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Pulsatile flow of an incompressible, inhomogeneous fluid in a smoothly expanded vascular tube
International Journal of Engineering Science 57 (2012) 1–10
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Pulsatile flow of an incompressible, inhomogeneous fluid in a smoothly expanded vascular tube Mani Shankar Mandal a,⇑, Swati Mukhopadhyay b, G.C. Layek b a b
Department of Mathematics, Krishnagar Govt. College, Krishnagar, Nadia 741101, India Department of Mathematics, The University of Burdwan, Burdwan 713104, India
a r t i c l e
i n f o
Article history: Received 30 August 2011 Received in revised form 27 February 2012 Accepted 1 April 2012
Keywords: Pulsatile flow Arterial flow Sudden expansion Haematocrit Blood viscosity Non-linear analysis Finite difference method
a b s t r a c t This paper aims to present pulsatile laminar flow of an incompressible, inhomogeneous fluid in an axi-symmetric smoothly expanded tube, modeled as artery, under some specific flow conditions. The flowing blood is an inhomogeneous fluid due to the presence of haematocrit (the percentage of total blood volume occupied by red blood cells (RBCs)). In the present study, the viscosity of flowing blood is assumed to be dependent on radius of the artery with a maximum value at the centerline of the artery. The governing nonlinear equations along with the appropriate boundary conditions are derived and are solved numerically using finite-difference method. The effects of inhomogeneities on flow quantities along with other flow parameters such as Reynolds number (Re), Strouhal number (St), expansion height (d) have been investigated. The numerical values of wall shear stress agree well with the available results of previously published works. It is seen that the value of wall shear stress and the corresponding length of flow separation increases significantly when the viscosity increases about the mean under both steady and pulsatile flow conditions. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The fluid viscosity is generally regarded as constant for some substances when the variations of pressure and temperature are negligible and is independent of the rate of the shear so long as this rate is not varied much. For certain mixture, emulsions etc., apparent coefficient of viscosity depends on the rate of shear as well as presence of particles (inhomogeneities) and the fluid is not a simple one (Dryden, Murnaghan, & Bateman, 1956). Inhomogeneities of flowing fluid have profound implications on the flow dynamics and have wide applications in geological flows, biological flows and many others. The flowing blood is very complex substance/fluid. Also, blood viscosity changes significantly with rate of shear in the flow. For flows in tubes it varies from zero at the centerline to maximum value at the wall. The endothelium plays a critical role in maintaining blood fluidity by balancing a natural tendency to clot in isolation with a set of counteracting mechanism (Anand, Rajagopal, & Rajagopal, 2003). Understanding of pulsatile fluid motion and the corresponding flow dynamics have important practical applications in branches of engineering and also in human cardiovascular system. The heart pumps blood intermittently through branches, between which cross sectional area may vary considerably. The expansions at branch points and the pulsatile nature of blood flow can form very complicated nature of flow (Ku, 1997). Several physical quantities have proposed in literature for measuring the risk zones in blood vessel. Observations have shown that one reason is the oscillations in blood flow during the
⇑ Corresponding author. Tel.: +91 9434556135; fax: +91 342 2530452. E-mail addresses: [email protected], [email protected] (M.S. Mandal). 0020-7225/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2012.04.002
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diastolic phase at every single heart beat. Thus clear understanding of pulsatile flow characteristics may help in the diagnosing of the arterial diseases. It is true that human blood may be considered as a Newtonian fluid for flow within the heart and the aorta. The Newtonian model of flowing blood is acceptable for high shear rate flow i.e. for flow through larger arteries. But under diseased conditions, blood exhibits remarkably non-Newtonian properties, even in larger arteries (Nakamura & Sawada, 1988). Blood is composed of fluid plasma and formed elements. The formed elements of blood are erythrocytes, leukocytes and platelets. Among them the presence of haematocrit is significant in the total blood volume and differs from person to person. The percentage volume of red cells is called haematocrit and is approximately 40–45% for an adult as reported by Oka (Oka, 1981). The typical ranges are 41.8–49% approximately for males and 38.6–45.6% for females (Jandl, 1996). In general, blood may be considered as incompressible and inhomogeneous fluid and the inhomogenity is mainly the result of haematocrit (Demiray, 2008). However, in the course of flow in larger arteries, the red blood cells in the vicinity of arterial wall move to the central region of the artery so that the haematocrit becomes quite low near the arterial wall, which results in lower viscosity in this region. Moreover, due to high shear rate near the arterial wall, the viscosity of blood is further reduced. The dependence of blood viscosity on increasing haematocrit was measured experimentally by Cinar, Demir, Pac, and Cinar (1999). From the data it has been shown that as haematocrit increases from 35% to 80%, viscosity increases from 4 to 14 times the viscosity of distilled water. The volume flow rate in comparison with Poiseuill’s law will vary significantly with the increasing level of haematocrit. Blood flows, particularly within the arteries, are sometimes disturbed possibly due to the irregular shape of the arteries as well as formation of plaque. It has been established that once a mild stenosis is developed, the altered haemodynamics may further influence the development of blockage. Flow through arteries also becomes complicated due to formation of aneurysm, a balloon like dilatation, found on the walls of a vessel where it has been weakened. Aneurysms are usually seen in arteries such as cerebral, carotid, thoracic, renal, abdominal, iliac, femoral, bronchial etc. It grows gradually as time elapses and grows faster as it becomes larger. It triggers the thrombus formation, extent of which has a correlation with the rate of growth of aneurysm and also with the degradation of vessel wall. Numerical investigations under steady and pulsatile flow conditions have been performed to calculate the flow quantities in particular, the computation of wall shear stress. Previous investigations had shown that pulsatile flows could essentially be controlled by two dimensionless parameters, the time mean Reynolds number and the Womersley number and might also be dependent on the flow rate (Budwig, Egelhoff, & Tavoularis, 1997; Tavoularis & Singh, 1999). The nonlinear separated vorticity modifies the boundary layer structure and its region of separation eventually changes the whole flow dynamics. The dynamics of this kind of steady and pulsatile flow phenomena and corresponding flow separation have been studied in detail by Pedrizzetti (1996) for homogeneous fluid. Recently, Layek and Mukhopadhyay (2008) investigated the effects of sudden smooth expansion on laminar flow in a circular tube in primitive variable approach based on Marker-and-Cell (MAC) method. But in their study, pulsatile nature of blood as well as the variable blood viscosity was disregarded. So, the objective of this study is to explore the combined effects of pulsatile flow conditions and variable viscosity due to the variation of haematocrit concentration in the flowing blood on flow through a tube having axi-symmetric sudden smooth expansion. A simplified quantitative analysis has been made taking into account blood viscosity to be varied only for haematocrit which is also varied only on the arterial radius. This may indicate qualitative analysis of blood flow dynamics with potential implications.
2. Mathematical model for flow problem The blood viscosity is mainly influenced by three factors: haematocrit, temperature and shear rate. High haematocrit can occur in arterial flow due to a variety of causes, for example: dehydration, polycythemia vera and exogeneous use of recombinant human erythropoietin (Stack & Berger, 2009 and reference therein). The red blood cells are the dominant contributor to the viscosity of blood. Blood becomes thicken significantly for greater haematocrit levels. As a result its rate of flow throughout the body becomes slow. This can increase the risk of tissue infarction. As haematocrit level increases due to several diseases, the blood viscosity also increases rapidly. Here blood is treated as an incompressible, inhomogeneous fluid and the viscosity is considered to be varied with the variation of haematocrit only. Lih (1975) proposed a mathematical relation for blood viscosity over the circular cross section of the vascular tube as below:
l ðr Þ ¼ lp 1 þ k 1
r R0
n ;
ð1Þ
where k is a constant depending on the value of the haematocrit at the core flow region of the tube, lp is the viscosity of plasma (assumed to be constant), n is a parameter determining the shape of the cell distribution in the flowing blood and R0 is the radius of the tube in the unexpanded portion. This viscosity variation describes a fluid that is inhomogeneous, and assumes that the streamlines follow the contour of the expansion during steady flow. Although, this assumption does not hold for pulsatile flow (as we will show in the results: refer to Fig. 11), it is a good starting point for an analysis on the pulsatile flow of an inhomogeneous fluid. Formula (1) indicates that viscosity increases as one moves from the wall towards the centre of the vascular tube where it has maximum value (Fig. 1).
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2 1.8
Viscosity
1.6 1.4 1.2 1
k =0.85,0.45,0.3,0.2,0 0.8
0
0.2
0.4
r
0.6
0.8
1
Fig. 1. Viscosity variation for different values of the haematocrit parameter k.
3. Shape of the smooth expansion Flows through sudden expansions are frequently encountered and are very interesting from practical point of view. Although the flow is complex, typically exhibiting separation, recirculation and reattachment regions, the detail description of flow features can provide an explanation of the post surgical complications occurring in some carotid arteries. For this reason, the shape of the smooth expansion has been chosen to be that of an operated human carotid and defined mathematically as (Pedrizzetti, 1996)
r0 ðzÞ ¼ 1 þ
d z 1 þ tanh ; 2 d
ð2Þ
where r0(z) denotes the radius of the tube in expanded region. Here d is the height of the expansion. A schematic diagram of the expanded tube considered in this analysis is given in Fig. 2. 4. Flow descriptions and governing equations The pulsatile flow of an incompressible viscous fluid, with constant density q and variable viscosity l⁄(r⁄) in an axi-symmetric vascular tube (modeled as artery) is considered. A smooth local expansion is mounted from the point z = z0. We assume the axis of the tube as the z-axis of a cylindrical system of coordinates (r⁄, z⁄, h⁄). The axial symmetry makes the flow independent of angular coordinate h⁄. Let R0 be the radius of the tube in the unexpanded portion and r0 ðz Þ defines the wall of the vascular tube. For pulsatile flow condition, the mean velocity at the inlet will be time-dependent. The following dimensionless quantities in this axi-symmetric two-dimensional unsteady flow are given as
z ¼ z =R0 ; r ¼ r =R0 ; r 0 ¼ r0 =R0 ; u ¼ u =U 0 ;
v ¼ v =U0 ;
t ¼ t =T; p ¼ p =qU 20 ;
ð3Þ
where p is the pressure, u and v are the velocity components along z and r -axes, respectively. The unsteady two dimensional Navier–Stokes equations of an incompressible fluid with variable viscosity may be written in dimensionless form as: ⁄
⁄
⁄
⁄
⁄
Fig. 2. Physical description of the flow problem (depicting dimensionless quantities).
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St
ou ouv ou2 uv op 1 þ þ þ ¼ þ ot oz Re or oz r
"
l1
( ) # o2 u 1 ou o2 u ol1 ou ov þ þ þ þ or 2 r or oz2 or or oz
ð4Þ
and
ov ov 2 ouv v 2 op 1 St þ þ þ ¼ þ or Re ot or oz r r
"
# ( ) o2 v 1 ov o2 v v ol1 ov l1 þ þ ; þ2 or 2 r or oz2 r 2 or or
ou ov r þ ¼ 0; oz or
ð5Þ
ð6Þ
and the variation of fluid viscosity (Eq. (1)) in terms of dimensionless coordinate becomes
l1 ¼ 1 þ kð1 rn Þ:
ð7Þ
Here the dimensionless parameter Re = U0R0/m is the flow Reynolds number, St = R0/U0T is the Strouhal number and k is the haematocrit parameter. The dimensionless Stokes stream function w(z, r, t) is defined as follows:
u¼
1 ow ; r or
v ¼
1 ow r oz
ð8Þ
and the corresponding azimuthal vorticity function x(z, r, t) as
x¼
ov ou : oz or
ð9Þ
Using (8), (9) and by cross-differentiation of the momentum equations (4) and (5), the pressure term is eliminated and we have the usual coupled equations for stream function and vorticity transport as
xr ¼
St
o2 w o2 w 1 ow ; þ oz2 or 2 r or
ox ox ox v x 1 þu þv ¼ Re ot oz or r
ð10Þ "
l1
! # o2 x o2 x 1 ox x ol1 ox x o2 l1 ou ov þ þ þ 2 þ : þ r or r 2 oz2 or 2 or or r or2 or oz
ð11Þ
At the inlet of the tube, the flow is assumed to be fully developed, that is, o x/ o z = o w/ o z = 0 and at the outlet, we have considered the flow field has no change which gives o 2x/ o z2 = o 2w/ o z2 = 0. A time dependent flow rate Q(t) is given at the upper wall of the tube. The flow symmetry gives the following conditions
w ¼ 0; x ¼ 0 at r ¼ 0:
ð12Þ
The conditions of ‘no slip’ at the tube wall requires that
ow=oz ¼ 0 ¼ ow=or
at r ¼ r0 :
ð13Þ
The stream function w at r = r0(z) (the tube wall) is
wðtÞ ¼ QðtÞ=2p;
ð14Þ
where the non-dimensional Q ðtÞ ¼ p2 ð1 þ sinð2ptÞÞ gives w(t) = 0.25(1 + sin 2pt). The vanishing normal velocity component at the tube wall gives that the stream function is constant along the wall at particular instant of time and also the zero value of the tangential velocity implies that the first order normal derivative and the second order mixed derivative of the stream function are zero (Batchelor, 1967; Pedrizzetti, 1996). These conditions of the stream function when used in Eq. (10) provide the wall vorticity at the upper wall. Using these conditions for stream function, the wall vorticity at r = r0(z) can be obtained which is most crucial in finding the flow quantities of the vorticitystream function formulation. A transformation of coordinate given by R = r/r0(z) is used. The equations for stream function (10) and the vorticity transport equation (11) are transformed to the new coordinate (R, z, t) and the transformed equations are given by
" # " # 2 2 o2 w R2 or 0 1 o2 w 2R or 0 o2 w 2R or0 R o2 r 0 1 ow þ ¼ Rr0 x; þ þ oz2 r 0 oz2 Rr 20 oR r20 oR2 r0 oz oRoz r 20 oz r 20 oz
ð15Þ
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St
" ! 2 2 ox ox R or 0 ox v ox v x 1 o2 x R or0 1 o2 x 2R or 0 o2 x ¼ l1 þ þ þu þ r0 r 0 oz oRoz ot oz r 0 oz oR r 0 oR Rr0 Re oz2 oz r 20 oR2 ! ! 2 # 2 2 2R or 0 R o r0 1 ox x 1 ol1 2 ox x 1 o l1 1 ou ov R or 0 ov þ þ þ þ 2 þ : r0 oz2 Rr20 oR R2 r 20 r 0 oR r 0 oR r 0 R r20 oz r 0 oR2 r 0 oR oz r 0 oz oR
ð16Þ
The transformed boundary conditions on the stream function w and vorticity x at R = 1 become
wðz; R ¼ 1; tÞ ¼ 0:25ð1 þ sin 2ptÞ;
ð17Þ
" 2 # 2 ! 1 or 0 o w xðz; R ¼ 1; tÞ ¼ 3 1 þ : oz r0 oR2 at R¼1
ð18Þ
5. Numerical method The transformed governing equations together with the initial and boundary conditions are solved numerically by using finite difference technique. A very efficient implicit technique, viz. Alternating Direction Implicit (ADI) method has been used to solve parabolic type vorticity transport equation. The elliptic stream function equation is discretised using central difference formula and the algebraic system is solved by using SLOR algorithm (Successive Line Over Relaxation Method) (Peyret & Taylor, 1982). The detail of derivations of the numerical discretization and stability can be found in Bandyopadhyay and Layek (2011). The grid-size Dz and DR and the selection of time step Dt are determined using usual numerical stability conditions, viz., CFL condition and restriction relative to viscous effects (Bandyopadhyay & Layek, 2011) Grid independence test has been carried out by taking the grid sizes as (0.075 0.05), (0.05 0.05), (0.05 0.02). Typical values of Dt = 0.0001 for the grid-size Dz = 0.05 and DR = 0.05 for Re = 100 to 1200 and St = 0.03 to 0.07 in pulsatile flow conditions are considered. 6. Results and discussions We now pay our attention for analyzing the flow characteristics in the region of expansion and the corresponding relations to the arterial diseases. The results obtained from this study may be useful to scientific community working on blood flow in the diseased artery in human cardio-vascular system. The important flow quantities like velocity components and wall shear stresses are computed using the developed numerical code based on above mentioned numerical method. It is noted that no significant differences in flow quantities are found. Finally, we have chosen the grid size (0.05 0.05) for the present computation. We are mainly interested to study the effects of haematocrit on blood flow under pulsatile conditions. For verification of the present code, the results obtained from this scheme based on stream function-vorticity are compared with the results obtained by Layek and Mukhopadhyay (2008) using Marker and Cell (MAC) method in primitive variable formulation and the comparison of wall shear stresses is displayed in Fig. 3. It is noticed that the wall shear stress curves match well. The velocity profiles in steady flow condition for Re = 400, d = 0.25, n = 2 at the axial position z = 1 for
3 2.5
wall shear stress
2
Study of Layek and Mukhopadhyay (2008)
1.5 1
k=0.85,0.45,0
0.5 0
−0.5 −1 −5
0
5
10
15
20
z Fig. 3. Comparison of wall shear stress distributions at steady flow rate for the Reynolds number (Re) = 400, expansion height = 0.25 with the result of Layek and Mukhopadhyay (2008).
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1.2 1 0.8
r
0.6 0.4
k=0,0.45,0.85
0.2 −0.2
0
0.2
u
0.4
0.6
0.8
1
Fig. 4. The velocity profile for steady flow conditions for the Reynolds number (Re) = 400 at the axial position z = 1 for d = 0.25, n = 2.
(a)
(b)
(c)
(d)
Fig. 5. Distribution of wall shear stress for pulsatile flow conditions for the Strouhal number (St) = 0.05, Reynolds number (Re) = 200, n = 2, d = 1 at (a) t = 0.25, (b) t = 0.50, (c) t = 0.75 and (d) t = 1.0.
different values of the haematocrit parameter (k) have been plotted in Fig. 4. It has been observed that the velocity increases and flattened with the increasing values of the haematocrit parameter (k). The arterial wall shear stress, in particular, oscillatory pattern of stress plays very crucial role in the formation and further development of arterial diseases (Zarins et al., 1983). Proper estimation of arterial stress is very difficult in this complex flow. No reliable method seems to be available for estimating wall shear stress. The numerical simulation, in this situation, can provide some insights. Fig. 3 illustrates the results of wall shear stress in steady flow condition for Re = 400, d = 0.25, n = 2 and various values of the haematocrit parameter (k). The value of wall shear stress and the corresponding length of flow separation increases significantly when the viscosity increases about the mean. Figs. 5(a)–(d) display the pattern of wall shear stress distribution in pulsatile flow conditions at different dimensionless times, say t = 0.25, 0.50, 0.75, 1.0. The magnitude of the peak value of wall shear stress increases at each time period as the haematocrit parameter (k) increases (that is the increase of the value of blood viscosity in the radial direction starting from the vessel wall). High wall shear stress may
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Fig. 6. Distribution of wall shear stress for pulsatile flow conditions for different values of expansion height (d), Strouhal number (St) = 0.05 and for the Reynolds number (Re) = 200 at t = 0.25, n = 2, k = 0.25.
wall shear stress
6
4
2
0
n =1 n =2 n =3
−2
−4 −5
0
5
z
10
15
Fig. 7. Distribution of wall shear stress for pulsatile flow conditions for different values of n, Strouhal number (St) = 0.05 and for Reynolds number (Re) = 200 at t = 0.25, d = 1, k = 0.25.
wall shear stress
15
Re=100 Re=300 Re=700 Re=1200
10
5
0
−5
−10 −5
0
z
5
10
15
Fig. 8. Distribution of wall shear stress for pulsatile flow conditions for different Reynolds number (Re) and for Strouhal number (St) = 0.05 at t = 0.25, d = 1, k = 0.25.
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Fig. 9. Distribution of wall shear stress for pulsatile flow conditions for different Strouhal number (St) and for the Reynolds number (Re) = 200 at t = 0.25, n = 2, d = 1, k = 0.25.
(a)
(b)
(c)
(d)
Fig. 10. The velocity profile for pulsatile flow conditions for the Strouhal number (St) = 0.05, Reynolds number (Re) = 200, n = 2, d = 1 at (a) t = 0.25, (b) t = 0.50, (c) t = 0.75 and (d) t = 1.0.
damage the vessel wall causing the intimal thickening. The places of zero vorticity indicate the locations of stagnation points as well as separation and reattachment points. It is noted that in the vicinity of the throat, the wall shear stress changes in sign in the down stream. This implies that a flow separation takes place and length of separation varies with time. At the peak flow time (i.e., t = 0.25), the wall shear stress takes the maximum negative peak and the separating region begin to grow. The variations of wall shear stresses and the corresponding formation of separating region may damage the endothelial layer of the artery. At t = 0.50, the maximum negative value decreases and the flow separation region grows continuously. An idea about the size of the recirculation eddies can be obtained from the negative shear stress regions i.e. the region of reversal flow. This is of pathological significance since the recirculation regions are the regions of low shear and may prolong the residing time of blood constituents. The plaque formation in an artery develops in the regions of low arterial wall shear stress. At t = 0.75, the wall shear stress is negative almost everywhere and the flow separation region increases. At
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M.S. Mandal et al. / International Journal of Engineering Science 57 (2012) 1–10 k=0
t =0.25 −2
0
2
4
6
8
10
12
t =0.50 −2
0
2
4
6
8
10
12
t =0.75 −2
0
2
4
6
k=0.3
k =0.2
8
10
12
2 1.5 1 0.5 0
−2
0
2
4
6
8
10
12
2 1.5 1 0.5 0
2 1.5 1 0.5 0 −2
2 1.5 1 0.5 0
−2
0
0
2
2
4
4
6
6
8
8
10
10
12
12
2 1.5 1 0.5 0
2 1.5 1 0.5 0
−2
−2
−2
0
0
0
2
2
2
4
4
4
6
6
6
8
8
8
10
10
10
12
12
12
2 1.5 1 0.5 0
2 1.5 1 0.5 0
2 1.5 1 0.5 0
Fig. 11. Pattern of stream lines in pulsatile flow conditions for the Reynolds number (Re) = 200, Strouhal number (St) = 0.05, n = 2 and d = 1 at different instant.
time t = 1, as the flow reaches its reattachment point, the value of wall shear stress gradually becomes positive in sign. Also the peak value of wall shear stress near the throat of the expansion region is much larger in pulsatile case than that of a steady case. In order to estimate the effect of expansion height (d) and the cell distribution parameter (n) of blood viscosity model, we like to present the results for the distributions of the wall shear stresses at time t = 0.25. Fig. 6 shows that variations of expansion height (d) have a little effect on wall shear stress. The peak value decreases with increasing values of expansion height (d). In Fig. 7, it is noticed that the peak value of wall shear stress increases to maximum negative value with increasing value of n. Fig. 8 illustrates the results of the distribution of the wall shear stress for different values of Reynolds number (Re) at time t = 0.25. It is observed that the magnitude of the peak values of wall shear stress increases with increasing values of Reynolds number (Re), but the length of separation decreases. Flow pulsation in terms of Strouhal number (St) has a significant impact on wall shear stress distribution which is given in Fig. 9. It reveals that the magnitude of the peak value of wall shear stress and the corresponding length of separation decreases with increasing value of Strouhal number (St). The peak of the shear stress is believed to cause severe damage to the arterial lumen which in turn helps in detecting the aggregation sites of platelets and may have several consequences in circulatory system. The fluctuation level in case of wall shear stress is noted for St = 0.03 and is of considerable interest. The variable shear stress can prevent endothelial cells from aligning in the direction of the flow and thereby making the intima more permeable to the entry of monocytes and lipoproteins. Atherosclerotic lesions are associated with oscillatory wall shear stress. The velocity profiles are of great interest since they can provide a detailed description of the flow field. Next, u-velocity profiles are plotted in a particular axial position at various times, viz. t = 0.25, 0.5, 0.75, 1 and also for different values of haematocrit parameter (k). The reversal flows in Figs. 10(a)–(d) clearly indicate the existence of flow separation region and reattachment points. Next, stream lines in pulsatile flow at different instants t = 0.25, 0.50, 0.75 for different values of haematocrit parameter (k) and for fixed value of Reynolds number (Re) = 200, Strouhal number (St) = 0.05 are presented in Figs. 11(a)–(c). It is clear from the figures that the larger vortices occur with increasing values of k for all times i.e. for t = 0.25, 0.5, 0.75.
7. Conclusions The pulsatile flow in a circular tube with a smooth expansion has been carried out numerically using finite-difference method in stream-function vorticity formulation taking into account that the fluid viscosity to be varied in the radial direction only due to the concentration of red cells in the flowing blood. The results obtained from this numerical code agreed well with the results in the reference (Layek & Mukhopadhyay, 2008) which is based on primitive variable formulation. Hence the present model would certainly provide better insight into the complex flow phenomena. Based on the predicted results the following conclusions can be drawn: (i) Viscosity increases with increasing haematocrit parameter k. (ii) Wall shear stress increases with increasing values of k (for fixed n) and for increasing n (for fixed k) for both steady and pulsatile flows. (iii) Under pulsatile flow conditions, flow separation region increases with time. (iv) Peak value of wall shear stress decreases with increasing height of expansion. (v) Length of separation decreases with increasing Reynolds number (Re) and Strouhal number (St). (vi) The velocities in the region of expansion can further complicate the downstream flow dynamics.
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Finally it can be concluded that potential improvement over the previous models has been made by incorporating pulsatile nature of blood flow and the haematocrit effect. Moreover, it is believed that the findings are very important in diagnosing the disease in its earlier stage. Acknowledgement Thanks are due to the honourable reviewers for their constructive suggestions which helped a lot for the improvement of the quality of the paper. References Anand, M., Rajagopal, K., & Rajagopal, K. R. (2003). A model incorporating some of the mechanical and biomechanical factors underlying clot formation and dissolution in flowing blood. Journal of Theo. Medicine, 5(3–4), 183–218. Bandyopadhyay, S., & Layek, G. C. (2011). Numerical computation of pulsatile flow through a locally constricted channel. Communion of Nonlinear Science Numerical and Simulation, 16, 252–265. Batchelor, G. K. (1967). An introduction to fluid dynamics. Cambridge University Press. Budwig, R., Egelhoff, C. L., & Tavoularis, S. (1997). Laminar pulsatile flow through an axisymmetric sudden expansion. ASME Journal of Fluids Engineering, 119, 208–211. Cinar, Y., Demir, G., Pac, M., & Cinar, A. B. (1999). Effect of hematocrit on blood pressure via hyperviscosity. American Journal of Hypertension, 12, 739–743. Demiray, H. (2008). Weakly nonlinear waves in a fluid with variable viscosity contained in a prestressed thin elastic tube. Chaos, Solutions Fractals, 36, 196–202. Dryden, H. L., Murnaghan, F. P., & Bateman, H. (1956). Hydrodynamics, Dover Publications, Inc. Jandl, J. H. (1996). Blood: Textbook of hematology (p. 53). Little, Brown and Company, Boston. Ku, D. N. (1997). Blood flow in arteries. Annual Review of Fluid Mechanics, 29, 399–434. Layek, G. C., & Mukhopadhyay, S. (2008). Laminar flow separation in an axi-symmetric sudden smooth expanded circular tube. Journal of Applied Mathematics and Computing, 28, 235–247. Lih, M. M. (1975). Transport phenomena in medicine and biology. New York: John Wiley, pp. 378–414. Nakamura, M., & Sawada, T. (1988). Numerical study on flow of non-Newtonian fluid through axi-symmetric stenosis. J. Biomech. Engg., 110, 247–262. Oka, S. (1981). Cardiovascular hemorheology. Cambridge, London: Cambridge University Press, pp. 28. Pedrizzetti, G. (1996). Unsteady tube flow over an expansion. Journal of Fluid Mechanics, 310, 89–111. Peyret, R., Taylor, T. D. (1982). Computational methods for fluid flow. Springer-Verlag. Stack, S. W., & Berger, S. A. (2009). The effects of high hematocrit on arterial flow- A phenomenological study of the health risk implications. Chemical Engineering Science, 64, 4701–4706. Tavoularis, S., & Singh, R. K. (1999). Vortex detachment and reverse flow in pulsatile laminar flow through axisymmetric sudden expansion. ASME Journal of Fluids Engineering, 121, 574–579. Zarins, Christopher K., Giddens, Don P., Bharadvaj, B. K., Sottiurai, Vikrom S., Mabon, Robert F., Glagov, Seymour, (1983). Carotid bifurcation atherosclerosis quantitative correlation of plaque localization with flow velocity profiles and wall shear stress, Circulation Research 53, 502–514.
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Pulsatile flow of an incompressible, inhomogeneous fluid in a smoothly expanded vascular tube Mani Shankar Mandal a,⇑, Swati Mukhopadhyay b, G.C. Layek b a b
Department of Mathematics, Krishnagar Govt. College, Krishnagar, Nadia 741101, India Department of Mathematics, The University of Burdwan, Burdwan 713104, India
a r t i c l e
i n f o
Article history: Received 30 August 2011 Received in revised form 27 February 2012 Accepted 1 April 2012
Keywords: Pulsatile flow Arterial flow Sudden expansion Haematocrit Blood viscosity Non-linear analysis Finite difference method
a b s t r a c t This paper aims to present pulsatile laminar flow of an incompressible, inhomogeneous fluid in an axi-symmetric smoothly expanded tube, modeled as artery, under some specific flow conditions. The flowing blood is an inhomogeneous fluid due to the presence of haematocrit (the percentage of total blood volume occupied by red blood cells (RBCs)). In the present study, the viscosity of flowing blood is assumed to be dependent on radius of the artery with a maximum value at the centerline of the artery. The governing nonlinear equations along with the appropriate boundary conditions are derived and are solved numerically using finite-difference method. The effects of inhomogeneities on flow quantities along with other flow parameters such as Reynolds number (Re), Strouhal number (St), expansion height (d) have been investigated. The numerical values of wall shear stress agree well with the available results of previously published works. It is seen that the value of wall shear stress and the corresponding length of flow separation increases significantly when the viscosity increases about the mean under both steady and pulsatile flow conditions. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The fluid viscosity is generally regarded as constant for some substances when the variations of pressure and temperature are negligible and is independent of the rate of the shear so long as this rate is not varied much. For certain mixture, emulsions etc., apparent coefficient of viscosity depends on the rate of shear as well as presence of particles (inhomogeneities) and the fluid is not a simple one (Dryden, Murnaghan, & Bateman, 1956). Inhomogeneities of flowing fluid have profound implications on the flow dynamics and have wide applications in geological flows, biological flows and many others. The flowing blood is very complex substance/fluid. Also, blood viscosity changes significantly with rate of shear in the flow. For flows in tubes it varies from zero at the centerline to maximum value at the wall. The endothelium plays a critical role in maintaining blood fluidity by balancing a natural tendency to clot in isolation with a set of counteracting mechanism (Anand, Rajagopal, & Rajagopal, 2003). Understanding of pulsatile fluid motion and the corresponding flow dynamics have important practical applications in branches of engineering and also in human cardiovascular system. The heart pumps blood intermittently through branches, between which cross sectional area may vary considerably. The expansions at branch points and the pulsatile nature of blood flow can form very complicated nature of flow (Ku, 1997). Several physical quantities have proposed in literature for measuring the risk zones in blood vessel. Observations have shown that one reason is the oscillations in blood flow during the
⇑ Corresponding author. Tel.: +91 9434556135; fax: +91 342 2530452. E-mail addresses: [email protected], [email protected] (M.S. Mandal). 0020-7225/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2012.04.002
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diastolic phase at every single heart beat. Thus clear understanding of pulsatile flow characteristics may help in the diagnosing of the arterial diseases. It is true that human blood may be considered as a Newtonian fluid for flow within the heart and the aorta. The Newtonian model of flowing blood is acceptable for high shear rate flow i.e. for flow through larger arteries. But under diseased conditions, blood exhibits remarkably non-Newtonian properties, even in larger arteries (Nakamura & Sawada, 1988). Blood is composed of fluid plasma and formed elements. The formed elements of blood are erythrocytes, leukocytes and platelets. Among them the presence of haematocrit is significant in the total blood volume and differs from person to person. The percentage volume of red cells is called haematocrit and is approximately 40–45% for an adult as reported by Oka (Oka, 1981). The typical ranges are 41.8–49% approximately for males and 38.6–45.6% for females (Jandl, 1996). In general, blood may be considered as incompressible and inhomogeneous fluid and the inhomogenity is mainly the result of haematocrit (Demiray, 2008). However, in the course of flow in larger arteries, the red blood cells in the vicinity of arterial wall move to the central region of the artery so that the haematocrit becomes quite low near the arterial wall, which results in lower viscosity in this region. Moreover, due to high shear rate near the arterial wall, the viscosity of blood is further reduced. The dependence of blood viscosity on increasing haematocrit was measured experimentally by Cinar, Demir, Pac, and Cinar (1999). From the data it has been shown that as haematocrit increases from 35% to 80%, viscosity increases from 4 to 14 times the viscosity of distilled water. The volume flow rate in comparison with Poiseuill’s law will vary significantly with the increasing level of haematocrit. Blood flows, particularly within the arteries, are sometimes disturbed possibly due to the irregular shape of the arteries as well as formation of plaque. It has been established that once a mild stenosis is developed, the altered haemodynamics may further influence the development of blockage. Flow through arteries also becomes complicated due to formation of aneurysm, a balloon like dilatation, found on the walls of a vessel where it has been weakened. Aneurysms are usually seen in arteries such as cerebral, carotid, thoracic, renal, abdominal, iliac, femoral, bronchial etc. It grows gradually as time elapses and grows faster as it becomes larger. It triggers the thrombus formation, extent of which has a correlation with the rate of growth of aneurysm and also with the degradation of vessel wall. Numerical investigations under steady and pulsatile flow conditions have been performed to calculate the flow quantities in particular, the computation of wall shear stress. Previous investigations had shown that pulsatile flows could essentially be controlled by two dimensionless parameters, the time mean Reynolds number and the Womersley number and might also be dependent on the flow rate (Budwig, Egelhoff, & Tavoularis, 1997; Tavoularis & Singh, 1999). The nonlinear separated vorticity modifies the boundary layer structure and its region of separation eventually changes the whole flow dynamics. The dynamics of this kind of steady and pulsatile flow phenomena and corresponding flow separation have been studied in detail by Pedrizzetti (1996) for homogeneous fluid. Recently, Layek and Mukhopadhyay (2008) investigated the effects of sudden smooth expansion on laminar flow in a circular tube in primitive variable approach based on Marker-and-Cell (MAC) method. But in their study, pulsatile nature of blood as well as the variable blood viscosity was disregarded. So, the objective of this study is to explore the combined effects of pulsatile flow conditions and variable viscosity due to the variation of haematocrit concentration in the flowing blood on flow through a tube having axi-symmetric sudden smooth expansion. A simplified quantitative analysis has been made taking into account blood viscosity to be varied only for haematocrit which is also varied only on the arterial radius. This may indicate qualitative analysis of blood flow dynamics with potential implications.
2. Mathematical model for flow problem The blood viscosity is mainly influenced by three factors: haematocrit, temperature and shear rate. High haematocrit can occur in arterial flow due to a variety of causes, for example: dehydration, polycythemia vera and exogeneous use of recombinant human erythropoietin (Stack & Berger, 2009 and reference therein). The red blood cells are the dominant contributor to the viscosity of blood. Blood becomes thicken significantly for greater haematocrit levels. As a result its rate of flow throughout the body becomes slow. This can increase the risk of tissue infarction. As haematocrit level increases due to several diseases, the blood viscosity also increases rapidly. Here blood is treated as an incompressible, inhomogeneous fluid and the viscosity is considered to be varied with the variation of haematocrit only. Lih (1975) proposed a mathematical relation for blood viscosity over the circular cross section of the vascular tube as below:
l ðr Þ ¼ lp 1 þ k 1
r R0
n ;
ð1Þ
where k is a constant depending on the value of the haematocrit at the core flow region of the tube, lp is the viscosity of plasma (assumed to be constant), n is a parameter determining the shape of the cell distribution in the flowing blood and R0 is the radius of the tube in the unexpanded portion. This viscosity variation describes a fluid that is inhomogeneous, and assumes that the streamlines follow the contour of the expansion during steady flow. Although, this assumption does not hold for pulsatile flow (as we will show in the results: refer to Fig. 11), it is a good starting point for an analysis on the pulsatile flow of an inhomogeneous fluid. Formula (1) indicates that viscosity increases as one moves from the wall towards the centre of the vascular tube where it has maximum value (Fig. 1).
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2 1.8
Viscosity
1.6 1.4 1.2 1
k =0.85,0.45,0.3,0.2,0 0.8
0
0.2
0.4
r
0.6
0.8
1
Fig. 1. Viscosity variation for different values of the haematocrit parameter k.
3. Shape of the smooth expansion Flows through sudden expansions are frequently encountered and are very interesting from practical point of view. Although the flow is complex, typically exhibiting separation, recirculation and reattachment regions, the detail description of flow features can provide an explanation of the post surgical complications occurring in some carotid arteries. For this reason, the shape of the smooth expansion has been chosen to be that of an operated human carotid and defined mathematically as (Pedrizzetti, 1996)
r0 ðzÞ ¼ 1 þ
d z 1 þ tanh ; 2 d
ð2Þ
where r0(z) denotes the radius of the tube in expanded region. Here d is the height of the expansion. A schematic diagram of the expanded tube considered in this analysis is given in Fig. 2. 4. Flow descriptions and governing equations The pulsatile flow of an incompressible viscous fluid, with constant density q and variable viscosity l⁄(r⁄) in an axi-symmetric vascular tube (modeled as artery) is considered. A smooth local expansion is mounted from the point z = z0. We assume the axis of the tube as the z-axis of a cylindrical system of coordinates (r⁄, z⁄, h⁄). The axial symmetry makes the flow independent of angular coordinate h⁄. Let R0 be the radius of the tube in the unexpanded portion and r0 ðz Þ defines the wall of the vascular tube. For pulsatile flow condition, the mean velocity at the inlet will be time-dependent. The following dimensionless quantities in this axi-symmetric two-dimensional unsteady flow are given as
z ¼ z =R0 ; r ¼ r =R0 ; r 0 ¼ r0 =R0 ; u ¼ u =U 0 ;
v ¼ v =U0 ;
t ¼ t =T; p ¼ p =qU 20 ;
ð3Þ
where p is the pressure, u and v are the velocity components along z and r -axes, respectively. The unsteady two dimensional Navier–Stokes equations of an incompressible fluid with variable viscosity may be written in dimensionless form as: ⁄
⁄
⁄
⁄
⁄
Fig. 2. Physical description of the flow problem (depicting dimensionless quantities).
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St
ou ouv ou2 uv op 1 þ þ þ ¼ þ ot oz Re or oz r
"
l1
( ) # o2 u 1 ou o2 u ol1 ou ov þ þ þ þ or 2 r or oz2 or or oz
ð4Þ
and
ov ov 2 ouv v 2 op 1 St þ þ þ ¼ þ or Re ot or oz r r
"
# ( ) o2 v 1 ov o2 v v ol1 ov l1 þ þ ; þ2 or 2 r or oz2 r 2 or or
ou ov r þ ¼ 0; oz or
ð5Þ
ð6Þ
and the variation of fluid viscosity (Eq. (1)) in terms of dimensionless coordinate becomes
l1 ¼ 1 þ kð1 rn Þ:
ð7Þ
Here the dimensionless parameter Re = U0R0/m is the flow Reynolds number, St = R0/U0T is the Strouhal number and k is the haematocrit parameter. The dimensionless Stokes stream function w(z, r, t) is defined as follows:
u¼
1 ow ; r or
v ¼
1 ow r oz
ð8Þ
and the corresponding azimuthal vorticity function x(z, r, t) as
x¼
ov ou : oz or
ð9Þ
Using (8), (9) and by cross-differentiation of the momentum equations (4) and (5), the pressure term is eliminated and we have the usual coupled equations for stream function and vorticity transport as
xr ¼
St
o2 w o2 w 1 ow ; þ oz2 or 2 r or
ox ox ox v x 1 þu þv ¼ Re ot oz or r
ð10Þ "
l1
! # o2 x o2 x 1 ox x ol1 ox x o2 l1 ou ov þ þ þ 2 þ : þ r or r 2 oz2 or 2 or or r or2 or oz
ð11Þ
At the inlet of the tube, the flow is assumed to be fully developed, that is, o x/ o z = o w/ o z = 0 and at the outlet, we have considered the flow field has no change which gives o 2x/ o z2 = o 2w/ o z2 = 0. A time dependent flow rate Q(t) is given at the upper wall of the tube. The flow symmetry gives the following conditions
w ¼ 0; x ¼ 0 at r ¼ 0:
ð12Þ
The conditions of ‘no slip’ at the tube wall requires that
ow=oz ¼ 0 ¼ ow=or
at r ¼ r0 :
ð13Þ
The stream function w at r = r0(z) (the tube wall) is
wðtÞ ¼ QðtÞ=2p;
ð14Þ
where the non-dimensional Q ðtÞ ¼ p2 ð1 þ sinð2ptÞÞ gives w(t) = 0.25(1 + sin 2pt). The vanishing normal velocity component at the tube wall gives that the stream function is constant along the wall at particular instant of time and also the zero value of the tangential velocity implies that the first order normal derivative and the second order mixed derivative of the stream function are zero (Batchelor, 1967; Pedrizzetti, 1996). These conditions of the stream function when used in Eq. (10) provide the wall vorticity at the upper wall. Using these conditions for stream function, the wall vorticity at r = r0(z) can be obtained which is most crucial in finding the flow quantities of the vorticitystream function formulation. A transformation of coordinate given by R = r/r0(z) is used. The equations for stream function (10) and the vorticity transport equation (11) are transformed to the new coordinate (R, z, t) and the transformed equations are given by
" # " # 2 2 o2 w R2 or 0 1 o2 w 2R or 0 o2 w 2R or0 R o2 r 0 1 ow þ ¼ Rr0 x; þ þ oz2 r 0 oz2 Rr 20 oR r20 oR2 r0 oz oRoz r 20 oz r 20 oz
ð15Þ
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St
" ! 2 2 ox ox R or 0 ox v ox v x 1 o2 x R or0 1 o2 x 2R or 0 o2 x ¼ l1 þ þ þu þ r0 r 0 oz oRoz ot oz r 0 oz oR r 0 oR Rr0 Re oz2 oz r 20 oR2 ! ! 2 # 2 2 2R or 0 R o r0 1 ox x 1 ol1 2 ox x 1 o l1 1 ou ov R or 0 ov þ þ þ þ 2 þ : r0 oz2 Rr20 oR R2 r 20 r 0 oR r 0 oR r 0 R r20 oz r 0 oR2 r 0 oR oz r 0 oz oR
ð16Þ
The transformed boundary conditions on the stream function w and vorticity x at R = 1 become
wðz; R ¼ 1; tÞ ¼ 0:25ð1 þ sin 2ptÞ;
ð17Þ
" 2 # 2 ! 1 or 0 o w xðz; R ¼ 1; tÞ ¼ 3 1 þ : oz r0 oR2 at R¼1
ð18Þ
5. Numerical method The transformed governing equations together with the initial and boundary conditions are solved numerically by using finite difference technique. A very efficient implicit technique, viz. Alternating Direction Implicit (ADI) method has been used to solve parabolic type vorticity transport equation. The elliptic stream function equation is discretised using central difference formula and the algebraic system is solved by using SLOR algorithm (Successive Line Over Relaxation Method) (Peyret & Taylor, 1982). The detail of derivations of the numerical discretization and stability can be found in Bandyopadhyay and Layek (2011). The grid-size Dz and DR and the selection of time step Dt are determined using usual numerical stability conditions, viz., CFL condition and restriction relative to viscous effects (Bandyopadhyay & Layek, 2011) Grid independence test has been carried out by taking the grid sizes as (0.075 0.05), (0.05 0.05), (0.05 0.02). Typical values of Dt = 0.0001 for the grid-size Dz = 0.05 and DR = 0.05 for Re = 100 to 1200 and St = 0.03 to 0.07 in pulsatile flow conditions are considered. 6. Results and discussions We now pay our attention for analyzing the flow characteristics in the region of expansion and the corresponding relations to the arterial diseases. The results obtained from this study may be useful to scientific community working on blood flow in the diseased artery in human cardio-vascular system. The important flow quantities like velocity components and wall shear stresses are computed using the developed numerical code based on above mentioned numerical method. It is noted that no significant differences in flow quantities are found. Finally, we have chosen the grid size (0.05 0.05) for the present computation. We are mainly interested to study the effects of haematocrit on blood flow under pulsatile conditions. For verification of the present code, the results obtained from this scheme based on stream function-vorticity are compared with the results obtained by Layek and Mukhopadhyay (2008) using Marker and Cell (MAC) method in primitive variable formulation and the comparison of wall shear stresses is displayed in Fig. 3. It is noticed that the wall shear stress curves match well. The velocity profiles in steady flow condition for Re = 400, d = 0.25, n = 2 at the axial position z = 1 for
3 2.5
wall shear stress
2
Study of Layek and Mukhopadhyay (2008)
1.5 1
k=0.85,0.45,0
0.5 0
−0.5 −1 −5
0
5
10
15
20
z Fig. 3. Comparison of wall shear stress distributions at steady flow rate for the Reynolds number (Re) = 400, expansion height = 0.25 with the result of Layek and Mukhopadhyay (2008).
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1.2 1 0.8
r
0.6 0.4
k=0,0.45,0.85
0.2 −0.2
0
0.2
u
0.4
0.6
0.8
1
Fig. 4. The velocity profile for steady flow conditions for the Reynolds number (Re) = 400 at the axial position z = 1 for d = 0.25, n = 2.
(a)
(b)
(c)
(d)
Fig. 5. Distribution of wall shear stress for pulsatile flow conditions for the Strouhal number (St) = 0.05, Reynolds number (Re) = 200, n = 2, d = 1 at (a) t = 0.25, (b) t = 0.50, (c) t = 0.75 and (d) t = 1.0.
different values of the haematocrit parameter (k) have been plotted in Fig. 4. It has been observed that the velocity increases and flattened with the increasing values of the haematocrit parameter (k). The arterial wall shear stress, in particular, oscillatory pattern of stress plays very crucial role in the formation and further development of arterial diseases (Zarins et al., 1983). Proper estimation of arterial stress is very difficult in this complex flow. No reliable method seems to be available for estimating wall shear stress. The numerical simulation, in this situation, can provide some insights. Fig. 3 illustrates the results of wall shear stress in steady flow condition for Re = 400, d = 0.25, n = 2 and various values of the haematocrit parameter (k). The value of wall shear stress and the corresponding length of flow separation increases significantly when the viscosity increases about the mean. Figs. 5(a)–(d) display the pattern of wall shear stress distribution in pulsatile flow conditions at different dimensionless times, say t = 0.25, 0.50, 0.75, 1.0. The magnitude of the peak value of wall shear stress increases at each time period as the haematocrit parameter (k) increases (that is the increase of the value of blood viscosity in the radial direction starting from the vessel wall). High wall shear stress may
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Fig. 6. Distribution of wall shear stress for pulsatile flow conditions for different values of expansion height (d), Strouhal number (St) = 0.05 and for the Reynolds number (Re) = 200 at t = 0.25, n = 2, k = 0.25.
wall shear stress
6
4
2
0
n =1 n =2 n =3
−2
−4 −5
0
5
z
10
15
Fig. 7. Distribution of wall shear stress for pulsatile flow conditions for different values of n, Strouhal number (St) = 0.05 and for Reynolds number (Re) = 200 at t = 0.25, d = 1, k = 0.25.
wall shear stress
15
Re=100 Re=300 Re=700 Re=1200
10
5
0
−5
−10 −5
0
z
5
10
15
Fig. 8. Distribution of wall shear stress for pulsatile flow conditions for different Reynolds number (Re) and for Strouhal number (St) = 0.05 at t = 0.25, d = 1, k = 0.25.
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Fig. 9. Distribution of wall shear stress for pulsatile flow conditions for different Strouhal number (St) and for the Reynolds number (Re) = 200 at t = 0.25, n = 2, d = 1, k = 0.25.
(a)
(b)
(c)
(d)
Fig. 10. The velocity profile for pulsatile flow conditions for the Strouhal number (St) = 0.05, Reynolds number (Re) = 200, n = 2, d = 1 at (a) t = 0.25, (b) t = 0.50, (c) t = 0.75 and (d) t = 1.0.
damage the vessel wall causing the intimal thickening. The places of zero vorticity indicate the locations of stagnation points as well as separation and reattachment points. It is noted that in the vicinity of the throat, the wall shear stress changes in sign in the down stream. This implies that a flow separation takes place and length of separation varies with time. At the peak flow time (i.e., t = 0.25), the wall shear stress takes the maximum negative peak and the separating region begin to grow. The variations of wall shear stresses and the corresponding formation of separating region may damage the endothelial layer of the artery. At t = 0.50, the maximum negative value decreases and the flow separation region grows continuously. An idea about the size of the recirculation eddies can be obtained from the negative shear stress regions i.e. the region of reversal flow. This is of pathological significance since the recirculation regions are the regions of low shear and may prolong the residing time of blood constituents. The plaque formation in an artery develops in the regions of low arterial wall shear stress. At t = 0.75, the wall shear stress is negative almost everywhere and the flow separation region increases. At
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M.S. Mandal et al. / International Journal of Engineering Science 57 (2012) 1–10 k=0
t =0.25 −2
0
2
4
6
8
10
12
t =0.50 −2
0
2
4
6
8
10
12
t =0.75 −2
0
2
4
6
k=0.3
k =0.2
8
10
12
2 1.5 1 0.5 0
−2
0
2
4
6
8
10
12
2 1.5 1 0.5 0
2 1.5 1 0.5 0 −2
2 1.5 1 0.5 0
−2
0
0
2
2
4
4
6
6
8
8
10
10
12
12
2 1.5 1 0.5 0
2 1.5 1 0.5 0
−2
−2
−2
0
0
0
2
2
2
4
4
4
6
6
6
8
8
8
10
10
10
12
12
12
2 1.5 1 0.5 0
2 1.5 1 0.5 0
2 1.5 1 0.5 0
Fig. 11. Pattern of stream lines in pulsatile flow conditions for the Reynolds number (Re) = 200, Strouhal number (St) = 0.05, n = 2 and d = 1 at different instant.
time t = 1, as the flow reaches its reattachment point, the value of wall shear stress gradually becomes positive in sign. Also the peak value of wall shear stress near the throat of the expansion region is much larger in pulsatile case than that of a steady case. In order to estimate the effect of expansion height (d) and the cell distribution parameter (n) of blood viscosity model, we like to present the results for the distributions of the wall shear stresses at time t = 0.25. Fig. 6 shows that variations of expansion height (d) have a little effect on wall shear stress. The peak value decreases with increasing values of expansion height (d). In Fig. 7, it is noticed that the peak value of wall shear stress increases to maximum negative value with increasing value of n. Fig. 8 illustrates the results of the distribution of the wall shear stress for different values of Reynolds number (Re) at time t = 0.25. It is observed that the magnitude of the peak values of wall shear stress increases with increasing values of Reynolds number (Re), but the length of separation decreases. Flow pulsation in terms of Strouhal number (St) has a significant impact on wall shear stress distribution which is given in Fig. 9. It reveals that the magnitude of the peak value of wall shear stress and the corresponding length of separation decreases with increasing value of Strouhal number (St). The peak of the shear stress is believed to cause severe damage to the arterial lumen which in turn helps in detecting the aggregation sites of platelets and may have several consequences in circulatory system. The fluctuation level in case of wall shear stress is noted for St = 0.03 and is of considerable interest. The variable shear stress can prevent endothelial cells from aligning in the direction of the flow and thereby making the intima more permeable to the entry of monocytes and lipoproteins. Atherosclerotic lesions are associated with oscillatory wall shear stress. The velocity profiles are of great interest since they can provide a detailed description of the flow field. Next, u-velocity profiles are plotted in a particular axial position at various times, viz. t = 0.25, 0.5, 0.75, 1 and also for different values of haematocrit parameter (k). The reversal flows in Figs. 10(a)–(d) clearly indicate the existence of flow separation region and reattachment points. Next, stream lines in pulsatile flow at different instants t = 0.25, 0.50, 0.75 for different values of haematocrit parameter (k) and for fixed value of Reynolds number (Re) = 200, Strouhal number (St) = 0.05 are presented in Figs. 11(a)–(c). It is clear from the figures that the larger vortices occur with increasing values of k for all times i.e. for t = 0.25, 0.5, 0.75.
7. Conclusions The pulsatile flow in a circular tube with a smooth expansion has been carried out numerically using finite-difference method in stream-function vorticity formulation taking into account that the fluid viscosity to be varied in the radial direction only due to the concentration of red cells in the flowing blood. The results obtained from this numerical code agreed well with the results in the reference (Layek & Mukhopadhyay, 2008) which is based on primitive variable formulation. Hence the present model would certainly provide better insight into the complex flow phenomena. Based on the predicted results the following conclusions can be drawn: (i) Viscosity increases with increasing haematocrit parameter k. (ii) Wall shear stress increases with increasing values of k (for fixed n) and for increasing n (for fixed k) for both steady and pulsatile flows. (iii) Under pulsatile flow conditions, flow separation region increases with time. (iv) Peak value of wall shear stress decreases with increasing height of expansion. (v) Length of separation decreases with increasing Reynolds number (Re) and Strouhal number (St). (vi) The velocities in the region of expansion can further complicate the downstream flow dynamics.
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Finally it can be concluded that potential improvement over the previous models has been made by incorporating pulsatile nature of blood flow and the haematocrit effect. Moreover, it is believed that the findings are very important in diagnosing the disease in its earlier stage. Acknowledgement Thanks are due to the honourable reviewers for their constructive suggestions which helped a lot for the improvement of the quality of the paper. References Anand, M., Rajagopal, K., & Rajagopal, K. R. (2003). A model incorporating some of the mechanical and biomechanical factors underlying clot formation and dissolution in flowing blood. Journal of Theo. Medicine, 5(3–4), 183–218. Bandyopadhyay, S., & Layek, G. C. (2011). Numerical computation of pulsatile flow through a locally constricted channel. Communion of Nonlinear Science Numerical and Simulation, 16, 252–265. Batchelor, G. K. (1967). An introduction to fluid dynamics. Cambridge University Press. Budwig, R., Egelhoff, C. L., & Tavoularis, S. (1997). Laminar pulsatile flow through an axisymmetric sudden expansion. ASME Journal of Fluids Engineering, 119, 208–211. Cinar, Y., Demir, G., Pac, M., & Cinar, A. B. (1999). Effect of hematocrit on blood pressure via hyperviscosity. American Journal of Hypertension, 12, 739–743. Demiray, H. (2008). Weakly nonlinear waves in a fluid with variable viscosity contained in a prestressed thin elastic tube. Chaos, Solutions Fractals, 36, 196–202. Dryden, H. L., Murnaghan, F. P., & Bateman, H. (1956). Hydrodynamics, Dover Publications, Inc. Jandl, J. H. (1996). Blood: Textbook of hematology (p. 53). Little, Brown and Company, Boston. Ku, D. N. (1997). Blood flow in arteries. Annual Review of Fluid Mechanics, 29, 399–434. Layek, G. C., & Mukhopadhyay, S. (2008). Laminar flow separation in an axi-symmetric sudden smooth expanded circular tube. Journal of Applied Mathematics and Computing, 28, 235–247. Lih, M. M. (1975). Transport phenomena in medicine and biology. New York: John Wiley, pp. 378–414. Nakamura, M., & Sawada, T. (1988). Numerical study on flow of non-Newtonian fluid through axi-symmetric stenosis. J. Biomech. Engg., 110, 247–262. Oka, S. (1981). Cardiovascular hemorheology. Cambridge, London: Cambridge University Press, pp. 28. Pedrizzetti, G. (1996). Unsteady tube flow over an expansion. Journal of Fluid Mechanics, 310, 89–111. Peyret, R., Taylor, T. D. (1982). Computational methods for fluid flow. Springer-Verlag. Stack, S. W., & Berger, S. A. (2009). The effects of high hematocrit on arterial flow- A phenomenological study of the health risk implications. Chemical Engineering Science, 64, 4701–4706. Tavoularis, S., & Singh, R. K. (1999). Vortex detachment and reverse flow in pulsatile laminar flow through axisymmetric sudden expansion. ASME Journal of Fluids Engineering, 121, 574–579. Zarins, Christopher K., Giddens, Don P., Bharadvaj, B. K., Sottiurai, Vikrom S., Mabon, Robert F., Glagov, Seymour, (1983). Carotid bifurcation atherosclerosis quantitative correlation of plaque localization with flow velocity profiles and wall shear stress, Circulation Research 53, 502–514.