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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
Russian Journal of Biomechanics
www.biomech.ac.ru
HYDRODYNAMIC LUBRICATION THEORY OF HUMAN JOINT GAP
K.Ch. Wierzchoiski
Department of Mechanical Engineering, Maritime Academy of Gdynia, Morska 83, PL-81225 Gdynia, Poland, e-mail: wierzch@wsm.gdynia.pl
Abstract. This paper shows the method of solution of systems of partial differential nonlinear, second order equations for axially symmetric and asymmetric synovial fluid flow in curvilinear orthogonal coordinates between two bone surfaces in human joint gap. We formulate theorems, which describe the unification method of solutions of partial differential nonlinear equations of synovial fluid flow.
Key words: hydrodynamic lubrication, synovial fluid flow in human joint gap
1. Introduction
This paper presents modeling and simulations for synovial fluid flow occurring in gap between two co-operating bone surfaces in human joint. The present paper gives analysis of solutions of systems for nonlinear, partial, differential equations for synovial fluid flow in human joint gap. Fig. 1 shows geometry of various human joints.
In the hip joint the spherical rotational bone and the pelvis bone create spherical gap (Fig. 1). In this gap between two co-operating bones synovial fluid flows [2, 5, 7-13, 15-19, 21]. The flow of this fluid is caused by motion of bone head. The theoretical considerations of non-Newtonian synovial fluid flow in thin joint gap taking into account boundary layer simplifications have practical applications in theory of lubrication in medicine [3, 20]. The consideration in the present paper enables to find synovial fluid flow parameters and carrying capacity force in joint gap between two co-operating bone surfaces in curvilinear orthogonal coordinates.
In opposition to achievements (see [7, 9, 12]) the results obtained in this paper show estimation of synovial fluid flow equations and by virtue of present results we can apply various geometry of bone surfaces in human joint considerations.
As contrasted with papers [1, 4, 6, 8, 10-12, 22] the present paper shows the unification and analytical method of solving a problem of lubrication in gaps of different joints (with spherical, parabolic, hyperbolic bone surfaces).
2. Formulation of the problem
The aims of the paper are
• to present the mathematical estimation of the terms of basic partial differential nonlinear equations of the second order for the fluid flow in thin joint gap between two bone surfaces for various geometries;
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
• to formulate the theorems which describe the unification and analytical method of solution of partial differential nonlinear equations for axially symmetric and asymmetric flow of synovial fluid in human joint gap.
3. Basic equations
In this section we show the basic equations describing the synovial fluid flow in joint gap. Equations of conservation of momentum and continuity equation being valid for the stationary flow of synovial compressible fluid between two non-rotational surfaces in the curvilinear orthogonal coordinates have form [18]:
f 1
grad—vv-vx rot v =divS', (1)
, 2 J
Fig. 1. Geometry of human joint: 1 - biobearing gap, 2 - spherical rotational bone head, 3 - two non rotational bones in the knee joint.
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Russian Journal of Biomechanics, Vol. 6, Jfe 1: 34-55, 2002
divv = 0,
(2)
and constitutive equations are as follows:
S = -pI + 2ilpTd (3)
where S is the stress tensor; J is the unit tensor; Td is the strain tensor and their components are Ту, ву, Sjj. We denote the following notations: p is the fluid density; r]p is the dynamic
viscosity of the synovial non-Newtonian fluid, strain component dependent; v is the fluid velocity „vector with components v;; p is the pressure; are the components of strain
tensor.
Geometrical dependencies between strain components and fluid velocity components in orthogonal curvilinear coordinates have form [14]:
з
Vi
hi d
ht da,
v
h,
+
hj d
hi dax
r \ v_±
KhJJ
+
dh,
k~i hA dak
i,j = 1,2,3,
(4)
hx, h2, h2 are Lame coefficients; ax, a2, аг are curvilinear orthogonal coordinates. We introduce equation (3) into the right hand side of equation (1). Thus we have:
(divS), =
1 dp 1
^ dat ghi
MdaJ
4
ghAj
h
J J
-it
z j=i
2?i
s% Ф,)2
' *1
daj
(5)
-я ' '
g *=l dak
where g = hxh2h2 and i = 1,2,3. Expanding the left hand side of equation (1) we obtain finally
o = diwS2y- 4 * * * 8
hk J
,W ^
grad---vxrotv
2 j i
-Z
7=1
1 dVj hj daj
8h
hthj dai
1 3
' +5a— S
ч h
v* dh,
hj k=i hk dak
VJ
(7)
4. Flow simulation for thin gap between two non-rotational bone surfaces
Case of flow 4.1. When the orthogonal curvilinear coordinates a,, a2, a3 coincide with curvature lines of the thin layer resting on non-rotational bone surface in human joint gap, where ax is the length direction, a2 is the perpendicular direction to the bone surface (in gap height direction), аъ is the width surface direction, then Lame coefficients for the thin layer surface with non-monotone curvatures are as follows:
hx=hx{ax,a2), h2= 1, s h2(ax,a3). (8)
Sketch of the proof. Let the vector equation of the surface have the following form (Fig. 2): 4
r0=r0(ax,a3). (9)
The position of any point in the direction of the normal vector n to the surface is determined as follows:
Яг =r0(ax,a3) + a2n(ax,a3), The square of the element’s length is given by
(10)
(dsf s(dqrf = |^ + «2
u(Z]
From Rodrigues law we have:
dn
[dax) + (da2) +
da-.
■ 4- (Xn
dn
\2
da
(da,)2. (11)
3 J
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
Fig. 3. Rotational spherical bone head surface with non- monotone generating line a3.
Fig. 4. Rotational bone surface with monotone generating line a3.
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
dn
1 drn dn 1 drn
dax Rx dax da3
(12)
where Rx, R3 denote radii of curvature in ax and аг directions. We substitute equation (12) in equation (11) and we take into account the thin layer simplifications (i.e. a2/Rx ->0, a2/R3 —>0), then we obtain:
h\ =
dr0(ax,a3)
da.
, h2 — 1, h3 —
dr0(ax,a3)
da-.
(13)
This remark completes the case of flow 4.11
5. Flow in thin gap between two rotational surfaces
Case of flow 5.1. If orthogonal curvilinear coordinates ax, a2, a3 are curvature lines
of thin layer of synovial fluid resting on rotational bone surface in human joint gap, where ax is the circumference direction, a3 is the generating line of rotational bone direction, a2 is the
gap height direction, then Lame coefficients for thin layer with non-monotone generating line are as follows:
h\ = hx (a3), h2=l, h3=h3 (a3) . (14)
Sketch of the proof. For the rotational surface the radius vector has the form:
r0 = iR(a3)cosax + jR(a3)sinax +kZ(a3), (15)
where i, j, к are the unit vectors in Cartesian system and the projections Z and R of the vector r0 indicated in Fig. 3 and Fig. 4 are the functions of a3. We put equation (15) in
equation (13) hence we obtain dependencies (14) what completes the proof of case of flow 5.1 И.
Case of flow 5.2. For monotone generating line of rotational surface (see Fig. 4) the Lame coefficients have the following form:
hx = hx{a3), h2 = 1, h3 = 1. (16)
Sketch of proof. For monotone generating line of rotational surface (see Fig. 4) dependencies (16) are true by virtue of basic theory of differential geometry what completes proof of case of flow 5.2И.
Lemma 5.1. Equations of conservation of momentum and continuity equation for incompressible, stationary synovial fluid flow in thin fluid layer in human joint resting on rotational bone surface with non-monotone generating line and for orthogonal, curvilinear coordinates ax, a2, аъ have the following form:
2 J)
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
t2j
+
(19)
2 rjp dh3 dv3 hi da3 da,
dal da2 da3
(Aiv3) = 0.
(20)
We have in length direction 0 < a] < 2n, in width direction bm<a3< bs and in gap height direction 0<a2<£(a{,a3), whereas bm, bs are constant limits of lubrication in ax, a3 directions.
Proof. We put equations (3)-(7) and (14) in equations (1) and (2), thus the conservation of momentum equations and continuity equation for incompressible, stationary synovial fluid flow in the thin layer resting on rotational surface with non-monotone generating line and in orthogonal, curvilinear coordinates ax, a2, a3 have the form
(17)-(20). This remark completes the proof of Lemma 5.1 Ш.
Lemma 5.2. Estimation of dimensionless terms with exactness from 0.10000 to 1.00000 with comparison to neglected terms of order of 0.0010, for equations of conservation of momentum and continuity equation (17)-(20), in curvilinear orthogonal coordinates ax, az, a3, for incompressible, stationary, and asymmetric synovial fluid flow in thin layer
resting on rotational bone surface of human joint gap, with non-monotone generating line, lead to the following basic equations:
o=-—-§£-+-3
\ dax
0 =
da2 dp da,
Sv,
da
2)
pv\ dhx 1 dp d f
hxh3 da3 Лз da3 da2
л 7 dv. . т d
0 = h3 1 + ^1^3 + -
dv-.
da
2 J
(W>
(21) (22)
(23)
(24)
Ubcl u“2
where in length, width and gap height directions we have:
0 < a, < 2n, bm < a3 < bs, 0 < a2 < s. (25)
Proof. System (21)-(24) describes four unknowns, namely three components of synovial fluid velocity vj(a],a2,a3) for i = 1, 2, 3 and pressure p = (ax,a3). If generating line of rotational thin layer surface in particular case is monotone function, then in (21)-(24) we have h3= 1.
Now we take into account axially asymmetric synovial fluid flow. We assume the following dimensionless values of Lame coefficients An, h3l, values of curvilinear
coordinates au, a2], a3], values of vector velocity components vn, v21, v31, pressure px [ chuamic viscosity щ . Dimensional values have then the following forms:
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Russian Journal of Biomechanics, Vol. 6, № 1:34-55,2002
h\ = Kh\ \ > — Л31 > (X\ =(Хц, a2 = ^Ra2x > cc3= R a3X ,
*
v, =t/vn, v2 = »F£/v2,, a3 s ^V31> P *ф-А» *7P 3 *7o*7i»
(26)
(27)
where Л is the radius i?j of the curvature in a, direction or radius of the rotational surface;
R* is the radius of the curvature in a3 direction or bearing length; *F = — «10-4 is the
R
v
dimensionless radial clearance; e is the gap height; U is the surface linear dimension velocity in a, direction; W is the surface linear dimension velocity in a3 direction; p*0 is the estimated value of dimension pressure; ij0 is the value of-dimensional dynamic viscosity of synovial fluid.
We insert dependencies (26), (27) in equations (17)-(20), hence we have:
Re*F
rvn dv,,'
dvn [ RW v3i dvn [ RW vnv31 dhn | _ EuRe 1 dp{ + hu daxx 3a2j R U A3j da3X R U AjjAjj da3x ^ 'F hxx 3ajj
2W2 d
Kx 3a, 1
m
dvn RW v31 dhu > R*U A31 da31 ,
da
Пх
21 V
^11
da
l + 'P"
1
21.
A„ da21
Их
dv
21
da
+ (28)
21 у
+4+
R
1 9 hxxHx A11 0 fVl,l W \ 0V„ j
A,2,A3I da3l _A31 da3X ^Aiy “1 ■ ■ ► U hn dan
0фе¥3) = _ EuRe + 0(xp2 ^.
¥ da1
(29)
l21
Re¥
+ W2
v„ av3,
+ v.
dv
31
WR v31 dv31 UR у,2, dhu EuRe R________________________1_ dpx
A,, dau dan UR* A31 da3l WR* AnA3, da3l J 4? R* A31 da3x
,2 UR d
1 d [ J y\ ’ UR hu d fvn'l , 1 dv3il
A„ Г WR* A31 3a31 ^11; An 3anJ
WR* da21
Их
VA3i da3i j
+
da
21
m
dv
31
da
+ 2W
21J
Я
1
АцА31 дазх 2*F2 R dhu U dvn R v31 dhn '
Km
dv
\
(30)
31
da
31 J
My
1 dh
31
//31 doc^
m
dv
31
da
31
АцА31 R
тЛх
da
31
W daxx R A31 da3l j
, 3vn dv2l UR d /. \
К ^ + KK +r-1 (^nv3i)-0
dau
da2j WR da3i
where Reynolds and Euler Numbers have the forms:
Res^£,Eus />0
Ho
U2p
and
T = 0(l0"3), 0<Re<l,^ = 0(l),
because W «U. Now we have two possibilities:
Eu =
EuRe r=r-
= л/Eu = 0
VRe
EuRe
'F
1
Uo.
if „* „ -Pq - Ho
Po pR2 ’ P° V2 ps2
(31)
(32)
(33)
(34)
and
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
Fig. 5. Boundary ranges of hydrodynamic pressure region on bone head in human joint.
*F EuRe . * p0 cotjR2
Eu = —,----------= 1 if pn = con, pn = —.
Re 'F Ио ' Уо 'F2 f2
(35)
The terms of inertia forces in equations (28)-(30) are multiplied by the factor ReT*. We neglect inertia forces terms and other terms (multiplied by factor ReT* or 'FB for я > 1, 'F s 1СГ3) which are of order of (l(T3)”< 0.001. Thus the system of equations (28)-(31) for axially asymmetric, isothermal, stationary synovial fluid flow in the film between two rotational surfaces with non-monotone generating line has in the curvilinear, orthogonal coordinates ax, a2, a3 the dimensional form (21)-(24). The term of centrifugal acceleration
of order of Rt'VU/W occurring in equation (30) cannot be negligibly small because
W «V. This term -
pvx dh\
h\h3 даъ
exists at the left hand side of equation (23). Gap height may
be a function of both variables a,, a3, i.e. e, = £х{щ,а3), where £«2*10 5 m. This remark completes proof of Lemma 5.2 И.
6. Boundary conditions
In human joint we have the following boundary assumptions for the pressure function *a,.a3),Fig.5[23]:
p(ax,a3 = bm) = pz(ax), p(ax,a3 = bs) = pw(ax),
* * (36)
p(ax =0,a3) = pz(a3), p(a, = 0,аъ) = pw(a3),
«here p.(ay =ae) = p*z(a3 =bm), pz(cc]=0) = p*z(a3=bm), p*w(a3 = bs) = pw(ax = ae),
pja~ =b!) = pw(a] - 0) and pz{ax), p*z(a3) mean pressure values at the inlet of the arocaladongap in a,, a3 directions, respectively; pw(ax), p*w(a3) are pressure values at the oadet of the gap in ax, a3 directions, respectively.
The boundary conditions for the synovial fluid velocity components, caused by the imbonal motion in a, direction of the surface, have the following form:
vx=cohx, v2 = 0, v3 = 0, for a2 = 0, (37)
■heir a is the angular velocity of the bone. The cartilage of human joint is motionless,
.f _ _
Moore
v, = 0, v2 = 0, v3 = 0, for a2 = s. (38)
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
Approximate formulae for the dynamic values for various shear rates have the following form:
ПР s77oo + f0"T7Г»По ~(Vo-Т1Х)®А + ... for О<02Я«1, (39)
и 1 + A-&
for other cases
VP =7=o+-—fg ~feo -Jloa)®A-(jlo-?ix)@2B + ..., (40)
1 + Л-0+.5-0
where 77^ and 70 mean the dynamic viscosity values of synovial fluid for large and small
shear rate values, in Pa s. Symbols A and В denote the coefficients, which were obtained by Wierzcholski [17] by virtue of experiments [2]. We have obtained A = 1.88307 s and
В - 0.00458 s2 for the normal human joint and also A = 0.03349 s and В = 0.00131 s2 for the pathological human joint. The shear rate has the form
f \t \
0O = 0
£ )
,© =
_ 5v,
da,
(41)
7. Properties of analytical solutions
Lemma 7.1. Involve solutions of the nonlinear partial differential system of the second order (21)-(24), for synovial fluid flow the velocity components and pressure in human joint gap resting on rotational bone surfaces with non-monotone generating line for the boundary conditions (36)-(38) and variable dynamic viscosity function (39) obtained from experiments, have the following form:
f ~ ' 1
, ч 1 dp
v,(a,,a2,a3) = —;—;—-f—a-,e
n \ j a2
4*7«A 5a, V £ )
+ o)ht
1 _<*г
У £ ) «2
+ -
2 Arj.
-Г (a2,p,Cl,A), (42)
v3(a„a2,a3) =
. . 1 f 5v, , 1 f d ,, u
г(а,,а2,а3) = -— J—-Ч/а2- — I~-{hxv3)da2, ft\ Q oax л,Лз 0 оаъ
sc 1
]a2 —
1 dp
VJh даъ
4 | ~L 1
fa,------da,- f-----
0 nPi(&) 0 7pi(A)
-da,
tlpii*)
da,
e 1
p
J n A
Vpi(b)
da-,
where the pressure function p satisfies equation:
da.
о
• jv,(a„a2,a3)da2
+ ■
1
/г3 (a3) dai
6
J/21(a3)v3(a1,a2,a3)da2
Vo
= 0.
Moreover
«2 _____________ -
T{a2,p,Cx,A)= (a2,p,C],A)da2—A- j-jE(a2,p,C{,A)da2,
0 p
Ща2>Р’С\,А) s
^ dp ^ a2
>1 5a, у + 2ЛС1(2,с0 -,„)+(^C,)2 +,o,
C| = -^,+5»—1 1
+ +2,„ +
flj j
Л 2hx dax As »
f - — yZ(a2,p,Cx,A)da2,
(43)
(44)
(45)
(46)
(47)
(48)
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
’lPt(A) = ’lpt(a2,PA'A) = ——Jp*
1 + A-^~ da 2
2щ - k{a2,p,Cx, A) 27oo- A(a2,p,Cx,A)’
(49)
j л I -• £
l(a2,p,Cx,A) = —-2a2)~ + ^E(a2,p,Cx,A) - - UE(a2,p,Cx,A)da2 .(50)
2hx da, J
for 0 < ax < In in the circumference direction, bm <a3 < bs in the width direction, and 0 < a2 < e in the gap height direction.
Proof. We integrate equation (21) with respect to the variable a2 and obtain:
Q, = tjp for Oj s — ^-a2 + C,.
da,
hx dax
(51)
*here C, means the integral constant. We put (40) and (41) into (51), thus we obtain the following algebraic equation:
Д|7оо®3 + (Л|/оо-5О1)©2 + (»7о-^П1)0-О1 = 0. (52)
For 0 < 02 В «1 we can simplify equation (51) which obtains the following form:
=0. (53)
In this case, the proper solution of equation (57) has the form:
dvi AQX -rj0 + ^{t]0 -AQx)2 + 4AQxfjm
0 =
dan
2 Arja
(54)
If A = 0, formula (54) has an indeterminacy point 0/0. We use the Г Hospital’s rule to recur, the limit of formula (54) as A tends to zero, i.e. when we consider the particular '• : nan case of the synovial fluid:
lim© = —. (55)
A^°
Let us find the solution for the small shear rates. We integrate equation (54) twice with -aspect to the variable a2. Hence we obtain the circumferential velocity component in the f:Lc-'*rr.g form:
az.a2) =
1
2?7co {2hx dax
l dp 2 ^
------a2 + Cxa2
Vo
j
I у________
a2+——yE(a2 ,p,Cx, A)da2 + C2 ,(56)
-re-e function ~(a2, p,Cx,A) has the form (47) and C,, C2 are integration constants.
The boundary conditions (37), (38) for the velocity component (56) have the :: -пг form:
v, (a2 = 0) = cohx, v, (a2 = e) = 0. (57)
We impose condition (57) on (56) and hence we obtain C2 = cohx and CX(A) in the ~ t '- ei form (48). We introduce the dependence (48) for constant Ci in solution (56), hence -e shii! get circumferential fluid velocity component in the form (42), where function Г rererrrmes formula (46) for 0 < ax < 2л, 0 <a2 < e, bm <a3 <bs.
Now from equation (23) we determine the velocity component v3. We neglect :emmf_gai acceleration term. Afterwards we integrate twice equation (23) with respect to the • —mme a-. Hence we obtain:
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
v3=-
1 dp
«2
a2
-da2 + C3 Г-
1
-doc2 + C4,
пЛдаг о " 0J rjpX{a2,p,Cx,A) z ^
where C3, C4 are integration constants. Dimensionless viscosity r\pX = rjp/pa0 has by virtue
of equations (39), (41), (58) form (49), whereas the function &(a2,p,Cx,A) determines formula (50). Now we impose the boundary conditions:
v3 («2 = 0) = 0, v3(a2 = e) = 0, (59)
on the longitudinal velocity component (58). Thus we obtain constants:
6 i
fa? —
1
C3 =
dp QJ *7,i(A)
da.
liJh
О
h
1
C4=0,
(60)
-da,
о V p\(^)
We substitute constants (60) into solution (58) hence we obtain the longitudinal velocity component of synovial fluid in form (44). Now we integrate once the continuity equation (24) with respect to the variable a2 and thus we obtain the radial velocity component of the synovial fluid in general form:
1 ardvt 1 °f 3 /. ч , ^
v2(ax,a2,a3) = —— )^da2-—- J-—{hxv3)da2+C5,
h\ oda\
hxh3 0Ja«3
(61)
where C5 is an integration constant. We impose the boundary condition v2(a2 - 0) = 0 on the solution (59) hence we obtain integration constant C5 = 0. Thus the radial velocity component has the form (43). Now we impose boundary condition v2(«2 =£•) = () on the solution (61) and we take into account the following identities:
d £r , . , srdvx(ax,a2
----Iv.(a„a2,a3)</a2 = I 1
n dal
da j
da.
О о a
\h](a3)v3(a],a2,ai)da2 = J-
da-.
tfo2 > (62)
(63)
‘з о о з
which are valid because v, (ax,a2 = e,a3) = 0 and v3(ax,a2 - £,a3) = 0 . Hence we obtain the modified Reynolds equation (49), which determines the unknown pressure function p.
This result completes the proof of Lemma 7.1 i.
Theorem 7.1. Approximate unknown particular solutions of the nonlinear partial differential system of the second order (21)-(24), for synovial non-Newtonian fluid velocity components and pressure in human joint gap resting on rotational bone surfaces with non-monotone generating line, are shown to be a sum of the following parts: the first part refers to the Newtonian properties of the synovial fluid, the second part is multiplied by the coefficient A and presents corrections caused by the non-Newtonian fluid properties, next
Л
parts are estimated by terms of order of A :
——-^-ss2 (l - s)+ah, (l - j)■-^i«(l - s)—^2 x hx 2p0 dax 8^ hx dax
+ o(a2),
v, =
2 o)hx + 1 clpm~) (l - 2^)
3?o [hx dax )
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
6 Vo
1 д2р 1
• + -
h, 3af 5«3
hx dp
\ h даъ)
+ A
cokxs 2
67«
(l-5)>
ilia v i [><0)1 1 1 d | eh,2 dpm ^ 1 dp(0) ds 1 dp(0) hx de
Л I |^2 hx dax l 5а, J 1 4 hxh3 da3 , A3 5a3 , hx dax dax 2h3 da3 h3 da3
Akxs2(). -5) (l-5> 1 d (е212др№Л 2 1 d i (V dpm 5р(0)У
4%VoVao hx dax ч h\ 5a, j I hxh3 da3 ф3 dax da3 J
(65)
l ds i dpm' 2, i ds i ф(0) i ар(0)] ~Ь >■
hx dax lAi 5a, J A3 5a3 A3 5a3 A, 5a,
o(a2),
v3=-
-L± JfLS£3(l _s)_ ^ _ s)_l3pT
2r)0 h3 a«3 8г/и ' h3 da3
, £2 й>Л, + 1 3pm 3 (l - 2s)
3% 1Л aa, J
o(j2),
(66)
p = p{0) + APw+o{a2), (67)
•btaere functions /?(0), jp(1) satisfy the following modified Reynolds partial differential sxanons:
1 d
hx dax
i J
i d
hx da i
^ф(0)^
7o 5a
(s2dp^
77o 5a,
+ ■
1 d
h3 aa3
Ajf3 5/? h3i70 5a
(0) Л
- 6ft>A,
зу
a^
dax
+ ■
i а
/г3 da3
1 а
- —со-
e2к, дрт 3
2 а«,
7оо 5а
^/г,£3 dp^ ^
Мо 5а3 у
f 1.2 Л
1 1 а
—со
1 7
4 А3 аа3
(0)Л
hxs кх dp М® da3
(68)
(69)
а-
1
л <2 — .------<кх =
г 50
М 0<«, <2*. 0<а2<е, Ьт <а3 <*,.
По
Proof. Formula (48) presents the equation with respect to the unknown constant C\(A). Гг осот constant Ci in analytical form we expand the right hand side of equation (48) in two s of Taylor series in the neighbourhood of the point A = 0. We obtain:
C, =-^^-cohx - + lim/(^,C1) +
2й, dax л->о V0 1
lim^CAQ A + o{a2), (70)
A^o dA 1! v ’
f(A,C,) = ^—— U~(a2,p,Cx,A)da2. c As J
(70*)
Since an indeterminate form of type 0/0 is obtained as A tends to zero, we use the TKospnal's rule to obtain the limits in the above formulae (70), (70*). In this way from vrr-ula (~0l we obtain the quadrate algebraic equation with respect to the constant CX(A). Tx -roper real root of this quadratic algebraic equation has the following form:
Vo
CX(A) =
2 Akx
i_£M_LJl _П(Л)
Vo h\ 5«1
(71)
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
where
skxA 1 dp
+ 4kxA
a>hx
(72)
70 h\ da\
As A tends to zero, we obtain indeterminate form of type 0/0 in CX(A). Hence, applying THospital’s rule, we find:
1 1 r)n mh.
(73)
r s, л \ 1 1 Ф a>hx
bm f(A,Cx) = --------~rj0 ■
Л->0
2 hx dax
Now we remove the constant Cx in the circumferential fluid velocity component (42), i.e. we remove constant Cx in function E(C,), see (47), and in function Г[Н(С,)], see (46). After such elimination we obtain:
A dp
E(a2,p,A) =
hx dax
a 2 —
£Л
2y
+ ^-(\-T1(A))+2tim -?j0
2k,
-i2
K\Vo
(74)
We use l’Hospital’s rule to obtain the limit of the last term at the right hand side of equation (46) as A tends to zero. In calculations we take into account equation (47). We obtain finally:
lim-
1
Г (a2,p,A) = -
277oo-7o 1 Ф
(o)
-a2e
a \
1__2
£ J
(75)
A~*°2Arin ' ' 4i/0^e hx dax v
Hence as A tends to zero, circumferential velocity component has the following classical form:
limv, =
1 1 dp
(o)
■a2s
a \ j a2
+ cohx
(У ^ 1 __2
J
V
(76)
j
o ‘ 2p0 hx dax
were /?(o) s= p(A = 0).
The circumferential velocity component (42) after the elimination C, in functions (E(C[) and Г[Н(С1)]) is expanded in two terms of Taylor series in the neighbourhood of the point A = 0 in the form:
lim V, +
Л-Й)
lim —
A—>o dA
1!
• +
(77)
Now we calculate the first derivative of function (42) with respect to A
dvx _ d dA dA
1
dp
sa-
^ a ^
47«^i da\
____________l dPm
2A2tjx 4r]xhx dax
\-=±
£ )
+ cohx
a Л
l__2_ £ j
+ ■
1
2 Arja
-T(a2,p,A)
Л*-Г
dA
(78)
a-,s
Г a ^
I ai
J
where
(79)
dE dE
■= \—4—da2 - — 1-Щ=йа2.
0J2VI £ 0J2Vs
From equations (46), (47) it follows that functions E, Г and their first derivatives with respect to A have in point A = 0 the following values:
3r_“r_ai
dA
E(A = 0) = rf,r(A = 0) = 0,
(80)
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
= 2(270,, -7„)
A=0
1 dp
(O)
8Г
da
J A=0
К Vo J
hx dax >)
a2 —
coh.
1 dP ( \
;—г—«2 («2 ~£\
hx dax
fd2E) = 2 ( 1 dpЛ 2 r a
{dA2} л=о u daxJ \
. a>h Л . 1 dp r a2 - £'
Ч- e 40 hx dax V
4
1Л,^=
6 Щ
(81)
+
—1 [2/7o + 4«2^i7o(2?7« - 7o)]• s )
The first derivative of function v, with respect to A (see (78)) has indeterminate form of type 0/0 as A tends to zero. Therefore we use once the Г Hospital’s rule to find the limit of function (78) in the following form:
lim
dvx
1
lim
8 T
1 dp
0)
л->о dA ArjK a->о qa2 Ar]Jix dax
-ea о
f a ^ 1 - —
£ J
(82)
We use the first derivative (80) and we calculate the second derivative in the following
'OCilil
д2Г
dA2
laf i i (dE)
2 J 0 2 H Vh {dA)
i a2s Vs йл2
da2
2 £•
J
1 1
rasV l
+ •
2 SVS v Й/4 J Vs <3/12
a2s
J»!
(83)
In equation (77) we introduce equations (76) and (82). Afterwards in formula (82) we sjbsctute the limits of function (83) as A tends to zero. To obtain these limits we must use (81), (80) and the value of the second derivative of function S with respect to A in .4 = 0. After the calculations we obtain finally an expression (64), which determines the crrcT-mferential velocity component.
The longitudinal velocity component (44) is expanded in two terms of Taylor series in ±c neighbourhood of the point A- 0 in the following form:
v3 (ax, a2, a3) = lim v3 (ck1 , a2, a3) +
A-> 0
lmA3
л-»о dA
A'
1!
(84)
It is easy to see by virtue of equation (47) that function (50) As ha м2,а3) = A(a2,p,Cx,A) tends to zero, if A tends to zero. Hence the longitudinal fiend velocity component (44), as A tends to zero, goes to the following classical form:
lim v3(ax,a2,a3) = (a2 -sa2). ■ (85)
л-»о 4т70й3 da3
Now we calculate the first derivative of function (44) with respect to A:
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
dv,
1 dp
dA rjxh3 da3
«2 д 1 «2
\a2 —-----— dct2- f
J 2dA « АЛЛ 2 J
d_
dA
MA)
da-,—
£c 1
J«2-----
Vp\(&)
-da-,
7Pi(A)
da-,
——</аД
7«A 5«3 0J r/pl(A)
\a-,
dA
Мд)
^ j £• i £
I------da, - -------da-, |
оЧ>(Д) oJ ^,(A) ^
d
dA
.»7pi(A)
(86)
da-,
e 1 f-i-
j n . Л
da.
Vo ^pi(A)
From equation (49) we obtain the first derivative of the reciprocal of viscosity function pp] with respect to A in the following form
,dA
d 1 1 1 CD 27» “A
dA _7pi(A)_ 1 ^ 1 _ 2щ - A _
_ д] 2^-70)
(2?7o - A)
f^ = W(A).
(87)
We substitute expression (87) in formula (86). Hence the first derivative of function v3 with respect to A has the following indeterminate form of type 0/0 as A tends to zero:
lim — = —l— lim ^
л-»о dA rjxh3 л-+о da3
£ 1
. „ \a2------da-,
(2 a2 J 1 j, (Д) 1
\a2W(A)da2- JW(A)da2^----------------
о 0 j- 1 .
0
^i(A)
lim
€ £ j £ £
. «2 , ja2fV(A)da2j—— da2-ja2—— da2\w{A)da2
Ф_ j_L_daJ_________оЧ<(Д> о MA) о
(88)
*7«Л A->oda3 0J T]p](A)
hn
-da-,
Vo *7pi(A)
To obtain limit of formula (88) we must calculate the limits of expressions W, see (87), and limit of the first derivative of A, see (50), with respect to A as A tends to zero:
Hra^ = JkZ%]±JP.(£_2 аг)-2п^
л-»о dA щ hy dax V e
lim W(A) -
A-+Q
{пгл~щ)
vl
(89)
(90)
(jhZJklL±.(e.2a2)+n.^
2щ hx dax V s
Now in equation (84) we introduce limits (85), (88) obtained by virtue of expressions (89), (90). After the calculations we obtain finally the longitudinal velocity component in the form (66).
We insert the longitudinal and circumferential fluid velocity components (64), (66) in equation (45). After calculations we equate the coefficients of the same power к of small
parameter Ak to zero. For k = 0 and k = \ we obtain the following forms of classical (68) and modified (69) Reynolds equations.
Equation (68) determines pressure function p(0) and equation (69) determines pressure corrections p(l), which are caused by non-Newtonian fluid properties.
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
We put the circumferential and longitudinal velocity components (64), (66) in the formula (43) and we take into account equations (68), (69), thus we obtain finally the form f 5»of the radial velocity component.
This result completes the proof of Theorem 7.1 Я. ч . .
8. Lame coefficients for some parabolic and hyperbolic coordinates-?
For thin layer synovial fluid resting on parabolic bone surface we have the following Lrx coefficients:
h, = a cos2 (Aa3), h2 = 1, h3 = Jl + 4a2A2 sin2(Aa3) cos(Aa3), A = -J— . (91)
b V a
We denote: a is the largest radius; ax is the smallest radius of parabolic surface; 2b is re bearing length; w = a-al. In this case we denote: ах=ф (circumference), a2 (gap z>t.zz: i. a, (longitudinal directions), respectively.
For thin layer synovial fluid resting on hyperbolic bone surface (see Fig. 6) we have re following Lame coefficients:
^1 + 4a2A2tg2(Aa3). (92)
A, =
cos2(Aa3)
, h2 — 1, h2 —
1
3 cos2(Aa3)
- -.ere
1 w
Ah- - , 0 < a, < In, 0 < a2 < s, a3 <
1
- arc cos,
(93)
b \ a ‘ * ' ' J| A \ a + w
'A'e make the following notations: a is the smallest radius of bone cross section; z = *- - j is the largest radius of bone cross section; w = ax -a; 2b is the joint length, s is me rrrers .onal value of gap height.
<*. Numerical examples for spherical bone surfaces and pressure distributions
?:r :hm layer oil resting on spherical bone surface we have the following Lame
oc
hx - Rsin—, h2 = 1, h2= 1, R
(94)
m-itr? У is me radius of sphere. We denote: ах=ф- the circumference direction; a2 = r -use pc zczz: direction; аг = 9 - the meridian direction.
Te^r:cer.cies between rectangular (x,y,z) and spherical (ax ~ф,а2=г,аг=9) -iiiranaoes mve the following form:
• f0) (9Л (9)
т = r sin — cos <p9 у = rsin — sin©, z = rcos —
UJ
, 0<r<R.
(95)
Lnsrcicai illustration of the centre of spherical bone head 0(0,0,0) and the centre of carriage in the point Oi(x - Aex,y- As2,z + Ae2) is presented in Fig. 7.
Emjsmec of spherical cartilage surface in point Qx(x - Asx,y - As2,z + Ae2) may be ш.г ng m me following form:
r-JLf ' - \y - A£2f +{z + As,)2 ={R + D + eminf, D = ^J7(A^yT(A^y.(96) * e ret dependencies (95) in (96), hence we obtain:
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
Fig. 6. Radial elbow joint in hyperbolic coordinates.
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
Л 2 f \ 2 f f „ \ \
a3 i sm — COS#?-A^ + r sin «3 sin <p - As2 + r COS a3 1 + A^3
.R) J ) ) l R J У
Gap height has the following form:
a3
-r- R.
(R + D + gmin )2 .(97)
(98)
We find r from (97) and put in formula (98). Hence gap height has finally the ::Gewm2 form:
a- I
<?.— = As, sin
R J
fa '
\Rj
fa'
cos#> + As2 sin — sin^-Ag3cos
v R J
fa'
\R;
-R +
-, Ag, sin
\
f a3 ^
~RJ
fa '
fa "
(99)
\R)
cos<p + Ag2sin — sin<p-Ag3cos — + (7? + gmin \R + 2D + gmjn ),
R j)
Reynolds equation (68) has the following form:
_8_
6<p
По SV’
+ R sin
Г1Л
yRj
d
89
g3 dP 70 99
(°) f _Q Л
sin
9
R)
= 6coR2 —sin2
d(p
f-1
kR)
(100)
«here 0 < a, =(p< 2ncx, 0 < с, < 1, <a3< . In numerical calculations we assume:
Ls = 5 цп, Ag2 = 5 pm , Ag3 = 5 pm, cx=l/2, radius of bone head R = 0.026575 m and ETTispc-eric pressure on a round of region Q(ax,a3) resting on bone head (see Fig. 8).
For a normal hip joint we take in calculations the smallest gap height gmin = 23 pm, Z - = 30 pm. Taking into account the angular velocity of bone head со = 1 s'1 and
f».—таг -alue of synovial fluid dynamic viscosity tj0 = 2.00 Pa , we obtain from (100) that
гоггоа-лис pressure p(o) has the maximal value 3.96-105 N/m2 «3.96at. Taking into acrccrx ±e angular velocity of bone head со = 0.1 s-1 and average value of synovial fluid m—.e—c -.-.scosity tj0 = 30.00 Pa, we obtain from (100) that hydrodynamic pressure p(o) has me га-длиа! value 5.43-105 N/m2 «5.43at, see Fig. 9. Lubrication surface has value
Lr2~ :cs- « 20.5 cm2.
rcr a pathological hip joint we take in calculations the smallest gap height = 4 J , i.e. D + gmin = 16 pm. Taking into account the angular velocity of bone head
a = Г arc average value of synovial fluid dynamic viscosity rjQ = 0.05 Pa, we obtain from - Ж mac hydrodynamic pressure p(0) has the maximal value 1.68 -105 N/m2 «1.68 at. ~Twg is: account the angular velocity of bone head со = 0.1 s-1 and average value of fwi chnamic viscosity rj0 =0.20Pa, we obtain from (100) that hydrodynamic
г : has the maximal value 1.27 ■ 105 N/m2 «1.27 at. лсое pressure distributions on bone head for normal and pathological human hip racr гаг are shown in Fig. 9 and Fig. 10.
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
Fig. 9. Two cases of pressure distribution in normal spherical hip joint gap. R = 0.026575 m,
D + £min = 30 pm , lubrication surface - 20.5 cm2; а) со = 0.1 s-1, rj0 = 30 Pa , pmax = 5.43 ■ 105 Pa , Cm = 909 N; b) a> = 1 s'1, J]0 = 2 Pa, Pmax =3.96 105 Pa, Clot =606N.
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55, 2002
/;, Pa
— 4-iO5
— 3105
— 2 105
b)
M Two cases of pressure distribution in pathological spherical hip joint gap. R = 0.026575 m, T - c —= i 6 pm , lubrication surface - 20.5 cm2; а) со = 0.1 s~', = 0.2 Pa, pmax = 1.27 • 105 Pa ,
Cw, = 56 N; b) © = 1 s_l, r/0 = 0.05 Pa, pmax = 1.68-105 Pa, C,0,=139N.
10. Final comments
Tbe present paper shows the method of determination of approximate solutions of ■щгж zc-dxear differential equations of non-Newtonian, asymmetric synovial fluid flow in a ill ill ill oc curring in human joint in curvilinear, orthogonal coordinates.
Tie presented method enables to obtain solutions in polynomial approximation form if 7iy.cc senes with increasing powers of small parameter of A obtained by experimental
ЖТ-
Iz particular case for symmetric flow we can by virtue of the presented theory find rui-.-c-. s?‘.unons in a simple form.
Acknowledgement
Tie iizhor thanks for the financial support by the Polish Grant KBN 8-T 1 IE-021-17.
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ГИДРОДИНАМИЧЕСКАЯ ТЕОРИЯ ТЕЧЕНИЯ СМАЗКИ В ЗАЗОРЕ
СУСТАВА ЧЕЛОВЕКА
К.Х. Вежхольский (Гдыня, Польша)
В статье предложен метод решения систем нелинейных дифференциальных уравнений в частных производных второго порядка, описывающих осесимметричное и несимметричное течение жидкости между двумя поверхностями кости в зазоре сустава
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Russian Journal of Biomechanics, Vol. 6, № 1: 34-55,2002
^ювека. Решения ведется в криволинейных ортогональных координатах. J ссс чулирован ряд теорем, которые позволяют унифицировать решение нелинейных -•л-оеренциальных уравнений при течении синовиальной жидкости. Библ. 23.
«Плечевые слова: гидродинамическая теория смазки, неньютоновские жидкости, згзовиальная жидкость, сустав человека, зазор сустава, решение
27 November 2002
55
