BULLETIN OF MATHEMATICALBIOLOGY
VOLUME37, 1975
A SEMI-EMPIRICAL MODEL F O R F L O W OF BLOOD AND O T H E R PARTICULATE SUSPENSIONS T H R O U G H N A R R O W TUBES
9 R. N. DAs and V. SESHADRI 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 hematecrit 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 hematocrit. The wall layer thickness is found to decrease 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 case of blood may be attributed to the deformability of the erythrocytes.
Human blood, from hemodynamical point of view, can be considered as a neutrally buoyant suspension of deformable particles, mainly erythrocytes (or red blood cells) in a Newtonian fluid called plasma. The percentage by volume of red blood cells (RBC) in blood, called hematocrit, varies with individuals and lies in the range of 40-45% . Blood, while flowing through larger vessels like arteries, behaves as a non-Newtonian fluid with shear thinning property. The characteristic equation for blood in these tubes can be described with reasonable accuracy by that of a Casson fluid (Merrill, 1969). However, in smaller vessels like arterioles and venules, where the diameter of the tubes is less than 500 ~m, blood cannot be treated as a continuum fluid and its two1. Introduction.
459
460
R . 1~r. D A S A N D V. S E S H A D R I
phase nature produces several artifacts. Since the dimensions of RBC 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 may be listed as follows: (i) The dependence of apparent viscosity on tube diameter. This effect was first observed by Fhhraeus 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 (Segrd 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 (Fhhraeus, 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 hematocrit 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 3 and concentration C/2 surrounding a central core of radius b ( = R o - 8) and concentration (or hematocrit) of C (see Figure 1). The wall layer concentration is not assumed
Power law profile in the core, U(r):A-Br n __
~'5
~
waJ L lo_ve_r~ = . . . . . . .
....... Core
lii
....
....
.....
dl__B .
.
.
.
.
.
Z
2R o 2 b - { ] b / R ; ;"
t-!
.....
~ C/2 Parabolic profile in fhe wall layer- -
CONCENTRATION
PROFILE
VELOCITY
F i g u r e l.
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 3/2 near the wall where ~ is the average particle diameter, due to "wall exclusion effect", a layer of reduced concentration must result (Vand, 1948). This effect may be further enhanced by whatever radial migration that 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 & 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. D A S A N D V. S E S H A D R I
particle diameter ratio varies from 5 to a maximum of 100, the concentration or hematocrit cannot be considered as a point function (i.e. defined at each point in the flow field). Also the rheological 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 that some averaging procedure has to be adopted to obtain the rheologieal 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 that of a suspension or blood of concentration C/2. It 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 b y Karnis et al. (1966). This is attributed to the fact that these particles are being dragged b y 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 (D/~). 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 D/a~ 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 axe 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, Cr~, as calculated from pressure drop flow rate data is defined as r
= r
~w, Dla) = 4~rQ,
(1)
A SEMI-EMPIRICAL MODEL FOR FLOW OF BLOOD where
463
= viscosity of suspending phase. Q = flow rate.
~r
average shear stress at the wall = Ap. R o 2L pressure drop in a tube of length L and radius R o. D=
diameter of the tube = 2R o.
C 0 = 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) = _ ~w___~r. /~s /~R0'
b ~< r ~< Ro,
(2)
/~s = effective viscosity of the suspension in the wall layer =
m1r Cr ~
= apparent relative fluidity of a suspension of concentration C/2
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~162 21~r
[1 - r2/R~];
b
VOLUME37, 1975
A SEMI-EMPIRICAL MODEL F O R F L O W OF BLOOD AND O T H E R PARTICULATE SUSPENSIONS T H R O U G H N A R R O W TUBES
9 R. N. DAs and V. SESHADRI 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 hematecrit 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 hematocrit. The wall layer thickness is found to decrease 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 case of blood may be attributed to the deformability of the erythrocytes.
Human blood, from hemodynamical point of view, can be considered as a neutrally buoyant suspension of deformable particles, mainly erythrocytes (or red blood cells) in a Newtonian fluid called plasma. The percentage by volume of red blood cells (RBC) in blood, called hematocrit, varies with individuals and lies in the range of 40-45% . Blood, while flowing through larger vessels like arteries, behaves as a non-Newtonian fluid with shear thinning property. The characteristic equation for blood in these tubes can be described with reasonable accuracy by that of a Casson fluid (Merrill, 1969). However, in smaller vessels like arterioles and venules, where the diameter of the tubes is less than 500 ~m, blood cannot be treated as a continuum fluid and its two1. Introduction.
459
460
R . 1~r. D A S A N D V. S E S H A D R I
phase nature produces several artifacts. Since the dimensions of RBC 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 may be listed as follows: (i) The dependence of apparent viscosity on tube diameter. This effect was first observed by Fhhraeus 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 (Segrd 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 (Fhhraeus, 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 hematocrit 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 3 and concentration C/2 surrounding a central core of radius b ( = R o - 8) and concentration (or hematocrit) of C (see Figure 1). The wall layer concentration is not assumed
Power law profile in the core, U(r):A-Br n __
~'5
~
waJ L lo_ve_r~ = . . . . . . .
....... Core
lii
....
....
.....
dl__B .
.
.
.
.
.
Z
2R o 2 b - { ] b / R ; ;"
t-!
.....
~ C/2 Parabolic profile in fhe wall layer- -
CONCENTRATION
PROFILE
VELOCITY
F i g u r e l.
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 3/2 near the wall where ~ is the average particle diameter, due to "wall exclusion effect", a layer of reduced concentration must result (Vand, 1948). This effect may be further enhanced by whatever radial migration that 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 & 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. D A S A N D V. S E S H A D R I
particle diameter ratio varies from 5 to a maximum of 100, the concentration or hematocrit cannot be considered as a point function (i.e. defined at each point in the flow field). Also the rheological 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 that some averaging procedure has to be adopted to obtain the rheologieal 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 that of a suspension or blood of concentration C/2. It 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 b y Karnis et al. (1966). This is attributed to the fact that these particles are being dragged b y 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 (D/~). 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 D/a~ 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 axe 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, Cr~, as calculated from pressure drop flow rate data is defined as r
= r
~w, Dla) = 4~rQ,
(1)
A SEMI-EMPIRICAL MODEL FOR FLOW OF BLOOD where
463
= viscosity of suspending phase. Q = flow rate.
~r
average shear stress at the wall = Ap. R o 2L pressure drop in a tube of length L and radius R o. D=
diameter of the tube = 2R o.
C 0 = 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) = _ ~w___~r. /~s /~R0'
b ~< r ~< Ro,
(2)
/~s = effective viscosity of the suspension in the wall layer =
m1r Cr ~
= apparent relative fluidity of a suspension of concentration C/2
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~162 21~r
[1 - r2/R~];
b
