Elastohydrodynamic - Journal de Physique II

J.

Phys.

II

France

(1996)

6

97-104

JANUARY

Elastohydrodynamic Lubrication Force The Bodies Motion of Hertzian Deformable J-H-

University Cambridge

of

Canlbridge, Department OHE, UK

1995,

June

2

Abstract.

Physics,

Laboratory,

Cavendish

on

is

calculated

classical

in the

theory of deformation deformation analytical in the and the velocity of approach

and and on

dependent

pressure

a

September1995,

15

Hertz's

with

of

elastohydrodynanlic

The

head

revised

Other topics in Disperse systems

PACS.46.90+s

force

Squeeze

the

Madingley

Road,

CB3

PACS.82.70-y

another

97

PAGE

Vliet

van

(Received

in

1996,

limit

accepted

13

mechanics

force

slightly

two

on

of small

lubrication

sizes

gap

theory.

An

bodies

defornlable

small

and

defornlations

expression

for

obtained. The effect of the gap size is the lubrication force is demonstrated.

viscosity

1995)

October

is

the

approaching one by combining force

lubrication

size, the Further, the

elasticity

gap

lubrication

calculated.

Introduction

1.

The hydrodynamics of cally important systems

bodies is of great experimental and practiinterest in many the surface forces scale, studies with mesoscopic in apparatus on e.g. of particles in liquids and in the rheology of suspensions Historically, and colloidal systems. elastohydrodynamic lubrication theory has been worked out in the context of understanding the workings of highly loaded gears and bearings, see e.g. iii. Here this theory is specialized elastic bodies approaching one another head on. to A hydrodynamic force between elastic bodies deforms these objects changing their two surface profile which in turn modification of the hydrodynamic force. If the leads to a distance between the surfaces is small enough, I.e. typically much smaller than the radii of curvature of the surfaces of the bodies, the Reynolds equation calculate the hydrodybe used to can namic

large

in

pressures

the

deformable a

gap

with

respect

to

ambient

pressure.

If the

deformations

not

are

too

approach [2,3] can be used to calculate the counteracting elastic pressures. Christensen recently Davis et al. [5] used the Hertzian approach to calculate the [4] and more deformation additional hydrodynamic numerically. The made and the assumptions pressure here to calculate the lubrication force analytically are an explicit reference to the surface profile of the deformed body (it is assumed to consist of two parts: a flat central region and a curved outside region which is not deformed), to the elastic profile (semi-elliptic) and to the pressure Sekivelocity of approach (constant in time and space). It should also be mentioned that and Leibler presented for tangential lubrication force deformable the expression moto on [6] an the

Hertzian

bodies. For the with

sake of

completeness

exponentially

an

Appendix. ©

Les

#ditions

de

and

dependent

Physique

1996

practical viscosity.

importance

the

The

of

results

lubrication this

exercise

force is also are

calculated

presented

in

the

JOURNAL

98

H~

h

Geometry

Fig. 1. H, Ho region

for

r

=

of

0, and

a

two

curved

gap

of size

H

before

surfaces h

PHYSIQUE

DE

N°1

r

~

(thick line) separated by a (thin line) with the radius of the

deformation

deformation

after

II

of

gap

flat

size

central

a.

'llheory

2.

Figure

In

sented.

1

tangentially

geometrical

principal

the

reduced

The

for

radius

parallel

for the surfaces of the solid bodies parameters are presurfaces with radii of Ri and R2 in a curvature

curved

two

orientation

defined

is

R

length

For

undeformed

compared expressed as

small

scales

surfaces

is

iii

as

~~(~(

=

radii

the

to

parabolic

a

Ho

where

surfaces

is

the

the

of closest approach equation is replaced by

where

h

denotes

deformation

the

deformability r

>

on

can

a

both

be

distance

the

is

Hertzian,

I-e-

distinguished

surfaces

for

r

=

flat

a

if both

0

position

at

surfaces =

r

surfaces

the

for

r

0.

=

deformed

For

~~~~

~~~

W(r)

surfaces

and

the

present
a analogous to the equations for the to the equations (17) and (19) for the viscosity case. constant Here

is

lubrication

the

calculated.

The

force

with

procedure

Rubr(r § a)

to

(~

=

dependent

pressure

a

do

l-

well

is

so

Ba*~

(C

=

known

Ba*~) log(C

Ba*~)

+ C

log(C)

°

3qoV °

Ae3xqoVR

~_

~~

Be$

R

a ~

~

R

Ce1+$1($)~+ ~j

(26)

JOURNAL

104

Rubr(r

>

a)

-~~ =

lxll

DE

PHYSIQUE

2/~arctan(§)

In r

order