Analysis of normal stress effects in a squeeze film porous bearing

Wear, 116 (1987)

ANALYSIS OF NORMAL POROUS BEARING N. M. BUJURKE,

Department (Received

231

237 - 248

STRESS

M. JAGADEESWAR

EFFECTS

IN A SQUEEZE

and P. S. HIREMATH

of Mathematics, Karnatak University, Dharwad 580003 September

19,1985;accepted

FILM

September

(India)

11, 1986)

Summary An analytical study of a porous bearing lubricated by a second-order fluid is considered. This investigation explains the working of general porous bearings and, in particular, describes the lubrication aspects of synovial joints. An approximate method for the solution of the governing fluid film equation and Darcy’s equation for a porous region is used. Exact expressions for dimensionless pressure, load capacity and response time are obtained. The load capacity and response time for the diseased joint decrease compared with the healthy joint. The decrease in permeability of cartilage enhances the load capacity.

1. Introduction An excellent review on squeeze film lubrication is given by Moore [ 11. Unlike solid bearings, which can be shown theoretically to carry infinite load, porous bearings have finite load-carrying capacity. There have been numerous analytical studies of porous bearings, these include work by Cameron et al. [ 21, Rouleau [3], Cusano [4], Murti [ 51, Prakash and Vij [6] and Wu [7]. They have confined their studies to newtonian fluids as lubricants. Most of the lubricants are non-newtonian, including the synovial fluids. McCutchen [8] briefly explained the lubrication aspects of synovial joints. The relative movement of the cartilage surfaces generates high pressure in the joint cavity, with an extremely low coefficient of friction. Experimental evidence that the tangential permeability is equal to the permeability normal to the surface suggests treating cartilage as an isotropic structure [9]. For mathematical simplicity the average of three cartilage layers of a cartilage matrix is modelled as a single porous layer. McCutchen [8] has explained the effect of normal stresses which make a significant contribution to joint lubrication. In the present paper a simplified model of knee-joint squeezing of a single porous surface, lubricated with a second-order fluid, has been considered. Besides explaining the various features of conventional porous 0043-1648/81/$3.50

@ Elsevier Sequoia/Printed

in The Netherlands

238

bearings, it helps in the understanding of the permeability effect of cartilage on synovial joint lubrication. We have Darcy’s equation for a porous matrix and the hydrodynamic equation in the fluid film region. The solution procedure is an elegant analyt&al method explained earlier by Bujurke [IO].

2. Mathematical

formulation

The model considered for the present problem is two-dimensional squeeze film lubrication between two rectangular surfaces. The geometry and coordinates of the problem are shown in Fig. l(b) [ 11, 121. All the conventions of lubrication problems are considered. The lower surface is the surface of a fixed porous matrix. The upper surface is a rigid plate which approaches the lower porous surface normally. The lubricant is a secondorder fluid. 2.1. Governing equations

2.1.1. Region 1 The governing

equations

of the fluid film region are [IO] au a3.4 ---++~++uay a3cay2

ah

au

w

ay

(1) -

1

P

ap= 2(2P

au a34 + 7) ay ay2

-

ay

(2)

aU+aU_ -0 ax

(3)

ay

Derivations

of eqns. (1) - (3) are given in Appendix

Femoral medlal epicondyle

POVJUS

Tlbiahedial) collateral

Rigid ba

liga

fiha (a)

(b)

Fig. 1. Rectangular plate model of the knee-joint.

B.

239

2.1.2. Region 2 The flow of a viscous fluid in a porous matrix is governed

au1 t- au1 dX

=0

(4)

aY

The velocity components Darcy’s law as

u1=-

by

-k,

ui

and

u1 in the porous

region

are given by

~PI

a0

ax

-kl

a~,

(5)

(6)

vl=xay where p1 is the pressure in the porous region. 2.1.3. Boundary conditions The boundary conditions for region 1 are u = 241

aty=O

(7a)

u=o

aty=h

(7b)

lJ=-V

aty=h

(7c)

u = u,

aty=O

(7d)

p=o

atx=+Q

(7e)

2

where p is the pressure for region 2 are

in the fluid film region. The boundary

conditions

atx=+a

PI= 0

(7f)

2

-aP1 0 ay I y=-_H, =

(7f4

P(K 0) = P,(X, 0)

(7h)

Equation y = 0.

(7h) implies

that

there

is continuity

of pressure

at the interface

2.1.4. Solution Using eqns. (5) and (6) in eqn. (4), we obtain a”P,

ax2

ah + -=

a9

0

Integrating eqn. (8) with respect using eqn. (7g), we obtain

(8) to y over the porous

layer thickness

and

(9) Assuming the porous layer (7h), eqn. (9) reduces to aP1 aY -1

=

_H

a=Pl _

thickness

H, to be very small and using eqn.

=-Hlg

(10)

l ax=

y=o

This approximation does not produce any significant error if the porous layer thickness is very small [ 61. From eqns. (7h) and (10) we have =-

(11)

-

(12) The solution

of eqns. (1) - (3) is attempted

u(x, y) = m(x)y2 + n(x)y +

in the form [lo] (13)

UO

where m(x) and n(x) are functions of x to be determined. and the boundary conditions (eqns. (7a) - (7d)), we obtain

(14)

u(x, y) = m(x)y2 + n(x)y + UI dm y3 v(x, Y) = -dx 3 -

Using eqn. (3)

dn y2 dJc T -

du, -&-Y

+ u1

(15)

where m(x) = -

n(x) =

6Vx - (3h + 6HI)ul

+ d,

h3

6Vx - (4h + GH$.+ + d, h=

(16)

(17)

and dr is an arbitrary constant to be determined. Using eqns. (14) - (17) in eqn. (l), we obtain

ap -1 -=2vm+p P

ax

(18)

The integral of eqn. (18) with respect to x gives

241

Ip(x,y) = 2YJmdx

+ 2/3mu, + m2y2(8P + 4Y) + mny(8P + 4Y)

P

3P+ 2Y +d +n2 2 i i

(19)

where d is a constant to be determined using the boundary conditions for the pressure. The average pressure distribution P across the film thickness is given by P 1 hp -=-dy P k”Ps en2

+ mnk(4P + 2y)

= 2~ mdx + Bflmu, + m2k2 f

3P + 2Y + d 2 i i

(20)

Owing to the boundary condition (eqn. (7e)), we have P = 0 at x = *a/2. Using these boundary conditions, the constants d and dI are obtained as d=-

3vVa2

-

9V2a2N1 k4

2k3 d, =

2ulN2

d12Nr

urdr

(21)

-h4-h4+h4N3

6u,hVN4

(22)

6VN,k - vk2

where N =P+2r 1 6 N3

= W3

N2 = (3/3 + 2y)k’ + 6(p+ y)hH, + 3(/3 + 2y)H,’ + Y)h

+ W3

The expression

j- = k3 + 6k,(k + W,) and the dimensionless p-

2y)H,

NG = (P + Y)k + (0 + 2Y)H,

for the average pressure distribution

YV

P

+

reduces to

1.5 -

(24)

pressure P is given by

Pho3 - (fi3 -I- 6&,(k + 2&))-’ ‘1.5 - F

pvVa2

i

(1 - 4-F2)

where

The d~ensionless

(23)

load @ carried per unit width is given by

(25)

242

=2

1

{h3+6k,(h+ti,)j

+

6k,(h + 2&)}-r(

‘(1.5-

F\(I-4z2)di

0

=

when

(6

1- F

j

0 = 0 and y = 0 the above equations

3. Response

reduce

to the newtonian

case.

time

Writing V = -dh/dt in eqn. (26), the response time E, for reducing the film thickness from an initial value ho to a final value h, is given in nondimensional form as Wh,‘At at=____+oL14 Who2 ’

{(b12h/b2) - 1}ri2 + (b,2h/b2)“2

= ~cP,a4b, (I,lg [ {(b12ho/bz) - 1)1’2 + (b,2ho/b2)1’2 I {( b12ho/b2) - 1)’ ‘2 + {(b,2ho/b2)

- l}“* + (b,2ho/b2)“2

{(b12h/b2) - 1)’ ’ -

{(b12h/b2) - 1}“‘2 + (b,2h/b,)“2 (27)

where v b _ Who3{h3 + 6k,(fi + u?,))-.’ b, = 12N, 2 6N,pa4

4. Results and conclusions The results exhibited in Figs. 2 - 5 are obtained after numerical computation of F, w and z for different values of the parameters involved. In Fig. 2 the nondimensional pressure is plotted against non-dimensional horizontal distance for two values of i;. The material parameters used in the calculations are given in Table 1 and the values of the permeability parameter k, chosen are 4.3 X lo-l3 and 7.65 X lo-l3 cm4 dyn’ s- ’ [9]. The nondimensional pressure peaks are higher for fluid samples with higher values of the material parameters and are found to increase considerably as i; decreases. Figure 3 shows the relationship between load capacity and h for different fluid parameters. The load capacity for a young synovial fluid is quite large compared with other fluids. We have analysed the influence of

243

Fig. 2. Variation in the non-dimensional axial pressure for different different fluid parameters A, B C and D given in Table 1.

102 0 0001

/

I

0.001

0.01

values of h and the

,

0.1

6

Fig. 3. Variation

in 6 with & for different

values of the fluid parameters.

permeability on load capacity and the results are shown in Fig. 4. For small values of h we observe that the load capacity increases with decreasing values of 12, for all the fluids considered. For calculating 5 we have chosen W = 40 X lo3 gf and kI = 7.65 X 1OWf3 cm4 dyn-’ s-l. Figure 5 shows that the approach time for diseased synovial joints is much less compared with that

244

Fig. 4. Variation in @ with 6 for different values of i, (for the parameters of fluid A).

1

I--t loL-

CI

looOr

01

D I

1

0 01

0 001

w=LOKg

h

Fig. 5. Non-dimensional response time for different values of the fluid parameters with different values of 6 (W = 40 X lo3 gf).

for the healthy joint. This may be one of the causes of inefficient functioning of synoviai joints. Also, eqn. (25) clearly shows that the pressure profiles are high for solid bearings. In a continuation of this work we are investigating a more realistic model of synovial joints where the upper surface is also a surface of porous matrix.

245 TABLE

1

Material parameters used in the calculations Fluid description

@O

@I

(P)

(PSI

A Normal young human synovial fluida

82

-975

B Normal old human synovial fluida

21.6

-24.1

48.2

18.5

-0.3

I.2

-0.025

0.05

C Polyisobutylene D Osteoarthritic aReference bReference

in ceteneb fluida

2.5

1950

13. 14.

Acknowledgments We are very much thankful to the referee for his valuable comments which greatly improved the quality of this paper and also to Professor N. Rudraiah, UGC-DSA Centre for Fluid Mechanics, for introducing us to this problem and the help rendered to undertake this work, References

10 11 12 13 14

D. F. Moore, A review of squeeze films, Wear, 8 (1965) 245 - 263. A. Cameron, V. T. Morgan and A. E. Stainsby, Critical conditions for hydrodynamic lubrication of porous metal bearings, Proc. Inst. Mech. Eng. London, 176 (1962) 761. W. T. Rouleau, Hydrodynamic lubrication of narrow press-fitted porous metal bearings, J. Basic Eng., 85 (1) (1963) 123. C. Cusano, Lubrication of porous journal bearings, J. Lubr. Technol., 94 (1) (1972) 69. P. R. K. Murti, Hydrodynamic lubrication of finite bearings, Wear, 19 (1972) 113. J. Prakash and S. K. Vij, Load capacity and time-height relations for squeeze films between porous plates, Wear, 24 (1973) 309 - 322. H. Wu, An analysis of the squeeze film between porous rectangular plates, Trans. ASME, F94 (1972) 64 - 68. C. W. McCutchen, in L. Sokoloff (ed.), The Joints and Synovial Fluids, Vol. 1, Academic Press, New York, 1978. P. A. Torzilli and Van C. Mow, On the fundamental fluid transport mechanics through normal pathological articular cartilage during function-II, J. Biomech., 9 (1976) 587 - 606. N. M. Bujurke, Slider bearing lubricated by a second-grade fluid with reference to synovial joints, Wear, 78 (1982) 355 - 363. P. N. Tandon and S. Jaggi, A model for the lubrication mechanism in knee-joint replacements, Wear, 52 (1979) 275 - 284. D. V. Davies and R. I. Coupland (eds.), Gray% Anatomy, Descriptive and Applied. Longmans Green, London, 1962. W. M. Lai, S. C. Kuei and V. C. Mow,J. Biomech. Eng., 100 (1978) 169. H. Markovitz, Normal stress measurements on polymer solutions, Proc. 4th Znt. Congr. on Rheology, Providence, RI, 1963, Part I, 1965, p. 189.

246

Appendix

A: Nomenclature

a ho h i; HI 6 k, f, N P PI P P t At IS? u, u u19 Vl V W w x> Y X

characteristic length of the bearing initial film thickness film thickness h/h,, non-dimensional film thickness thickness of the porous surface HI/ho, non-dimensional thickness of porous surface permeability of porous bearing nondimensional permeability Who2, nondimensional parameters defined in eqn. (23), i = 1, 2, 3, 4 pressure in fluid film region 1 pressure in porous region 2 average pressure in fluid film region 1 Pho3/pvVa2, nondimensional average pressure time response time Who2At/@,a4, nondimensional response time velocity components in fluid film region 1 velocity components in porous region 2 dh/dt, velocity of approach load capacity Who3/pvVa4, nondimensional load capacity Cartesian coordinates x/a, nondimensional x coordinate

P Y

@,/p, viscoelasticity a2/p, cross viscosity so/p = cc/p, coefficient of viscosity density of the fluid material constants of the fluids (i = 0, 1,2)

V P

@i

Appendix

B

The relation the fluid motion given in ref. Bl. S = -PI

between the Cauchy stress tensor of an incompressible homogeneous

+ aoAl + QlA2 + Q2A12

components second-order

Sij and fluid is (Bl)

where p is the pressure and -pI is the spherical stress due to the incompressibility of the fluid. Al and A2 are the first two Rivlin-Ericksen tensors given by A,=gradV+(gradV)T

(B2)

A2=A,+A,gradV+(gradV)TA,

(B3)

247

where V is the velocity field, CD0is the viscosity and Qp, and apz are the two normal stress moduli usually defined as the coefficient of viscoelasticity and the cross-viscosity respectively. The dot in eqn. (B3) denotes the material time derivative. The constitutive relation (eqn. (Bl)) can be considered as the second-order approximation of a simple fluid [Bl], it could also be taken as an exact model for some fluid. Neglecting body forces, the system of field equations governing the motion of an incompressible second-order fluid is given by div V = 0

(B4)

divS=p’C;

(B5)

where p is the density of the fluid and $’ is the total derivative of V. For plane steady flow the explicit forms of eqns. (B4) and (B5) in Cartesian coordinates are au au -+-_=o ax ay

(B6)

and au Uax

au +u--=-ay

1 ap -

P

av au u-...- -$-~_Z--ax ax

ax

fvV2U+fik1+yS1

1 ap +“V2v+/3kz++y& P ay

respectively, where u and u are the velocity directions respectively, Also (9, v=_.__

@$l

(W components

in the x and y

,=%

P

V2=$

(B7)

P +

a2 ay2

k,=2 + 2

k2=a$

a au ---+-_ ay i ax +2-

au

au au

ay

ax ay,,i

a%

+

aY2 a aU au +2------___ ax ( ax ay

au au ax ay. i

(B9)

a_,$ +,a”

ay

8=u$

+ug

We consider e=-

ho L

the dimensionless fi=-

U

fi=-.-

V

EV

scheme

V

E2L p=--(BlO) L EL Q’OVP where L is the characteristic length and v is the characteristic velocity. Since the minimum film thickness ho is very small compared with the length L, the dimensionless parameter is a small quantity. Introducing the relations of eqn. (BlO) into eqns. (B6) - (B8) and collecting the terms of order 1/e2, which are dominant in these equations, the simplified governing equations of the problem are obtained and the dimensional forms of these equations

,2

jk”

are

a2u au au a% +p - +4ay2 ay axay t ay2 ax

1 ap

a2u

- --=“p ax

+2y

;

a% i axay --

1 ap ay = 2(2p + y) g

au ay

a3u axay2

+u -+--++-

a2y au ay2 ay

Bl

1

(Bll i

$

(B12 1

au au _+-_=o ay ay Reference

a% a?3

(Bl3)

for Appendix B

B. D. Coleman and W. Nell, Arch. Ration.

Mech. Anal., 6 (1960)

355.