic load support coefficient: (a) h0 =R ¼ 5 107 and (b) h0 =R ¼ 5 106 .
zones: (i) almost no effect exists at small values of k2 (large slip zone); (ii) the hydrodynamic force increases with the proportionality coefficient in medium levels of k2 (small slip zone); and (iii) no effect exists again at large k2 (no slip zone). In the situation under study, the proportionality coefficient gives rise to almost the same effect on wall slip and fluid load support when the dimensionless initial shear stress at surface 2 ranges from 0 to 105 . 4. Comparison with experiments Comparison of the hydrodynamic force predicted by the present work with those measured by SFA [8] for the hydrodynamic force of a squeeze film flow of deionized water is given in Fig. 9. The solid surfaces were atomically smooth micas coated with a methyl-terminated closepacked monolayer of condensed octadecyltriethoxysiloxane (OTE). On such a poorly wetting surface, the advancing contact angle of water was about 110 C. The theoretical prediction agrees reasonably with the experimental data in a large range of shear rate. After the
minimum fluid film thickness h0 is less than about 150 nm (h0 =R 7:5 106 ), wall slips start at both of the surfaces and it is as if the hydrodynamic force reached a saturation level beyond which only a limited amount of further increase was possible. The measured hydrodynamic load support coefficient, f , at a large film thickness is slightly larger than the theoretical prediction. This may be due to the experimental error. At a small film thickness (h0 o5 nm), however, the measured data are slightly smaller than those predicted. The sources of experimental error in this area may be more complex than that in the large film thickness. One of the most probable explanations is that the driving velocity, v, induced from the amplitude of oscillation in SFA system, was reduced by compliance of the apparatus at the lesser values of film thickness. On the other hand, the continuous media assumption used in the theoretical analysis may meet a problem for a very small film thickness. However, our numerical results show that the elastic deformation effect of the solid surfaces is so small that the hydrodynamic force changes no more than 0.1% (h0 45 nm). In the inset of Fig. 9 SLM predictions using Eq. (13) are also shown, best fitting f either in small h0 or in large h0 . Evident is the big deviation of SLM from the experiments. The theoretical prediction of the present paper is also compared with the experimental data measured by Craig et al. [7] using AFM technique (see Fig. 10). A good agreement is obtained again. Since no effect of the fluid pressure on wall slip was reported in the literature [7,8,32], we did not consider the effect of proportionality coefficient on wall slip (k1 ¼ k2 ¼ 0).
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870 150 No-slip
SLM: b=12nm 100 F (nN)
Experiment[7]
50 Theory
0 0.0
0.1
0.2
0.3
0.4
0.5
1/h0 (nm-1)
Fig. 10. Comparison of the predicted hydrodynamic force with those measured by AFM technique [7]. Both of the surfaces of the approaching sphere (silica) and the flat wall (mica) were coated with a 15.1 nm thickness of gold film. The fluid is an aqueous sucrose with viscosity Z ¼ 38:9 mPa s. tL ¼ t0 ¼ 300 Pa, R ¼ 10:4 mm, v ¼ 21:6 mm=s.
5. Discussions In the present paper, we investigate the effect of wall slip on the isothermal squeeze film flow of Newtonian fluid between spherical hydrophobic surfaces based on the concept of interfacial limiting shear stress [see Eq. (9)]. Apart from the work of Smith [24] and Bair and Winer [25–27], several other authors [33–37] have also studied the limiting shear stress of lubricants. The limiting shear stress consists of two important parameters: the initial limiting shear stress, t0 , and the proportionality coefficient, k. In a wetting interface of oil and steel, the reported initial shear strength usually is very large, varying from 0.16 to 8 MPa [27,31,35], but the results should be treated with care because it is a difficult parameter to measure. The proportionality coefficient, k, has been examined in several ways and by a few research groups [31,35–37] with the results that range from about 0.007 to 0.15 and the variation is found to be temperature-dependent. For a nonwetting or partially wetting surface, the limiting shear stress is very small and was reported in only a few papers [9,21,22,32]. It was found to depend on surface wettability, surface roughness, fluid viscosity, etc. A small value of the limiting shear stress of about 0.3 Pa was reported for the interface of deionized water and atomically smooth micas (r.m.s roughness 0:2 nm) coated with a methyl-terminated close-packed monolayer of condensed OTE [32]. However, the limiting shear stress is about 0.6 Pa for the interface of tetradecane and the same solid surface. The limiting shear stress may decrease with surface contact angle [32] and fluid viscosity [10]. The roughness effect on the limiting shear stress is not so clear now. Some researchers [5,9,32] reported that surface roughness inhibits wall slip or increases the limiting shear stress, but others [10] reported that it increases wall slip. In Section 4 for
liquid slippage on a nonwetting or partially wetting surface, the effect of proportionality coefficient was not considered, i.e., the limiting shear stress equals the initial limiting shear stress. However, due to the small value of the limiting shear stress, we believe the proportionality coefficient is very small for a nonwetting or partially wetting surface. Therefore, some researchers [16,19,20,22] neglected its effect. Spikes and Granick [22] also discussed a similar slip problem with a different viewpoint. They developed a new equation for Newtonian fluid flow in the presence of wall slip. Slip is envisaged to occur only when a critical surface shear stress is reached, and once slip starts, it is described by the slip length model. But they think the critical shear stress is pressure-independent. It is to be noted that flow visualization of lubricants at steel gap (highly wetting surface) at high pressure (up to 1 GPa) and shear stress (order of 10 MPa) by Bair et al. [38–40] showed that the slip occurred at the midplane within the film, where the temperature and shear rate were localized, but not at the wall. At such a high pressure the viscosity–pressure relation has to be considered and thus the lubricant viscosity may be so high that it looks like a solid behavior (near the glass transition of the lubricant). However, the fluid pressure studied here is very small. For example, the calculated maximum fluid pressures at h0 ¼ 5 nm and 8 nm in Fig. 9 are about 2.8 kPa and 2.2 kPa, respectively. The match of slippage properties of two surfaces significantly affects the hydrodynamic load support coefficient of a squeeze film between spherical surfaces. Thus in both AFM and SFA measuring systems, the slip should be carefully deduced from measurements of the hydrodynamic load support coefficient. The same group of the load support coefficients may give rise to different explanations for the slip mechanism if the range of shear rate is not very large. For example, if we fit the experimental data in Fig. 10 only after the separation h0 410 nm (1=h0 ¼ 0–0.1), both the constant slip length model and the limiting shear stress model can fit the experimental data pretty well. However, in the case of large range of separation (large range of shear rate) the slip length model cannot agree with the experiment as shown in Fig. 10. We also noted that some researches, for example Charlaix’s group [41,42] using an SFA measuring system, did not report a shearrate-dependent slip behavior. We believe that one of the important reasons may be the small shear rate range under their study. Of course, other mechanisms, such as surface contamination, roughness, nanobubbles or small thickness of gaseous film, match of surface slippage properties, etc., may also affect the slip behavior. The mechanism study of wall slip is out of the scope of the present paper. In this paper, we did not consider the elastic deformation of the solid surfaces. In the case of high squeeze velocity and without wall slip, effect of elastic deformation of the solid surfaces may need studying [43]. In a very small film thickness, the elastic deformation and the surface force may play a role in determining the total interaction force
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[44]. Both experimental measurements (SFA and AFM) and our theoretical prediction may give a little large error for a very small film thickness, i.e., h0 o3–5 nm. However, for the hydrophobic surfaces with small squeeze velocity and large wall slip as we studied here, effect of the elastic deformation on the wall slip and hydrodynamic force should be negligible. This has been supported by the experimental measurements of squeeze film of spherical hydrophobic surfaces with wall slips [7–11].
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7. Acknowledgments The authors would like to express their high appreciation to V.S.J. Craig for his original experimental data supplied for Fig. 10. This research was supported by NSFC (10421002, 10332010, 10272028) and SRFDP (20030141013).
References 6. Conclusions The wall slip in isothermal squeeze film of a smooth spherical surface and a flat surface is studied in the present paper. Wall slip for a Newtonian fluid flowing at a solid surface is analyzed using the limiting shear stress model and complementary algorithm. The wall slip velocity is controlled by the limiting shear stress at the liquid–solid interface. It is found that good agreements exist between the present theoretical predictions and some existing experimental observations of squeeze film flow for nonwetting or partially wetting surfaces, especially at high shear rates. The numerical results show that after the maximum surface shear stress reaches the limiting shear stress, a wall slip occurs and then spreads out upon increasing the squeezing velocity or decreasing the film gap (increasing shear rate). After the wall slip starts, the hydrodynamic force increases very slowly with further decrease of the film gap. The hydrodynamic load support coefficient decays with a negative exponential function of the surface limiting shear stress and film thickness. For two surfaces with the same slip property and negligible proportionality coefficient of the limiting shear stress, the following empirical equation is given for prediction of the hydrodynamic load support coefficient: f ¼ 1 e2:26ðtL R=ZvÞ
2=3
ðh0 =RÞ
.
If h0 =R is kept unchanged, f is reduced with increasing Zv, but increased with increasing R and tL . In other words, if h0 =R is kept unchanged, we predicted that wall slip increases with increasing Zv, but reduces with increasing R and tL . In the squeeze film flow of spherical surfaces with wall slip, surface shear strength, surface curvature radius, fluid viscosity, the separation and squeezing velocity of two surfaces are coupled together to affect the slip behavior. The possible effect of pressure on wall slip is also discussed using a proportionality coefficient of the limiting shear stress. It is found that pressure starts to suppress wall slip after the proportionality coefficient reaches a critical threshold, but almost no effect is found when the proportionality coefficient is very small. Finally, it is found that the match of slippage properties of the two surfaces in a spherical squeeze film system is another important factor affecting the wall slip behavior.
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