B KrLLE TIl~ O F ~ATHEI~ A TICAL BIOLOGY VOLUME
38, 1 9 7 6
T H E E F F E C T O F MICROSTI:~UCTUEE ON THE RHEOLOGICAL P R O P E R T I E S OF B L O O D
[] C. K. K A v a National Tsing-Hua University, I-Isin-Chu, Taiwan, China [] A. C. EaI~G~SN Princeton University, Princeton, N.J., U.S.A.
The micromorphic theory of Eringen is applied to study the tube flow of blood. The blood is considered to be a deformable suspension, with constitutive relations of the form of those of simple microfluids. By means of energy consideration, a relation is established betweerL the local concentration parameter and the measure of rotationality involving both macroand raicromotions. The tube flow problem is then solved with some analyses on viscosity coefficients and boundary conditions. The results obtained indicate an integrated explanation of various important physical phenomena associated with blood flow, such as the tube size dependence of the apparent viscosity and the non-uniform concentration distribution over a tube cross section.
1. Introduction. Rheologically n o r m a l h u m a n blood is a macroscopically continuous suspension of p a r t i c u l a t e m a t t e r s ; red cells, white cells, a n d platelets, in a microscopically continuous fluid, t h e b l o o d p l a s m a (Houssay, 1955). Of t h e particulate m a t t e r s , t h e red cells are d o m i n a n t , occupying a v o l u m e fraction (hematoerit, co) of 3 5 - 5 0 % of the blood. T h e h u m a n red cell is a small a n d e x t r e m e l y flexible disc h a v i n g a n a v e r a g e d i a m e t e r a b o u t 8 p m (micron) a n d thickness a b o u t 2 #m. I t has a s.g. 1.085, which a p p r o x i m a t e l y equals t h a t of t h e blood p l a s m a w i t h a s.g. of 1.035. Therefore, i n rheological studies, it is customary to consider the b l o o d as a c o n c e n t r a t e d suspension of n e u t r a l l y b u o y a n t deformable particles in a viscous fluid. 135
136
C . K . K A N G A N D A. C. E R I N G E N
When blood is flowing in smaller vessels, the apparent viscosity decreases with decreasing tube radius (the Fahraeus-Lindqvist effect (Haynes, 1961). This phenomena has been observed also in slurry flows and was designated as the sigma-phenomena (Scott-Blair, 1958). Several attempts exist to explain this phenomena: Dix and Scott-Blair (1940) proposed that the flowing laminae has a finite thickness, thus in calculating the flow rate, a summation of flow layers should be employed instead of integration. B y this process, they managed to obtain a tube-size dependence. Others (Vand, 1948) assumed the existence of a particle-free layer with a smaller viscosity adjacent to the tube wall, which causes an increase of flow rate in smaller tubes. Nevertheless, no theory appears to exist encompassing explanations of these phenomena along with other flow properties. Another peculiar aspect of the blood flow is the radial redistribution of red cells in tube flows (Segr6 and Silberberg, 1962). For a dilute suspension, starting from a uniform initial distribution of particles over the cross section, a necklacelike particle aggregation has been observed at about 0.6 radius from the tube center (the Segr6-Silberberg phenomenon). In concentrated suspensions, axial accumulation occurs resulting in reduction of cells or even establishment of a particle-free zone along the tube wall. Theoretical studies have focused attention on the motion of a single particle in viscous shear flow (the migration phenomenon (Goldsmith and Mason, 1967), but no quantitative results appear to be available for the explanation of these phenomena inherent to concentrated suspensions such as the blood. Experiments (Kuroda et al., 1958) also have shown that the deformability of red cells has a pronounced effect on the apparent viscosity. At normal conccntration, the suspension with hardened cells will exhibit a viscosity about l0 times higher than that of the normal blood. Again, the lack of a quantitative analysis does not permit the incorporation of the deformability effects, on a rational basis, in the calculations. The present study employs the micromorphic theory of Eringen (1964), (1967), (1972) to analyse the flow of blood. As a supplement to the microfiuid model, in section 3, we introduce, for neutrally buoyant suspensions, a relation between concentration parameter and flow rotationality. In section 4, we estimate the values of viscosity coefficients and discuss the forms of boundary conditions. A tube flow problem is then solved in section 5. The velocity and rotation profiles obtained are compared with certain experiments in section 6. The present model also provides quantitative calculations for the axial migration phenomena, the Fahraeus-Lindqvist effect, and the effects of cell deformability.
2. Rdsurad of the microfluid theory.
Eringen (1964) introduced the notion of
MICROSTRUCTURE ON THE RHEOLOGICAL PROPERTIES OF BLOOD
137
simple microfluids to describe t h e b e h a v i o r of some fluent m e d i a which can s u p p o r t stress m o m e n t s , b o d y m o m e n t s , a n d possess local spin inertia t h a t can influence the flow properties. T h e g o v e r n i n g balance laws of the m i c r o m o r p h i c mechanics (Eringen, 1964; 1967) are listed below :* conservation of m a s s
ap --
at
+(pv~);~
=
0
(2.1)
conservation of micro-inertia
aikm at + ikm;r vr-irmvkr --i~rvmr = 0
(2.2)
balance of linear m o m e n t u m
tkz;~ + p(ff - ~ ) = 0
(2.3)
balance of first m o m e n t stresses ,~Im;~ + tmZ _ smz + p ( l l m _ ~zm) = 0
(2.4)
conservation of energy
p~ = tk~vz;~ + (s ~l - tk~)vZk + 2k~mVlm;k + qk;k+ ph the e n t r o p y i n e q u a l i t y
ph
>f o.
(2.6)
The e n t r o p y i n e q u a l i t y (2.6) is p o s t u l a t e d to be v a l i d for all i n d e p e n d e n t processes. I n these equations : p = ikm= v~ = t kz = ff = s k* = 2k~m = pm= 5Zm =
m a s s density m i c r o i n e r t i a tensor velocity vector stress tensor b o d y force per u n i t mass micro stress a v e r a g e tensor t h e first stress m o m e n t tensor t h e first b o d y m o m e n t per u n i t m a s s spin inertia
* H e r e , v kz r e p l a c e s vz k d e f i n e d i n ( E r i n g e n , 1967).
138
C. K . K A N G A N D A. C. E R I N G E N
internal energy density per unit mass ykl .~ gyration tensor
of the body
fl ~ = heat vector directed outward
h = heat source per unit mass t l = entropy per unit mass T = absolute temperature. Throughout this paper, a semi-colon (;) is employed to indicate covariant partial differentiation, and a superposed dot (') to indicate the material time derivative, e.g. vk
;1 = Vk;~ .6
I It} rl
vr'
~k~- av~z ~t
.6 ~kl'mVm'
where {r~t}is the Christoffel symbol of the second kind formed by the metric tensor g~z of the spatial coordinates x ~. The spin inertia is given b y _
irm(
r+
(2.7)
Eringen also gave a set of constitutive equations for simple microfluids. For a non-heat conducting medium, these are the relations between t klm, t ~, s ~Z, and the objective deformation rate measures
d~l = ½(v~;z + v z:k ), b~ -~ v~z+ vz; ~ = v~l + dkz - (o~z,
(2.8)
t~klra ~ "¢kl;m.
Of these, d is the deformation rate tensor and b and a are called micro-deformation rate tensor and gyration gradient, respectively. For the present work, we reproduce only the linear constitutive relations of simple microfluids: t~l = [ - n .6 )~vdrr + to(brr - dr)]5~l + 2pd~1.6 2fl0(b~ - d~z) Jr 2/~l(bt ~ - d~), s~z = [ - ~ + !lvdrr .6 17o(brr - drr)]57cz.6 2/~dkz.6 ~2(b~z.6 b~~ - 2d~) ,
(2.9)
)/~lm = (~lamrr .6 ~2armr .6 ~Sarrm)Skl + (~dalrr -~ ysarlr -t- 76arrl)5~ m
.6]~7a
lcr rlc r It r .6]~8a r - 6 ~ 9 a r )glm
.6?lOa~m nu?lla
k ml
+ ~12a~ .6 ~ s a ~ .6 ~ a a ~ ~ + ~ a ~ where 5~ is the unit tensor, g~ is the metric tensor, iv, t0, /tv, p0, pl, ~v, ~0, ~ , 7~ to 7~5 are the viscosity coefficients which are functions of temperature in general and where u is the thermodynamic pressure defined b y u ~ - Oe/~p-~[,.
MICROSTRUCTURE
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139
Although the balance laws and constitutive equations given above form a determinate system (with suitably chosen geometry and boundary conditions) for simple microfluids, to apply the theory to suspensions, additional considerations are needed. In section 3 we make use of the physical concepts of the local volume elements to introduce a concentration parameter d into the theory.
3. Concentration parameter. In classical continuum mechanics, material points of a body are considered to be geometrical points embodied with continuous fields~ The matter is assumed to be continuous no matter how small the element of volume m a y be (Batchelor, 1967). For small enough volume elements, this hypothesis breaks down and the individuality of substructure (grains, molecules, etc.) affect the physical outcome drastically. Consideration must therefore be made to take into account the effect of local structure of bodies in the treatment of physical phenomena having characteristic dimensions that compare with those of the inner structure of bodies. Thus for structured or micro-continua, the material points of the body m a y be considered to be small deformable bodies approximated b y orientable points, i.e., points with a triad of deformable directors attached to it. The orientation of the material points can be characterized b y measuring the rotation, shearing, and stretch of the directors. This is the physical model underlying the theory of micromorphic contina (Eringen and S,uhubi, 1964; Eringen, 1964, 1967, 1972). If the directors are considered to be rigid, then we have the micropolar media. The material points of a micropolar continuum are, roughly speaking, small rigid particles. Mathematically, these directors introduce additional degrees of freedom besides the displacement or velocity field of the classical continua. In the micromorphic fluids, the extra degrees of freedom are the gyration tensor Ykl , whose skew-symmetric part viii represents local intrinsic angular velocity of the substructure, and the symmetric part ~(~z) are the local stretchings and shearings (Eringen, 1964). The suspensions deviate structurally from an ideal micromorphic medium b y not having moment of inertia associated with those points in the fluid region. Therefore a triad of directors are attached only to suspended particles b u t not to the points of fluids. Nevertheless a concentrated suspension system at rest (volume concentration of the suspended particles Co > 0.1) could be regarded as a microfluid with the proper provision that its material point is a combination of the suspended particle with its surrounding fluid. Observations indicate that when a particle is suspended in a viscous fluid, the rotation and deformation of the particle is felt by the surrounding fluid. Inside some 'region of influence', the fluid will display, at least instantaneously, a mean motion with its center of rotation
140
C . K . KANG AND A, C. EI~INGEN
coinciding with that of the particle. We, therefore, construct the one-particlecell model as the 'structured point' in the continuum approach. Our continuum is filled with particles surrounded by effective fluid masses. This model should be valid for suspensions in which the diffusion of particles is not important, that is, local velocities of fluid and particle are almost the same. This is about the case for neutrally buoyant particles (Happel and Brenner, 1965). For simplicity, one may consider the region of influence spherical so that the structured point consists of a fluid sphere enclosing a core particle. Doublets, multiplets, could be included by regarding these aggregates as larger particles, although under higher shear rates, the possibility of forming an aggregation is small (Wayland,
1967).
F i g u r e 1.
C o o r d i n a t e s y s t e m s in AV
Let R~(x, t) denote the radius of the sphere of influence consisting of some fluid mass surrounding a particle (Figure 1). If the particle volume and the total volume of influence are, respectively, AV~ and AV, the local volume ratio may be defined b y d(x, t) = AV~o[AV. To have a rough idea about the magnitude of d, consider a sphere-in-sphere configuration with a particle of radius/~1. When R1/R2 varies from 1.0 to 0.2, d --- (RI[R~) 3 varies from 1.0 to 0.008, that is, from 'all particle' to dilute suspensions. According to the micromorphic theory, the velocity and acceleration in a volume element are
where v ° is the velocity of the mass-center of the particle, v (kz) is the stretchingshearing and VCkZ]is the spin of the particle with respect to its mass center. The particle may be regarded to be spinning at v[~z]-~o~z and to be stretching-
IVIICROSTRUCTURE
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141
shearing at v(~)-dk~ with respect to the suspending fluid mass. Hence in a suspension, the motion of the particle relative to the fluid mass can be described
by the relative quantities ~'~Icl ~ Y [ k l ] - (-Okl
(3.1)
Dk~ =-- v(kz)--dkz.
Before we deduce an explicit formula for the concentration parameter ~ given by d(x, t) ~_ A V V / A V ( x , t) in terms of a variable local volume AV, a proper measure of local rotationality must be introduced. Truesdell and Toupin (1960) have shown that a proper description of local rotation in classical fluid mechanics is given by the kinematic vorticity number (0~]clCOkl~½ ( t r t°2~ ½ WK-- \ ~ ] = \trdZ/
(3.2)
which is a quantitative measure of the amount of rotation in a motion, indicating the relative importance of local vorticity and deformation. Motivated with this, in the micromorphie fluid mechanics we introduce the 'rotationality' by
= \t~/
L-(vt~l)-dkz)(v - - ~
--L--d~_l
"
(3.3)
Here ~ is an objective quantity. It is a combination of the invariants of motion, and is a measure of the relative intensity of rotation and stretching-shearing parts of the motion. From the kinematical considerations of W K and the intuitive idea that, in a suspension, faster rotating particles should influence more fluid mass than the slower ones, it follows that the concentration 6 will decrease with increasing rotations provided the particles are volume-preserving. It is therefore natural to conjecture that the local volume ratio ~ may be related to the local rotationality ~, although the explicit functional form of ~ in terms of ~ is yet to be found. In classical studies dilute suspensions were modelled after the problem of a single particle motion in a viscous fluid. Since the size of particles is very small, the problem is usually considered to fall into the region of low Reynolds numbers, or creeping flows. By determining the viscous energy dissipation due to the presence of a particle, one is then able to calculate the effective viscosity for a dilute suspension. However, for non-dilute suspensions (Co /> 0.1), the particle interactions are important. We now investigate the effects of particle crowding by means of the classical method of 'cage model' analysis. Let a viscous fluid be filled into the space between a rigid sphere of radius R1 and a concentric hollow spherical shell of radius Re (Figure 2). Suppose that the inner sphere rotating with an angular velocity ~oj and the outer shell is fixed to
142
C . K . KA_NG ANY) A. C, E R I I ~ G E N
Figure 2. A structure model for the effect of crowding
simulate the effects of other particles (Happel and Brenner, 1965). I t can be shown that (of. Landau, 1959) the velocity field between the spheres is given by
R1R 2 v~ - .R2a------a_ R1
-
1--d\p 3
1
1 _~
e~j~cojxk
-1)
(3. 1
where ~ = (R1/R~) 3, o9~ _~ e~k~oj. Hence the rotational part of energy dissipation in the fluid due to the presence of a wall can be obtained from e°) ~ - 2P f V (v~d+vj,~)2dV
(1-~).
2
3
= ,~R~ tr ea2fl(~). Simha et al. (cf. Brenner and I-Iappel, 1965) have shown that the stretchingshearing deformation will introduce essentially the same type of energy dissipation due to the presence of the wall, i.e.,
ea = fl,pRal tr a211 + al~ + a2~ 5/~ +...] = upR~ tr d~f2(~).
(3.6)
Thus the total energy dissipation due to particle crowding may be written in the form
e(d) = e~+ea = u#R~[fl(~) tr ~2+f2(~ ) tr d ~]
(3.7)
where f l , f2 are usually monotically increasing functions of ~. The energy dissipation of a rigid spherical particle in an unbounded shear flow is obtained
IVIICROSTRUCTURE
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RHEOLOGICAL
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by Jeffery as 20
e0 =
_~ tr d2"
(3.8)
- ~ - 7~#/~ 1
Therefore, the ratio of the dissipated energy, when the particle crowding is taken into account, to that of a free suspended particle is a function of the local volume par ameter e(e)/eo =
3 [- . ~ t r0)2 1 [fl(c) +f2(e)_ = F2(~)+-]~l(e)W2.
(3.9)
Observing Ie-col < 1, we assume that e(d)/eo may be expanded into a Taylor series about e = Co, i.e., e(a)leo -- F (co) + ( 8 -
+
(e- co) F (co) +... 2
+ W2¢[ El(c°) + (e-c°)Fl(c°)+
(e-c°)------~22Fl(c°)+' " 2 "]
Upon inverting this relation, one notices that the local volume ratio could be determined by evaluating the ratio of local energies. A natural criterion may now be introduced to deduce an explicit relation between e and W~. To this end we posit that the local concentration takes the values for which e(e)/eo will be a minimum subject to the conservation of particles in the suspension, i.e. global restriction of e : ¢(e) -
fv (e-Co)pdV
= 0
or
fvdpdV = copoVo.
(3.10)
By using the method of Lagrange's multiplier to find the minima of e(e)/eo, we have
d A d~b 0 d-~ (e/eo)+ de =
(3.11)
where ~tis the constant Lagrange's multiplier. Using (3.9) ~nd (3.10) this gives
)~
=
-
r ~ t ~ 2 (poVo)--1 [F2(c)+FI(C)WK].
(3.12)
Employing the power series expansions of F~ and F~ for small e-Co and neglecting higher order terms than 2 in (e - Co),this gives ~--C0
=
Iv F2(c0) + F I/i( c o ) W2K + ' "
144
C. I~. IZANG A N D A. C. E R I N G E N
This relation can be written as a polynomial in W~, i.e.,
kpo Vo + F'~(co)
61 ~- CO--
~
-
(3.13)
F~(c0) 1
P/
C
- [~p0VoFl(0) [F~(c0)] 2
tr
!
+ Fl(c0)F2(c0)-
Fl(co)F2(co)]. ¢
it
Since F1 and F~ are monotonically increasing functions of d, from (3.12) it follows that 2 < 0. Other than this restriction, clearly ~1 and 62 can be positive or negative. However, as we have stated in the beginning of this section, higher vortieity means lower d; if we retain terms only to the first order of WE, we will have, for WE ~ 1, with 61 > 0, 62 < 0. This relation (3.13) is a much simplified formula for the determination of local concentration ~ in classical fluid mechanics. In the micromorphie formulation, this suggests that we employ the relation = Cl-C~ 2
(3.14)
where Cl > 0, c2 > 0, and ~ is the micro-rotationality defined b y (3.3). Therefore, neutrally buoyant suspensions may be treated as microfluids, with the non-uniformity of the local volume characterized b y (3.14). In calculations, the accuracy of (3.14) has been satisfactory. The positive constants el, c2 can be determined b y using (3.10) together with dynamic considerations, i.e. drag and/or lift forces on particles considered in classical migration studies (cf. Giesekus, 1962; Bretherton, 1962). One source of information about el, c~. is probably through the study of particle motions, adjacent to a boundary (Goldman et al., 1967 ; Mason et al., 1965), which should provide boundary values for & At present, such studies are not conclusive. 4. Viscosity coej~cients and boundary conditions (1) The viscosity coe~cients. The constitutive relations for a simple microfluid suspension have the forms
t~ = - ~ 0 ~ + 2~d~ + ~,%, + ~ (~~'~,~-o~'~'~
~
= - i o , ~ + 2~d~ + 2~('~)
(4.1)
r
~" .
+(]~Tv/Cr;r+ $8
yrk
,
k
k
;r)glm+ 710v l;m+711VL;l+ 719,Vl ;m
MICROSTRT_TCTUI%E
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where the incompressibility conditions dk~ = v~k = 0 have been employed, and
2 # - - 2Pv+~,
~1--= 2(p0+#1),
~ = 2(#0-Pl).
(4.2)
The entropy inequality for non-heat-conducting fluids has the form: Sklvl;k-[-)'klmYlm;k +(8 m 1 - t m l) v 1ra >~ O,
(4.3)
valid for all independent processes. From (4.1) and (4.3) one can obtain restrictions on the viscosity coefficients. The important ones relevant to tube flow are listed below : p /> 0,
2 ~ 2 - - ~ 1 >I ~: ~> 0,
?i-[- ])2-[- "'" -[- ])15 >/ 0, 714 I> O,
~]14 >I 715,
4 # ( ~ 2 - - ½ ~ 1 ) 2 - - ( ~ 1 / 2 ) 2 ~> 0 ~/2 + ? n + ?z4 >i 0
~)14 I> 710-}-713,
(4.4)
~14 I> 712,"" •
Since for different initial suspension concentration Co'S we have in effect different materials, the constitutive coefficients should depend on co, as well as on the particle shape, deformability, and the fluid viscosity. The determination of the explicit functional forms of all coefficients is a difficult problem. Some crude estimates are possible and are found to be adequate for the present purposes. In dilute suspensions, it is shown (Batchelor, 1970) that the mean stress tensor given b y 1
tij = -p5~i + 2pd~j + ~ ~n [4~#C~tk~dkz + ½(C~i~ + sllk)Lk]
(4.5)
takes the form 1 tlj = -- 195ij + 2#(dij - v (4i)) + -~ ~n [4~#Cig~(dkl - v (kz)) + ½(C~jk+ ~ijk)Lk] + 2,uv(tj)
(4.6)
upon a linear substitution for deformable, volume preserving particles (Goddard and Miller, 1967). I-Iere the asymmetric part of the stress tensor is related only to the imposed couple L~. This is because the suspension is dilute. For concentrated suspensions, due to the effect of crowding, the asymmetric stress should exist irrespective to the imposed couple, and this is true in the micro fluid formulation (4.1). Although (4.6) is not directly applicable to concentrated suspensions, some information can be obtained b y comparing it with the stress tensors of the present continuum formulation (4.1). Since the magnitude of Cijk~ is that of the particle volume, and the summation is taken over all particles D
146
C . K . KA_NG AIqD A. C. ]ERII~GEI~
in the suspension, V-1ZC~j~ will be of order co. By comparing the corresponding coefficients ofdk~, v(kz), and v[k~]- o)kl in these expressions, we conclude t h a t : (a) magnRudes of p, K, ~1, in general, increase with increasing Co. (b) ~1, ~2 and p are of different signs, i.e. ~1/# < 0, ~2/~ < 0, for substructures of equal volume at least. (c) ~2, ~ are of the same order, and # is either of the same or higher order and depends on co. As for order of magnitudes of the coefficients ~, in the moment-stress constitutive equations, since the classical problem of suspension is t h a t of low Reynolds number flows, the significant length dimension must be t h a t of the particle size. Which means t h a t (7/#) ½, a length scale, should be 10 -4 cm to 10-a cm for concentrated blood. Apparently, ~/# decreases slightly with increasing co as the local volume decreases. The deformability and rigidity information of particles are contained in the coefficients ~1, ~ and ~ respectively. Since the effective rotation is measured by ~[k~]- ~k~, and deformation by v(k~)- d ~ , at fixed #, higher ~ represents t h a t a larger portion of stress is produced by rotation, hence a more rigid structure. Higher [~l, ]~l represents t h a t a larger portion of stress is produced by deformation, hence more flexible structure. The precise rates at which ]~1, 1~2], (~) will increase (decrease) with deformability (rigidity) are, of course, unknown.
(2) The boundary conditions. The general boundary conditions for the micromorphic theory m a y be stated in the form of surface loads, on a part St of the boundary S, as tikn~ -- T,, ~klmnm = A~l, and velocities, as v~ = Vk, v~ = ivan, on the remainder Sv of S. Of course, other mixtures are also possible. A wellposed problem requires the proof of the uniqueness theorem which is not available even in classical fluid dynamics, For simple micro-fluids, the boundary conditions on a rigid boundary may be taken as follows : (i) v~ = 0, i.e. no slip velocity exists at bounding surfaces. (ii) For the skew-symmetric part of the gyration, one m a y either assume v[~z] = 0, i.e. perfect adherence; or moment free condition, 2[~Z]mnm -~ 0 (Eringen, 1966); or a given fixed gyration, v[kl] -- v[0kl] (Ariman, 1970). Other intermediate types of boundary conditions are also possible, e.g. v[~z] = s~o~z, where 0 ~< s ~< 1 (Acre et al., 1965). The symmetric part of the gyration v(~z)is assumed to have a given value on walls. This represents a prescribed rate for the microstretch. The determination of the proper boundary conditions for the suspension flows is a complicated topic. By considering the motion of a particle adjacent to
MICROSTRUCTURE
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147
a wall in shear flow, we could obtain some relevant information on the nature of boundary conditions. The essential classical results are : (a) The particle can never be in real contact with the wall (Goldman et al., 1967). This means that the concentration parameter ~ at the wall is either zero or small. (fl) The particle rolls with slip along the wall. For a rigid sphere, the expressions obtained b y perturbation methods are ° --
--
- - ~ I - - - - ~
16
16 (4.7)
U/hF ~_2 1 - 1 6 \ h / _ ] _ _ 2
1---e16
where ~ and U are the angular and linear velocities of the particle, 2F is the shear rate of the flow, R1 is the radius of the particle, and h is the distance between the particle center and the wall. (?) Migration away from the wall exists for non-slender non-spherical particles (Mason et al., 1965). In pipe flows with no imposed moment stresses, since the only cause of particle rotation and deformation is the macro-velocity field, a tensorial formulation in terms of the velocity gradient for the boundary conditions m a y be assumed, i.e.
~ij]w = ~.ldrrsi] -}- 22dii -}- ~3(9~j]w
(4.8)
with 0 ~< [211, 1~21, 12~1 < 1, and the r s depend on the initial concentration co, and interactions of particles and fluid in a complicated way. Therefore, together with the non-slip condition v~[w = 0 for velocity, the boundary conditions employed in the present study are of type (ii), or, ,~(~j) = 22d~j
(4.9)
at the wall. The magnitude of 2% can be estimated b y comparing with the expressions (4.7), and we have
lit21--
5 1---co 16
[23[
(15 ) ----c0 16
(1--co)
(4.10)
148
C.K. KANG AND A. C. EI~INGEN
physically, the microspin v[~jl is equal to the particle rotation multiplied b y a crowding factor (1-co), and the micro-stretch v(~j) is related to the slip. I t is intuitively clear that the crowding has no pronounced effects on slips. Finally, v11, v22 should be related to the migration velocity, which is usually small in steady flows of concentrated suspensions, hence is not very important.
5. The solution of the tube flow of blood. Blood is assumed to flow steadily in a tube of radius R. We employ the cylindrical coordinate system x 1 = r, x 2 = 0, x 3 = z, with z along the tube axis. Components of tensors will be indicated b y numerical indices and physical components b y alphabetical subscripts. All field quantities v ~, v~, are zero except v 3, v½, v31, P,8 and v 3, vzs, v~ are assumed to be functions of r only, while 1o is a function of z. The equations of motion (with conservations of mass and micro-inertia satisfied identically), are t~;~ = 0 ,,l'~Im:/c+
t~m-- s~m = 0.
(5.1)
Employing ~he constitutive equations given in 4, the relevant equations become 1 T
t l ' 3 + - t ½ = p , 8 - - P. £1'13 +
(Al13- 2 ~3) + tal-s81
0
~.1'38 q-
(~131--)~ 32) + t13--813
0
(~.2)
where ()'-_- d( ) , d½ = -1v ,3 = . I w', o~1~ = --lw, dr 2 2 2 v l = vl a
:
Vla
=
V 13
_
Vrz, v31
_~_
v l
--__ V 3 1
= VSl __ Vzr"
After some manipulations, we obtain
(l~+ 2) rw" + ~lrv(rz) + ~rVEr~j= r2P/2, Kl[~;r+r-l~#_(~l~+r-~
" -1 Vrz' (:~+r-2),r~] )vzr]+K2[vrz+r
K3[vzr+r-lVzr-(~+r-2)Vzr]+K4[v;z+r-Zv~z-(~+r-2)vrz] = t¢Pr/(2#+ ~)
0
(5.3)
M I C R O S T R U C T U R E ON T H E RI-IEOLOGICAL P R O P E R T I E S O F B L O O D
149
where KI = 71-}-713+]~15+74-}-712-]-714 >i 0,
K2 =
72+711+714-{-75+710+715 i> 0,
K 3 = 71+])13+715-(74+712+714) ~ 0,
K4---~
=
72-{'-~11-]"714--(75+710"~-'~"15) -
~2 =
/> 0,
( ~ 1 - 2~2)/K1,
__ ( ~ 1 - -
(5.4)
2~2)/K2,
a~ = - ~(2# + ~1)/K3(2# + ~), a42 = ~(2#- ~i)/K4(2/~+ to). This set of equations have solutionsof the form
Yzr = alll(o:r) + a2Ii(flr) + a3r ,
(5.5)
vrz = blll(~r) + b2Ii(/3r) + b3r s.atisfying the b o u n d a r y conditions w'(0) = 0, vrz(0) = Vzr(0) = 0, where Ii(kr) is the modified Bessel function of order 1. Equations (5.3) and the b o u n d a r y conditions
V(rz)(R) = - )4w'(R) ~r~l(R) = ~3w'(R)
(5.6)
are used to determine the integration constants a l , a2, a3, bi, b2, b3, a, ft. I t can be shown t h a t a2 and f12 are the two positive roots of
x2(g2K3 - K 1 K 4 ) - x[(a~ + ~ ) g 2 g 3 - (a~ + a~)K1K4] + ~ a 2 K 2 K 3 - a 2l %2K 1 K 4 = O.
(5.7)
The existence of these roots can be proved b y considering the restrictions on viscosity coefficients or b y physical arguments. Together with the b o u n d a r y condition w(R) = O, we have the solutions in non-dimensional form
w / w . = 1 _ p 2 + b_.L1( ~ 1 - ~)fl + ~I--F tC [ I o ( L ) - I o ( L p ) ] L 2 p + lc , b~ ( ~ 1 - ~.)f2+ ~i+ ~ [ I o ( M ) - I o ( M p ) ] . -t-~r 2#+ V(rz)R/w. = ½[b~(1 + f l ) I i ( L p ) + b~(1 +fz)II(Mp)] V[rz]R/w, = p + ½[b~(1 - I x ) I I ( L p ) + b~(1 - f 2 ) I i ( M p ) ]
(5.8)
150
C. E:. K A N G A N D A. C. ] ~ R I N G E N
w . =-- --P~2/4p, p =-- r/R, L - al:I, M ~ fiR, fl -- - K J - ~ ) / K 3 ( ~ - ~ ) ,
f~ =- -K~(Z ~ - ~2)/K1(~
~-~),
b~ = - [( - 2).2/g1) -t- (2),3/g2) ~- (1/g~.)][(h~/ff2) - (hl/gl)] -1 ,
;) 1 g2 =-- ½Ii(L){(1-f1)[l+A3~(l~--}-
2)-ll-t-(1--I-fl)23~l(#+2)-l},
The above results indicate that the micro-effects are pronounced only when L, M 4 50, which implies that the geometric length is less than or about equal to 50 times of the natural characteristic length of the suspension. I t must be remarked that the dependence of p on concentration co is not derivable in the continuum approach.
6. Results and discussions (1) Velocity and particle rotation profiles.
Figure 3 shows the theoretical velocity and rotation (spin) profiles calculated for 400/o hematocrit blood flowing in 40 and 70 ttm tubes. When compared with the experimental findings of Bugliarello and Sevflla (1970), we have the following observations: (a) In the present theory, the velocity profiles are slightly flatter than the classical parabolic shapes and this agrees well with the experiments. The profiles calculated agree with experimental values better than those of model I (Newtonian-parabolic) and close to those of model I I (two-fluid model) of Bugliorello
MICROSTRUCTURE
OX
THE
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BLOOD
151
and Sevilla. Also, the profiles approach to parabolic shape as the tube size increases, and when the tube size decreases, they approach to the partial-plugflow type. Much more flattened profiles were obtained b y Kirwan (1970), and Ariman (1970), b y using the micropolar fluid theory, however, they have introduced large boundary values for spin or stretchings. (b) For co = 0.4, according to results of section 4, we choose ~13 = ½ (1 -~¢c0) (1 - Co), 22 = - ½(1 - ~¢c0). With this the spin tensor v[kz] in the present theory gives proper description of particle rotations. Figure 3 also shows that the calculated gyration agrees with the measurements of Bugliorello et al. The angular velocity of particles is usually lower than the fluid spin. In fact, calcula-~'[m] R -w ~R W~
l.O
0.5
I
1.0
/
'/ 0.8
/
,/q
'
1.5 / / ' o Exp-[BUGLIARELLOa z~// SEVILLA,Fig,5J, 40fire /zP~-70/zm l w'R
CLASSICAL--"~
0,6 r/R
"\
0.4-70ffmJ- 'w,
....
0,2 I
0.2 Figure 3.
°.4
o "\
o.s W W~
o ° :o'X,~70/.Lm
\w !.o
oO\\ - \\ ,.4.
,.6
The velocity, vortieity a n d part~iele rogat~ion profiles
tions and measurements both show that the angular velocity is about half of those of the fluid vorticity for co = 0.4. This seems to indicate that ~2, ~a we have chosen are quite reasonable. Other constants employed for Figure 3 are listed in Table 1. (c) It is interesting to observe that the calculated plasmatic layer thickness 5 (defined in (3) as the place were ~ = ct/2, ct is the true tube hematocrit) agrees quite well with the experimental findings (Figure 5), that is, 5 / R ,,, 0.1 for 40 #m tube, and 5 / R ~ 0.05 for 70 #m tube. The concentration profiles show the shape of the axial migration phenomena. (2) T h e apparent viscosity. Experimentally observable sigma-phenomena are also manifested in the present study. Since the total volume flow rate Q
152
O. "K. K A N G A N D A. C. E R I I ~ G E N
which is independent of the cell redistribution, is given by
Q-Qp+Qf=
f
[ew+(1-e)w]dV= fv wdV
(6.1)
or l
Q = 2uR 2
f
w(p)p dp = 2uR2w,Q
o
b~ ( ~ - ~ ) f l + ; z + ~ {½10(L) - L-1II(L) } ~ ~+ ~2Z+~
(6.2)
+ _b~(;~-~)A+;~+~ {lZ0(~)- ~-lZl(M) }. M
2#+
The apparent viscosity is defined in terms of the total volume flow rate by
~zRaP ~l a
=
(6.3)
SQ
Hence for the non-dimensional apparent viscosity ~ a we have ~/a
1
(6.~)
where p is a function of Co. The dependence of ~ a/# on the tube size is through the variables L -~ aR, M =- fiR. For a concentrated suspension of rigid particles (22 = 0), in a typical case when 23 = 0, we obtain
b'~ = O, b~ = -1/I1(M),
fl = f2 = - 1 ,
L= M
(6.5)
and Q = k
2x - MII(M) 2# + 1
[~I0(M)--~-lZl(M)].
(6.6)
Thus tla/p can only be greater t h a n unity, as noticed by Cowin (1972). Therefore, the reduction of apparent viscosity in smaller tubes is a result of deformability of the sub-structure, which is manifested in the coefficients ~1, ~2. Computations show t h a t only when ~1/# is negative (as was observed in section 4), sigma-phenomena can be observed. Also, the magnitudes of ~'s increase with initial concentration co, which means t h a t the more concentrated a suspension is, the more likely its substructures will deform. Constants selected in all cal-
!VIICROSTRUCTURE ON TI-IE RHEOLOGICAL PROPERTIES OF BLOOD
153
culati0ns are listed in Table I below. I t should be remarked t h a t these are b u t one possible set of coefficients. Table I centipoise
~o 0.8 0.7 0.6 0.5 0.4 0.3 0.2
~ 5.4 4.3 3.5 2.9 2.8 1.8 1.5
10 - 6 c m 2 10 - 6 c m 2 10 - 6 c m 2 10 - 6 c m "~
- ~,lz 1.87 1.67 1.88 1.08 0.75 0.55 0.45
~
38, 1 9 7 6
T H E E F F E C T O F MICROSTI:~UCTUEE ON THE RHEOLOGICAL P R O P E R T I E S OF B L O O D
[] C. K. K A v a National Tsing-Hua University, I-Isin-Chu, Taiwan, China [] A. C. EaI~G~SN Princeton University, Princeton, N.J., U.S.A.
The micromorphic theory of Eringen is applied to study the tube flow of blood. The blood is considered to be a deformable suspension, with constitutive relations of the form of those of simple microfluids. By means of energy consideration, a relation is established betweerL the local concentration parameter and the measure of rotationality involving both macroand raicromotions. The tube flow problem is then solved with some analyses on viscosity coefficients and boundary conditions. The results obtained indicate an integrated explanation of various important physical phenomena associated with blood flow, such as the tube size dependence of the apparent viscosity and the non-uniform concentration distribution over a tube cross section.
1. Introduction. Rheologically n o r m a l h u m a n blood is a macroscopically continuous suspension of p a r t i c u l a t e m a t t e r s ; red cells, white cells, a n d platelets, in a microscopically continuous fluid, t h e b l o o d p l a s m a (Houssay, 1955). Of t h e particulate m a t t e r s , t h e red cells are d o m i n a n t , occupying a v o l u m e fraction (hematoerit, co) of 3 5 - 5 0 % of the blood. T h e h u m a n red cell is a small a n d e x t r e m e l y flexible disc h a v i n g a n a v e r a g e d i a m e t e r a b o u t 8 p m (micron) a n d thickness a b o u t 2 #m. I t has a s.g. 1.085, which a p p r o x i m a t e l y equals t h a t of t h e blood p l a s m a w i t h a s.g. of 1.035. Therefore, i n rheological studies, it is customary to consider the b l o o d as a c o n c e n t r a t e d suspension of n e u t r a l l y b u o y a n t deformable particles in a viscous fluid. 135
136
C . K . K A N G A N D A. C. E R I N G E N
When blood is flowing in smaller vessels, the apparent viscosity decreases with decreasing tube radius (the Fahraeus-Lindqvist effect (Haynes, 1961). This phenomena has been observed also in slurry flows and was designated as the sigma-phenomena (Scott-Blair, 1958). Several attempts exist to explain this phenomena: Dix and Scott-Blair (1940) proposed that the flowing laminae has a finite thickness, thus in calculating the flow rate, a summation of flow layers should be employed instead of integration. B y this process, they managed to obtain a tube-size dependence. Others (Vand, 1948) assumed the existence of a particle-free layer with a smaller viscosity adjacent to the tube wall, which causes an increase of flow rate in smaller tubes. Nevertheless, no theory appears to exist encompassing explanations of these phenomena along with other flow properties. Another peculiar aspect of the blood flow is the radial redistribution of red cells in tube flows (Segr6 and Silberberg, 1962). For a dilute suspension, starting from a uniform initial distribution of particles over the cross section, a necklacelike particle aggregation has been observed at about 0.6 radius from the tube center (the Segr6-Silberberg phenomenon). In concentrated suspensions, axial accumulation occurs resulting in reduction of cells or even establishment of a particle-free zone along the tube wall. Theoretical studies have focused attention on the motion of a single particle in viscous shear flow (the migration phenomenon (Goldsmith and Mason, 1967), but no quantitative results appear to be available for the explanation of these phenomena inherent to concentrated suspensions such as the blood. Experiments (Kuroda et al., 1958) also have shown that the deformability of red cells has a pronounced effect on the apparent viscosity. At normal conccntration, the suspension with hardened cells will exhibit a viscosity about l0 times higher than that of the normal blood. Again, the lack of a quantitative analysis does not permit the incorporation of the deformability effects, on a rational basis, in the calculations. The present study employs the micromorphic theory of Eringen (1964), (1967), (1972) to analyse the flow of blood. As a supplement to the microfiuid model, in section 3, we introduce, for neutrally buoyant suspensions, a relation between concentration parameter and flow rotationality. In section 4, we estimate the values of viscosity coefficients and discuss the forms of boundary conditions. A tube flow problem is then solved in section 5. The velocity and rotation profiles obtained are compared with certain experiments in section 6. The present model also provides quantitative calculations for the axial migration phenomena, the Fahraeus-Lindqvist effect, and the effects of cell deformability.
2. Rdsurad of the microfluid theory.
Eringen (1964) introduced the notion of
MICROSTRUCTURE ON THE RHEOLOGICAL PROPERTIES OF BLOOD
137
simple microfluids to describe t h e b e h a v i o r of some fluent m e d i a which can s u p p o r t stress m o m e n t s , b o d y m o m e n t s , a n d possess local spin inertia t h a t can influence the flow properties. T h e g o v e r n i n g balance laws of the m i c r o m o r p h i c mechanics (Eringen, 1964; 1967) are listed below :* conservation of m a s s
ap --
at
+(pv~);~
=
0
(2.1)
conservation of micro-inertia
aikm at + ikm;r vr-irmvkr --i~rvmr = 0
(2.2)
balance of linear m o m e n t u m
tkz;~ + p(ff - ~ ) = 0
(2.3)
balance of first m o m e n t stresses ,~Im;~ + tmZ _ smz + p ( l l m _ ~zm) = 0
(2.4)
conservation of energy
p~ = tk~vz;~ + (s ~l - tk~)vZk + 2k~mVlm;k + qk;k+ ph the e n t r o p y i n e q u a l i t y
ph
>f o.
(2.6)
The e n t r o p y i n e q u a l i t y (2.6) is p o s t u l a t e d to be v a l i d for all i n d e p e n d e n t processes. I n these equations : p = ikm= v~ = t kz = ff = s k* = 2k~m = pm= 5Zm =
m a s s density m i c r o i n e r t i a tensor velocity vector stress tensor b o d y force per u n i t mass micro stress a v e r a g e tensor t h e first stress m o m e n t tensor t h e first b o d y m o m e n t per u n i t m a s s spin inertia
* H e r e , v kz r e p l a c e s vz k d e f i n e d i n ( E r i n g e n , 1967).
138
C. K . K A N G A N D A. C. E R I N G E N
internal energy density per unit mass ykl .~ gyration tensor
of the body
fl ~ = heat vector directed outward
h = heat source per unit mass t l = entropy per unit mass T = absolute temperature. Throughout this paper, a semi-colon (;) is employed to indicate covariant partial differentiation, and a superposed dot (') to indicate the material time derivative, e.g. vk
;1 = Vk;~ .6
I It} rl
vr'
~k~- av~z ~t
.6 ~kl'mVm'
where {r~t}is the Christoffel symbol of the second kind formed by the metric tensor g~z of the spatial coordinates x ~. The spin inertia is given b y _
irm(
r+
(2.7)
Eringen also gave a set of constitutive equations for simple microfluids. For a non-heat conducting medium, these are the relations between t klm, t ~, s ~Z, and the objective deformation rate measures
d~l = ½(v~;z + v z:k ), b~ -~ v~z+ vz; ~ = v~l + dkz - (o~z,
(2.8)
t~klra ~ "¢kl;m.
Of these, d is the deformation rate tensor and b and a are called micro-deformation rate tensor and gyration gradient, respectively. For the present work, we reproduce only the linear constitutive relations of simple microfluids: t~l = [ - n .6 )~vdrr + to(brr - dr)]5~l + 2pd~1.6 2fl0(b~ - d~z) Jr 2/~l(bt ~ - d~), s~z = [ - ~ + !lvdrr .6 17o(brr - drr)]57cz.6 2/~dkz.6 ~2(b~z.6 b~~ - 2d~) ,
(2.9)
)/~lm = (~lamrr .6 ~2armr .6 ~Sarrm)Skl + (~dalrr -~ ysarlr -t- 76arrl)5~ m
.6]~7a
lcr rlc r It r .6]~8a r - 6 ~ 9 a r )glm
.6?lOa~m nu?lla
k ml
+ ~12a~ .6 ~ s a ~ .6 ~ a a ~ ~ + ~ a ~ where 5~ is the unit tensor, g~ is the metric tensor, iv, t0, /tv, p0, pl, ~v, ~0, ~ , 7~ to 7~5 are the viscosity coefficients which are functions of temperature in general and where u is the thermodynamic pressure defined b y u ~ - Oe/~p-~[,.
MICROSTRUCTURE
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139
Although the balance laws and constitutive equations given above form a determinate system (with suitably chosen geometry and boundary conditions) for simple microfluids, to apply the theory to suspensions, additional considerations are needed. In section 3 we make use of the physical concepts of the local volume elements to introduce a concentration parameter d into the theory.
3. Concentration parameter. In classical continuum mechanics, material points of a body are considered to be geometrical points embodied with continuous fields~ The matter is assumed to be continuous no matter how small the element of volume m a y be (Batchelor, 1967). For small enough volume elements, this hypothesis breaks down and the individuality of substructure (grains, molecules, etc.) affect the physical outcome drastically. Consideration must therefore be made to take into account the effect of local structure of bodies in the treatment of physical phenomena having characteristic dimensions that compare with those of the inner structure of bodies. Thus for structured or micro-continua, the material points of the body m a y be considered to be small deformable bodies approximated b y orientable points, i.e., points with a triad of deformable directors attached to it. The orientation of the material points can be characterized b y measuring the rotation, shearing, and stretch of the directors. This is the physical model underlying the theory of micromorphic contina (Eringen and S,uhubi, 1964; Eringen, 1964, 1967, 1972). If the directors are considered to be rigid, then we have the micropolar media. The material points of a micropolar continuum are, roughly speaking, small rigid particles. Mathematically, these directors introduce additional degrees of freedom besides the displacement or velocity field of the classical continua. In the micromorphic fluids, the extra degrees of freedom are the gyration tensor Ykl , whose skew-symmetric part viii represents local intrinsic angular velocity of the substructure, and the symmetric part ~(~z) are the local stretchings and shearings (Eringen, 1964). The suspensions deviate structurally from an ideal micromorphic medium b y not having moment of inertia associated with those points in the fluid region. Therefore a triad of directors are attached only to suspended particles b u t not to the points of fluids. Nevertheless a concentrated suspension system at rest (volume concentration of the suspended particles Co > 0.1) could be regarded as a microfluid with the proper provision that its material point is a combination of the suspended particle with its surrounding fluid. Observations indicate that when a particle is suspended in a viscous fluid, the rotation and deformation of the particle is felt by the surrounding fluid. Inside some 'region of influence', the fluid will display, at least instantaneously, a mean motion with its center of rotation
140
C . K . KANG AND A, C. EI~INGEN
coinciding with that of the particle. We, therefore, construct the one-particlecell model as the 'structured point' in the continuum approach. Our continuum is filled with particles surrounded by effective fluid masses. This model should be valid for suspensions in which the diffusion of particles is not important, that is, local velocities of fluid and particle are almost the same. This is about the case for neutrally buoyant particles (Happel and Brenner, 1965). For simplicity, one may consider the region of influence spherical so that the structured point consists of a fluid sphere enclosing a core particle. Doublets, multiplets, could be included by regarding these aggregates as larger particles, although under higher shear rates, the possibility of forming an aggregation is small (Wayland,
1967).
F i g u r e 1.
C o o r d i n a t e s y s t e m s in AV
Let R~(x, t) denote the radius of the sphere of influence consisting of some fluid mass surrounding a particle (Figure 1). If the particle volume and the total volume of influence are, respectively, AV~ and AV, the local volume ratio may be defined b y d(x, t) = AV~o[AV. To have a rough idea about the magnitude of d, consider a sphere-in-sphere configuration with a particle of radius/~1. When R1/R2 varies from 1.0 to 0.2, d --- (RI[R~) 3 varies from 1.0 to 0.008, that is, from 'all particle' to dilute suspensions. According to the micromorphic theory, the velocity and acceleration in a volume element are
where v ° is the velocity of the mass-center of the particle, v (kz) is the stretchingshearing and VCkZ]is the spin of the particle with respect to its mass center. The particle may be regarded to be spinning at v[~z]-~o~z and to be stretching-
IVIICROSTRUCTURE
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141
shearing at v(~)-dk~ with respect to the suspending fluid mass. Hence in a suspension, the motion of the particle relative to the fluid mass can be described
by the relative quantities ~'~Icl ~ Y [ k l ] - (-Okl
(3.1)
Dk~ =-- v(kz)--dkz.
Before we deduce an explicit formula for the concentration parameter ~ given by d(x, t) ~_ A V V / A V ( x , t) in terms of a variable local volume AV, a proper measure of local rotationality must be introduced. Truesdell and Toupin (1960) have shown that a proper description of local rotation in classical fluid mechanics is given by the kinematic vorticity number (0~]clCOkl~½ ( t r t°2~ ½ WK-- \ ~ ] = \trdZ/
(3.2)
which is a quantitative measure of the amount of rotation in a motion, indicating the relative importance of local vorticity and deformation. Motivated with this, in the micromorphie fluid mechanics we introduce the 'rotationality' by
= \t~/
L-(vt~l)-dkz)(v - - ~
--L--d~_l
"
(3.3)
Here ~ is an objective quantity. It is a combination of the invariants of motion, and is a measure of the relative intensity of rotation and stretching-shearing parts of the motion. From the kinematical considerations of W K and the intuitive idea that, in a suspension, faster rotating particles should influence more fluid mass than the slower ones, it follows that the concentration 6 will decrease with increasing rotations provided the particles are volume-preserving. It is therefore natural to conjecture that the local volume ratio ~ may be related to the local rotationality ~, although the explicit functional form of ~ in terms of ~ is yet to be found. In classical studies dilute suspensions were modelled after the problem of a single particle motion in a viscous fluid. Since the size of particles is very small, the problem is usually considered to fall into the region of low Reynolds numbers, or creeping flows. By determining the viscous energy dissipation due to the presence of a particle, one is then able to calculate the effective viscosity for a dilute suspension. However, for non-dilute suspensions (Co /> 0.1), the particle interactions are important. We now investigate the effects of particle crowding by means of the classical method of 'cage model' analysis. Let a viscous fluid be filled into the space between a rigid sphere of radius R1 and a concentric hollow spherical shell of radius Re (Figure 2). Suppose that the inner sphere rotating with an angular velocity ~oj and the outer shell is fixed to
142
C . K . KA_NG ANY) A. C, E R I I ~ G E N
Figure 2. A structure model for the effect of crowding
simulate the effects of other particles (Happel and Brenner, 1965). I t can be shown that (of. Landau, 1959) the velocity field between the spheres is given by
R1R 2 v~ - .R2a------a_ R1
-
1--d\p 3
1
1 _~
e~j~cojxk
-1)
(3. 1
where ~ = (R1/R~) 3, o9~ _~ e~k~oj. Hence the rotational part of energy dissipation in the fluid due to the presence of a wall can be obtained from e°) ~ - 2P f V (v~d+vj,~)2dV
(1-~).
2
3
= ,~R~ tr ea2fl(~). Simha et al. (cf. Brenner and I-Iappel, 1965) have shown that the stretchingshearing deformation will introduce essentially the same type of energy dissipation due to the presence of the wall, i.e.,
ea = fl,pRal tr a211 + al~ + a2~ 5/~ +...] = upR~ tr d~f2(~).
(3.6)
Thus the total energy dissipation due to particle crowding may be written in the form
e(d) = e~+ea = u#R~[fl(~) tr ~2+f2(~ ) tr d ~]
(3.7)
where f l , f2 are usually monotically increasing functions of ~. The energy dissipation of a rigid spherical particle in an unbounded shear flow is obtained
IVIICROSTRUCTURE
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RHEOLOGICAL
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143
by Jeffery as 20
e0 =
_~ tr d2"
(3.8)
- ~ - 7~#/~ 1
Therefore, the ratio of the dissipated energy, when the particle crowding is taken into account, to that of a free suspended particle is a function of the local volume par ameter e(e)/eo =
3 [- . ~ t r0)2 1 [fl(c) +f2(e)_ = F2(~)+-]~l(e)W2.
(3.9)
Observing Ie-col < 1, we assume that e(d)/eo may be expanded into a Taylor series about e = Co, i.e., e(a)leo -- F (co) + ( 8 -
+
(e- co) F (co) +... 2
+ W2¢[ El(c°) + (e-c°)Fl(c°)+
(e-c°)------~22Fl(c°)+' " 2 "]
Upon inverting this relation, one notices that the local volume ratio could be determined by evaluating the ratio of local energies. A natural criterion may now be introduced to deduce an explicit relation between e and W~. To this end we posit that the local concentration takes the values for which e(e)/eo will be a minimum subject to the conservation of particles in the suspension, i.e. global restriction of e : ¢(e) -
fv (e-Co)pdV
= 0
or
fvdpdV = copoVo.
(3.10)
By using the method of Lagrange's multiplier to find the minima of e(e)/eo, we have
d A d~b 0 d-~ (e/eo)+ de =
(3.11)
where ~tis the constant Lagrange's multiplier. Using (3.9) ~nd (3.10) this gives
)~
=
-
r ~ t ~ 2 (poVo)--1 [F2(c)+FI(C)WK].
(3.12)
Employing the power series expansions of F~ and F~ for small e-Co and neglecting higher order terms than 2 in (e - Co),this gives ~--C0
=
Iv F2(c0) + F I/i( c o ) W2K + ' "
144
C. I~. IZANG A N D A. C. E R I N G E N
This relation can be written as a polynomial in W~, i.e.,
kpo Vo + F'~(co)
61 ~- CO--
~
-
(3.13)
F~(c0) 1
P/
C
- [~p0VoFl(0) [F~(c0)] 2
tr
!
+ Fl(c0)F2(c0)-
Fl(co)F2(co)]. ¢
it
Since F1 and F~ are monotonically increasing functions of d, from (3.12) it follows that 2 < 0. Other than this restriction, clearly ~1 and 62 can be positive or negative. However, as we have stated in the beginning of this section, higher vortieity means lower d; if we retain terms only to the first order of WE, we will have, for WE ~ 1, with 61 > 0, 62 < 0. This relation (3.13) is a much simplified formula for the determination of local concentration ~ in classical fluid mechanics. In the micromorphie formulation, this suggests that we employ the relation = Cl-C~ 2
(3.14)
where Cl > 0, c2 > 0, and ~ is the micro-rotationality defined b y (3.3). Therefore, neutrally buoyant suspensions may be treated as microfluids, with the non-uniformity of the local volume characterized b y (3.14). In calculations, the accuracy of (3.14) has been satisfactory. The positive constants el, c2 can be determined b y using (3.10) together with dynamic considerations, i.e. drag and/or lift forces on particles considered in classical migration studies (cf. Giesekus, 1962; Bretherton, 1962). One source of information about el, c~. is probably through the study of particle motions, adjacent to a boundary (Goldman et al., 1967 ; Mason et al., 1965), which should provide boundary values for & At present, such studies are not conclusive. 4. Viscosity coej~cients and boundary conditions (1) The viscosity coe~cients. The constitutive relations for a simple microfluid suspension have the forms
t~ = - ~ 0 ~ + 2~d~ + ~,%, + ~ (~~'~,~-o~'~'~
~
= - i o , ~ + 2~d~ + 2~('~)
(4.1)
r
~" .
+(]~Tv/Cr;r+ $8
yrk
,
k
k
;r)glm+ 710v l;m+711VL;l+ 719,Vl ;m
MICROSTRT_TCTUI%E
O N TI-IE R H E O L O G I C A L
PROPERTIES
OF BLOOD
145
where the incompressibility conditions dk~ = v~k = 0 have been employed, and
2 # - - 2Pv+~,
~1--= 2(p0+#1),
~ = 2(#0-Pl).
(4.2)
The entropy inequality for non-heat-conducting fluids has the form: Sklvl;k-[-)'klmYlm;k +(8 m 1 - t m l) v 1ra >~ O,
(4.3)
valid for all independent processes. From (4.1) and (4.3) one can obtain restrictions on the viscosity coefficients. The important ones relevant to tube flow are listed below : p /> 0,
2 ~ 2 - - ~ 1 >I ~: ~> 0,
?i-[- ])2-[- "'" -[- ])15 >/ 0, 714 I> O,
~]14 >I 715,
4 # ( ~ 2 - - ½ ~ 1 ) 2 - - ( ~ 1 / 2 ) 2 ~> 0 ~/2 + ? n + ?z4 >i 0
~)14 I> 710-}-713,
(4.4)
~14 I> 712,"" •
Since for different initial suspension concentration Co'S we have in effect different materials, the constitutive coefficients should depend on co, as well as on the particle shape, deformability, and the fluid viscosity. The determination of the explicit functional forms of all coefficients is a difficult problem. Some crude estimates are possible and are found to be adequate for the present purposes. In dilute suspensions, it is shown (Batchelor, 1970) that the mean stress tensor given b y 1
tij = -p5~i + 2pd~j + ~ ~n [4~#C~tk~dkz + ½(C~i~ + sllk)Lk]
(4.5)
takes the form 1 tlj = -- 195ij + 2#(dij - v (4i)) + -~ ~n [4~#Cig~(dkl - v (kz)) + ½(C~jk+ ~ijk)Lk] + 2,uv(tj)
(4.6)
upon a linear substitution for deformable, volume preserving particles (Goddard and Miller, 1967). I-Iere the asymmetric part of the stress tensor is related only to the imposed couple L~. This is because the suspension is dilute. For concentrated suspensions, due to the effect of crowding, the asymmetric stress should exist irrespective to the imposed couple, and this is true in the micro fluid formulation (4.1). Although (4.6) is not directly applicable to concentrated suspensions, some information can be obtained b y comparing it with the stress tensors of the present continuum formulation (4.1). Since the magnitude of Cijk~ is that of the particle volume, and the summation is taken over all particles D
146
C . K . KA_NG AIqD A. C. ]ERII~GEI~
in the suspension, V-1ZC~j~ will be of order co. By comparing the corresponding coefficients ofdk~, v(kz), and v[k~]- o)kl in these expressions, we conclude t h a t : (a) magnRudes of p, K, ~1, in general, increase with increasing Co. (b) ~1, ~2 and p are of different signs, i.e. ~1/# < 0, ~2/~ < 0, for substructures of equal volume at least. (c) ~2, ~ are of the same order, and # is either of the same or higher order and depends on co. As for order of magnitudes of the coefficients ~, in the moment-stress constitutive equations, since the classical problem of suspension is t h a t of low Reynolds number flows, the significant length dimension must be t h a t of the particle size. Which means t h a t (7/#) ½, a length scale, should be 10 -4 cm to 10-a cm for concentrated blood. Apparently, ~/# decreases slightly with increasing co as the local volume decreases. The deformability and rigidity information of particles are contained in the coefficients ~1, ~ and ~ respectively. Since the effective rotation is measured by ~[k~]- ~k~, and deformation by v(k~)- d ~ , at fixed #, higher ~ represents t h a t a larger portion of stress is produced by rotation, hence a more rigid structure. Higher [~l, ]~l represents t h a t a larger portion of stress is produced by deformation, hence more flexible structure. The precise rates at which ]~1, 1~2], (~) will increase (decrease) with deformability (rigidity) are, of course, unknown.
(2) The boundary conditions. The general boundary conditions for the micromorphic theory m a y be stated in the form of surface loads, on a part St of the boundary S, as tikn~ -- T,, ~klmnm = A~l, and velocities, as v~ = Vk, v~ = ivan, on the remainder Sv of S. Of course, other mixtures are also possible. A wellposed problem requires the proof of the uniqueness theorem which is not available even in classical fluid dynamics, For simple micro-fluids, the boundary conditions on a rigid boundary may be taken as follows : (i) v~ = 0, i.e. no slip velocity exists at bounding surfaces. (ii) For the skew-symmetric part of the gyration, one m a y either assume v[~z] = 0, i.e. perfect adherence; or moment free condition, 2[~Z]mnm -~ 0 (Eringen, 1966); or a given fixed gyration, v[kl] -- v[0kl] (Ariman, 1970). Other intermediate types of boundary conditions are also possible, e.g. v[~z] = s~o~z, where 0 ~< s ~< 1 (Acre et al., 1965). The symmetric part of the gyration v(~z)is assumed to have a given value on walls. This represents a prescribed rate for the microstretch. The determination of the proper boundary conditions for the suspension flows is a complicated topic. By considering the motion of a particle adjacent to
MICROSTRUCTURE
O N TIlE R H E O L O G I C A L
PROPERTIES
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147
a wall in shear flow, we could obtain some relevant information on the nature of boundary conditions. The essential classical results are : (a) The particle can never be in real contact with the wall (Goldman et al., 1967). This means that the concentration parameter ~ at the wall is either zero or small. (fl) The particle rolls with slip along the wall. For a rigid sphere, the expressions obtained b y perturbation methods are ° --
--
- - ~ I - - - - ~
16
16 (4.7)
U/hF ~_2 1 - 1 6 \ h / _ ] _ _ 2
1---e16
where ~ and U are the angular and linear velocities of the particle, 2F is the shear rate of the flow, R1 is the radius of the particle, and h is the distance between the particle center and the wall. (?) Migration away from the wall exists for non-slender non-spherical particles (Mason et al., 1965). In pipe flows with no imposed moment stresses, since the only cause of particle rotation and deformation is the macro-velocity field, a tensorial formulation in terms of the velocity gradient for the boundary conditions m a y be assumed, i.e.
~ij]w = ~.ldrrsi] -}- 22dii -}- ~3(9~j]w
(4.8)
with 0 ~< [211, 1~21, 12~1 < 1, and the r s depend on the initial concentration co, and interactions of particles and fluid in a complicated way. Therefore, together with the non-slip condition v~[w = 0 for velocity, the boundary conditions employed in the present study are of type (ii), or, ,~(~j) = 22d~j
(4.9)
at the wall. The magnitude of 2% can be estimated b y comparing with the expressions (4.7), and we have
lit21--
5 1---co 16
[23[
(15 ) ----c0 16
(1--co)
(4.10)
148
C.K. KANG AND A. C. EI~INGEN
physically, the microspin v[~jl is equal to the particle rotation multiplied b y a crowding factor (1-co), and the micro-stretch v(~j) is related to the slip. I t is intuitively clear that the crowding has no pronounced effects on slips. Finally, v11, v22 should be related to the migration velocity, which is usually small in steady flows of concentrated suspensions, hence is not very important.
5. The solution of the tube flow of blood. Blood is assumed to flow steadily in a tube of radius R. We employ the cylindrical coordinate system x 1 = r, x 2 = 0, x 3 = z, with z along the tube axis. Components of tensors will be indicated b y numerical indices and physical components b y alphabetical subscripts. All field quantities v ~, v~, are zero except v 3, v½, v31, P,8 and v 3, vzs, v~ are assumed to be functions of r only, while 1o is a function of z. The equations of motion (with conservations of mass and micro-inertia satisfied identically), are t~;~ = 0 ,,l'~Im:/c+
t~m-- s~m = 0.
(5.1)
Employing ~he constitutive equations given in 4, the relevant equations become 1 T
t l ' 3 + - t ½ = p , 8 - - P. £1'13 +
(Al13- 2 ~3) + tal-s81
0
~.1'38 q-
(~131--)~ 32) + t13--813
0
(~.2)
where ()'-_- d( ) , d½ = -1v ,3 = . I w', o~1~ = --lw, dr 2 2 2 v l = vl a
:
Vla
=
V 13
_
Vrz, v31
_~_
v l
--__ V 3 1
= VSl __ Vzr"
After some manipulations, we obtain
(l~+ 2) rw" + ~lrv(rz) + ~rVEr~j= r2P/2, Kl[~;r+r-l~#_(~l~+r-~
" -1 Vrz' (:~+r-2),r~] )vzr]+K2[vrz+r
K3[vzr+r-lVzr-(~+r-2)Vzr]+K4[v;z+r-Zv~z-(~+r-2)vrz] = t¢Pr/(2#+ ~)
0
(5.3)
M I C R O S T R U C T U R E ON T H E RI-IEOLOGICAL P R O P E R T I E S O F B L O O D
149
where KI = 71-}-713+]~15+74-}-712-]-714 >i 0,
K2 =
72+711+714-{-75+710+715 i> 0,
K 3 = 71+])13+715-(74+712+714) ~ 0,
K4---~
=
72-{'-~11-]"714--(75+710"~-'~"15) -
~2 =
/> 0,
( ~ 1 - 2~2)/K1,
__ ( ~ 1 - -
(5.4)
2~2)/K2,
a~ = - ~(2# + ~1)/K3(2# + ~), a42 = ~(2#- ~i)/K4(2/~+ to). This set of equations have solutionsof the form
Yzr = alll(o:r) + a2Ii(flr) + a3r ,
(5.5)
vrz = blll(~r) + b2Ii(/3r) + b3r s.atisfying the b o u n d a r y conditions w'(0) = 0, vrz(0) = Vzr(0) = 0, where Ii(kr) is the modified Bessel function of order 1. Equations (5.3) and the b o u n d a r y conditions
V(rz)(R) = - )4w'(R) ~r~l(R) = ~3w'(R)
(5.6)
are used to determine the integration constants a l , a2, a3, bi, b2, b3, a, ft. I t can be shown t h a t a2 and f12 are the two positive roots of
x2(g2K3 - K 1 K 4 ) - x[(a~ + ~ ) g 2 g 3 - (a~ + a~)K1K4] + ~ a 2 K 2 K 3 - a 2l %2K 1 K 4 = O.
(5.7)
The existence of these roots can be proved b y considering the restrictions on viscosity coefficients or b y physical arguments. Together with the b o u n d a r y condition w(R) = O, we have the solutions in non-dimensional form
w / w . = 1 _ p 2 + b_.L1( ~ 1 - ~)fl + ~I--F tC [ I o ( L ) - I o ( L p ) ] L 2 p + lc , b~ ( ~ 1 - ~.)f2+ ~i+ ~ [ I o ( M ) - I o ( M p ) ] . -t-~r 2#+ V(rz)R/w. = ½[b~(1 + f l ) I i ( L p ) + b~(1 +fz)II(Mp)] V[rz]R/w, = p + ½[b~(1 - I x ) I I ( L p ) + b~(1 - f 2 ) I i ( M p ) ]
(5.8)
150
C. E:. K A N G A N D A. C. ] ~ R I N G E N
w . =-- --P~2/4p, p =-- r/R, L - al:I, M ~ fiR, fl -- - K J - ~ ) / K 3 ( ~ - ~ ) ,
f~ =- -K~(Z ~ - ~2)/K1(~
~-~),
b~ = - [( - 2).2/g1) -t- (2),3/g2) ~- (1/g~.)][(h~/ff2) - (hl/gl)] -1 ,
;) 1 g2 =-- ½Ii(L){(1-f1)[l+A3~(l~--}-
2)-ll-t-(1--I-fl)23~l(#+2)-l},
The above results indicate that the micro-effects are pronounced only when L, M 4 50, which implies that the geometric length is less than or about equal to 50 times of the natural characteristic length of the suspension. I t must be remarked that the dependence of p on concentration co is not derivable in the continuum approach.
6. Results and discussions (1) Velocity and particle rotation profiles.
Figure 3 shows the theoretical velocity and rotation (spin) profiles calculated for 400/o hematocrit blood flowing in 40 and 70 ttm tubes. When compared with the experimental findings of Bugliarello and Sevflla (1970), we have the following observations: (a) In the present theory, the velocity profiles are slightly flatter than the classical parabolic shapes and this agrees well with the experiments. The profiles calculated agree with experimental values better than those of model I (Newtonian-parabolic) and close to those of model I I (two-fluid model) of Bugliorello
MICROSTRUCTURE
OX
THE
RHEOLOGICAL
PROPERTIES
OF
BLOOD
151
and Sevilla. Also, the profiles approach to parabolic shape as the tube size increases, and when the tube size decreases, they approach to the partial-plugflow type. Much more flattened profiles were obtained b y Kirwan (1970), and Ariman (1970), b y using the micropolar fluid theory, however, they have introduced large boundary values for spin or stretchings. (b) For co = 0.4, according to results of section 4, we choose ~13 = ½ (1 -~¢c0) (1 - Co), 22 = - ½(1 - ~¢c0). With this the spin tensor v[kz] in the present theory gives proper description of particle rotations. Figure 3 also shows that the calculated gyration agrees with the measurements of Bugliorello et al. The angular velocity of particles is usually lower than the fluid spin. In fact, calcula-~'[m] R -w ~R W~
l.O
0.5
I
1.0
/
'/ 0.8
/
,/q
'
1.5 / / ' o Exp-[BUGLIARELLOa z~// SEVILLA,Fig,5J, 40fire /zP~-70/zm l w'R
CLASSICAL--"~
0,6 r/R
"\
0.4-70ffmJ- 'w,
....
0,2 I
0.2 Figure 3.
°.4
o "\
o.s W W~
o ° :o'X,~70/.Lm
\w !.o
oO\\ - \\ ,.4.
,.6
The velocity, vortieity a n d part~iele rogat~ion profiles
tions and measurements both show that the angular velocity is about half of those of the fluid vorticity for co = 0.4. This seems to indicate that ~2, ~a we have chosen are quite reasonable. Other constants employed for Figure 3 are listed in Table 1. (c) It is interesting to observe that the calculated plasmatic layer thickness 5 (defined in (3) as the place were ~ = ct/2, ct is the true tube hematocrit) agrees quite well with the experimental findings (Figure 5), that is, 5 / R ,,, 0.1 for 40 #m tube, and 5 / R ~ 0.05 for 70 #m tube. The concentration profiles show the shape of the axial migration phenomena. (2) T h e apparent viscosity. Experimentally observable sigma-phenomena are also manifested in the present study. Since the total volume flow rate Q
152
O. "K. K A N G A N D A. C. E R I I ~ G E N
which is independent of the cell redistribution, is given by
Q-Qp+Qf=
f
[ew+(1-e)w]dV= fv wdV
(6.1)
or l
Q = 2uR 2
f
w(p)p dp = 2uR2w,Q
o
b~ ( ~ - ~ ) f l + ; z + ~ {½10(L) - L-1II(L) } ~ ~+ ~2Z+~
(6.2)
+ _b~(;~-~)A+;~+~ {lZ0(~)- ~-lZl(M) }. M
2#+
The apparent viscosity is defined in terms of the total volume flow rate by
~zRaP ~l a
=
(6.3)
SQ
Hence for the non-dimensional apparent viscosity ~ a we have ~/a
1
(6.~)
where p is a function of Co. The dependence of ~ a/# on the tube size is through the variables L -~ aR, M =- fiR. For a concentrated suspension of rigid particles (22 = 0), in a typical case when 23 = 0, we obtain
b'~ = O, b~ = -1/I1(M),
fl = f2 = - 1 ,
L= M
(6.5)
and Q = k
2x - MII(M) 2# + 1
[~I0(M)--~-lZl(M)].
(6.6)
Thus tla/p can only be greater t h a n unity, as noticed by Cowin (1972). Therefore, the reduction of apparent viscosity in smaller tubes is a result of deformability of the sub-structure, which is manifested in the coefficients ~1, ~2. Computations show t h a t only when ~1/# is negative (as was observed in section 4), sigma-phenomena can be observed. Also, the magnitudes of ~'s increase with initial concentration co, which means t h a t the more concentrated a suspension is, the more likely its substructures will deform. Constants selected in all cal-
!VIICROSTRUCTURE ON TI-IE RHEOLOGICAL PROPERTIES OF BLOOD
153
culati0ns are listed in Table I below. I t should be remarked t h a t these are b u t one possible set of coefficients. Table I centipoise
~o 0.8 0.7 0.6 0.5 0.4 0.3 0.2
~ 5.4 4.3 3.5 2.9 2.8 1.8 1.5
10 - 6 c m 2 10 - 6 c m 2 10 - 6 c m 2 10 - 6 c m "~
- ~,lz 1.87 1.67 1.88 1.08 0.75 0.55 0.45
~
