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Russian Journal of Biomechanics
MODELING OF PERISTALTIC AND PULSATING FLOWS OF BLOOD
S.R. Ismagilova*, A.A. Lezhneva*, D.V. Stolyarov**, N.V. Shakirov**
* Perm State Technical University, 29a, Komsomolskii Prospect, 614600, Perm, Russia
** Institute of Continuous Media Mechanics, 1, Korolyov Street, 614013, Perm, Russia, e-mail: [email protected]
Abstract. In this paper, the motion of viscoelastic liquids in cylindrical channels of constant and variable cross-section is investigated. The Theological properties are described in terms of nonlinear integral model with the memory function of exponential type. The model is characterized by relaxation time, retardation time and effective viscosity that depends on the shear rate. The viscosity of human blood is approximated by a four-parametric function. Two problems are examined: peristaltic transfer of a fluid through a tube and pulsating flow in a fixed channel. Solutions to these problems are found by the method of expansion in terms of a small parameter. The influence of rheological properties on the averaged flow rate is analyzed. The role of the elastic and inertial effects and the role of the shape of the viscosity vs. shear rate curve are discussed.
Keywords: blood, viscosity, viscoelasticity, peristaltic flow, pulsating flow
A flow of physiological fluids has long attracted the scientists' interest. Periodic and sporadic pumping of fluids or fluid-solid mixtures through muscular tubes by means of peristaltic waves is a widely employed biological transport mechanism. Examples include the passage of urine from kidneys to bladder, the movement of chyme in the gastrointestinal tract and the flow of blood in the vessels.
Many investigations on peristaltic flow of Newtonian fluids have been carried out. In the earlier works [1-4] the flows of viscous incompressible fluids initiated by slight wavy motions of the tube walls were computed by the method of disturbances. These studies have focused almost exclusively upon cases in which the ratio of wave amplitude to wavelength is zero or very small and inertial effects are insignificant. It was found that the mean rate of flow excited by peristaltic motions of the walls is proportional to the square of the relative amplitude. It was shown [3] that pumping against positive pressure gradient may lead to initiation of the reverse flow. This phenomenon is of particular interest to urologists because it may provide a reasonable explanation for the bacteria motion from urinary bladder to kidney. In addition, in [2] the authors discuss the effect of the inertia forces and wall curvature on the flow.
The use of numerical methods for calculation of peristaltic liquid transportation allowed us to take into account the influence of the inertia forces, the amplitude finiteness and the length of the running peristaltic wave on the flow. In works [5-7] the finite-difference method was used to solve the Navier-Stokes and continuity equations. The equations of motion are based on a moving coordinate system translating longitudinally at a constant speed. In [8] a detailed analysis of peristaltic pumping was made by the method of finite
Introduction
elements. It was shown that the presence of erythrocytes changes the velocity profile in the vicinity of these cells. It was found that the shear stress developed along the peristaltic wall increases dramatically with Reynolds number.
Since many biological liquids including blood show viscoelastic behavior, simulation of peristaltic motions of such media is a challenging problem [9-11].
The pipe flow of a viscoelastic liquid under a pulsating pressure has been a fascinating problem for rheologists who wish to test their favourite constitutive equations. This problem has been the focus of extensive investigation in numerous works. The statement of the problem, the methods of its solution and the basic results are discussed in greater detail in works [12-14].
The aim of the present work is to approximate the curves of the human blood flow and to model peristaltic and pulsating flows of viscoelastic liquid of the integral type.
i 1. Rheological characteristics of blood
From the rheological point of view, human blood is a suspension system with two structural arrangements of erythrocytes, non-aggregated cells and an aggregation of cells in the form of rouleaux of other clumps, suspended in plasma as a continuous phase.
To describe the rheological properties of human blood we use a refined Walters-Fredricson model, which in the case of a shear flow is presented as
<t) = *(yfey(th (1)
o \ k\ J
(i = 1'2>
(1 + flj)
Here t and y are stress and shear rate, Xl and X2 are relaxation and retardation times. The effective viscosity is well approximated by the following equations:
^ - 'Hoc +(^0 ~^oo)exp(~(^y)*)> (3)
=-' r| = my , (4)
l + |ay|
where X0i9 bi9 r|0, r^, a, (3, m and n are characteristic parameters as defined in the model.
This model is applied with a good approximation under low values of the shear rates i. e. in rouleaux formation area.
Fig. 1 gives experimental results [15-17], which show the dependence of viscosity on the shear rate.
Curve designated by number 1 was taken from work [15] (hematocrit # = 0.4, temperature T = 3TC). Curve 2 (// = 0.43, r = 22°C) corresponds to experimental results obtained by Thurston [16]. Curve 3 was plotted by data reported in [17] (// = 0.42±0.04, 7 = 37°C). These three curves were used to construct the averaged curve No. 4, which was approximated by equation (3).
Since the zero-shear rate viscosity r|0 was evaluated with an error the other defined parameters have also an error. For example, an adequate approximation of curve 4 is obtained by formula (3) in which the constants are assumed to be equal to: r|0 =93.56mPa-s,
^loo = 4.327mPa• s, a = 2.050s, k = 1 (curve 5) and r|0 =113.87 mPa-s, =4.327mPa-s, a = 5.088 s, £ = 0.5 (curve 6).
Shear rate, s 1 Fig. 1. Steady shear viscosity data for the human blood.
We may select other combinations of rheological constants to approximate the flow curves with the specified accuracy. In this case the values of T]0can differ considerably from
one another. For stationary flows with these constants the results of calculations will agree fairly well, whereas in dynamic processes the results may be different. In Section 3 we will show the effect of the accuracy of the flow curve approximation on the value of the flow rate in the pulsating flow.
2. Peristaltic transport 1
Consider an axisymmetric flow of linear viscoelastic fluid in a circular cylindrical tube with a sinusoidal wave of small amplitude travelling along its wall (Fig. 2). We assume that the fluid is so viscous that the inertia forces are negligible compared with extra stresses and the wave length is much greater than the channel radius. The equation of tube motion is taken as
H{z\t) = a + bcos^{z'-ct), ' " (5)
A
where a is radius of the undisturbed tube, b is amplitude of the wave, X is wave length, c is wave speed, t is time.
We consider the pressure drop at the tube ends as the known time function
P{zutyP{z29t) = P{t),
and the distance from the inlet to the outlet of the tube as a multiple of the wave length z2~z] = nX.
Fig. 2. Geometry of the peristaltic flow.
To study the problem, we will transform the stationary coordinates, z', r to moving coordinates z, r, which move with the wave velocity c in the positive z' direction, as follows:
r -r , z-z'-ct, U — V_' — C , V' — Vr' ,
where w , v, vz* and vr> are velocity components in the corresponding systems. We assume that the particles of the wavy wall move strictly up and down
^ = v = 0 at r = 0, or
v = ¥L = u I
dt 11
dH
=" dz
at r = H .
(6)
The peristaltic flow of the Walters-Fredricson liquid is described by the system of equations
Tlo i
X22r\0
t(,v)exp
r \ t-s
X
ds,
2 7
(7)
(8) (9)
where x and y are shear stress and shear rate, u and v are flow velocity, and r|, Xx and X2 are rheological constants.
By integrating momentum equation (9) and substituting the determined stress in rheological equation (7) we define the shear rate
dp rkx lrdp r Xx-X2
7
-f
3z 2t\X2 J & 2r| xl
exp
í \ t-s
ds
-2 J
and the axial velocity
u -
r2-H2
4T1
dp X] 'rdp A,, - X2
r
dz I, J
dz
X^
exp
f \
t-s
X2 j
ds
Substituting formula (10) in continuity equation (8) and using boundary conditions (6) give the expression for pressure gradient
1 d
16r| dz
dp 'rdp A,] — X2 dzX2 J dz xl
exp
/ \ \
t-s ds
V J J
= H
ÔH dt
From this equation one can readily obtain the pressure gradient
dp dz
= i6ii
K+Ajt) x2 ^ lçFw + A(s) - X2
HA
J-
H*
K
/ \ 1
-exp t-s ds\
V J
(ii)
where
ÔH
dz.
F~ ' i" 3,
An arbitrary function of integration A{t) is derived from equation
t-s
X
ds.
2 J
The flow rate within the time interval rbeing equal to the period of wall
pulsation is defined by
TH
Q~2nj^rudrdt.
(12)
o o
The calculations for the peristaltic flow initiated by the movement of the wall according to law (5) have been made using formulae (10)-(12). It has been found that the flow rates of the linear viscoelastic and Newtonian (X} -X2) liquids coincide. Similar results were obtained for Maxwell [9] and Giesekus [11] models. The analysis made in work [9] also indicated that fluctuations of the flow rate and pressure differ in phase.
A comparison of calculations made for Maxwell and Walters-Fredricson models demonstrated that the phase shift for the three-parameter model is less than for the two-parameter model 1
3. Pulsating flow
We consider here the problem of flow of non-Newtonian liquids in a straight tube of circular cross section under the influence of the sinusoidal pressure gradient
= /?5(l + ecosco/). • v (13)
Here ps is mean pressure drop per unit length of the pipe, co and s are frequency and
amplitude of the oscillation, t is time. The value of e is assumed to be small enough in order
to neglect the terms of order s3 in the following analysis.
The tube was assumed to be infinitely long so that the entry and exit effects can be ignored.
In the case under consideration, the relevant form for the physical component of the velocity vector is
vr =0, v0 = 0, v2 =v(r,f), (14)
which automatically satisfies the equation of continuity. The stress equation of motion is given by:
+ (15)
where p is fluid density and t denotes shear stress.
We obtain a solution to this problem only for small e, using the perturbation analysis, by expanding all necessary terms in powers of 8.
The relevant quantities may be expanded in the following way
v(r, t) = V, (r)[l Hh 8V, (t) + 82V2 (i) + o(83 )] ,
(16)
In this work our main concern is the effect of the pulsatile pressure gradient on the mean flow rate. This quantity may be expressed as
j V® R
q = co J271 jvz(r,t)rdrdt. (17)
0 0
A more convenient expression for our purposes is obtained by integrating Eq. (17) by parts giving
1/co R
q — -co J 7irz{r,t)r2drdt.
0 0
The steady state flow rate qs is given by
We now define the relative difference in the mean flow rate of pulsatile and stationary flows with the same pressure gradient by I, where
Substituting formula (1) in the equilibrium equation and using expansion in terms of the parameter (16), we obtain for small Raynolds numbers the following equation for the relative difference in the flow rate:
R
/-8
2 0
Jto yldr
(18)
where (j) denotes time averaging:
co 271
27T/CÖ
1 A? co2
\q(t)dt, /( <a)= 01 2, (p(y0) = ri(y0)y0, ri 1 + ^V
02
(y2)to =/W
2(• \d <p(y0) <P lYo) ^
d(p(to)
. )
(19)
41
/(C0)S2
3.0 2.5 2.0 1.5 1.0 0.5
i i t $ ill* 1 1 s i 1 1
: * 1 „ w ** «. « ™ «« * 5. i I i I / V M \ / f X" / - - « -I - ~ - ~ ~ - - i f 11 \ ! : 2 \ 1 r-Vr-- 1 \ ! i \ 1 \ \ V
\ S \ \ 1 * f 1 I
\ 1 1 i \ \
1 2 3 4 1A_3 n
ps>\0 Pa
Fig. 3. Relative increase in the flow rate versus average pressure gradient.
Fig. 4. Relative increase in the flow rate versus frequency of pressure oscillation.
For some types of liquids the integrals in (18) are defined analytically. In particular, for the linear viscoelastic liquid (л = г1о> ^ = const) 1 = 0, and for the power law liquid
(r| = my", Xi = const) the relative flow rate is equal to I = £ ^ /(ш).
4 n
The typical results of calculation made with the use of Eqs. (3), (18) and (19) are shown in Fig. 3. Curves 1 and 2 were obtained using expression (3) in which we assumed that T] q = 190 mPa • s, r|oo =50mPa-s, G = 0.04s, ¿ = 0 (curve 1); r|o =180mPa's>
= 60 mPa • s, G = 0.05 s , к = 0 (curve 2). The results of calculation for a power law flow
(m = 34mPa-cn; n = 0.61) are designated by 3. It is readily seen that the accuracy of the flow curve approximation has a considerable effect on I.
The effect of the inertia terms in the equation of motion on the percentage increase of the flow rate is shown in Fig, 4. The calculations were made using formulae (1, 2, 4, 18, 19), in which the parameters were taken as: a = 0.3 s , ß = 0.32, X0l = 4 s, r^ = axf = b = 0. In all
calculations the tube radius was assumed to have the same value R = 1.6-10~3 m. From this
follows that up to the frequency со < 5 s"1 one can use the results of calculation made without taking into account the inertia terms.
Conclusions
The effect of rheological liquid parameters, the averaged pressure gradient, the oscillation amplitude and frequency on the mean flow rate has been investigated. The analysis shows that the averaged flow rate of linear viscoelastic liquid agrees with that of Newtonian liquid. The main reason for the increase of the flow rate in the pulsating flow is a decrease of the effective viscosity with the growth of the shear rate (nonlinearity of the flow curve). The accuracy of the flow curve approximation essentially affects the results of computation of the relative flow rate.
References
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5. BROWN T.D., HUNG Т.К. Computational and experimental investigations of two-dimensional nonlinear peristaltic flows. J of Fluid Mechanics, 83(2): 249-272,1977.
6. ЛЕВИНА Г.В. Перистальтическое течение при конечных числах Рейнольдса. Известия Академии наук СССР. Механика жидкости и газа, № 3: 11-15, 1985 (in Russian).
7. ЗАИКО В.М., СТАРОБИН И.М., УТКИН A.B. Численное моделирование движения вязкой жидкости (крови) в трубке с активно деформирующейся стенкой. Механика композитных материалов (Рига), № 3: 515-523, 1979 (in Russian).
8. TONG P., VAWTER D. An Analysis of Peristaltic Pumping. Transactions of ASME. J Appl Mech, 39, 857-862,1972.
9. КОЗЛОВ Л.Ф., ВОРОПАЕВ Г.А., КУЗЬМЕНКО A.M. Течение вязко-упругой жидкости в трубке с деформирующейся стенкой. Прикладная механика (Киев), 14(6): 84-89, 1978 (in Russian).
10. BÖHME G. FRIEDRICH R. Peristaltic flow of viscoelastic liquids. J Fluid Mech, 28(1): 109-122, 1983.
11. KANAKA RAJU K., RATHNA DEVANATHAN, Peristaltic motion of a non-Newtonian fluid. Part II. Visco-elastic fluid. Rheol. Acta, 13: 944-948,1974.
12. BARNES H.A, TOWNSEND P., WALTERS K. On pulsating flow of non-Newtonian liquids. Rheologica Acta, 10(3), 517-527,1971.
13. PHAN-THIEN N. On pulsating flow of polymeric fluids. J of Non-Newtonian Fluid Mechanics, 4(3): 167-176, 1978.
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17. ANTONOVA N., VELCHEVA I. Characteristic parameters derived from whole blood viscometry in cerebrovascular disease. Russian Journal of Biomechanics, 3(1): 37-46, 1999.
МОДЕЛИРОВАНИЕ ПЕРИСТАЛЬТИЧЕСКОГО И ПУЛЬСИРУЮЩЕГО ТЕЧЕНИЙ КРОВИ
С.П. Исмагилова, A.A. Лежнёва, Д.В. Столяров, Н.В. Шакиров
(Пермь, Россия)
Исследуется движение упруговязких жидкостей в цилиндрических каналах постоянного и переменного сечений. Реологические свойства описываются нелинейной интегральной моделью с функцией памяти экспоненциального типа. Модель характеризуется временем релаксации, временем ретардации и эффективной вязкостью, зависящей от скорости сдвига. Вязкость человеческой крови аппроксимируется четырёхпараметрической функцией. Исследуются две задачи: перистальтический перенос жидкости в трубе и пульсирующее течение в неподвижной трубе. Для решения этих задач используется метод разложения по малому параметру. Анализируется влияние реологических свойств на величину среднего за период расхода. Библ. 17.
Ключевые слова: кровь, вязкость, вязкоупругость, перистальтическое течение, пульсирующее течение
!
Received 01 February 2002
