Effect of body acceleration on pulsatile flow of Herschel – Bulkley fluid through an inclined mild stenosed artery

Effect of body acceleration on pulsatile flow of Herschel – Bulkley fluid through an inclined mild stenosed artery

Dr. G.Neeraja,

Associate Professor,Dept. Of Mathematics, M.S.R.I.T., Bangalore.India.

Vidya. K,

Assistant Professor, Dept. of Mathematics, SDMIT, Ujire, South Canara(Dist), Karnataka.

Abstract

The pulsatile flow of Herschel-Bulkley fluid through an inclined stenosed artery under the influence of body acceleration is studied. The non linear equations governing the flow are solved using perturbation method. The effects of pulsatility, inclination of artery and body acceleration of blood on velocity, flow rate, wall shear stress are discussed. It is noticed that the effect of the stenosis is to reduce flow rate. The impact of body acceleration and inclination of the artery is to enhance the velocity of the blood flow.

Key words : Pulsatile flow, Body acceleration, Herschel-Bulkley fluid, Stenosed artery.

1. Introduction.

   Human body many a times get disturbed by external accelerations .Due to this, in the long run, health problems such as loss of vision, head ache , increase in pulse rate causes, on account of disturbances in blood flow [Burton et al ,1974[3];Hialt ct al .,1969[20]]. Therefore , for long and short term exposures of human body to such acceleration, it is desirable to set a standard..If the response of the human body to such accelerations is understood properly, the controlled accelerations can be used for therapeutic treatments, development of new diagnostic tools and for better designing of protective pads (Amtzenius et al,1972[1];Verdouw et al.,1973[33]).

   There are many evidences that vascular fluid dynamics plays a major role in the development and progression of arterial stenosis. It is quite common to find arteries are narrowed by the development of atherosclerotic plaques that protrude into the lumen, resulting arterial stenosis. When an obstruction developed in an artery, one of the most serious consequences is the increased resistance and the associated reduction of the blood flow to the particular vascular bed supplied by the artery. Thus the presence of a stenosis leads to the serious circulatory disorder. A knowledge of the flow characteristics in the vicinity of a stenosis may help to further understand some major complications which can arise such as , an in- growth of tissue in the artery, the development of a coronary thrombosis, the weakening and bulging of the artery downstream from the stenosis, etc. Investigation of the role of hydrodynamic factors in the development of the above complications provides relevance to the analysis of flow through a modeled arterial stenosis.

   Due to physiological importance of body acceleration many theoretical investigations have been carried out for the flow of blood under the influence of body acceleration with and without stenosis. Sud and Sekhon (1985)[34] studied the pulsatile flow of blood through a rigid circular tube subject to body acceleration, treating blood as a Newtonian fluid. Misra and Sahu (1988)[24] analyzed the flow of blood through large arteries under the action of periodic body acceleration. Belardinelli et al.(1989)[2] proposed mathematical models for various forms of body acceleration. Usha and Prema (1999) [32] studied the pulsatile flow of particle-fluid suspension model of blood under the presence of periodic body acceleration. Using Laplace and Henkel tansforms Elshehawey et al. (2000)[16] studied the effect of body acceleration on pulsatile flow of blood through a porous medium by treating blood as a Newtonian fluid .Later El-shahed (2003)[15] extended this study for a stenosed porous medium.

   1

   In all these investigations blood is modeled as a Newtonian fluid. It is reported that the rheological properties of blood and its flow behavior through tubes of varying cross section play an important role in understanding the diagnosis and treatment of many cardiovascular diseases (Fry, 1968[18], Dimenfass, 1977[11];Caro,1981[4]).It is well known that blood being a suspension of cells, behaves as a non Newtonian fluid at low shear rates and during its flow through small blood vessels, especially in diseased states when clotting effects in small arteries are present. Experiments conducted on blood (Scott Blair,1959[19];Cokelet et al.,1965[6]) with varying heamatocrit, anticoagulants, temperature, etc suggested that the behavior of blood at low shear rates can be best described by Casson model(Charm and Kurland

   ,1965[5]; Merill and Pelletier,1967[23]).Chaturani and Palaniswamy (1990a:1990b)[8],[9] analyzed the pulsatile flow of blood under the influence of periodic body acceleration by assuming blood as a Casson fluid and also a power law fluid by using finite difference scheme. Sarojamma and Nagarani (2002)[27] studied the flow of a Casson fluid in a tube filled with porous medium under periodic body acceleration with application to artificial organs. Mandal et al,.(2007)[22]

   Fig 1.Geometry of the inclined stenosed artery

   Consider the pulsatile flow of blood in presence of externally imposed periodic body acceleration in an inclined artery with mild stenosis. We consider the flow is axially symmetric, laminar, fully developed where the flowing blood is modeled as a Herschell-bulkley fluid

   .Following Young (1968) , the stenotic protuberance is assumed to be an axisymmetric surface generated by a cosine curve. The geometry of the stenosis is as shown in the figure 1 and is given by

   ]

   developed a two dimensional mathematical model to study the effect of extremely imposed periodic body

   acceleration on non Newtonian blood flow through an

   R(z)

   R0 1

   cos

   z

   2z0

   for z

   2z0

   to z

   2z0

   (1)

   elastic stenosed artery where the blood is characterized by the generalized power-law model. P.Nagarani and G.Sarojamma (2008)[28] analyzed the effect of body acceleration on pulsatile flow of Casson fluid through a mild stenosed artery. D.S.Sanker and Ahmad Izani Md.Ismail (2009)[12] did a comparative study on Two- fluid mathematical models for blood flow in stenosed arteries by considering Casson as well as Herschel- Bulkley fluids.

   In view of the above, a mathematical model is developed to study the pulsatile flow behavior of blood in an inclined artery under stenotic condition subject to both the pulsatile pressure gradient due to normal heart action and of periodic body acceleration . Blood is modeled as a Herschel-Bulkley fluid by properly accounting for yield stress of blood .The combined effect of pulsatality, stenosis, body accelertion, inclination of the artery on the flow parameters is illustrated graphically in results and discussion section.

2. Mathematical formulation

   R0 otherwise

   where 4 z0 is the length of the stenotic region, 2 is the maximum proturberance of the stenotic form of the artery wall and R0 is the radius of the normal artery.

   The periodic body acceleration F( t ) in the axial

   direction is given by

   F (t ) a0 cos(wbt ) , (2a)

   where a0 is its amplitude, wb 2 fb , fb is its frequency in Hz, the lead angle of F( t ) with respect to the heart action. The frequency of body acceleration fb is assumed to be small, so that wave effects can be neglected .The pressure gradient at any z may be

   represented as follows.

   p A A cos(w t ) , (2b)

   z 0 1 p

   where

   A0 is steady component of the pressure gradient,

   Introducing the non-dimensional variables,

   A1 is amplitude of the fluctuating component and u z

   wp =2

   f p ,

   f p is the pule frequency.Both

   A0 and

   u

   A0 R

   , z

   0

   2 /4 0

   , t w p`t , ,

   R

   R

   0 0

   A1 are functions of z .It can be shown that the radial

   ,

   H , R(z)

   R(z) , r

   r ,

   velocity is very small in magnitude so that it may be neglected for problem with mild stenosis.

   A0 R 0 / 2

   A0 R

   0 /2

   R 0 R 0

   The specified momentum equation for the flow in cylindrical coordinate system is given by

   a a 0 , e A0

   A1 , w

   A0

   w b ,

   0

   w p

   H (2 / R0

   A )n 1

   u p 1

   0

   F = 4

   0 / gA R 2

   (6)

   t z r

   r (r

   rz )

   F (t )

   0 0

   g sin al

   (3a)

   Here

   A0 is steady component of the pressure gradient,

   p

   =0 (3b)

   R 0 is the radius of the normal artery, w p is the

   z

   where r and z denote the radial and axial coordinates

   frequency of oscillation of the pulsatile flow and then

   A R 2

   respectively and denotes density, u axial velocity

   0 0

   4

   represents the central line velocity in a

   of blood, t time, p pressure and the shear stress and al be the small angle of inclination.For Herschell- bulkley fluid the relation between shear stress and shear rate is given by (Herschell-Bulkley 1929),

   Poiseuille flow.

   The non dimensional momentum equation 3(a)becomes

   H )

   1/ n ( u 1/ n

   r

   H if H

   (4a)

   2 u 4 (1

   t

   ecost)

   4a cos (wt

   ) sin al F

   u

   2 )

   r r rz

   (7)

   0 if H

   r

   (4b)

   Where

   R 2

   2 0 0 , is Womersley frequency

   Where u is the total velocity, p is the pressure, t is

   the time , H is the yield stress and H is the coefficient of viscosity for Herschell-bulkley fluid

   .Equation (5b) corresponds to the vanishing of the velocity gradient in the region where the shear stress is

   p

   parameter.

   Equation (4) can be rewritten as

   1

   1 u n

   less than the yield stress which impliesa plug flow

   if ,

   (8a)

   whenever H .However ,the fluid behavior is indicated whenever H .

   2 r

   (8b)

   u 0 if

   r

   The boundary conditions appropriate to the problem under study are the no slip condition

   The boundary conditions (5a,5b) reduce to

   1. u= o at r=R(z) (9a)

      (i) u=0 at r

      R(z)

      (5a)

   2. isfiniteatr=0 (9b)

   The geometry of the stenosis in non-dimensional form is given by

   and(ii) and is finite at r = 0 (5b)

   R(z) 1

   1 COS

   z

   2z0

   for z

   2z0 to z

   2z0

   The plug core velocity equation (18) as

   u 0 p

   can be obtained from

   1 otherwise

   (10)

   u0 p

   A(R n 1

   R0 p

   n 1 )

   B ( R n

   R0 p )

   (19)

3. Method of solution

   On using perturbation method, the velocity u, shear

   Neglecting the terms of O( 2 ) and higher powers of

   n

   in equation (11c) R0 p can be obtained from (14) as

   stress, plug core radius Rp and plug core velocity u p 2

   are expanded as follows in terms of 2 (where 2

   1

   <<1)

   R0 p

   f (t) k

   (20)

   u(z, r,t)

   u0 (z, r,t)

   2u (z, r,t)

   ………. ……….

   (11a)

   Using Equation (13) we get the solution for 1 as,

   u

   z, r,t 0 z, r,t

   2 (z, r,t)

   ………. ……….

   . (11b)

   1 a7 r

   a6 r

   a3 r

   (21)

   1

   n 1

   n 2

   u p (z, r, t)

   u0 p

   (z, r, t)

   2 (z, r, t)

   ………(11c)

   Where

   1 p

   Rp (z,r,t)

   R0 p

   (z,r,t)

   2 (z,r,t)

   ………. ..

   (11d) a1

   ( 1) n 1

   f (t) n 1 f

   1(t) ,

   R

   1p

   n a R n 1 n a

   a 1 , a 1 ,

   Substituting (11a)and (11b) in Equation (7) and equating the constant term and 2

   2 2(n 1)

   (n 1)

   3 (n

   a a R n

   1)(n 3)

   term we get

   a , a

   1 4 ,

   r 0 2 r 1

   ecos t

   a cos ( t

   ) sin al

   4 f (t) 5 2

   a a

   r 4F

   (12)

   a 1 4

   , a 7 a2

   a5 ,

   u 6 n 2

   0 2 r

   t r r 1

   (13)

   Similarly using Equations (17) and (18)we can obtain the solution for , u 1 and u 1 p as

   Integrating Equation ( 12) and using boundary condition (9b) we obtain

   u a 16

   1 2n 1

   R 2n 1

   r 2n 1

   a 15

   2n 2

   R 2n 2

   r 2n 2

   0 f (t) r

   (14)

   a 9 R n r n a 13

   R n 1

   r n 1

   (22)

   Where

   f (t) 1 e cost

   a cos( t

   ) sin al

   n n 1

   a a

   4F (15)

   u 16 R 2n 1 R 2n 1 15 R 2n 2 R

   2n 2

   Substituting (11a) and (11b) in (8) we get

   1 p 2n 1

   0 p 2n 2

   0 p

   (23)

   u0 n 1

   a 9 R n rR n a 13

   R n 1 R n 1

   r 2 0

   ( 0 n )

   .(16)

   n

   where a = 2n

   0 p n 1

   f (t)n 2 (n 1)

   1. p

      , a = a a ,

      u1 2 n

      n 2 (

      (n 1) )

      8

      a10 = – a6 a8 ,

      n 1

      9 7 8

      a11 = – a3a8 ,

      r 0 1 0

      a12 = – 2n

      f (t) , a13 =

      a7 a12 ,

      (17)

      a = – a a

      , a = – a a

      , a = a a .

      Integrating Equation(16),using the relation (14) and the

      14 6 12 15

      12 3 16

      11 14

      boundary condition (9a) we obtain Using equation (11),the total velocity distribution and

      0

      u A(R n 1

      r n 1)

      B ( R n

      r n )

      (18)

      shear stress can be written as

      Where

      A 2(

      1. n 1

      n

      f (t) n

      ,

      1

      B 2(

      1) n 1

      f (t) n 1

      u A(R n 1

      r n 1)

      B ( R n

      r n ) + 2

      It is generally observed that the typical value

      a 16

      2n 1

      R 2n 1

      r 2n 1

      a 15

      2n 2

      R 2n 2

      r 2n 2

      of n for blood flow is taken to be lies between 0.9 and

      1.1 and in this paper we have taken typical value of n as

      a 9 R n r n a 13

      n n 1

      2

      0 1 r R

      R n 1

      r n 1

      (24)

      0.95.Since the typical values for non-dimensional yield

      stress for blood are between 0.02 and 0.04, the value

      f (t)R

      2 a R

      a R n 1

      a R n 2

      (25)

      of yield stress has been takes as 0.04 for Herschell-

      7

      6

      3

      The second approximation plug core radius

      R1 p

      can

      Bulkley fluid.Thestenosisheight

      s

      =0.2, Womersley

      be obtained by neglecting terms of 0( 4 ) and higher powers of in Equation (11c)as

      parameter = 0.1and as f = 2.0

      R1 p

   2. R0 p

   f (t)

   (26)

   Axial velocity profiles at the centre of the stenosis z = 0 with body acceleration a, constant

   With the help of Equations (26),(20) and (11c), R p

   can be given by

   pressure gradient e and inclined angle al are shown in

   R k 2

   2

   a R a R n 1

   a R n 2

   Fig 2. It is observed that body acceleration parameter a

   p f (t)

   7 0 p

   6 op

   3 0 p

   (27)

   brings qualitatative as

   The volumetric flow rate Q is given by

   R( z )

   Q(t)

   4 r u(z, r, t)dr

   0

   (28)

4. Results and discussion

   The wall shear stress is a physiologically important quantity which plays an important role in determining aggregate sites of platelets and is given

   by

   well as quantitative changes in velocity profiles.(Fig 2 (a)).

   Fig 2.(a) Variation of velocity for different values of body acceleration parameter a.

   Fig 2.(b).Variation of velocity for different values inclinationangle al.

   Magnitude of velocity increases with inclination angle al increases.(Refer Fig 2 (b)), but it decreases with increases in pressure gradient parameter e.(Refer Fig 2.(c)).

   Fig 3 (a), (b) and (c) as well as Fig 4 (a), (b), (c) depics plug core velocity profiles and wall shear

   stress profiles with axial difference..We have seen symmetry at the centre of the stenosis in both the profiles.

   Fig 2.(c). Variation of velocity for different values of pressure gradient parameter e.

   .

   Fig 3 (a). Variation of plug core velocity for different values of body acceleration parameter a.

   Fig 3 (b). Variation of plug core velocity for different values of inclination angle al.

   6

   Fig 3 (c).Variation of plug core velocity for different values of pressure gradient parameter.

   Fig 4(a).Variation of wall shear stress for different values of body acceleration parameter a

   Fig 4 (b). Variation of wall shear stress for different values of inclination angle al.

   Fig 4 (c). Variation of wall shear stress for different values of pressure gradient parameter

   The volumetric flow variation with the stenosis height s , for for different values of a, al, e is shown in the tables 1 and 2.As e increases , the volumetric flow decreases.Also when the stenosis height s increases, the volumetric flow quantitatively decreases which is true with the physiological condition.The inclination of the artery does not effect the flow rate at the vicinity of the stenosis.As the body acceleration parameter increases, the volumetric flow also increases, but the

   7

   volumetric flow drastically decreases as stenosis height increases.

   Table 1.

   s e = 0-5 e=0.6 e =0.7

   List of symbols

   represents the dimenmsional quantities

   a0 amplitude of body acceleration.

   A0 steady component of pressure gradient

   A1 amplitude of fluctuating component of

   0 3.39791

   1.41234

   3.28022

   5

   1.36360

   3.14634

   2

   1.30816

   presure gradient

   A ratio of amplitude of body acceleration with steady component of pressure gradient given in Eq. (6)

   0.1

   4

   	0.45599	0.44035	0.42256
   0.2	4	6	5

   0.09298

   9

   0.08983

   6 e ratio of amplitude of fluctuating component with steady component of pressure gradient given in Eq. (6)

   f(t) function defined in Eq. (15) F(t) body acceleration function

   0.3 4

   4 0.08625

   fb frequency of the body acceleration

   0.4

   0.00622

   3 0.00602

   Table 2

   0.00578

   9

   f p pulse frequency P pressure

   Q(t) volumetric flow rate

   r radial coordinate

   s a = 0 a = 1 a = 2

   R(z) radius of the stenosed artery

   0 3.39791

   1.41234

   3.43586

   1

   3.47127

   9

   1.44272

   R0

   R0 p

   R

   radius of the normal artery

   first approximation plug radius second approximation plug radius

   0.1

   4 1.42806

   8 1 p

   	0.45599	0.46103	0.46574	Rp	radius of plug core velocity
   0.2	4	7	4	T	Time
   	0.09298		0.09494	u	axial velocity of the fluid
   0.3	4	0.094	8	u0	first approximation velocity

   0.00622

   0.00628

   u0 p

   first approximation plug velocity

   0.4 3

   9 0.00635

   u1 second approximation velocity

5. Conclusions

By using perturbation analysis assuming that the

u1 p

u p

first approximation plug velocity

plug core velocity

Womersley frequency parameter is small, the

z axial coordinate

pulsatile flow of blood with periodic body acceleration under the presence of stenosis in an

4zo

length of the stenotic region

inclined artery is studied by modeling blood as a Herschell-bulkley fluid .It is observed that the body acceleration parameter, inclination and radius of stenosis are the strong parameters influencing the flow qualitatively and quantitatively..The presence and increase of the protuberance is found to reduce the magnitude of the velocity. The effect of stenosis is to reduce the flow rate and the presence of body acceleration and inclination of artery are to increase the flow rate. The body acceleration and inclination of artery are found to reduce the flow resistance.

Greek letters

lead angle of body acceleration function with respect to heart action

density

flow resistance

non-dimensional yield stress of the fluid ratio of b with p

Shear stress of the fluid

viscosity of the fluid

Womersley frequency parameter

0 first approximation shear stress

8

1 second approximation shear stress b 2fb

p 2fp

y yield stress of the fluid

2 maximum protuberance of the stenotic form of the artery wall

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