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Second-grade fluid film lubrication in hertzian contacts with reference to synovial joints
Wear, 84 (1983)
261 - 274
261
SECOND-GRADE FLUID FILM LUBRICATION IN HERTZIAN CONTACTS WITH REFERENCE TO SYNOVIAL JOINTS
K. M. NIGAM,
N. M. BUJURKE,
Centre for Atmospheric I I001 6 (India) (Received
September
M. P. SINGH and K. MANOHAR
and Fluids Sciences,
28,1981;
Indian Institute of Technology,
New Delhi
in revised form March 3,1982)
Summary
An analysis of squeeze film lubrication between two approaching surfaces is presented with reference to normally loaded human joints. A second-grade fluid represents the synovial fluid in the region between the approaching surfaces. Analytical expressions for the pressure distribution, load-carrying capacity and time of approach between any two film thicknesses are derived. The normalized pressure as well as the average pressure increase with an increase in the non-newtonian dimensionless parameter IV,. With normal young synovial fluid the pressure developed in the joint cavity is considerably greater than with normal old and osteoarthritic synovial fluid. Pressure variation is least around the centre and exit of the surfaces and largest in the intermediate region. The rate of variation in the intermediate region is large for larger values of the minimum film thickness ho. The rate of variation in the load-supporting capacity is smaller for larger values of ho in contrast with the intermediate range of values of ho for which it is larger. The times of approach for osteoarthritic and old normal synovial fluid are smaller than the time of approach for healthy normal young synovial fluid and this partly accounts for increased cartilage wear in old osteoarthritic patients.
1. Introduction
A weight-bearing synovial joint consists of two mating bones covered with cartilage and synovial fluid in the cavity between the bones. Synovial fluid is clear and colourless, or slightly yellowish, and in a human joint there is about 0.2 - 0.5 ml. Biochemically, it is a dialysate of blood plasma composed of approximately one-third of a protein concentration of plasma and a polymer, hyaluronic acid, which endows the synovial fluid with interesting rheological characteristics. Ogston and Stanier [ 11, Dintenfass [2], Davies [ 31 and Balazs and Gibbs [4] reported the non-newtonian behaviour of synovial fluid. Many workers [5 - 71 have made theoretical and experimental investigations of the rheological characteristics of synovial fluid and provided 0043-1648/83/0000-0000/$03.00
0 Elsevier Sequoia/Printed
in The Netherlands
262
useful information concerning the appropriate material constants and functions for various rheological fluid models proposed for synovial joint lubrication. The human joint may be considered as a load-bearing system without wear and with a very low coefficient of friction. The synovial fluid works as a lubricant in this load-bearing system. Fein [ 81 and Torzilli [9] suggest squeeze film lubrication in synovial joints where large contact areas exist. As the articular surfaces impinge during the loading phase, the squeeze film action prevents cartilage-cartilage interaction and avoids surface damage. In the present study the articulation of a joint is modelled as a sphere approaching a plane surface. A second-grade fluid is taken as the lubricant in the joint cavity. The geometry of such a system is approximately congruent to the hip joint. Although various modes of lubrication occur under different operating conditions and may occur simultaneously in most situations during various cycles of articulation, squeeze film lubrication is considered as it is effective in joints where the area of contact between the two surfaces is a maximum as in standing and jumping.
2. Formulation
of the problem
For the general class of incompressible homogeneous fluids the Cauchy stress Sij and the fluid motion are assumed according to [lo] S = --PI+
&A, + @iA, + Wi2
where the spherical stress pl is due to incompressibility, first two Rivlin-Ericksen tensors defined by Al = grad V + (grad V)T
second-grade to be related (1)
Al and A2 are the (2)
A, =A, +A, grad V + (grad V)TAI
(3) where V is the fluid velocity, & is the viscosity, @i and 42 are the two normal stress moduli which are generally termed the coefficient of viscoelasticity and the cross-viscosity respectively. The dot in eqn. (3) denotes the material time derivative. While the constitutive relation (1) can be considered as the second-order approximation of a simple fluid [lo], it may also be taken as an exact model for some fluid because relation (1) is properly invariant. In this sense it is now appropriate to call it a relation for a second-grade fluid instead of a relation representing a fluid of second order. Thus the secondgrade fluid will be an exact model of non-newtonian fluid behaviour and can be studied in its own right [ 111. If body forces are neglected, the system of field equations governing the motion of an incompressible fluid of second grade is given by div V = 0
(4)
div S = p e
(5)
263
where p is the density and v is the total derivative of V. For plane steady flow the explicit forms of eqns. (4) and (5) in Cartesian coordinates are
au
au
“ax+va;;=-,a,
1
aP
+vV*U+/3&+y~I
(7)
and
au a~ u~+vay=---ay
1
ap
-+vv2v+j3K2+yS2
where u and v are the velocity components tively. Also, v=
(8) in the x and y directions respec-
p$
4% P
‘i72= d’ $ a2 ax2 @
and a2A KI=~~
a*A + ayz
(10)
264
Rigid
sphere
L-4 Fig. 1. A model
for a synovial
joint.
The problem considered is that of the steady flow of an incompressible second-grade fluid between the plane and a sphere impinging on it (Fig. 1). This model is congruent to the hip joint under normal loading conditions. The moving boundary is described by h = h,(t) + ;
(11)
where h,(t) is the minimum film thickness which approaches the plane with a velocity
and R is the radius of the sphere
dho u0=--
dt
We use the following EZ-
non-dimensional
ho
jj=
L
scheme:
_z_ ue
(12)
where L is the characteristic length, U is the characteristic velocity and P is the viscosity. Since the film thickness ho is very small compared with the length L, the nondimensional parameter E = ho/L is a small quantity. If eqn. (12) is introduced to eqns. (6) - (8) and terms of order l/e2 are collected, the dominating terms give the equations (13)
--1 aj = __1 a2fi R, a2 R, ap aa a*ii +2__+fjap ay*
3a*iiaii
ap a% ay3
a2
+4~
a*ii a3ii +ii -+--+ ay agag a;Yaji2
as7 aii ap2 aj? (14)
265
(15) where R, = UL/v equations are
au
is the Reynolds
number.
The dimensional
forms of these
a~ + - =o
(16)
a3c ay 1
ap
p
ax
--
a%
a%
=v--++
ay2
au
3---+4---
i ay2 ax
av a% +2--+vay ay2 1
ap=wP+Y)ayay2 au
pay
au a%
ay axay
a+.4
+u -+--+
axay2
a% au a9 ay
a% au a3.4 +2y------ayJ i ay axay
(17)
ah (18)
The assumption dp/ay = 0 is not made, in contrast with the newtonian in which ap/ay = 0. The appropriate boundary conditions are u=O
and
u=O
andv=v,
p=O
at x=+L
3. Solution
v=O
case
at y=O at y=h
(19)
of the problem
Tanner [12] showed that for a plane steady flow the newtonian velocity distribution also satisfies the flow equations of the second-grade fluid. Hence a solution of eqns. (17) and (18) is sought in the form u(x, Y) = Y2rn@) + yn(x) By using the continuity we obtain v(x,y)=-_-_-
(20)
equation
y3 dm
y2 dn
3d.x
2dx
--
(eqn. (15)) and boundary
conditions
(19)
(21)
and m(x) = ;(6
VOX+ c)
(22) n(x) = - L(6vox h2
+ c)
266
where c is an arbitrary constant to be determined. Substitution of the expressions for u and u from eqns. (20) and (21) in eqns. (17) and (18) gives 1 dP - - = 2vm + 4(2p + Y)y(2ymm’
P ax
1
+ nm’ + mn’) + (30 + 2y)nn’
ap
- - = 4(2p + y)m(2my + n) P ay where primes denote differentiation with respect to x. Solution of eqns. (23) and (24) for p gives P(X9 Y) _ P
2vsm dx + 4(2p + y)my(my
30 + 2Y n2 + d + n) + -.2
where d is an arbitrary constant. To determine boundary conditions (19) are used in the form
the constants
(23)
(24)
(25) c and d the
h
Jp dy = 0
at x = ?L
(26)
0
which gives c=O 6vovR d= Z12
(27)
6(p+ 2y)uo2L2 214
where zl=ho+
&
(28)
Thus the pressure distribution P(X, Y) = 6vovR(~ _ P
-
;)
in the film region is + 14Wfl
6@+ 2y)uo2L2
+ 18(3P + 2y)u()2xZ h4 The load-Carrying
W= jjp(x, -L
becomes
0
capacity
y1dyd.x
+h:)u:x’Y(;
W defined
q4 by
_ 1) +
(29)
267
(30)
where
For a constant applied load, the time-height from eqn. (30) in the form tz-t1=
relation can be obtained
t
where Rzt4A _ 4 x 2’12h0
L32
--
RLz12
2
2
and ho(tt) and hO(t2) are the values of ho at any times tr and fz+ The dimensionless average pressure distribution, d~ension~ess foadcarrying capacity and dimensio~l~ss time of approach may be written as follows: (34)
=6(Z1-l)“2
IV,-1
21-l
* -g&3&3
and
2 ROWI -
-
tan-‘((Z*
-
lp)
+
f. 1
14&* - 8Z1 + 16) +
4v1+ R&l2
2)
(35)
268 l/2
m2
T=
i P@
t
+ WP02
t
fdffo) =2J fdH0) - f4Wo)K dHo
(36)
H”(t)
where R
22 L R1=
E= h0 L
R”=L
_!!
“1
zl=
ho
H=k
Z2’
ho N
n
“2
Ho=-
ho(td
ho
J+2y -u
vho
ho(h)
RP
O
Mn = {Wp@ + 2y)}“2
fsW0)= oK12f4(Ho) + ~f3(Ho,Y
f~(Ho)=(~~‘2tan-l~(~~‘2~ +3& f3W
= z&
Wh2 -2E
JW'r3 +-.-"-_-_-_
22
4eHo
e3
(37)
4. Discussion In addition to the usual d~ensionl~ss numbers of classical lub~cat~on theory additional dimensionless parameters N, and M, are used which are of interest in second-grade fluid theory. These parameters characterize the additional rheological effects of the synovial fluid, For N, = 0 and M, = 0 in eqns. (34) - (36) squeeze film characteristics of classical newtonian theory are obtained. For the computation of numerical results the values of the parameters V, /3 and y for different situations, taken from ref. 4 and given in Table 1, are used in the calculation of N, and M,. Thus different values of N, and M, represent different situations such as a normal synovial fluid sample or an old-age sample. The velocity u. of approach is taken as 0.030 48 cm s-l [ 131, Figure 2 shows the normalized pressure fi,, in the film region, which is defined as the ratio of the non-dimensional average pressure @ to the nondimensional newtonian pressure (fi for N, = 0) at 5 = 0, uers’susthe horizontal distance f for fixed values of R, L and ho and for various values of N,, With second-grade fluids the normalized pressure is greater than that of the cor-
269
TABLE 1 Values of #o, #i and $2 appearing in eqn. (I) for different Batazs and Gibbs f4 f f
synovial fluid samples (from
Fluid sample
Description
$0 (P)
@I (P $1
42 (P $1
: c
Normal young old fluidfluid Usteoarthritic fluid
82.0 21.6 2.5
--975.0 -24.1 -0.025
1950.0 48.2 OX351
Fig. 2, The normalized dimensionless (R = 0.5; z = 0.5; ho = 0.01).
axial pressure distribution
for various vrrlues of N,
responding newtonian fluid (Nn = 0). Figures 3 - 5 show the non-dimensional average pressure J uersus horizontal distance along the joint bed, For fixed R, L and ho the variation in J for various values of N, is given in Fig. 3. Figure 4 shows the variation in 8 with respect to the minimum film thickness he for fixed values of N,, R and L, As expected, p increases rapidly with the decrease in hW Figure 5 presents the variation in p with respect to Z for various values of R and L but with fixed IV, and he. An increase in R corresponds to an increase in p and an increase in L corresponds to a decrease in IS. From Figs. 2 - 5 the rate of variation in g is least in the central and the exit
210
Fig. 3. The dimensionless L = 0.5; ho = 0.01).
axial pressure
Fig. 4. The dimensionless R = 0.5; L = 0.5).
axial pressure
distribution
distribution
for various
values of N,
(R = 0.5;
for various values of ho (N, = 3.567;
271
Fig. 5. The dimensionless 3.567; ho = 0.01).
axial pressure
distribution
for various
values of R and L (N, =
regions of the bearing surface and is largest in the intermediate region, which is expected physically because at 2 = 0 the pressure has a maximum value whereas it has a minimum value at the exit. From Fig. 4, the rate of variation in p in the intermediate region is larger for larger values of the film thickness ho and the pressure build-up in the cavity is larger for normal young synovial fluid than for normal old and osteoarthritic synovial fluid. In Fig. 3 the different values of N, represent these situations. Figure 6 shows the variation in the non-dimensional load-carrying capacity m versus e. p increases as he decreases and W has considerably higher values than in the corresponding newtonian fluid case (IV, = 0). This increase in load-carrying capacity caused by the rheological characteristics of synovial fluid is most desirable because it enables natural joints to support loads of two to three times the body weight. The rate of variation in ff is smaller for smaller values of ho and for larger values of ho in contrast with the intermediate range of values of h,, for which it is larger. Figure 7 shows the time-height relation for various values of the nondimensional parameter M,. The time of approach increases as the film thickness ratio Ho decreases. For the osteoarthritic and normal old synovial fluid samples the time of approach is substantially lower than for the normal young synovial fluid. This in turn leads to severe damage of the cartilage structure by wear.
2-
I c
201
100
Fig. 7. The time--height relation for various values of M, in synovial fluid samples a - c (R = 0.5; L = 0.5).
Fig. 6. The dimensionless load-carrying capacity @ us. E for synovial fluid samples a - c (R = 0.5;L = 0.5).
a.
I -
300
0
0.1L.
400
F
0.5
500
0.6
273
5. Conclusion From the analysis, it may be concluded that larger normal stress effects (larger values of N,, and M,) imply more pronounced deviations from the newtonian case and the modelling of synovial fluid as a second-grade fluid results in a sufficiently high build-up of pressure, which is desirable because the pressure peaks are very high in normal joints. Thus the present study is physically relevant and is a useful beginning for an approach to more realistic models. However, for a more realistic model of the human joint the effect of cartilage structure should also be considered. This is at present being studied.
Nomenclature A,,
A2
fl,
f2
f3,
f4,
fs
h ho H Ho L ;$I, P 3 PllOrm R R, t T u, ” u UO
V W w
px:;, v E P P
the first two Rivlin-Erickson tensors functions of ho (defined in eqn. (33)) functions of Ho (defined in eqn. (37)) film thickness (defined in eqn. (11)) minimum film thickness dimensionless film thickness ratio of film thicknesses at any two times characteristic length functions of x dimensionless parameters (defined in eqn. (37)) pressure in the film region dimensionless pressure (defined in eqn. (34)) normalized pressure distribution radius of the sphere Reynolds’ number time of approach dimensionless time of approach (defined in eqn. (36)) velocity components in the x and y directions respectively characteristic velocity velocity of approach velocity vector load-carrying capacity dimensionless load-carrying capacity (defined in eqn. (35)) coordinates viscosity coefficients (defined in eqn. (9)) ho/L, dimensionless parameter newtonian viscosity density
References 1 A. G. Ogston and J. E. Stanier, The physiological functions of hyaluronic acid in synovial fluid: viscous elastic and lubricant properties, J. Physiol., 119 (1953) 244. 2 L. Dintenfass, Rheology of complex fluids and some observations on joint lubrication, Fed. Proc., Fed. Am. Sot. Exp. Biol., 25 (1965) 1054. 3 D. V. Davies, Synovial fluid as a lubricant, Fed. Proc., Fed. Am. Sot. Exp. Biol., 25 (1965) 1069.
274 4 E. A. Balazs and D. A. Gibbs, The rheological properties and biological function of hyaluronic acid. In E. A. Balazs (ed.), Chemistry and Molecular Biology of the Intercellular Matrix, Vol. 3, Academic Press, New York, 1970, pp. 1241 - 1253. 5 B. Block and L. Dintenfass, Rheological study of human synovial fluid, Aust. N.Z. J. Surg., 33 (1963) 108. 6 S. C. Kuei and V. C. Mow, The rheological model for synovial fluids, Trans. 23rd Annu. Meet. of the ORS, Vol. 2, Orthopaedic Research Society, 1977, p. 136. 7 W. M. Lai, S. C. Kuei and V. C. Mow, Rheological equations for synovial fluids, J. Biomech. Eng., 100 (1978) 169. 8 R. S. Fein, Are synovial joints squeeze film lubricated?, Proc., Inst. Mech. Eng., London, 181 (1967) 125. 9 P. A. Torzilli, The lubrication of human joints. In D. G. Fleming and B. N. Feinberg (eds.), Handbook of Engineering in Medicine, Chemical Rubber Co., Cleveland, OH, 1976, pp. 225 251. 10 B. D. Coleman and W. Noll, An approximation theorem for functionals with applications in continuum mechanics, Arch. Ration. Mech., 6 (1960) 355. 11 J. E. Dunn and R. L. Fosdick, Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Ration. Mech. Anal., 3 (1974) 191. 12 R. I. Tanner, Plane creeping flows of incompressible second-order fluid, Phys. Fluids, 9 (1968) 1246. 13 V. C. Mow, Effects of viscoelastic lubricant on squeeze film lubrication between impinging spheres, J. Lubr. Technol., 90 (1968) 113.
261 - 274
261
SECOND-GRADE FLUID FILM LUBRICATION IN HERTZIAN CONTACTS WITH REFERENCE TO SYNOVIAL JOINTS
K. M. NIGAM,
N. M. BUJURKE,
Centre for Atmospheric I I001 6 (India) (Received
September
M. P. SINGH and K. MANOHAR
and Fluids Sciences,
28,1981;
Indian Institute of Technology,
New Delhi
in revised form March 3,1982)
Summary
An analysis of squeeze film lubrication between two approaching surfaces is presented with reference to normally loaded human joints. A second-grade fluid represents the synovial fluid in the region between the approaching surfaces. Analytical expressions for the pressure distribution, load-carrying capacity and time of approach between any two film thicknesses are derived. The normalized pressure as well as the average pressure increase with an increase in the non-newtonian dimensionless parameter IV,. With normal young synovial fluid the pressure developed in the joint cavity is considerably greater than with normal old and osteoarthritic synovial fluid. Pressure variation is least around the centre and exit of the surfaces and largest in the intermediate region. The rate of variation in the intermediate region is large for larger values of the minimum film thickness ho. The rate of variation in the load-supporting capacity is smaller for larger values of ho in contrast with the intermediate range of values of ho for which it is larger. The times of approach for osteoarthritic and old normal synovial fluid are smaller than the time of approach for healthy normal young synovial fluid and this partly accounts for increased cartilage wear in old osteoarthritic patients.
1. Introduction
A weight-bearing synovial joint consists of two mating bones covered with cartilage and synovial fluid in the cavity between the bones. Synovial fluid is clear and colourless, or slightly yellowish, and in a human joint there is about 0.2 - 0.5 ml. Biochemically, it is a dialysate of blood plasma composed of approximately one-third of a protein concentration of plasma and a polymer, hyaluronic acid, which endows the synovial fluid with interesting rheological characteristics. Ogston and Stanier [ 11, Dintenfass [2], Davies [ 31 and Balazs and Gibbs [4] reported the non-newtonian behaviour of synovial fluid. Many workers [5 - 71 have made theoretical and experimental investigations of the rheological characteristics of synovial fluid and provided 0043-1648/83/0000-0000/$03.00
0 Elsevier Sequoia/Printed
in The Netherlands
262
useful information concerning the appropriate material constants and functions for various rheological fluid models proposed for synovial joint lubrication. The human joint may be considered as a load-bearing system without wear and with a very low coefficient of friction. The synovial fluid works as a lubricant in this load-bearing system. Fein [ 81 and Torzilli [9] suggest squeeze film lubrication in synovial joints where large contact areas exist. As the articular surfaces impinge during the loading phase, the squeeze film action prevents cartilage-cartilage interaction and avoids surface damage. In the present study the articulation of a joint is modelled as a sphere approaching a plane surface. A second-grade fluid is taken as the lubricant in the joint cavity. The geometry of such a system is approximately congruent to the hip joint. Although various modes of lubrication occur under different operating conditions and may occur simultaneously in most situations during various cycles of articulation, squeeze film lubrication is considered as it is effective in joints where the area of contact between the two surfaces is a maximum as in standing and jumping.
2. Formulation
of the problem
For the general class of incompressible homogeneous fluids the Cauchy stress Sij and the fluid motion are assumed according to [lo] S = --PI+
&A, + @iA, + Wi2
where the spherical stress pl is due to incompressibility, first two Rivlin-Ericksen tensors defined by Al = grad V + (grad V)T
second-grade to be related (1)
Al and A2 are the (2)
A, =A, +A, grad V + (grad V)TAI
(3) where V is the fluid velocity, & is the viscosity, @i and 42 are the two normal stress moduli which are generally termed the coefficient of viscoelasticity and the cross-viscosity respectively. The dot in eqn. (3) denotes the material time derivative. While the constitutive relation (1) can be considered as the second-order approximation of a simple fluid [lo], it may also be taken as an exact model for some fluid because relation (1) is properly invariant. In this sense it is now appropriate to call it a relation for a second-grade fluid instead of a relation representing a fluid of second order. Thus the secondgrade fluid will be an exact model of non-newtonian fluid behaviour and can be studied in its own right [ 111. If body forces are neglected, the system of field equations governing the motion of an incompressible fluid of second grade is given by div V = 0
(4)
div S = p e
(5)
263
where p is the density and v is the total derivative of V. For plane steady flow the explicit forms of eqns. (4) and (5) in Cartesian coordinates are
au
au
“ax+va;;=-,a,
1
aP
+vV*U+/3&+y~I
(7)
and
au a~ u~+vay=---ay
1
ap
-+vv2v+j3K2+yS2
where u and v are the velocity components tively. Also, v=
(8) in the x and y directions respec-
p$
4% P
‘i72= d’ $ a2 ax2 @
and a2A KI=~~
a*A + ayz
(10)
264
Rigid
sphere
L-4 Fig. 1. A model
for a synovial
joint.
The problem considered is that of the steady flow of an incompressible second-grade fluid between the plane and a sphere impinging on it (Fig. 1). This model is congruent to the hip joint under normal loading conditions. The moving boundary is described by h = h,(t) + ;
(11)
where h,(t) is the minimum film thickness which approaches the plane with a velocity
and R is the radius of the sphere
dho u0=--
dt
We use the following EZ-
non-dimensional
ho
jj=
L
scheme:
_z_ ue
(12)
where L is the characteristic length, U is the characteristic velocity and P is the viscosity. Since the film thickness ho is very small compared with the length L, the nondimensional parameter E = ho/L is a small quantity. If eqn. (12) is introduced to eqns. (6) - (8) and terms of order l/e2 are collected, the dominating terms give the equations (13)
--1 aj = __1 a2fi R, a2 R, ap aa a*ii +2__+fjap ay*
3a*iiaii
ap a% ay3
a2
+4~
a*ii a3ii +ii -+--+ ay agag a;Yaji2
as7 aii ap2 aj? (14)
265
(15) where R, = UL/v equations are
au
is the Reynolds
number.
The dimensional
forms of these
a~ + - =o
(16)
a3c ay 1
ap
p
ax
--
a%
a%
=v--++
ay2
au
3---+4---
i ay2 ax
av a% +2--+vay ay2 1
ap=wP+Y)ayay2 au
pay
au a%
ay axay
a+.4
+u -+--+
axay2
a% au a9 ay
a% au a3.4 +2y------ayJ i ay axay
(17)
ah (18)
The assumption dp/ay = 0 is not made, in contrast with the newtonian in which ap/ay = 0. The appropriate boundary conditions are u=O
and
u=O
andv=v,
p=O
at x=+L
3. Solution
v=O
case
at y=O at y=h
(19)
of the problem
Tanner [12] showed that for a plane steady flow the newtonian velocity distribution also satisfies the flow equations of the second-grade fluid. Hence a solution of eqns. (17) and (18) is sought in the form u(x, Y) = Y2rn@) + yn(x) By using the continuity we obtain v(x,y)=-_-_-
(20)
equation
y3 dm
y2 dn
3d.x
2dx
--
(eqn. (15)) and boundary
conditions
(19)
(21)
and m(x) = ;(6
VOX+ c)
(22) n(x) = - L(6vox h2
+ c)
266
where c is an arbitrary constant to be determined. Substitution of the expressions for u and u from eqns. (20) and (21) in eqns. (17) and (18) gives 1 dP - - = 2vm + 4(2p + Y)y(2ymm’
P ax
1
+ nm’ + mn’) + (30 + 2y)nn’
ap
- - = 4(2p + y)m(2my + n) P ay where primes denote differentiation with respect to x. Solution of eqns. (23) and (24) for p gives P(X9 Y) _ P
2vsm dx + 4(2p + y)my(my
30 + 2Y n2 + d + n) + -.2
where d is an arbitrary constant. To determine boundary conditions (19) are used in the form
the constants
(23)
(24)
(25) c and d the
h
Jp dy = 0
at x = ?L
(26)
0
which gives c=O 6vovR d= Z12
(27)
6(p+ 2y)uo2L2 214
where zl=ho+
&
(28)
Thus the pressure distribution P(X, Y) = 6vovR(~ _ P
-
;)
in the film region is + 14Wfl
6@+ 2y)uo2L2
+ 18(3P + 2y)u()2xZ h4 The load-Carrying
W= jjp(x, -L
becomes
0
capacity
y1dyd.x
+h:)u:x’Y(;
W defined
q4 by
_ 1) +
(29)
267
(30)
where
For a constant applied load, the time-height from eqn. (30) in the form tz-t1=
relation can be obtained
t
where Rzt4A _ 4 x 2’12h0
L32
--
RLz12
2
2
and ho(tt) and hO(t2) are the values of ho at any times tr and fz+ The dimensionless average pressure distribution, d~ension~ess foadcarrying capacity and dimensio~l~ss time of approach may be written as follows: (34)
=6(Z1-l)“2
IV,-1
21-l
* -g&3&3
and
2 ROWI -
-
tan-‘((Z*
-
lp)
+
f. 1
14&* - 8Z1 + 16) +
4v1+ R&l2
2)
(35)
268 l/2
m2
T=
i P@
t
+ WP02
t
fdffo) =2J fdH0) - f4Wo)K dHo
(36)
H”(t)
where R
22 L R1=
E= h0 L
R”=L
_!!
“1
zl=
ho
H=k
Z2’
ho N
n
“2
Ho=-
ho(td
ho
J+2y -u
vho
ho(h)
RP
O
Mn = {Wp@ + 2y)}“2
fsW0)= oK12f4(Ho) + ~f3(Ho,Y
f~(Ho)=(~~‘2tan-l~(~~‘2~ +3& f3W
= z&
Wh2 -2E
JW'r3 +-.-"-_-_-_
22
4eHo
e3
(37)
4. Discussion In addition to the usual d~ensionl~ss numbers of classical lub~cat~on theory additional dimensionless parameters N, and M, are used which are of interest in second-grade fluid theory. These parameters characterize the additional rheological effects of the synovial fluid, For N, = 0 and M, = 0 in eqns. (34) - (36) squeeze film characteristics of classical newtonian theory are obtained. For the computation of numerical results the values of the parameters V, /3 and y for different situations, taken from ref. 4 and given in Table 1, are used in the calculation of N, and M,. Thus different values of N, and M, represent different situations such as a normal synovial fluid sample or an old-age sample. The velocity u. of approach is taken as 0.030 48 cm s-l [ 131, Figure 2 shows the normalized pressure fi,, in the film region, which is defined as the ratio of the non-dimensional average pressure @ to the nondimensional newtonian pressure (fi for N, = 0) at 5 = 0, uers’susthe horizontal distance f for fixed values of R, L and ho and for various values of N,, With second-grade fluids the normalized pressure is greater than that of the cor-
269
TABLE 1 Values of #o, #i and $2 appearing in eqn. (I) for different Batazs and Gibbs f4 f f
synovial fluid samples (from
Fluid sample
Description
$0 (P)
@I (P $1
42 (P $1
: c
Normal young old fluidfluid Usteoarthritic fluid
82.0 21.6 2.5
--975.0 -24.1 -0.025
1950.0 48.2 OX351
Fig. 2, The normalized dimensionless (R = 0.5; z = 0.5; ho = 0.01).
axial pressure distribution
for various vrrlues of N,
responding newtonian fluid (Nn = 0). Figures 3 - 5 show the non-dimensional average pressure J uersus horizontal distance along the joint bed, For fixed R, L and ho the variation in J for various values of N, is given in Fig. 3. Figure 4 shows the variation in 8 with respect to the minimum film thickness he for fixed values of N,, R and L, As expected, p increases rapidly with the decrease in hW Figure 5 presents the variation in p with respect to Z for various values of R and L but with fixed IV, and he. An increase in R corresponds to an increase in p and an increase in L corresponds to a decrease in IS. From Figs. 2 - 5 the rate of variation in g is least in the central and the exit
210
Fig. 3. The dimensionless L = 0.5; ho = 0.01).
axial pressure
Fig. 4. The dimensionless R = 0.5; L = 0.5).
axial pressure
distribution
distribution
for various
values of N,
(R = 0.5;
for various values of ho (N, = 3.567;
271
Fig. 5. The dimensionless 3.567; ho = 0.01).
axial pressure
distribution
for various
values of R and L (N, =
regions of the bearing surface and is largest in the intermediate region, which is expected physically because at 2 = 0 the pressure has a maximum value whereas it has a minimum value at the exit. From Fig. 4, the rate of variation in p in the intermediate region is larger for larger values of the film thickness ho and the pressure build-up in the cavity is larger for normal young synovial fluid than for normal old and osteoarthritic synovial fluid. In Fig. 3 the different values of N, represent these situations. Figure 6 shows the variation in the non-dimensional load-carrying capacity m versus e. p increases as he decreases and W has considerably higher values than in the corresponding newtonian fluid case (IV, = 0). This increase in load-carrying capacity caused by the rheological characteristics of synovial fluid is most desirable because it enables natural joints to support loads of two to three times the body weight. The rate of variation in ff is smaller for smaller values of ho and for larger values of ho in contrast with the intermediate range of values of h,, for which it is larger. Figure 7 shows the time-height relation for various values of the nondimensional parameter M,. The time of approach increases as the film thickness ratio Ho decreases. For the osteoarthritic and normal old synovial fluid samples the time of approach is substantially lower than for the normal young synovial fluid. This in turn leads to severe damage of the cartilage structure by wear.
2-
I c
201
100
Fig. 7. The time--height relation for various values of M, in synovial fluid samples a - c (R = 0.5; L = 0.5).
Fig. 6. The dimensionless load-carrying capacity @ us. E for synovial fluid samples a - c (R = 0.5;L = 0.5).
a.
I -
300
0
0.1L.
400
F
0.5
500
0.6
273
5. Conclusion From the analysis, it may be concluded that larger normal stress effects (larger values of N,, and M,) imply more pronounced deviations from the newtonian case and the modelling of synovial fluid as a second-grade fluid results in a sufficiently high build-up of pressure, which is desirable because the pressure peaks are very high in normal joints. Thus the present study is physically relevant and is a useful beginning for an approach to more realistic models. However, for a more realistic model of the human joint the effect of cartilage structure should also be considered. This is at present being studied.
Nomenclature A,,
A2
fl,
f2
f3,
f4,
fs
h ho H Ho L ;$I, P 3 PllOrm R R, t T u, ” u UO
V W w
px:;, v E P P
the first two Rivlin-Erickson tensors functions of ho (defined in eqn. (33)) functions of Ho (defined in eqn. (37)) film thickness (defined in eqn. (11)) minimum film thickness dimensionless film thickness ratio of film thicknesses at any two times characteristic length functions of x dimensionless parameters (defined in eqn. (37)) pressure in the film region dimensionless pressure (defined in eqn. (34)) normalized pressure distribution radius of the sphere Reynolds’ number time of approach dimensionless time of approach (defined in eqn. (36)) velocity components in the x and y directions respectively characteristic velocity velocity of approach velocity vector load-carrying capacity dimensionless load-carrying capacity (defined in eqn. (35)) coordinates viscosity coefficients (defined in eqn. (9)) ho/L, dimensionless parameter newtonian viscosity density
References 1 A. G. Ogston and J. E. Stanier, The physiological functions of hyaluronic acid in synovial fluid: viscous elastic and lubricant properties, J. Physiol., 119 (1953) 244. 2 L. Dintenfass, Rheology of complex fluids and some observations on joint lubrication, Fed. Proc., Fed. Am. Sot. Exp. Biol., 25 (1965) 1054. 3 D. V. Davies, Synovial fluid as a lubricant, Fed. Proc., Fed. Am. Sot. Exp. Biol., 25 (1965) 1069.
274 4 E. A. Balazs and D. A. Gibbs, The rheological properties and biological function of hyaluronic acid. In E. A. Balazs (ed.), Chemistry and Molecular Biology of the Intercellular Matrix, Vol. 3, Academic Press, New York, 1970, pp. 1241 - 1253. 5 B. Block and L. Dintenfass, Rheological study of human synovial fluid, Aust. N.Z. J. Surg., 33 (1963) 108. 6 S. C. Kuei and V. C. Mow, The rheological model for synovial fluids, Trans. 23rd Annu. Meet. of the ORS, Vol. 2, Orthopaedic Research Society, 1977, p. 136. 7 W. M. Lai, S. C. Kuei and V. C. Mow, Rheological equations for synovial fluids, J. Biomech. Eng., 100 (1978) 169. 8 R. S. Fein, Are synovial joints squeeze film lubricated?, Proc., Inst. Mech. Eng., London, 181 (1967) 125. 9 P. A. Torzilli, The lubrication of human joints. In D. G. Fleming and B. N. Feinberg (eds.), Handbook of Engineering in Medicine, Chemical Rubber Co., Cleveland, OH, 1976, pp. 225 251. 10 B. D. Coleman and W. Noll, An approximation theorem for functionals with applications in continuum mechanics, Arch. Ration. Mech., 6 (1960) 355. 11 J. E. Dunn and R. L. Fosdick, Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Ration. Mech. Anal., 3 (1974) 191. 12 R. I. Tanner, Plane creeping flows of incompressible second-order fluid, Phys. Fluids, 9 (1968) 1246. 13 V. C. Mow, Effects of viscoelastic lubricant on squeeze film lubrication between impinging spheres, J. Lubr. Technol., 90 (1968) 113.