BIORHEOLOGY, 28; 207-219,1991 0006-355X/91 $3.00 + .00 Printed in the USA. Copyright (c) 1991 Pergamon Press pic. All rights reserved.
GENERALIZED DISPERSION IN A SYNOVIAL FLUID OF HUMAN JOINTS N.Rudraiah l , S.R.Kasiviswanathan 2 and p.N.Kaloni 2 I.Gulbarqa University, 2.Department of Mathematics, Gulbarga-585 106. University of Windsor India Windsor, Ontario, Canada (Received 20.4.1990; accepted in revised form 3.2.1991 by Editor N. Ohshima) ABSTRACT An unsteady convective diffusion in a synovial fluid of human joints modeled as a power-law fluid is studied using the generalized dispersion model of Gill and Sankarasubramanian [12J. The contributions of convection and diffusion, and pure convection on the dispersion of nutrient are investigated in detail. It is shown that the effect of decrease in non-Newtonian parameter is to decrease the dispersion coefficient. The mean concentration distribution appears to increase as the non-Newtonian parameter decreases upto a certain value of the axial distance. Beyond this point, however, the reverse pattern is observed. INTRODUCTION The biomechanics of the knee joint, called the synovial joint, plays a significant role in the study of human locomotion. A synovial joint consists of load bearing bone whose ends are covered by cartilages lubricated by synovial fluid. The anotomical dimensions of the cavity and the location of the source and sink of the synovial fluid are well explained. in the reviews given by Engin [1], Dowson and wright [2J and Torzilli [3}. The main mechanism of the driving force here is the pressure gradient generated across the tissues. The synovial fluid impergnates movable ~oints. of the body and is obtained in the capsules of the ioints in different volumes (roughlv about 0.2 ml). This fluid, although comp,ositionally bears some resemblances to blood plasma, lacks all the clotting agents such as fibrinoqen. It is believed that the·fluid performs the following two major functions [IJ: (i) It serves as a lubricant between cartilage surface (ii) It carries out metabol ic funct ions by providing nutrients to the cartilage From a biomechanics point of view, the most important constituent of synovial fluid is the non-sulphated mucopolysaccharide hyaluronic acid. Studies made by Palfrey and white [41 reveal that the viscosity of this synovial fluid decreases with increasin9 shear rate. Thus, the synovial fluid displays Key words: Power-law, G~neralized dispersion, coefficient. 207
Dispersion
208
SYNOVIAL FLUID OF HUMAN JOINTS
Vol. 28, Nos. 3/4
a non-Newtonian behaviour governed by a power law type constitutive equation. The remarkable performance of the lubrication of load bearing human ioints is well known, but the dispersion of nutrients from the synovial fluid to the articular cartilage has not been given much attention. This is important in the design of artificial joint to replace the disordered human ioints. This aspect is considered in this paper. Recently Nigam et al [5) and Buiurke and Jayaraman [6) have made theoretical study of the lubrication aspect of synovial fluid bv treating svnovial fluid as a micropolar fluid and a couple stress fluid, respectively. The metabolic function of synovial fluid has not received much attention. This function is important to understand normal and abnormal synovial joint motion, especially if one seeks some of the leading causes of the degenerative ioint disease. The concentration of the hyaluronic acid molecules and other high molecular weight substances in the synovial fluid observed by Maroudas et al (7) and Walker et al (8) may be responsible to disperse the nutrients to the cartilage. The object of this paper is, therefore, to develop a theoretical model based on the dispersion mechanism of Taylor (9J and Gill and Sankarasubramanian [101 to account for the transport of nutrients from synovial fluid to articular cartilage. Considerable amount of theoretical and experimental work has been done on dispersion in Newtonian fluids using Taylor [ 9) , Aris (11) and Gill and Sankarasubramanian [10) approaches. The results obtained by using the Newtonian fluid model are very restrictive. Previous analytical work on dispersion in Power law fluid i5 limited to Taylor [9J analysis. Fan and Hwana [12J extended Taylor's dispersion procedure to power law fluids and showed that an axiallv dispersed model can be used to represent the area mean concentration of the solvent. It is known that Taylor's analysis holds for liquids with large Peclet numbers (Pe 200), so that the results are valid asvmptoticallv for large dimensionless time (see [13) [14)). In the studv of dispersion of nutrients in synovial fluid it is advantageous to determine the dispersion coefficient in power law fluid valid for all time. The study of this, using the generalized dispersion model of Gill and Sankarasubramanian [10), is the objer:t of this paper. The motivation for this study stems from a desire to develop a theoretical model to transport nutrients from synovial fluid to articular cartila~e. This paves the way to develop better techniaues for the reconstructive ;oint surgery and to desiqn aporooriate mechanical components for partial or total ioint replacement. For this purpose, the dominant dispersion coefficient for combined convection and diffusion and for pure convection alone is determined. It is shown that the effect of decrease in power law parameter is to oecrE'ase the dispersion coefficient. Deo and Vasudeva r 15) have used a flow method based on the central line concentration to measure the diffusion coefficient in power law fluids. Therefore, to facilitate the measurement of diffusion coefficient in the synovial
209
SYNOVIAL FLUID OF HUMAN JOINTS
Vol. 2B. Nos. 3/4
fluid we also compute the mean concentration distribution. It is shown that the mean concentration distribution increases as the non-Newtonian parameter decreases uoto a certain point in the axial distance, x. Reverse trend occurs beyond this point. FORMULATION OF THE PROBLEM The physical configuration of the problem considered in this paper is shown in Figure 1. It consists of an incomoressible power-law fluid flowing between two parallel plates separated by a distance 2h apart. Although the actual morohologv is three dimensional, we consider onlv two dimensional flows because of the assumption that the lenath of the cavity is much larqer than the width. y=h
Fig. 1.
v = - h
Physical configuration of the problem.
A Cartesian coordinate frame is embedded in the lower plate with the x-axis coinciding with the direction of the flow and y-axis oriented normal to the olates. In a power-law fluid the deviatoric stress T can be expressed as l' f (1T d) in which
~ is the second
invariant of the deformation rate tensor defined
as
(1)
D .. 0 .. , 1)
1J
where
( 2)
Dij For power-law fluids we write [16J n-J. 1T
which
are
of
interest
in
this
work
--
) 2 "1 = K( ~ 2
with K, the consistency index. The eauation in steady, fully developed power-law fluid reduces to n-l dp a (K Idv au I ~) = 0, - dx + dv 3y where p is the pressure, in the x-direction.
and
u
is
the
of motion for a a channel, thus
velocity
(4 )
of
the
fluid
VELOCITY DISTRIBUTION
The solution of flll
W = and
0 at y
u = 0 at v
=
0
hi
(4) satisfying the boundary conditions ( 5)
210
SYNOVIAL FLUID OF HUMAN JOINTS
is
Vol. 28, Nos. 3/4
n+1
(1-1'1 n
u
where
),
( 6)
hn+ J
n
= + n+l (- - -
B
and n
K
= v/h.
(7)
We remark that to satisfy the no-slip boundary conditions (n+l )/n has to be even. This means that the above result is true for a ~hear thinning fluid only (n
GENERALIZED DISPERSION IN A SYNOVIAL FLUID OF HUMAN JOINTS N.Rudraiah l , S.R.Kasiviswanathan 2 and p.N.Kaloni 2 I.Gulbarqa University, 2.Department of Mathematics, Gulbarga-585 106. University of Windsor India Windsor, Ontario, Canada (Received 20.4.1990; accepted in revised form 3.2.1991 by Editor N. Ohshima) ABSTRACT An unsteady convective diffusion in a synovial fluid of human joints modeled as a power-law fluid is studied using the generalized dispersion model of Gill and Sankarasubramanian [12J. The contributions of convection and diffusion, and pure convection on the dispersion of nutrient are investigated in detail. It is shown that the effect of decrease in non-Newtonian parameter is to decrease the dispersion coefficient. The mean concentration distribution appears to increase as the non-Newtonian parameter decreases upto a certain value of the axial distance. Beyond this point, however, the reverse pattern is observed. INTRODUCTION The biomechanics of the knee joint, called the synovial joint, plays a significant role in the study of human locomotion. A synovial joint consists of load bearing bone whose ends are covered by cartilages lubricated by synovial fluid. The anotomical dimensions of the cavity and the location of the source and sink of the synovial fluid are well explained. in the reviews given by Engin [1], Dowson and wright [2J and Torzilli [3}. The main mechanism of the driving force here is the pressure gradient generated across the tissues. The synovial fluid impergnates movable ~oints. of the body and is obtained in the capsules of the ioints in different volumes (roughlv about 0.2 ml). This fluid, although comp,ositionally bears some resemblances to blood plasma, lacks all the clotting agents such as fibrinoqen. It is believed that the·fluid performs the following two major functions [IJ: (i) It serves as a lubricant between cartilage surface (ii) It carries out metabol ic funct ions by providing nutrients to the cartilage From a biomechanics point of view, the most important constituent of synovial fluid is the non-sulphated mucopolysaccharide hyaluronic acid. Studies made by Palfrey and white [41 reveal that the viscosity of this synovial fluid decreases with increasin9 shear rate. Thus, the synovial fluid displays Key words: Power-law, G~neralized dispersion, coefficient. 207
Dispersion
208
SYNOVIAL FLUID OF HUMAN JOINTS
Vol. 28, Nos. 3/4
a non-Newtonian behaviour governed by a power law type constitutive equation. The remarkable performance of the lubrication of load bearing human ioints is well known, but the dispersion of nutrients from the synovial fluid to the articular cartilage has not been given much attention. This is important in the design of artificial joint to replace the disordered human ioints. This aspect is considered in this paper. Recently Nigam et al [5) and Buiurke and Jayaraman [6) have made theoretical study of the lubrication aspect of synovial fluid bv treating svnovial fluid as a micropolar fluid and a couple stress fluid, respectively. The metabolic function of synovial fluid has not received much attention. This function is important to understand normal and abnormal synovial joint motion, especially if one seeks some of the leading causes of the degenerative ioint disease. The concentration of the hyaluronic acid molecules and other high molecular weight substances in the synovial fluid observed by Maroudas et al (7) and Walker et al (8) may be responsible to disperse the nutrients to the cartilage. The object of this paper is, therefore, to develop a theoretical model based on the dispersion mechanism of Taylor (9J and Gill and Sankarasubramanian [101 to account for the transport of nutrients from synovial fluid to articular cartilage. Considerable amount of theoretical and experimental work has been done on dispersion in Newtonian fluids using Taylor [ 9) , Aris (11) and Gill and Sankarasubramanian [10) approaches. The results obtained by using the Newtonian fluid model are very restrictive. Previous analytical work on dispersion in Power law fluid i5 limited to Taylor [9J analysis. Fan and Hwana [12J extended Taylor's dispersion procedure to power law fluids and showed that an axiallv dispersed model can be used to represent the area mean concentration of the solvent. It is known that Taylor's analysis holds for liquids with large Peclet numbers (Pe 200), so that the results are valid asvmptoticallv for large dimensionless time (see [13) [14)). In the studv of dispersion of nutrients in synovial fluid it is advantageous to determine the dispersion coefficient in power law fluid valid for all time. The study of this, using the generalized dispersion model of Gill and Sankarasubramanian [10), is the objer:t of this paper. The motivation for this study stems from a desire to develop a theoretical model to transport nutrients from synovial fluid to articular cartila~e. This paves the way to develop better techniaues for the reconstructive ;oint surgery and to desiqn aporooriate mechanical components for partial or total ioint replacement. For this purpose, the dominant dispersion coefficient for combined convection and diffusion and for pure convection alone is determined. It is shown that the effect of decrease in power law parameter is to oecrE'ase the dispersion coefficient. Deo and Vasudeva r 15) have used a flow method based on the central line concentration to measure the diffusion coefficient in power law fluids. Therefore, to facilitate the measurement of diffusion coefficient in the synovial
209
SYNOVIAL FLUID OF HUMAN JOINTS
Vol. 2B. Nos. 3/4
fluid we also compute the mean concentration distribution. It is shown that the mean concentration distribution increases as the non-Newtonian parameter decreases uoto a certain point in the axial distance, x. Reverse trend occurs beyond this point. FORMULATION OF THE PROBLEM The physical configuration of the problem considered in this paper is shown in Figure 1. It consists of an incomoressible power-law fluid flowing between two parallel plates separated by a distance 2h apart. Although the actual morohologv is three dimensional, we consider onlv two dimensional flows because of the assumption that the lenath of the cavity is much larqer than the width. y=h
Fig. 1.
v = - h
Physical configuration of the problem.
A Cartesian coordinate frame is embedded in the lower plate with the x-axis coinciding with the direction of the flow and y-axis oriented normal to the olates. In a power-law fluid the deviatoric stress T can be expressed as l' f (1T d) in which
~ is the second
invariant of the deformation rate tensor defined
as
(1)
D .. 0 .. , 1)
1J
where
( 2)
Dij For power-law fluids we write [16J n-J. 1T
which
are
of
interest
in
this
work
--
) 2 "1 = K( ~ 2
with K, the consistency index. The eauation in steady, fully developed power-law fluid reduces to n-l dp a (K Idv au I ~) = 0, - dx + dv 3y where p is the pressure, in the x-direction.
and
u
is
the
of motion for a a channel, thus
velocity
(4 )
of
the
fluid
VELOCITY DISTRIBUTION
The solution of flll
W = and
0 at y
u = 0 at v
=
0
hi
(4) satisfying the boundary conditions ( 5)
210
SYNOVIAL FLUID OF HUMAN JOINTS
is
Vol. 28, Nos. 3/4
n+1
(1-1'1 n
u
where
),
( 6)
hn+ J
n
= + n+l (- - -
B
and n
K
= v/h.
(7)
We remark that to satisfy the no-slip boundary conditions (n+l )/n has to be even. This means that the above result is true for a ~hear thinning fluid only (n
