Study on the effect of non-Newtonian nature of blood flowing through an elastic artery with slip condition

Abstract

A mathematical model is developed in the analysis for studying blood flow through an elastic artery with the consideration of slip velocity at the inner wall of the artery. Power law fluid model have been utilized in the study to account for the presence of red cells (erythrocytes) in plasma. The governing equations of Power law fluid model is solved analytically with slip and other appropriate boundary conditions. The effect of elastic nature of blood and power index on longitudinal velocity of blood are presented graphically for the model under consideration. The other expressions derived analytically and graphically reveal considerable alterations in flow characteristics due to the presence of elastic property of blood vessel and velocity slip at the wall.

Full text

Study on the effect of non-Newtonian nature of blood
flowing through an elastic artery with slip condition
1Saktipada Nanda, 2B. Basu Mallik, 2Chetna Singh, 2Santanu Das, 3Sayudh Ghosh,
3Shibaprasad Bhattacharya, 3Shyam Sundar Chatterjee
1Department of Electronics and Communication Engineering, Institute of Engineering &
Management, Salt Lake Electronics Complex, Kolkata – 700091. West Bengal, India
2Department of Basic Science & Humanities, Institute of Engineering & Management, Salt Lake
Electronics Complex, Kolkata – 700091. West Bengal, India
3Department of Mechanical Engineering, Institute of Engineering & Management, Salt Lake
Electronics Complex, Kolkata – 700091. West Bengal, India
Abstract: A mathematical model is developed in the analysis for studying blood flow
through an elastic artery with the consideration of slip velocity at the inner wall of the
artery. Power law fluid model have been utilized in the study to account for the
presence of red cells (erythrocytes) in plasma. The governing equations of Power law
fluid model is solved analytically with slip and other appropriate boundary conditions.
The effect of elastic nature of blood and power index on longitudinal velocity of blood
are presented graphically for the model under consideration. The other expressions
derived analytically and graphically reveal considerable alterations in flow
characteristics due to the presence of elastic property of blood vessel and velocity slip
at the wall.
Keywords: Elastic artery, non- Newtonian fluid, velocity slip, fluid index parameter,
velocity profile, shear stress
INTRODUCTION
The cause and development of many arterial diseases are related to the flow characteristics of blood and
the mechanical behaviour of blood vessel walls. A wide variety of investigations have been conducted
based on experimental, analytical and numerical techniques to characterize and explain some
phenomenological aspects of the flow characteristics of blood in the human artery / vein. In most of the
investigations, the Newtonian model of blood (single phase homogenous viscous fluid) were
considered. Experimental observations reveal that blood being predominantly a suspension of
erythrocytes in plasma may be better represented by non-Newtonian model under certain conditions.
The flows i) through narrow arteries ii) at low shear rates iii) in diseased state.
It has been pointed out that the flow behaviour of blood in small diameter tubes (less than 0.2
mm) at less than  shear rate can be reasonably represented by a power law fluid. It is further
revealed that blood may be fairly closely represented by Harschel-Bulkley model at low shear rates
when flowing through a tube of diameter 0.095 mm or less.

The study of anatomy and physiology of an organic system and mechanical behaviour of the
blood vessel walls confirms the elastic property of human artery. Artery may be modelled as an elastic
tube whose radius varies with a pulsating pressure. Due to the ejection of blood by the heart at a certain
velocity, pressure and flow waves are generated and propagated and this phenomenon confirms the
elastic nature of the artery. Also, a number of studies (both analytical and experimental) on the blood
flow have established the presence of slip (velocity discontinuity) at the flow boundaries. As a result of
the presence of slip, the effective viscosity is reduced.
A review of the literature of pulsatile flow of blood in an elastic tube indicate the contributions
of Johnson et. al.1, Gupta and Agarwal 2 , Dash and Mehta 3, Pedrizzetti et. al 4., Mandal 6, Sankar and
Hemlatha 7. In a recent study Kumar,S 11 has considered the non-Newtonian blood flow through an
elastic artery. The analysis is performed under Power law fluid model. The effect of elastic nature of
arterial wall on longitudinal velocity of blood is analysed for both models of blood under consideration.
In all the above mentioned studies, traditional no-slip boundary conditions have been employed.
However, a number of studies (theoretical and experimental) on blood flow, in general, suggested the
likely presence of slip (a velocity discontinuity) at the flow boundaries (or, in their immediate
neighbourhood). Biswas and Chakraborty 10, Verma et. al 12 have developed mathematical models for
blood flow through elastic arterial segment, by taking a velocity slip condition at the constricted wall.
Thus, it seems that consideration of a velocity slip in the study of blood flow modelling will be quite
rational.
It gives us an opportunity to consider the problem of blood flow through an elastic artery. The
non-Newtonian characteristics of blood is described by Power law fluid model. As the effects of
velocity slip on the flow variables are found to be reasonable so the velocity slip condition at the vessel
wall is given due consideration in the analysis. An extensive quantitative analysis is carried out by
performing numerical computations of the desired quantities having more physiological significance to
explain the effects of power index (n) and elastic nature of arterial wall on velocity profile for the model
under consideration. Also, the variation of volume flow rate, shear stress on velocity profile is studied
taking appropriate values of the material constants and other parameters.
MATHEMATICAL FORMULATION OF THE PROBLEM
In the present study a non-Newtonian blood flow is considered through an elastic artery. The artery is
considered as a long cylindrical tube and the axis is taken along the axis of z. The flow is considered to
be axisymmetric, laminar and fully developed. The schematic diagram of the vessel is given in Fig. 1.
Fig.1: Diagram of Arterial segment
p0

As considered by Mazumdar 9, the transmural pressure difference of the blood vessel is taken as
])(.[ 0
pzprTh 
---------------- (1)
where h is the wall thickness, r is the radius of the tube,
0
p
and p are the exterior and interior pressure
respectively, 
0
p
is the transmural pressure difference, T is the tension per unit length and per unit
thickness of the tube.
By the Hooke’s law the tension T is given by

--------------------- (2)
where at  the tension becomes zero at the equilibrium position and E is the Young’s modulus.
From (1) and (2) we have,
 
󰇛󰇛󰇜󰇜
 ------------------------ (3)
The governing equation of motion of a Newtonian fluid in cylindrical co-ordinate system 󰇛󰇜 is
given by
󰇣

󰇡
󰇢󰇤
 ------------ (4)
where µ is viscosity,  is the axial velocity,  the pressure and  the radius of the tube.
By Newton’s law of viscosity

 ------------------- (5)
From eqn. (4) and (5) we have

󰇛󰇜
------------- (6)
Integrating eqn. (6) we have


--------------- (7)
Here  considered to be finite at .
From (7) it is easy to say that .
To find the flow field completely one use Power Law model for shear stress and shear rate relation.
The constitutive equation of Power Law model is given by
󰇛󰇜󰇛
󰇜
n
----------------- (8)

where  is the shear stress,  is the shear strain rate and is the apparent viscosity of the fluid.
The slip velocity is considered at the wall  of the tube i.e. 
METHOD OF SOLUTION
Introducing the non-dimensional variables,



 --------------- (9)
where, 

 ---------------(10)
and, 
 ---------------------(11)
are the velocity and shear stress respectively.
Using (3) and (10) the non-dimensional form of constitutive equation be
 ------ (12)
Therefore, the constitutive equation of flow of a Newtonian fluid in elastic tube is
 
󰇛󰇛󰇜󰇜
 ----------------- (13)
The constitutive equation of Power Law in non-dimensional form becomes

󰇡
󰇢
󰇛󰇜
------------- (14)
The boundary condition in non-dimensional form is reduced to
------------------- (15)
We know 


 .
Using eqn (3), (13) and (14), the above expression gives

󰇛󰇜󰇡
󰇢

󰇛󰇛󰇜󰇜󰇡
󰇢󰇛󰇜
 
where 󰇡
󰇢
 --------------(16)

󰇛󰇛󰇜󰇜
󰇛󰇜------------ (17)

Then,
󰇛󰇜󰇡
󰇢󰇧󰇛󰇜󰇡
󰇢
󰇡
󰇢󰇨
Using eqns (3), (17) & boundary condition u=us at r=1 we get,



󰇛󰇛󰇜󰇜󰇡
󰇢 ------------------- (18)
The volumetric flow rate is given by



󰇛󰇜





Using τ = r for  we get
 
󰇡
󰇢
 ------------------- (19)
NUMERICAL RESULTS AND DISCUSSION
For the computational purpose, the following values of different material constants and other parameters
collected from standard literatures have been used.
Apparent viscosity of blood ( 󰇜
Fluid index parameter (n) = 1.05 – 1.50
Wall shear stress (󰇜
Slip velocity (󰇜
In order to have an estimate of the qualitative and quantitative effects of the various parameters involved
in the analysis, it is necessary to quantify the flow velocity (u), volume flow rate (Q) and yield stress in
the elastic vessel for the model under consideration. In each case, the presence of slip (velocity
discontinuity) at the flow boundaries are given due weightage. Computer codes are developed and the
graphs are plotted using MATLAB 8.5. The range of the values of fluid index parameter n is (1.05,
1.50).

Fig.2 Variation of Volumetric flow rate (Q) versus shear stress (󰇜 for different values of fluid
index parameter (n)
Fig. 2 demonstrates the variation of volumetric flow rate Q with shear stress τ0 taking n in the range of
1.05-1.50. The flow pattern is almost similar in the range of values of n. The flow rate gradually
decreases with the increase in the value of τ0 which is significant. As the value of n increases, the values
of Q become closer at τ0 = 0.2-0.8 and finally coincide for n = 1.45 and 1.50.
Fig.3. Variation of flow velocity (u) with radial distance (r) for different values of shear stress ()
The variation of flow velocity (u) with radial distance r for = 0.2, 0.4, 0.6, 0.8 and 1 is exhibited in
Fig. 3. It is observed that the flow velocity decreases with the increase of r i.e. when we approach
towards the artery wall which establishes the effect of slip (velocity discontinuity) in the arterial wall.
For different values of τ0, u almost converge at r = 1 (arterial wall).
0.768
0.744 0.729 0.72 0.712 0.706 0.701 0.697
0.605 0.568 0.547 0.533 0.523 0.514 0.507 0.501
0.484 0.442 0.419 0.404 0.392 0.383 0.375 0.369
0.387 0.344 0.321 0.306 0.294 0.285 0.278 0.272
0.328 0.286 0.263 0.249 0.238 0.229 0.222 0.216
0.278 0.237 0.216 0.202 0.192 0.184 0.178 0.172
0.236 0.197 0.177 0.164 0.155 0.148 0.142 0.137
0.212 0.174 0.156 0.144 0.135 0.128 0.123 0.118
0.178 0.143 0.126 0.116 0.108 0.102 0.097 0.093
0.16 0.127 0.111 0.101 0.094 0.088 0.084 0.08
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Volumetric Flow Rate(Q)
Shear Stress(τ0)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.2 0.4 0.6 0.8 1 1.2
Flow Velocity (u)
Radial Distance (r)
τ˳=0.1 τ˳=0.2 τ˳=0.4 τ˳=0.6 τ˳=0.8 τ˳=1

Fig.4 Variation of Flow velocity (u) with radial distance (r) with different values of slip velocity
(us)
Fig. 4 illustrates the variation of flow velocity at different points of r with slip velocity (us) = 0.0, 0.01
and 0.05. For r = 0 to 0.1 i.e. towards the centre of the artery, a reverse velocity is observed for the
given values of.
At each case, the velocity is more for less value of  and achieve maximum value at  (no slip).
So the slip velocity significantly alters the flow pattern. For more slip velocity, flow is more restricted.
Fig.5 Variation of Volumetric flow rate (Q) with fluid index parameter (n) for different values of
shear stress (τ0)
The fluid index parameter has a commendable role in the volumetric flow rate for admissible values of
shear stress displayed in Fig.5. As n increases (0.1-0.8), the volumetric flow rate shows decreasing
behaviour for all values of shear stress and almost converge at low shear.
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 0.2 0.4 0.6 0.8 1 1.2
Flow velocity(u)
Radial distance(r)
us us=0.01 us=0.05 Linear (us=0.05)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0246810 12
Volumetric Flow rate(Q)
Fluid index parameter(n)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Series9

CONCLUSION
The theoretical analysis led to following relevant conclusions:
I. The consideration of elastic property of artery has a significant role in the flow behaviour of
blood.
II. Rheological parameters like fluid index parameter and slip velocity have a commendable role
in the blood flow pattern.
III. Consideration of velocity slip in the arterial wall significantly alters the flow pattern.
IV. Results obtained through the theoretical study may be utilized in the clinical treatment of the
haematological patients.
V. Further careful investigations are suggested to address the problem more realistically and to
overcome the restrictions imposed on the present study.
REFERENCES
1.
Johnson, G.A., H.S. Borovetz and J.L. Andersons, “A model of pulsatile flow in a uniform
deformable vessel,” Journal of Biomechanics,
25(1),
91-100 (1992).
2.
Gupta, R.C. and B.P. Agarwal, “Non-newtonian fluid flow development in a circular pipe,”
Fluid Dynamics Research,
12
, 203-213 (1993).
3. Dash, R.K. and K.N. Mehta, “Casson fluid flow in a pipe filled with a homogenous porous
medium,” Int. J. Engg. Sci., 34
,
1145-1156, (1996).
4. Pedrizzetti, G., F. Domenichini, A. Tortoriello and L. Zovatto, “Pulsatile flow inside moderately
elastic arteries, its modeling and effects of elasticity,” Computer Methods in Biomechanics
and Biomedical Engineering,
5(3)
, 219-231 (2002).
5. McDonald, D.A., Blood flow in arteries, Edward Arnold Publishers, London (1974).
6. Mandal, P.K., “An unsteady analysis of non- Newtonian blood flow through tapered arteries
with a stenosis,” International Journal of Non-Linear Mechanics,
40
, 151-164 (2005).
7. Sankar, D.S. and K. Hemalatha, “Pulsatile flow of Herschel-bulkley fluid through catheterized
arteries- a mathematical model,” Applied Mathematical Modelling,
31
, 1497-1517(2007).
8. Sankar D. S. and U., Lee, “Mathematical modelling of pulsatile flow of non-Newtonian fluid in
stenosed arteries,” Commun. in Nonlinear Sci. Numer. Simul.
14,
2971 -2981 (2009).
9. Mazumdar, J.N., Biofluid Mechanics, World Scientific Publishing Co. Singapore,(1992).
10. Biswas, D. and U.S. Chakraborty, “Pulsatile flow of blood in a constricted artery withg body
acceleration”, Appl.Math.
4
,329-342 (2009).
11. Kumar, S., “Study of blood flow using Power law and Harschel-Bulkey non Newtonian fluid
model through elastic artery”, Proceedings of ICFM 2015, 229-235(2015).

12.Verma N.K., Shailesh Mishra, S. Siddqui,R.S. Gupta., “Effect of slip veloci ty on blood flow
through a catheterized artery”, Applied Mathematics,2,764-770(2011).
13. Basu Mallik B, Nanda S, et.al. “Pulsatile Flow of Casson fluid in mild stenosed artery with
periodic body acceleration and slip condition”, Sch. J. Eng. Tech.; 1(1), 27-38 (2013).
14. Basu Mallik B, Nanda S, et.al. “A non- Newtonian fluid model for blood flow using power law
through an atherosclerotic arterial segment having slip velocity”, IJPCBS 3(3), 752-760 (2013)