Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

48, 1, pp. 191-205, Warsaw 2010

ANALYSIS OF CONTACT MIXING WITH BOUNDARY AND

HYDRODYNAMIC FLUID FILMS CONSIDERING

BOUNDARY SLIPPAGE

Yongbin Zhang

Zhejiang Jinlei Electronic and Mechanical Co., Zhejiang Province, P.R. China

e-mail: engmech1@sina.com

An analysis is presented for amicro-contactwhere boundary and hydro-
dynamic fluid films simultaneously occur considering boundary slippa-
ge appearance at the upper contact surface in the boundary film area.
The contact is one-dimensional, composed of twoparallel plane surfaces,
which are respectively rough rigid with rectangular projection in profile
and ideally smooth rigid. In the outlet zone of the contact a boundary
film occurs, and in the inlet zone of the contact a conventional hydrody-
namic fluid film emerges. In the boundary film area, the film slips at the
upper contact surface due to the limited shear stress capacity of the film-
contact interface,while the filmdoes not slip at the lower contact surface
due to the shear stress capacity of the film-contact interface, which is
large enough. In the boundary film area, the viscosity and density of the
film are varied across the film thickness due to the film-contact interac-
tions, and their effective values are used inmodelling which depends on
the boundary film thickness. In the fluid film area, the film does not slip
at either of the contact surfaces.

Key words: boundary film, nanometer-scale thin film, fluid film, boun-
dary slippage

1. Introduction

In engineering applications, two solid surfaces often slide one against another.
The contact between themusually needs to generate the load-carrying capaci-
ty with the ability of low friction and anti-wear. To achieve this requirement,
the contact is usually lubricated with fluids. Since the solid contact surface is
actually rough, when in carrying the load and fluid lubrication, the contact
between two solid surfaces is actually usually quitemixed.The author pointed
out that in a hydrodynamic lubricated line or point, concentrated contacts,



192 Y. Zhang

different contact regimes frequently simultaneously occur in different areas of
the contact in modern industry (Zhang, 2004b, 2006b). Such contact regimes
are conventional hydrodynamic lubricated contact, physical adsorbed boun-
dary layer contact, chemical boundary layer contact and fresh solid material
to material contact. Experiments and theories showed that these four contact
regimes often simultaneously occur in the hydrodynamic concentrated contact
with medium or heavy loads (Zhang, 2004a, 2006a,b; Begelinger and Gee de,
1974, 1976; Tabor, 1981).

Inmodelling of the hydrodynamic concentrated contact, however, the stu-
dy goes backward to the experimental findings. It was pointed out that before
1990s the theoretical modelling of the hydrodynamic concentrated contact be-
longs to the classical modelling (Zhang, 2006b). In that modelling, although
the contact surfaces were considered as rough, the whole contact between the
two surfaces were treated as the conventional hydrodynamic lubricated con-
tact, where the fluid film is relatively thick, see for example Goglia et al.
(1984), Lubrecht et al. (1988). The theoretical modelling of the hydrodynamic
concentrated contact at the end of the last century and in the begining of this
century belongs to the modern modelling (Zhang, 2006b). In that modelling,
thewhole contact of the two surfaces is considered as consisted of different con-
tact parts where different contact regimes respectively occur. These contact
regimesmaybe the conventional hydrodynamic lubricated contact and surface
asperity to asperity contact, see for example Jiang et al. (1999), Holmes et al.
(2003). They may be also the conventional hydrodynamic lubricated contact
and physical adsorbed boundary layer contact, see Zhang (2004a), Zhang et
al. (2003), Zhang and Lu (2003). It was pointed out that in the futuremodel-
ling of the hydrodynamic concentrated contact, conventional hydrodynamic
lubricated contact, physical adsorbed boundary layer contact, chemical boun-
dary layer contact and fresh solidmaterial tomaterial contactmay need to be
simultaneously considered and treated as occurring in different contact parts
depending on the operating condition (Zhang, 2006b). In the recentmodelling
of the hydrodynamic contact, conventional hydrodynamic lubricated contact,
physical adsorbed boundary layer contact and direct surface asperity to aspe-
rity contact between two surfaces were simultaneously considered and treated
as occurring in different parts of the contact depending on the film thickness
(Zhang, 2007a,b,c). It is a more advanced modelling of the contact. As that
modelling showed, it is admitted that the physical adsorbed boundary layer
contact and direct surface asperity to asperity contact both actually occur in
micro-contact areaswith a very small contact width,which are irregularly and
discretely distributed in the realistic hydrodynamic contact.



Analysis of contact mixing... 193

The boundary slippage is a phenomenon occurring in the hydrodyna-
mic contact when the film-contact interfacial shear stress exceeds the shear
strength of the film-contact interface. Experiments (Bonaccurso et al., 2003;
Pit et al., 2000;Craig et al., 2001; ZhuandGranick, 2001) andmolecular dyna-
mics simulations (Thompson andTroian, 1997; Sun and Ebner, 1992) showed
that the film-contact interfacial slippage can actually occur in a practical hy-
drodynamic contact not only of non-wetting or partially wetting systems but
also of completely wetting systems due to the weak interaction strength and,
then, the low shear strength of the film-contact interface. Experiments and
molecular dynamics simulations showed that in micro-contact with boundary
films the slippage at the film-contact interface needs to be considered in de-
veloping its load-carrying capacity theory. Bymolecular dynamics simulation,
Thompson andTroian (1997) showed that in themicro-contact with boundary
films, the boundary condition at the film-contact interface should be generally
considered as slippage. They showed that this boundary slippage is a result
of inefficient momentum transfer at the film-contact interface. It was deter-
mined by the interaction strength between the contact and the film. It was
experimentally shown by Zhu and Granick (2001) that the film-contact in-
terfacial slippage reduces the load-carrying capacity of boundary films in the
micro-contact. Craig et al. (2001) experimentally found the slippage at the
film-contact interface in the micro-contact with boundary films by using the
Atomistic Force Microscope measurement. They found that the degree of the
film-contact interfacial slippage in the micro-contact is increased with both
contact surface speed and film viscosity. Sun and Ebner (1992), Cieplak et al.
(2001) and Cheikh and Koper (2003) also observed the slippage occurring at
the film-contact interface in the general micro-contact with boundary films,
respectively by molecular dynamics simulation and experiments. They all fo-
und that the film-contact interfacial slippage is determined by the interaction
strength between the contact and the film. It is generally agreed that a weak
adhesion strength between the contact and the film or a repulsive wall yield
film-contact interfacial slippage, while a strong film-contact attraction gives
no slippage at the film-contact interface.

The present paper analytically investigates the boundary slippage effect in
the micro-contact mixing with boundary and hydrodynamic fluid films con-
sidering the film-contact interfacial slippage in the boundary film area. The
contact is one-dimensional, formed by two parallel rigid plane surfaces. The
upper contact surface is roughwith rectangular ridges in profile and the lower
contact surface is ideally smooth. The boundary film is formed between the
ridge of the upper contact surface and the lower contact surface. The fluid



194 Y. Zhang

film is formed between the dent of the upper contact surface and the lower
contact surface. The analytical approach proposed by the author and his col-
leagues (Zhang, 2006c,d; Zhang et al., 2003) is used for the boundaryfilm, and
conventional hydrodynamic analysis is used for the fluid film.

2. Contact model

The micro-contact studied in the present paper is one-dimensional and iso-
thermal, formed between two parallel rigid plane surfaces. The upper contact
surface is stationary and rough with rectangular ridges in profile. The lower
contact surface ismoving and ideally smooth. In themicro-contact, the boun-
dary film occurs in the outlet zone and the hydrodynamic fluid film occurs in
the inlet zone. In the boundary film area, the film slips at the upper contact
surface due to the limited shear stress capacity of the film-contact interface,
and it does not slip at the lower contact surface due to the shear stress ca-
pacity of the film-contact interface, which is large enough. The viscosity and
density properties of the boundary film are considered. In the fluid film area,
the film does not slip at either of the contact surfaces. The fluid is entrained
from the fluid film area into the boundary film area.
Figure 1 shows the profile of this micro-contact. In the figure, hb is the

boundary film thickness, ha is the hydrodynamic fluid film thickness, la is the
width of the fluid filmarea, lb is thewidth of the boundary filmarea, and u is
the speed of the lower contact surface.

Fig. 1. Simulatedmicro-contact; B1 – boundary film area, B2 – hydrodynamic fluid
film area

3. Analysis

The analysis of the pressures and carried loads of the micro-contact shown in
Fig.1 are respectively made with and without slippage assumed at the upper



Analysis of contact mixing... 195

contact surface in the boundary film area, i.e. the B1 sub-zone. The analyses
are based on the following assumptions:

(a) The flow is one-dimensional;

(b) The pressure across the film thickness is constant;

(c) The film inertia is negligible;

(d) The operating condition is steady-state.

These assumptions are usually realistic. Also, in the present analysis, the bo-
undary film shear elastic modulus effect is neglected (Zhang, 2009a,b).

The used coordinates are shown in Fig.1. The pressure boundary condi-
tions of the micro-contact are

p|x=0 =0 p|x=(lb+la) =0 (3.1)

3.1. No slippage at the upper contact surface in the boundary film area

The approach proposed by the author and his colleagues (Zhang, 2006c;
Zhang et al., 2003) is used to analyse the boundary film behaviour in the
B1 sub-zone in Fig.1. The approach needs to incorporate both the molecu-
lar dynamics effect and the non-continuum effect of the boundary film. The
boundary film non-continuum effect is described by the flow factor approach
(Zhang and Lu, 2005; Zhang, 2006d) which incorporates the boundary film
discontinuity and inhomogeneity effects across the film thickness. According
to the previous simulation for the same contact, the value of the flow fac-
tor θv depicting the boundary film non-continuum effect is very small (no
more than 1.01) (Zhang, 2008). It means that in the present analysis, the
value of θv can be taken as unity and the boundary filmnon-continuum effect
is negligible. The boundary filmmolecular dynamics effect is described by the
following equivalent continuum rheological model (Zhang, 2004a,b; Zhang et
al., 2003)

γ̇=
τ

η
eff
bf
(p,hb)

for |τ|<τl

τ = sgn(γ̇)τl for |τ| ­ τl

(3.2)

where τ is the shear stress, γ̇ – shear strain rate, η
eff
bf
– boundary film effec-

tive viscosity, and τl – limiting shear stress. In the present case, within the
boundary film and at the two contact surfaces τl =+∞. The model assumes
that slippage does not occur either within the boundary film or at the two
contact surfaces.



196 Y. Zhang

The boundary film viscosity is predicted by the following equation (Zhang
et al., 2003; Zhang and Lu, 2003)

η
eff
bf
(hb)=Cy(hb)ηc for 0<

hb
hcr,bf

< 1 (3.3)

where ηc is constant representing the average viscosity of the continuum film
at the pressure of the boundary film area, hcr,bf is the critical boundary film
thickness, and Cy is expressed as

Cy(hb)= a0+a1
( hb
hcr,bf

)

−1
+a2
( hb
hcr,bf

)

−2
(3.4)

where a0, a1 and a2 are constants.
The boundary film density is expressed as (Zhang et al., 2003; Zhang and

Lu, 2003)

ρ
eff
bf
(hb)=Cq(hb)ρc for 0<

hb
hcr,bf

< 1 (3.5)

where ρc is constant representing the average density of the continuum film
at the pressure of the boundary film area, and Cq is expressed as

Cq(hb)= g0+g1
hb
hcr,bf

+g2
( hb
hcr,bf

)2
+g3
( hb
hcr,bf

)3
(3.6)

where g0, g1, g2 and g3 are constants.
The Reynolds equation in the boundary film area is

qm =−
uhbρ

eff
bf

2
−
ρ
eff
bf
h3b

12η
eff
bf

dp

dx
(3.7)

where qm is the mass flow through the contact.
Define

λb,bf =−12
(

qm+
1

2
uhbρ

eff
bf

)

ηc
Cy

ρ
eff
bf
h3
b

SolvingEq. (3.7) byusing the boundary condition expressed byEq. (3.1) gives
the pressure in the boundary film area as follows

p=λb,bfx (3.8)

The Reynolds equation in the B2 sub-zone in Fig.1 is

qm =−
1

2
ρcuha−

ρch
3
a

12ηc

dp

dx
(3.9)



Analysis of contact mixing... 197

Define

λa,ρ =−12ηc
1

h3a

(qm
ρc
+
1

2
uha
)

SolvingEq. (3.9) byusing the boundary condition expressed byEq. (3.1) gives
the pressure in the B2 sub-zone as follows

p=λa,ρ(x− la− lb) (3.10)

From the constraint condition that the pressure at x = lb in Fig.1 is
continuous, the mass flow through the contact qm is obtained by solving the
constraint equation on the pressure at x= lb as follows

qm =−
uρcφ2(hb+haφ1)

2(1+φ1φ2)
(3.11)

where φ1 = rl/(r
3
hCy), φ2 =Cq. Here rl = la/lb and rh =ha/hb.

The carried load (per unit contact length) by themicro-contact is derived
as

w1 =
λb,bfl

2
b −λa,ρl

2
a

2
(3.12)

The pressure at x= lb is

pno−slip|x=lb =λb,bflb (3.13)

3.2. Slippage at the upper contact surface in the boundary film area

In this case, theboundaryfilmnon-continuumeffect is negligible due to the
value of θv approaching unity. The boundary film molecular dynamics effect
is still described by Eq. (3.2). For this case, in Eq. (3.2), within the boundary
film and at the film-lower contact surface interface τl = +∞, while at the
film-upper contact surface interface τl = τs. Here, τs is the shear strength
of the film-upper contact surface interface. The model assumes that slippage
does not occur either within the boundary film or at the film-lower contact
surface interface, but can occur at the film-upper contact surface interface if
the magnitude of the shear stress at the film-upper contact surface interface
exceeds the value of τs.

The shear strength τs of the film-upper contact surface interface is predic-
ted by the following equation (Zhang et al., 2003; Zhang and Lu, 2003)

τs(hb)=C
i
taol(hb)τs,c for β0 ¬

hb
hcr,bf

< 1 (3.14)



198 Y. Zhang

where τs,c is constant representing theaverage shear strengthof the continuum
film-upper contact surface interface at the pressure of the boundary film area,
and Citaol is expressed as

Citaol(hb)= d
i
0+d

i
1

( hb
hcr,bf

)

−1
+di2

( hb
hcr,bf

)

−2
(3.15)

where di0, d
i
1 and d

i
2 are constants.

The Reynolds equation for the boundary film area is

qm =B0+B1
dp

dx
(3.16)

where qm is the mass flow through the contact and

B0 =
ρ
eff
bf
τsh
2
b

2η
eff
bf

−ρ
eff
bf
uhb B1 =−

ρ
eff
bf
h3b

2η
eff
bf

The solution of Eq. (3.16) is

p=
qm−B0
B1

x+ c (3.17)

From the boundary condition p(0) = 0, it is solved that c=0. The pressure
in the boundary film area is then expressed as

p=
qm−B0
B1

x (3.18)

At x= lb, the film pressure is

pslip|x=lb =
(qm−B0)lb
B1

(3.19)

According to Section 3.1, it is solved from the fluid filmarea that at x= lb
the film pressure is pslip|x=lb =−λa,ρla. It is then equated that

qm−B0
B1

lb =−λa,ρla (3.20)

It is solved from Eq. (3.20) that

qm =
uhaρc

(

6rl+2Cyr
2
h−

τsharh
uηc

)

−12rl−
2Cyr

3

h

Cq

(3.21)

The load per unit contact length carried by the micro-contact is

w2 =
qm−B0
2B1

l2b −
1

2
λa,ρl

2
a (3.22)



Analysis of contact mixing... 199

3.3. Performance parameters of the contact

Define the relative reduction of the carried load of the contact due to the
boundary slippage as

rw =
w1−w2
w1

(3.23)

The value of rw can reflect the influence of the boundary slippage on the
load-carrying capacity of the contact.

Define the relative reduction of the pressure at x= lb due to the boundary
slippage as

rp =
pno−slip|x=lb −pslip|x=lb

pno−slip|x=lb
(3.24)

The value of rp can reflect the influence of the boundary slippage on the local
pressure in the contact.

Define the relative increase of the mass flow through the contact due to
the boundary slippage as

Iq =
|qm,slip|− |qm,no−slip|

|qm,no−slip|
(3.25)

where qm,no−slip is calculated from Eq. (3.11) and qm,slip is calculated from
Eq. (3.21).

4. Results

The values of rw, rp and Iq are respectively calculated for two cases. Case 1
represents the heavy load and high sliding speed operating condition. Case 2
represents the light load and low sliding speed operating condition. The ope-
rational parameter values for these two cases are respectively listed as follows:

Case 1: ηc = 4000Pas, u = 100m/s, la + lb = 1µm, hcr,bf = 20nm,
ha−hb =20nm, rl =0.5, ρc =960kg/m

3, τs,c =1∼ 100MPa.

Case 2: ηc = 0.1Pas, u = 0.01m/s, la + lb = 1µm, hcr,bf = 20nm,
ha−hb =20nm, rl =0.5, ρc =960kg/m

3, rh =11 (hb =2nm).

These two caseshave the sameboundaryfilmpropertyparameter values shown
in Table 1.



200 Y. Zhang

Table 1.Boundary film property parameter values

β0 a0 a1 a2 g0 g1

0.075 1.0822 −0.1758 0.0936 1.30 −1.0654

g2 g3 d
i
0 d

i
1 d

i
2

1.3361 −0.571 0.9726 0.0261 1.3158 ·10−3

4.1. Case 1

It is found that forCase 1 the values of rw, rp and Iq are not dependent on
the interfacial shear strength τs,c but obviously depend on the boundary film
thickness hb, when the value of τs,c ranges between 1MPa and 100MPa. This
may be a common result for the heavy load and high sliding speed condition.
It may be drawn that in the heavy load and high sliding speed condition the
reductions of the carried load and the local pressure of the contact and the
increase of the mass flow through the contact due to the boundary slippage
in the boundary film area are practically not determined by the boundary
film-contact interfacial shear strength but determined by the boundary film
thickness.

Figure 2 plots the values of rw, rp and Iq against the dimensionless boun-
dary film thickness obtained for Case 1. The values of rw and rp are nearly
linearly increased with the increase of the boundary film thickness hb and
strongly influenced by the boundary film thickness. The value of rw appears
considerable. It means that the boundary slippage considerably reduces the
load-carrying capacity of the micro-contact.

Fig. 2. Plots of the values of rw, rp and Iq against the dimensionless boundary film
thickness obtained for Case 1



Analysis of contact mixing... 201

Researches may also care about the value of rp at a small boundary film
thickness. It is shown in Fig.2 that the value of rp is increased from 0.12 to
0.3 when the boundary film thickness hb is increased from 2nm to 5nm. It
is found that when the boundary film thickness is below 5nm, the boundary
slippage occurring in the boundary film area can cause considerable reduc-
tions of the local pressure in the micro-contact. This reduction may be able
to considerably influence the contact surface elastic deformations and then
the local film thickness when the contact surfaces are elastic and the boun-
dary film thickness is on the film molecule size scale. Figure 2 suggests that
when the boundary film thickness is below 5nm and the boundary film is
considered in themixed contactmodelling, themodel should also consider the
boundary film-contact interfacial slippage, probably occurring in the contact
and influencing the local film thickness.

Figure 2 shows that theboundaryslippage in theboundaryfilmarea causes
themass flow through the contact increased nearly one times in the condition
of heavy load and high sliding speed for all boundary film thicknesses. At a
small boundary film thickness, this increase is more significant.

4.2. Case 2

Figure 3 plots values of rw, rp and Iq against the interfacial shear
strength τs,c obtained for Case 2. It is found that in the condition of a li-
ght load and low sliding speed, the values of rw, rp and Iq strongly depend on
the boundary film-contact interfacial shear strength. These values are shown
to be linearly increasedwith the reduction of the boundary film-contact inter-
facial shear strength. In the condition of the light load and low sliding speed,
the boundary film interfacial slippage can also considerably reduce the load-
carrying capacity of themicro-contact. Although the values of rp are shown to
be small, the local pressure reductions under these rp values due to the boun-
dary slippage are usually able to considerably influence the local film thickness
by changing the local contact surface elastic deformations when the contact
surfaces are elastic and the boundary film thickness is on the film molecule
size scale. In the mixed contact modelling, for the condition of the light load
and low sliding speed, when the boundary film thickness is below 5nm and
the boundary film is considered, the boundary film-contact interfacial shear
strength and the boundary film-contact interfacial slippage need to be con-
sidered. For the light load and low sliding speed condition, the reduction of
the boundary film-contact interfacial shear strength is shown to significantly
increase the mass flow through the contact.



202 Y. Zhang

Fig. 3. Plots of the values of rw, rp and Iq against the interfacial shear strength τs,c
obtained for Case 2

5. Conclusions

The present paper analytically investigates the effect of the boundary film-
contact interfacial slippage on the local pressure, carried load and mass flow
of the micro-contact. The contact is a mixed contact. In its outlet zone there
occurs a physical adsorbedboundary layer, and in its inlet zone a conventional
hydrodynamic fluid film. The contact is formed between two parallel rigid
plane surfaces.Theupper contact surface is roughwith rectangular projection.
The lower contact surface is ideally smooth. The boundary film is assumed to
slip at the upper contact surface but not slip at the lower contact surface. It
is also assumed not to slip within the film. The conventional hydrodynamic
fluid film is assumed not to slip at either of the contact surfaces.

It is found that when the boundary film thickness is below 5nm, the
boundary slippage occurring in the boundary film area can cause considerable
reductions of the local pressure in the micro-contact. This reduction may be
able to considerably influence the contact surface elastic deformations and,
then, the local film thickness when the contact surfaces are elastic and the
boundary film thickness is on the filmmolecule size scale. It is suggested that
when the boundary film thickness is below 5nm and the boundary film is
considered in themixed contactmodelling, themodel should also consider the
boundaryfilm-contact interfacial shear strengthandtheboundaryfilm-contact
interfacial slippage, which probably occurs in the contact and influences the
local film thickness.

Project support: Natural Science Foundation of Jiangsu Province, China

(BK2008189).



Analysis of contact mixing... 203

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Analysis of contact mixing... 205

Analiza kontaktu mieszanego z uwzględnieniem poślizgu brzegowego

w strefie hydrodynamicznego filmu

Streszczeie

Analiza zaprezentowana w pracy dotyczy zagadnienia mikro-kontaktu w strefie
jednoczesnego występowania brzegowego i hydrodynamicznego filmu płynu z możli-
wością poślizgu w strefie brzegowej na górnej powierzchni kontaktu. Rozważany kon-
takt jest jednowymiarowy i odnosi się do dwóch równoległych płaskich powierzchni,
z których jedna jest sztywna i szorstka w przekroju prostopadłym, a druga tak samo
sztywna lecz idealnie gładka.Wobszarzewyjścia ze strefy kontaktu pojawia się brze-
gowy film płynu, natomiast w strefie wejścia konwencjonalny film hydrodynamiczny.
W strefie brzegowej film ulega poślizgowi na górnej powierzchni wskutek ograniczo-
nej zdolności przenoszenia naprężeń stycznychpomiędzy płynema ścianką, natomiast
na powierzchni dolnej poślizg nie występuje. W strefie brzegowej filmu jego lepkość
i gęstość zmienia się wzdłuż grubości wskutek interakcji płynu z powierzchnią. Efek-
tywne wartości użyte wmodelowaniu kontaktu zależą od grubości filmu brzegowego.
W regularnej strefie film nie ulega poślizgowi na żadnej z omawianych powierzchni.

Manuscript received May 7, 2009; accepted for print July 23, 2009