A semi-empirical model for flow of blood and other particulate suspensions through narrow tubes

BULLETIN OF MATHEMATICAL BIOLOGY V O L U M E 37, 1975

A SEMI-EMPIRICAL MODEL FOR FLOW OF BLOOD AND OTHER PARTICULATE SUSPENSIONS THROUGH NARROW TUBES

•

R. N. DAS and V. SES~XD~ Department of Applied Mechanics, Indian Institute of Technology, New Delhi 110029

A semi-empirical model applicable to the flow of blood and other particulate suspensions through narrow tubes has been developed. I t envisages a central core of blood surrounded by a wall layer of reduced hematocrit. With the help of this model the wall layer thickness and extent of plug flow may be calculated using pressure drop, flow rate and hematocrit reduction data. I t has been found from the available data in the literature that for a given sample of blood the extent of plug flow increases with decreasing tube diameter. Also for a flow through a given tube it increases with hematoerit. The wall layer thickness is found to deerease with increase in blood hematocrit. A comparison between the results of rigid particulate suspensions and blood reveals that the thicker wall layer and smaller plug flow radius in the ease of blood may be attributed to the deformability of the erythroeytes.

1. Introduction. H u m a n blood, f r o m h e m o d y n a m i c a l p o i n t of view, c a n be considered as a n e u t r a l l y b u o y a n t suspension o f d e f o r m a b l e particles, m a i n l y e r y t h r o c y t e s (or red blood cells) in a N e w t o n i a n fluid called plasma. T h e percentage b y v o l u m e of red blood cells (RBC) in blood, called h e m a t o c r i t , varies w i t h individuals a n d lies in t h e range of 4 0 - 4 5 % . Blood, while flowing t h r o u g h larger vessels like arteries, behaves as a n o n - N e w t o n i a n fluid w i t h shear thinning p r o p e r t y . T h e characteristic e q u a t i o n for blood in these t u b e s can be described w i t h reasonable a c c u r a c y b y t h a t of a Casson fluid (Merrill, 1969). H o w e v e r , in smaller vessels like arterioles a n d venules, w h e r e t h e d i a m e t e r of the t u b e s is less t h a n 500 f~m, blood c a n n o t be t r e a t e d as a c o n t i n u u m fluid and its two459

460

1%. N. DAS A N D V. S E S H A D R I

phase nature produces several artifacts. Since the dimensions of I~BC are not negligible compared to dimensions of the tubes through which blood is flowing, blood has to be treated as a two-phase fluid. The important artifacts that have been observed in flow of blood and other particulate suspensions through narrow tubes m a y be listed as follows: (i) The dependence of apparent viscosity on tube diameter. This effect was first observed b y F~hraeus and Lindquist (1931) and is known by their names. As the diameter of the tube is reduced below 300 ~m, the apparent viscosity of blood has been observed to decrease with decreasing tube diameter. (ii) Existence of a wall layer. A thin layer near the tube wall which is of reduced concentration or completely devoid of particles has been observed in narrow tubes. This has been attributed to "wall exclusion effect" (Maude and Whitmore, 1956) and to hydrodynamic forces which cause the particles to migrate radially toward the tube axis (Segr~ and Silberberg, 1961). (iii) Hematocrit defect. The radial migration of particles causes an increase in their average velocity as compared to that of the suspending fluid. This results in the reduction of actual hematocrit within the tube as compared to the hematocrit of blood entering or leaving the tube (F£hraeus, 1929; Seshadri and Sutera, 1968). (iv) Changes in the velocity profile. A central core where plug flow occurs has been observed in narrow tubes. The extent of plug flow increases with decreasing tube diameter (Goldsmith and Mason, 1969). The above anomalous effects have been observed, although to a varying degree in both blood and rigid particulate suspensions. Several mechanisms that have been able to account partially for the above artifacts have been proposed b y different authors. The measurement of wall layer thickness and velocity profile in narrow tubes is very difficult and requires extensive experimental setup. However, measurements of flow rate, pressure drop and hematocrit are relatively easier to make. Hence, in this article an attempt has been made to develop a semi-empirical model of flow which takes into account the anomalous effects. With the help of this model, velocity profile, and wall layer thickness may be calculated from experimentally measured values of flow rate, pressure drop and hcmatocrit defect.

2. Proposed Model of Flow.

Blood flow in narrow tubes is characterized b y small Reynolds number and hence fluid inertia can be neglected as compared to viscous forces. Also the curvature of the blood vessels does not affect the flow pattern within the tube due to small inertia forces. Thus the actual value of Reynolds number is unimportant. The tube can be considered as rigid since

A SEMI-EMPIRICAL MODEL F O R F L O W OF BLOOD

461

in micro-circulation the vessel walls derive rigidity from surrounding tissues and the changes in the diameter D = 2R o of the vessels due to pressure fluctuations are negligible. The proposed model envisages a wall layer of thickness 6 and concentration C/2 surrounding a central core of radius b ( = R o - 6) and concentration (or hematocrit) of C (see Figure 1). The wall layer concentration is not assumed

Power IGw profile in the core, U(r)=A-Br n

k,.b,]

Woll t ~

~c~

CONCENTRATION

__~___

Porobolic profile in the well leyer--~-~

PROFILE

VELOCITY

F i g u r e 1.

P r o p o s e d flow m o d e l

PROFI LE

to be zero, since in the range of physiological concentrations (40%) the radial migration of particles is very much inhibited due to extreme crowding and hence a wall layer of pure fluid is unlikely to occur. This has been supported by experimental observations (Karnis et al., 1966). However, since roughly half of the sphere centres are excluded from a region of the thickness d/2 near the wall where d is the average particle diameter, due to "wall exclusion effect", a layer of reduced concentration must result (Vand, 1948). This effect m a y be further enhanced by whatever radial migration t h a t can take place, which tends to increase the thickness of the region of reduced concentration. Taking the above facts into account the wall layer has been assumed to have an average concentration C/2 and thickness 6. Lih (1969) has proposed a model in which concentration and the viscosity in the wall layer have been assumed to vary continuously. However, direct observations (Karnis et al., 1966; Seshadri and Sutera, 1968) reveal that at physiological concentrations the wall layer has a thickness of only one or two particle diameters. Further, the axial distribution of the particles in the wall layer has been observed to be non-uniform. Since we are interested in the flow of suspensions and blood through tubes where tube to

462

R. :N. ]:)AS A N D V. S E S H A D R I

particle diameter ratio varies from 5 to a maximum of 1O0, the concentration or hematocrit cannot be considered as a point function (i.e. defined at each point in the flow field). Also the rheologieal equation relating viscocity to concentration m a y not be valid under the conditions existing in the wall layer where the size of particles is large compared to wall layer thickness. These considerations indicate t h a t some averaging procedure has to be adopted to obtain the rheological behaviour of the suspension in the wall layer. The logical procedure would be to consider each particle near the wall separately and sum up the effect of individual particles on the wall shear stress and strain rate (Brenner and Bungay, 1971). However, this procedure is very cumbersome and an alternate procedure could be to define an effective viscosity in the wall layer which would take the presence of particles into account. In the present model this effective viscosity is taken to be t h a t of a suspension or blood of concentration C/2. I t has to be noted that this procedure is suitable if we are only interested in predicting the gross behaviour of the suspension. Further the particles in the wall layer have been observed to travel much faster than the average fluid in the wall layer by Karnis et al. (1966). This is attributed to the fact t h a t these particles are being dragged by the particles in the faster moving core. Thus, in this model, the particles in the wall layer are assumed to travel with the velocity of suspension at the edge of the wall layer. The core has been assumed to be of radius b ( = R 0 - 8) and uniform concentration C. The velocity profile in the core has been observed to depend not only on the concentration but also on the tube to particle diameter ratio (DieT). A central region near the tube axis where the velocity of the suspension is constant and velocity gradient is zero, has been observed to exist (Karnis et al., 1966). The extent of this region of plug flow seems to be a function of Did and concentration C. It is observed to increase with decreasing tube diameter and increasing concentration. To account for this effect the velocity profile in the core is assumed to be of the form u(r) = A - Br ", where A, B and n are constants. The constants A and B are evaluated from the boundary condition at the edge of the wall layer. The power law index n is equal to 2 in the case of Poiseuille profile. Increase in the value of n makes the profile more blunt, thereby effectively increasing the extent of plug flow. Also n has to be greater than unity for shear stress at the tube axis to vanish. Now the equations governing the flow can be derived as follows. The apparent relative fluidity, ¢}, as calculated from pressure drop flow rate data is defined as 1 0, = CT(

D/J)

=

(1)

A S E M I - E M P I R I C A L M O D E L F O R F L O W OF B L O O D

463

where ~f = viscosity of suspending phase. Q = flow rate. ~w = average shear stress at the wall =

Ap. R o 2L

Ap = pressure drop in a tube of length L a n d radius R o. D = diameter of the tube = 2R o. C O = concentration (or hematocrit) of the suspension entering the tube. Based on the foregoing arguments, we can write for the wall layer,

du(r____) = dr where

-r(r) I~

~r. i~sRo'

b <~ r <~ R o,

(2)

~ = effective viscosity of the suspension in the wall layer =

Cr(~') = apparent relative fluidity of a suspension of concentration C/2 ~

F

in the absence of a n y wall effect. Integrating (2) and using no-slip condition at the wall, we get for the velocity profile in the wall layer,

u(r) = R°~~¢r(e/2) [1 - r2/R~)]; 21~r

b <~ r <. R o.

(3)

The velocity at the edge of the core is,

ub = u(r = b) = R°ewCr(C/2) [1 - r2] -, 2~r

y = b/R o.

(4)

The volume flow rate within the anmllus is

(2(Ro-b) = f : o 27rru(r) dr = ~R~¢r(e/2)4~tf

[1

-- y212.

(5)

NOW in the core, we have

u(r)

= A -- B r n ;

0 < r < b.

(6)

The constants A and B are evaluated b y equating velocity and velocity gradient at the edge of the core. The flow rate in the core, Qb, is obtained b y integration and is given by,

= I ~ 2~ru(r) dr = Jo Now, total flow rate Q is given b y

2(n + 2)/x~

72 + - n

- ~4 .

(7)

464

1%. N. ])AS A N D V. SESHADI~I

Substituting various values from (1), (5) and (7) and simplifying we get the relation, 2V2 (11 + 2)¢1 = (n + 2)~br(C]2)(1 - ~,2)2 + 2nsbr(C/2)(72- + -~-

-

~,~ ) •

(s)

Further, in steady state the quantities of two phases of suspensions flowing into the tube should be equal to those flowing past any cross section of the tube. The volume of particles flowing in the wall layer per unit time is (C]2)u~cr(R~ - b 2) where us is the velocity of the particles. In this model, it has been assumed that us = ub. Now CQb is the volume of particles flowing per unit time in the core. The total volume should be equal to volume of particles entering the tube. Thus we have the relation CoQ = C(2~ + ( C / 2 ) % ~ ( R ~ - b~).

Substiuting for u b from (4) and simplifying we get the relation C ~ C 0-

(9)

continuity of the suspending phase also gives the same relation. Now the average tube concentration, C, is defined as,

The

C.rrR~ = C~rb2 + (C]2)Tr(R~ - 52) or~ C ~ ~

20

1 +, ~ 2

Putting C = Co, we get the relation C Co

1 +~2 2

(to)

C is the concentration that one would measure if the flow is stopped and the suspension trapped within the tube is analyzed. C o is the concentration of the suspension or blood either entering the tube or flowing out. Equations (8) and (10) can be used to calculate the two unknowns n and $ b y using the experimentally measured values of ¢2, C and C o. Also ¢r(C/2) is obtained from either empirical formulae or experimental data. This is the fluidity of a suspension of concentration C/2 in a large tube where wall effects are absent. The wall layer thickness 8 is given b y 3 = Re(1 - ~). The plug flow radius r c is defined as that radius where the velocity is 0.99 times that at the tube axis. This is a well known criterion for boundary layer thickness. The dimensionless plug flow radius, R, is given by: rc R -= Re

antilog

{log (A/B) - 2} n "

(11)

A SEMI-EMPII~ICAL MODEL FOR FLOW OF BLOOD

465

R can be calculated from the value of power law index n and dimensionless core radius y. 3. Results and Discussions. Figures 2-5 show the results obtained from the model by using the experimental data on neutrally buoyant suspension of rigid spherical particles (Seshadri and Sutera, 1970). The fluidity of the wall layer, Cr(C') is calculated from the empirical formula (Thomas, 1965). 1 10.05C '2 + 0.00273 e x p (16.6C')]

Cr(C') = [1 + 2.5C' +

(12)

This has been found to agree with a large set of experimental observations of various authors up to concentrations of 40% (Seshadri and Sutera, 1970). D/c]=I0.65

[,C --

0.8-CK 0.6

r ir~ / Aj / D...~ . Expe { Kom:: feOt'o~a{u:~ ) 8

0.4

0.2 I

5

I0

t

15 Reservoir

I

20

,l

25

concentration,

I

I

50 Co

55

~

40

I

45

%

Flow of rigid particulate suspensions: Variation of R with Co

Figure 2.

[.0-LO

"~,z~

0.8--

. D/d-: [ 0 . 6 5

0.6-0.4'-0.2 0

I

5

L

I0

I

Ib Reservoir

Figure 3.

,, I

20

I

25

concentration,

1

,[

50 Co

55

I

40

I

45

%

Flow of rigid particulate suspensions: Variation of ~/cl wi~h Oo

In

466

R.N. DAS AND V. SESHADRI

r~

O.E--

Co= 40%

._~ 0.6--

z& I ~

o

C°=225%

0.#-

E 0.2--

IO

30

20

40

114

Diamei-er rario, D/d

Figure 4. Flow of rigid particulate suspensions: Variation of R with D/d 1.2

/Co=22'5 %

I.C 6o 0.8 ~'

o.c

~ 0.4 0.2

~ 10

20

30

40

50

1[4

Diamefer rdfio, D/d

Figure 5. Flow of rigid particulate suspensions: Variation of ~/(~ with Did these calculations the experimentally measured values of Ap, L, R o, D/d, C o, C and Q are used to predict wall layer thickness (3) and plug flow radius (R). Figure 2 shows the variation of plug flow radius, R, with concentration, C o, for diameter ratios 5.33 and 10.65. I t is seen from the figure t h a t the flow becomes more and more plugged with increasing concentration. Also, at a given concentration the extent of plug flow is more at the lower diameter ratios. These results are in qualitative agreement with experimental observations (Karnis et al., 1966). Figure 3 shows the variation of wall layer thickness with reservoir concentration. I t is observed t h a t the wall layer thickness decreases with increasing

A SEMI-EMPIRICAL MODEL F O R F L O W OF BLOOD

467

concentration. This can be attributed to inhibition of radial migration of particles at higher concentration. Figures 4 and 5 show the variation of R and 8/d as a function of Did at two different concentrations 22.5% and 40%. I t is seen that the extent of plug flow decreases with increasing diameter ratio. This shows that tile wall effect decreases for a given suspension as the tube diameter is increased. In order to apply the model to the flow of blood through narrow tubes, one has to take all the measurements required (e.g. ¢r1, Co, 0 and 0) simultaneously as was done b y Seshadri and Sutera in the case of rigid particulate suspensions (Seshadri and Sutera, 1970). However such measurements are not available in the literature at present, although several investigators have measured hematocrit defect and pressure drop-flow rate relation separately. Some of these measurements have been used to calculate 8 and R for blood flow in the thesis b y i~. N. Das (Da~, 1973). We shall give here only a comparison between the results obtained for rigid particulate suspensions and RBC suspensions in isotonic saline solution at a hematoerit of 40~o. The hematocrit defect data on I~BC suspension have been taken from Davis and Hochmuth (1969), and the pressure drop data have been obtained from Doody (1969). The wall layer fluidity is obtained by using the relation, 1

Cr(C') = [1 + 2C' + 0.S85C '2]

(13)

This seems to give a good fit to experimental data obtained by Doody in large tubes. Figures 6 and 7 show the comparison between the flow of rigid particulate suspensions and I~BC suspensions. It is observed that the flow is less plugged Rigid purficulafe suspensions

rr

0.8

" - - - ~ ~

RBC in Isofonic suline

0.6 8

o.4 O. ~

I0

20

50

Diomefer ret-ie,

40

4

D/d,

F i g u r e 6. Comparison of results of rigid p a r t i c u l a t e suspensions and I~BC suspensions in isotonic saline: V a r i a t i o n of R w i t h Did.

Go

=

40%

468

R. :N. ])AS AND V. SESHADRI 2.25 RBC in isotonic saline

c~

o -

-

\.

~

1.75

1,25

fe

suspensions

0.75-

0,25 --

I

10

I

2O Diameter ratio,

I

50

|o

D/{]

Figure 7. Comparison of results of rigid particulate suspensions and RBC suspensions in isotonic s~line: Variation of 3/~/with Did. Co = 40% a n d wall l a y e r thickness is larger in t h e ease of R B C suspensions as c o m p a r e d to rigid p a r t i c u l a t e suspensions. This can be a t t r i b u t e d to flexibility of red b l o o d cells. This conclusion is s u p p o r t e d b y t h e e x p e r i m e n t a l o b s e r v a t i o n s (Goldsmith a n d Mason, 1969) where it was o b s e r v e d t h a t t h e d e f o r m a b l e particles m i g r a t e m o r e readily as c o m p a r e d to rigid particles. Since a decrease in plug flow radius a n d a n increase in wall l a y e r thickness causes a reduction in resistance to flow, we can conclude t h a t R B C flexibility p l a y s a m a j o r role in r e d u c i n g t h e flow resistance of blood in n a r r o w tubes.

4. Concluding RemarIcs. I t is seen t h a t t h e p r o p o s e d m o d e l gives predictions which are in q u a l i t a t i v e a g r e e m e n t w i t h e x p e r i m e n t a l o b s e r v a t i o n s of several investigators. H o w e v e r , f u r t h e r justification of t h e model can o n l y be o b t a i n e d w h e n m o r e h e m a t o c r i t defect a n d pressure d r o p - f l o w r a t e d a t a for blood are available. Currently, a t t e m p t is being m a d e to o b t a i n s i m u l t a n e o u s m e a s u r e m e n t s of all r e l e v a n t quantities (viz. pressure d r o p - f l o w r a t e a n d h e m a t o e r i t defect) in the case of blood flow t h r o u g h n a r r o w tubes. These results w o u l d be able to show q u a n t i t a t i v e l y t h e role of I~BC flexibility in reducing resistance t o flow in n a r r o w tubes. APPENDIX b = Radius of core. C = Concentration in the core. Go = Concentration of the suspension entering or leaving the tube. = Average concentration in the tube. = Average diameter of the suspended particles. D = 2Ro = Diameter of tube. D -:_ = Diameter ratio. d

A SEMI-EMPIRICAL MODEL F O R F L O W OF BLOOD

469

n = I n d e x of p o w e r l a w v e l o c i t y profile. r = :Radial d i s t a n c e f r o m t u b e axis. r c = P l u g flow r a d i u s i n t h e core. R = N o n - d i m e n s i o n a l i z e d plug core r a d i u s = rc/Ro. = Flow rate through the tube. = F l o w r a t e t h r o u g h t h e core. QRo- b = F l o w r a t e ~ h r o u g h t h e wall-layer a n n u l u s . u(r) = V e l o c i t y of s u s p e n s i o n a t r a d i a l d i s t a n c e r f r o m t h e t u b e axis. q~s = V e l o c i t y of t h e particles i n t h e wall layer. ~b = V e l o c i t y a~ t h e edge of core, i.e. r = b. = T h i c k n e s s o f wall-layer. V = b/Ro, N o n - d i m e n s i o n a l i z e d core radius. = A p p a r e n t r e l a t i v e fluidity of suspension. ¢~(0/2) = A p p a r e n t r e l a t i v e fluidity of a s u s p e n s i o n of c o n c e n t r a t i o n , C/2. ¢,(c') = F l u i d i t y o f a s u s p e n s i o n of c o n c e n t r a t i o n C ' o b t a i n e d f r o m s e m i - e m p i r i c a l formula. /~r = Coefficient of viscosity of t h e s u s p e n d i n g fluid. /~s = E f f e c t i v e v i s c o s i t y of t h e s u s p e n s i o n i n t h e wall layer. Ap = P r e s s u r e d r o p o v e r a t u b e l e n g t h , 15. 5w = A v e r a g e s h e a r stress a t t h e t u b e wall. LITERATURE B r e n n e r , H. a n d P. M. B u n g a y . 1971. " R i g i d - P a r t i c l e a n d L i q u i d D r o p l e t Models of R e d Cell M o t i o n i n C a p i l l a r y T u b e s . " Fedn Proc., 30, 1565-1576. Das, R . N . 1973. " A n A n a l y t i c a l Model for t h e F l o w of B l o o d a n d P a r t i c u l a t e Suspensions t h r o u g h N a r r o w T u b e s . " M.Tech. D i s s e r t a t i o n , I n d i a n I n s t i t u t e of T e c h n o l o g y , Delhi. Doody, C.N. 1969. " T h e Flow of H u m a n E r y t h r o e y t e S u s p e n s i o n s in S m a l l Glass Capillaries." M.S. Thesis, W a s h i n g t o n U n i v e r s i t y . F ~ h r a c u s , 1~. 1929. " T h e S u s p e n s i o n S t a b i l i t y of t h e B l o o d . " Physiol. Rev., 9, 241-274. - a n d T. L i n d q u i s t . 1931. " T h e Viscosity of t h e B l o o d in N a r r o w Capillary T u b e s . " A m . J. PhysioL, 96, 562-568. G o l d s m i t h , 14. L. a n d S. G. Mason. 1964. " T h e M i c r o r h e o l o g y of D i s p e r s i o n s . " I n Rheology, Theory and Applications (Ed. F. 1~. E i r i c h ) , Vol. 4, 85-249. L o n d o n : Academ i c Press. H o c h m u t h , ]~. M. a n d D. O. Davis. 1969. " C h a n g e s i n H e m a t o c r i t for B l o o d F l o w in N a r r o w T u b e s . " Bibl. anat., 10, 59-65. K a r n i s , A., H . L. G o l d s m i t h a n d S. G. Mason. 1966. " T h e K i n e t i c s of F l o w i n g Disp e r s i o n s . " J. Colloid Interface Sci., 22, 531-553. Lih, M . M . 1969. " A M a t h e m a t i c a l Model for A x i a l M i g r a t i o n of S u s p e n d e d P a r t i c l e s in Tube Flow." Bull..Math. Biophys., 31, 143-156. M a u d e , A. D. a n d t%. L. W h i t m o r e . 1956. " T h e W a l l Effect a n d t h e V i s c o m e t r y of S u s p e n s i o n s . " Brit. J. Appl. Phys., 7, 98-102. Merrill, E . W . 1969. " R h e o l o g y of B l o o d . " Physiol. Rev., 49, 863-888. Segr6, G. a n d A. Silberberg. 1961. " l ~ a d i a l P a r t i c l e D i s p l a c e m e n t s in Poiseuille F l o w of S u s p e n s i o n s . " Nature, 189, 209-210. Seshadri, V. a n d S. P. Sutera. 1968. " C o n c e n t r a t i o n C h a n g e s of S u s p e n s i o n s of R i g i d Spheres F l o w i n g t h r o u g h N a r r o w T u b e s . " J. Colloid Interface Sci., 27, 101-110. - and - 1970. " F l o w of S u s p e n s i o n s T h r o u g h T u b e s . " TraJ~s. Soc. Rheol., 14, 351-373.

470

R. N. DAS

AND

V. SESI-IADRI

Thomas, D.G. 1965. "Transport Characteristics of Suspensions V I I I . " J. Colloid. Sci., 20, 267-277. Vand, V. 1948. "Viscosity of Solutions and Suspensions." J. Phys. Colloid. Chem., 52, 277 321. RECwIVEI) 4-12-74 REVISE]) 12-3-74