adomian decomposition method for

Journal of Mechanics in Medicine and Biology Vol. 17, No. 1 (2017) 1750007 (28 pages) c World Scientific Publishing Company ° DOI: 10.1142/S0219519417500075

ADOMIAN DECOMPOSITION METHOD FOR MAGNETOHYDRODYNAMIC FLOW OF BLOOD INDUCED BY PERISTALTIC WAVES

J. Mech. Med. Biol. 2017.17. Downloaded from www.worldscientific.com by 181.215.222.126 on 05/29/18. For personal use only.

G. C. SHIT*,†, N. K. RANJIT*,‡ and A. SINHA*,§ *Department of Mathematics Jadavpur University, Kolkata - 700032, India † gopal [email protected] ‡ [email protected] §[email protected]

Received 20 August 2015 Revised 30 October 2015 Accepted 10 January 2016 Published 5 April 2016 The present investigation deals with the application of Adomian decomposition method (ADM) to blood flow through an asymmetric non-uniform channel induced by peristaltic wave in the presence of magnetic field and the velocity slip at the wall. The ADM is applied with an aim to avoid any simplifications and restrictions, which changes non-linearity of the problem as well as to provide analytical solution. The blood flowing through the vessel is assumed to be Newtonian and incompressible with constant viscosity. The analytical expressions for the axial velocity component, streamlines and wall shear stress are presented. The numerical results of these physical quantities are obtained for different values of the Reynolds number, wave number and Hartmann number. The results obtained for different values of the parameters involved in the problem under consideration show that the flow is appreciably influenced by the presence of slip velocity as well as magnetic field. From this study, we conclude that the assumption of long wavelength and low Reynolds number overestimates the flow characteristics even for a small change in the parameters. Keywords: Peristaltic flow; slip velocity; magnetic field; asymmetric channel; adomian decomposition method.

1. Introduction Peristalsis is an important mechanism for mixing and transporting fluids, which is generated by a progressive wave of area contraction and expansion moving along the length of the vessel. Physiological fluids are, in general, pumped by the continuous periodic muscular oscillation of the ducts. These oscillations are caused by the progressive transverse contraction waves that propagate along the length of the ducts. Peristalsis is the mechanism of fluid transport, which occurs generally from a † Corresponding

author. 1750007-1


G. C. Shit, N. K. Ranjit & A. Sinha

region of lower pressure to higher pressure when a progressive wave of area contraction and expansion travels along the flexible wall of the vessel. The physical phenomena, like urine transport from kidney to bladder through the ureter, movement of chyme in the gastrointestinal tract, the movement of spermatozoa in the ducts afferent of the male reproductive tract and the ovum in the fallopian tube, the locomotion of some worms, transport of lymph in the lymphatic vessels and vasomotion of small blood vessels, such as arterioles, venules and capillaries involves the peristaltic motion. In addition, peristaltic pumping occurs in many practical applications involving biomechanical systems. Moreover, by using the principle of peristalsis, some biomechanical instruments, e.g., heart–lung machine and blood pump machine have been designed. Several investigators Pozrikidis1 and Fung and Yih2 have pointed out that the mechanism of peristaltic flows are used for the transport of blood within small blood vessels or can be used to design artificial blood devices. Moreover, they suggested that the arterioles and venules in some preparations are seen to change their diameters periodically. Although the spatial wave form of such a vasomotion has not been ascertained, it is conceivable that pumping is involved. The first attempt regarding the peristaltic flow has been made by Latham.3 Later on, various investigators (cf. Burns and Parkes,4 Shapiro et al.5 and Srivastava and Srivastava6) have studied the peristaltic transport of viscous incompressible fluid in an axisymmetric tube under long wavelength and low Reynolds number assumptions. Further studies on peristaltic transport phenomena have been made by Misra and Pandey7,8 and Hayat et al.9,10 by taking into consideration various nonNewtonian fluid models. Mishra and Rao,11 Misra et al.12 have investigated the flow in an asymmetric channel generated by peristaltic waves propagating along the walls with different amplitudes and phases. Peristaltic transport in a two-dimensional channel, filled with a porous medium in the peripheral region and a Newtonian fluid in the core region studied by Mishra and Rao13 under the assumptions of long wavelength and low Reynolds number. Maiti and Misra and Misra and Maiti14,15 carried out the peristaltic motion of blood in the micro-circulatory system by taking into account the non-Newtonian nature of blood and the non-uniform geometry of the micro-vessels with the assumption of the low Reynolds number and long wavelength. The assumptions of long wavelength and low Reynolds number imply simplification of the non-linear terms by ignoring inertia part. However, Hussain et al.16 solve their problem using perturbation solution techniques by keeping those of non-linear terms. Maiti and Misra17 have presented a theoretical study of peristaltic flow of a fluid in a porous channel. They have also obtained series solution to the streamline and pressure gradient as power series in terms of the small amplitude ratio. The slip condition plays an important role in shear skin, spurt and hysteresis effects. The fluids that exhibit boundary slip have also important applications in polishing valves of artificial heart and internal cavities. The no-slip condition is inadequate in many situations when one considers fluids exhibiting macroscopic wall slip which is, in general, governed by the relation between the slip velocity and 1750007-2


Adomian Decomposition Method for MHD Flow of Blood Induced by Peristaltic Waves

traction. The main concern of this study is to explore the reality of blood slip near the vessel wall. In microscopic observations, the red cells of blood maintaining sliding contact with the wall, where the shear rate is low. The results of Bloch18 reported that the red cells occupying some portion touching the wall with 1–2 (micron) depth, while the experimental studies of Bennett19 show that this depth should be about 1.4 (micron) with respect to vessel wall. They pointed out that the velocity of cells containing wall could be about 10–80 micron/velocity. Finally, they concluded that blood slip status is a function of the slip characteristics of both red cell and plasma components. The slip condition also arises when there is an elastic nature of the boundary wall. In this regard, Ali et al.20 have investigated the slip effects on peristaltic transport on MHD fluid flow with variable viscosity. They observed that the slip parameter has a reducing effect on the formation of bolus as well as the pumping phenomena. Mekheimer et al.21 examined the effects of Hall current and slip condition on the MHD flow induced by sinusoidal peristaltic wavy wall in two-dimensional viscous fluid through a porous medium. Misra and Shit22 and Sinha and Shit23 extensively studied the role of slip velocity on blood flow through arterial stenosis under different situations. The magnetohydrodynamical behavior of blood flow has an enormous application in physiological systems. The flow characteristic of an electrically conducting fluid changes when exposure to a magnetic field. It is well known that the biological effects on patient when exposed to a high static magnetic field in Magnetic resonance imaging (MRI) are still somehow unclear to the biomedical engineers and scientists because of its lack of information. The magnetic field is applied during surgery to control the excess of bleeding as well as during (MRI) in order to get better resolution of the scan images. Nowadays most of the MRI machines use 1.5 T as a strength of the magnetic field, which is 105 times the earth’s magnetic field strength. However, medical scientists/engineers want to increase the magnetic field strength from 1.5 T to 3 T or more upto 8 T not only in MRI but also in the hyperthermic treatment of cancer therapy (cf. Raymond et al.24 and Jekic et al.25). When a high static magnetic field is imposed to a human body, it needs to have clear information/conclusion for medical disorders of health such as cardiovascular diseases, determination of blood pressure, flow rate, effects of motion-induced currents, voltage measurements in ECG signals, chemical reaction, and thermal responses. The effect of magnetic field on peristaltic transport of physiological fluids have been examined by several investigators.12,26,27 The pulsatile flow of hydromagnetic blood through a porous saturated tapered artery with overlapping stenosis under the influence of periodic acceleration has been carried out by Zaman et al.28 Shit et al.29 presented a comprehensive theoretical study on peristaltic mechanism of magneto-micropolar fluid in an asymmetric channel by taking into account the effect of induced magnetic field. Very recently, Shit and Roy30 examined the effect of induced magnetic field on blood flow in the constricted channel. Misra et al.31 have investigated the mathematical modeling of a blood flow in a porous vessel having double stenoses in the presence of an external magnetic field. All these studies 1750007-3



carried out with the assumption that the inertia term neglected because of the long wavelength and low Reynolds number assumptions. These studies observed that the fluid velocity decreases and pressure gradient increases with increasing magnetic field strength. Being motivated by the above mentioned studies, we have investigated the blood flow problem induced by the peristaltic motion in a situation where the inertial effect has been taken into account. As an illustrative example, blood flows in the microvessel, obeying the properties of Newtonian fluid model. In our study, we have not neglected the non-linear terms in the governing equation by assuming as usual long wavelength and low Reynolds number assumptions. Mathematically, it is very difficult to find analytical solutions to the non-linear problems. Some of the studies presented the analytical solutions either using perturbation technique or power series solution with respect to the small parameter. With an aim to get better and accurate results, we have used a powerful method known as decomposition method which has been developed by Adomian,32,33 and provides analytical approximations to a wide class of non-linear ordinary and partial differential equations. The Adomian decomposition method (ADM) involves separating the equation under investigation into linear and non-linear portions. The linear operator representing the linear portion of the equation is inverted and the inverse operator is then applied to the equation. The non-linear portion is decomposed into a series of Adomian polynomials. This method generates a solution in the form of a series whose terms are determined by a recursive relationship using these Adomian polynomials. The ADM has been successfully applied to solve non-linear equations in studying many interesting problems arising in applied sciences and engineering,34–37 and is usually characterized by its higher degree of accuracy. The rapid convergence of the series solution obtained by ADM has been discussed by Cherrualt and Adomian38 and provides an insight into the characteristics and behavior of the solution as in the case with the closed form solution. It has some distinct advantages that it is computationally convenient and provides analytical solutions without having perturbation, linearization or the massive computation. In this paper, we have analyzed the effects of magnetic field, velocity slip at the wall, wave number as well as Reynolds number on the blood flow characteristics such as axial velocity component, streamline pattern and wall shear stress (WSS). The numerical results are shown graphically and discusses some important phenomena. The paper is organized as follows: The problem under consideration is formulated in Sec. 2. The solution to the problem by ADM is given briefly in Sec. 3. The important results for velocity distribution, pumping and trapping phenomena are studied in Sec. 4 and some final remarks are given in Sec. 5. 2. Formulation of the Problem Let us consider the peristaltic motion of blood flowing through an asymmetric and non-uniform two-dimensional channel. Although almost all peristalsis occur in tube 1750007-4



particularly in the physiological system, for the sake of simplification as well as finding analytical solution of the problem, we have approximated a tube flow by considering channel flow through a two-dimensional cross-section of the tube that leads to a rectangular cartesian coordinate system. For low Reynolds number flow in the physiological system, the estimates of the fluid dynamical behavior are approximately represented by the channel flow. In this context, we have been motivated by the works of Fung and Yih,2 Shapiro and Latham39 and Pozrikidis1 wherein they considered the flow induced by peristaltic motion of the walls of a twodimensional channel. The channel walls propagate along the X 0 -direction with a wave speed c having wavelength and different amplitudes a1 and a2 at the upper and lower walls. We assume that the channel walls are asymmetric caused by phase difference ð0 Þ and inclined at an angle with the line parallel to X 0 -axis. The Y 0 -axis is taken in the transverse direction. The geometry of the physical problem (cf. Fig. 1) given by y0 ¼ h 01 and y0 ¼ h 02 representing respectively the upper and lower walls as 2 0 ðX ct0 Þ h 01 ðX 0 ; t0 Þ ¼ d1 þ ðX 0 ct0 Þ tan þ a1 cos ð1Þ on the upper boundary and h 02 ðX 0 ; t0 Þ

2 0 ðX ct0 Þ þ ¼ d2 ðX ct Þ tan a2 cos 0

0

ð2Þ

on the lower boundary; where d1 and d2 are the constant lengths from the central line to the upper and lower walls of the channel and measures the non-uniformity of the channel.

2

Y α

1

d1

φ

a1

X

0

d2

a2 1

λ

2 1.0

0.5

0.0

0.5

Fig. 1. A physical sketch of the problem. 1750007-5

1.0


The equations of motion for unsteady flow of an incompressible Newtonian fluid with externally imposed magnetic field are


r0 q0 ¼ 0; @q0 0 r0 Þq0 þ ðq ¼ r0 p0 þ r02 q0 þ J0 B0 ; @t0

ð3Þ ð4Þ

where q0 ¼ ðU 0 ; V 0 ; 0Þ is the velocity vector, p0 is the fluid pressure, the fluid density, the dynamic viscosity of the fluid, B0 ð0; B0 ; 0Þ the magnetic field vector in which a constant magnetic field of strength B0 is applied perpendicular to the blood flowing along X 0 -direction. The last term appearing in Eq. (4) is the magnetic body force per unit volume in which J0 ¼ ðE0 þ V0 B0 Þ represents the Lorentz force arising from Maxwell’s equations of electromagnetism. Due to the imposition of magnetic field in an electrically conducting fluid, an electromagnetic force (EMF) arises in the direction perpendicular to both the fluid motion and applied magnetic field and thereby induces electric as well as magnetic field. Since the electrical conductivity of blood is very small, magnetic Reynolds number becomes 1 and hence we have neglected the induced magnetic field and electric field. In our model, no external electric field is applied. It is further noted that the problem is formulated in the laboratory frame of reference, where all flow quantities such as velocity, pressure and the movements of walls (representing by Eqs. (1)–(4)) are time dependent. Due to the time dependence of the channel wall, in the laboratory frame ðX 0 ; Y 0 Þ, the flow is unsteady. Since the coordinate frame moving in the wave speed c away from a fixed frame ðX 0 ; Y 0 Þ, the boundary shape becomes stationary and hence the flow in the wave frame is steady. The following are the coordinate transformations to represent the problem in the wave frame of reference ðx0 ; y0 Þ only. x0 ¼ X 0 ct0 ;

y0 ¼ Y 0 ;

u0 ðx0 ; y0 Þ ¼ U 0 c;

v0 ðx0 ; y0 Þ ¼ V 0 ;

ð5Þ

where ðu0 ; v0 Þ and ðU 0 ; V 0 Þ are the respective velocity components in the laboratory and wave frames. Hence forward all the flow quantities are derived in the wave frame. Using the coordinate transformations (5), the governing Eqs. (3) and (4) extracted in two-dimensional cartesian coordinate system as @u0 @v0 þ ¼ 0; @x0 @y0 0 @u0 @p0 0 0 @u þ r02 u0 B 20 ðu0 þ cÞ; ðu þ cÞ 0 þ v ¼ @x @y0 @x0 0 @v0 @p0 0 0 @v ðu þ cÞ 0 þ v þ r02 v0 ; ¼ @x @y0 @y0 where r02

@2 @x02

@ þ @y 02 . 2

1750007-6

ð6Þ ð7Þ ð8Þ


For the present problem, the velocity-slip boundary conditions are taken as ðu0 þ cÞ þ sl

@u0 ¼ 0 on y0 ¼ h 01 ðx0 Þ; @y0

ð9Þ

ðu0 þ cÞ sl

@u0 ¼ 0 on y0 ¼ h 02 ðx0 Þ; @y0

ð10Þ

where sl denotes the slip length and prime ð0 Þ denotes the variables in dimensional form. We express the following non-dimensional variables by defining


x¼

x0 ;

y¼

y0 ; d1

u¼

u0 ; c

v0 ; d1 c

v¼

h2 ðxÞ ¼

h 02 ðx0 Þ d1

t¼

;

ct0 ;

p¼

¼

d 21 p0 ðx0 Þ c

0

cd1

;

h1 ðxÞ ¼

:

h 01 ðx0 Þ ; d1 ð11Þ

Using the dimensionless variables defined in (11) and introducing the dimensionless stream function , Eqs. (7) and (8) reduce to @ @ @p Re x þ r2 y Ha2 ð y þ 1Þ; ð12Þ ¼ y @x @y y @x @ @ @p 3 y 2 r2 x ; Re ð13Þ ¼ x @y @x x @y is given by u ¼

where the non-dimensional stream function

@ @y ,

and v ¼ @@x ,

@ @ r2 2 @x 2 þ @y2 and the dimensionless parameters that appear in (12) and (13) are qffiffiffi defined as Re ¼ cd1 the Reynolds number, ¼ d1 the Wave number, Ha ¼ B0 d1 2

2

the Hartmann number. Now, eliminating the pressure between Eqs. (12) and (13), the resulting equation can be written as Re½

yr

2

x

xr

2

y

¼ r4 Ha2

yy :

ð14Þ

The instantaneous volumetric flow rate in the fixed frame is given by Z h0 1 Q¼ U 0 ðX 0 ; Y 0 ; t0 Þdy0 ;

ð15Þ

where h 01 and h 02 are functions of X 0 and t0 . The rate of volume flow in the wave frame is found to be given by Z h0 1 q¼ u0 ðx0 ; y0 Þdy0 ;

ð16Þ

h 02

h 02

where h 01 , and h 02 are functions of x0 alone. 1750007-7


We note that h1 ðxÞ and h2 ðxÞ represent the dimensionless form of the peristaltic channel walls given by the equations of the form, x tan þ a cosð2xÞ; x tan h2 ðxÞ ¼ d b cosð2x þ Þ;

h1 ðxÞ ¼ 1 þ

ð17Þ ð18Þ

where a ¼ ad11 , b ¼ ad21 , d ¼ dd21 are constants. Using the transformations (5) into Eqs. (15) and (16), the relation between Q and q can be obtained as J. Mech. Med. Biol. 2017.17. Downloaded from www.worldscientific.com by 181.215.222.126 on 05/29/18. For personal use only.

Q ¼ q þ cðh1 h2 Þ: The time mean flow over a period T at a fixed position X 0 is defined as Z T 1 Qdt: Q0 ¼ T 0 Using (19) in (20), the flow rate Q0 has the form, Z T 1 Q0 ¼ qdt þ cðh1 h2 Þ ¼ q þ cd1 þ cd2 : T 0

ð19Þ

ð20Þ

ð21Þ

The non-dimensional form of Eq. (21) is given by ¼ F þ 1 þ d; where ¼

Q0 cd1

and F ¼

ð22Þ

q cd1 ,

such that Z h 1 @ dy ¼ ðh1 Þ ðh2 Þ: F¼ h2 @y

ð23Þ

The boundary conditions (9) and (10) for the dimensionless stream function ðx; yÞ can be put mathematically as @ @2 þ 2 ¼ 1 on y ¼ h1 ; @y @y

ð24Þ

@ @2 2 ¼ 1 on y ¼ h2 ; @y @y

ð25Þ

in addition; ¼

¼

F on y ¼ h1 ; 2

F on y ¼ h2 ; 2

ð26Þ ð27Þ

where ¼ ds1l is defined as the slip parameter. Since Eq. (14) is fourth order differential equation in , it must be assigned four boundary conditions for . The boundary conditions (24) and (25) directly derived from the velocity boundary conditions (9) and (10). However, the additional boundary conditions (26) and (27) are expressed in terms of the volumetric flow rate. 1750007-8


3. Solution by ADM


Equation (14) is a non-linear partial differential equation and the exact solution of this equation is not always possible. This equation can be solved by using traditional techniques which result in massive numerical computations. In the recent years, a lot of attention has been devoted to the study of ADM to investigate various scientific models including non-linear phenomena. In this section, we will solve the problem by the ADM. @2 In order to employ the ADM, let us consider L @y 2 (the second order differential operator), and therefore Eq. (14) can be written as L2 ¼ ReN þ Ha2 L 4

@4 @2 2 ðL Þ; @x4 @x2

ð28Þ

xr

ð29Þ

where N

2 yr

¼

x

2

y

represents non-linear terms. Since L2 is a fourth-order differential operator, L2 is a fourth-fold integration operator defined by Z yZ yZ yZ y L2 ðÞ ¼ ðÞdydydydy: ð30Þ 0

0

0

0

If we operate both sides of Eq. (28) by the inverse operation L2 ðÞ, then we get @4 @2 ¼ 0 þ L2 ReN þ Ha2 L 4 4 2 2 ðL Þ ; ð31Þ @x @x where

0

is the solution of the homogeneous equation ¼ 0;

ð32Þ

c1 ðxÞy3 c2 ðxÞy2 þ þ c3 ðxÞy þ c4 ðxÞ: 6 2

ð33Þ

L2

0

which yields 0

¼

The integrating constants c1 ; c2 ; c3 and c4 involved in Eq. (33) are to be determined from the given boundary conditions (24)–(27). Next, we decompose and N into the following forms: ¼

1 X

n

n

ð34Þ

n¼0

and N

¼

1 X

n Pn ;

ð35Þ

n¼0

where Pn are Adomian’s special polynomials. The parameter used in (34) and (35) is not a perturbation parameter, it is only used for grouping the terms of different 1750007-9


orders. Then, the parameterized form of (31) gives @4 @2 ¼ 0 þ L2 ReN þ Ha2 L 4 4 2 2 ðL Þ : @x @x

ð36Þ


Now, we substitute (34) and (35) into (36) and then equating the like-power terms of on both sides of the resulting expression, we get 4 @2 n 2 2 4@ RePn þ Ha L n 2 2 ðL n Þ ; ð37Þ nþ1 ¼ L @x4 @x where n ¼ 0; 1; 2; . . . . Once the component 0 is determined, the other components of such as 1 ; 2 ; 3 , etc. can be easily determined from (37). Further we take parameterized decomposition of 0 in the following way: 0

¼

1 X

0;n :

n

ð38Þ

n¼0

Substitution of (34), (35) and (38) into (36) gives the double decomposition components of and these are given by 4 @2 n 2 2 4@ RePn þ Ha L n 2 2 ðL n Þ ; ð39Þ nþ1 ¼ 0;nþ1 þ L @x4 @x where n ¼ 0; 1; 2; . . . : Since the expression for 0 contains the constants c1 ; c2 ; c3 and c4 , the parameterized decomposition forms of all these constants are given by c1 ¼

1 X

n c1;n ;

c2 ¼

n¼0

1 X n¼0

n c2;n ;

c3 ¼

1 X

n c3;n ;

c4 ¼

n¼0

1 X

n c4;n :

ð40Þ

n¼0

Now if we substitute (38) and (40) into (33) and then if we compare the like powers of on both side of the resulting expression, we get 0;nþ1

¼

c1;nþ1 y3 c2;nþ1 y2 þ þ c3;nþ1 y þ c4;nþ1 : 6 2

ð41Þ

The relations together with (39) and (41) give the components of . The constant involved in each n will be determined by their respective boundary conditions. The polynomials P0 ; P1 ; . . . ; Pn are Adomian’s polynomials and these are defined in the following way that P0 P0 ð 0 Þ, P1 P1 ð 0 ; 1 Þ, P2 P2 ð 0 ; 1 ; 2 Þ; . . . ; Pn Pn ð 0 ; . . . ; n Þ. In order to determine these polynomials, we substitute (34) and (35) into (29) and then comparison of the terms of like power of on both sides of the resulting equations give the following set of Adomians’s polynomials: @ 0 @ 0 @ 0 @ 0 P0 ¼ r2 r2 ; ð42Þ @y @x @x @y 1750007-10


@ 1 @ 0 @ 1 2 @ 0 2 @ 1 2 @ 0 P1 ¼ r þ r r @y @x @y @x @x @y @ 0 @ 1 r2 ; @x @y

ð43Þ

and so on. Again substitution of (34) into the boundary conditions (24)–(27) gives the boundary conditions for the respective components 0 ; 1 ; . . . ; etc. as follows:


@ 0 @2 0 þ

¼ 1 on y ¼ h1 ; @y @y2 @ 0 @2 0

¼ 1 on y ¼ h2 ; @y @y2 0

ð44Þ

F ¼ 0 on y ¼ h1 ; 2 ¼

F0 on y ¼ h2 ; 2

and @ n @2 n þ

¼ 0 on y ¼ h1 ; @y @y2 @ n @2 n

¼ 0 on y ¼ h2 ; @y @y2 n

F ¼ n on y ¼ h1 ; 2 ¼

Fn on y ¼ h2 ; 2

where n being any positive integer. Using the boundary conditions (44) in (33) we get 0

¼

ð45Þ

0

as follows:

c1 ðxÞy3 c2 ðxÞy2 þ þ c3 ðxÞy þ c4 ðxÞ; 6 2

ð46Þ

where the expressions for the co-efficient c1 ; c2 ; c3 and c4 are given in Appendix A. The expression for 1 can be obtained from (39) and (41) by putting n ¼ 0 and performing the operation of the inverse operator L2 , we get m 1 y3 m 2 y 2 þ þ m ¼ y þ m 1 3 4 6 2 2 Z yZ yZ yZ y c1 y þ c2 y þ c3 þ Re 2 0 0 0 0

1750007-11



3 @ y3 @3 y2 @3 @3 þ þ 2 c c c y þ c @x3 1 6 @x3 2 2 @x3 3 @x3 4 @ @ @ y3 @ y2 @ @ c1 y þ c2 c1 þ c2 þ c3 y þ c4 þ @x @x @x 6 @x 2 @x @x 2 2 2 2 @ y @ @ þ 2 c2 y þ 2 c3 þ c1 2 c @x2 1 2 @x @x 4 3 @ y @4 y2 @4 @4 þ þ þ Ha2 ðc1 y þ c2 Þ 2 c c c y þ c @x4 1 6 @x4 2 2 @x4 3 @x4 4 2 2 @ @ 2 c1 y þ 2 c2 dydydydy: 2 @x @x Using the first two components

0

and

¼

0

1,

þ

the approximate solution is given by 1

þ ...

Once we determined the stream function, then it can be easily obtained the expression for axial velocity components u ¼ @@y as u¼

@ 0 @ þ 1 þ ... @y @y

Similarly, the expression for wall shear is determined from the following relation: 2 @ w ¼ : @y2 y¼h1 or h2

4. Results and Discussion The analytical expressions for the stream function, axial velocity and WSS are derived in the previous section. The numerical results corresponding to the axial velocity, stream function and WSS have been computed with the help of MATHEMATICA software subject to the following data a ¼ b ¼ 0:5; d ¼ 1:0; ¼ 2:4; ¼ 0:0; ¼ 0:1; ¼ 0 ; 5 ; 10 ; 15 ; ¼ 0:0; 0:2; 0:4; 0:6; Ha ¼ 0; 1; 2; 4; 6; ¼ 0:1; 0:5; 0 :7; 0:9; Re ¼ 10; 30; 50; 70: 4.1. Velocity distribution Figure 2 represents the comparison of axial velocity with the results obtained in the present study and the results of Kothandapani and Srinivas,40 who have neglected the non-linear terms by assuming long wavelength and low Reynolds number. For the purpose of comparison, both the studies have been naturally brought to the same platform by considering appropriate values of the parameters. It is interesting to note from this figure that our results are in good agreement with those when the wave number ! 0 and Reynolds number Re ! 0. However, for ¼ 0:3 and Re ¼ 30, the present result shows significant discrepancy in the axial velocity. 1750007-12

Adomian Decomposition Method for MHD Flow of Blood Induced by Peristaltic Waves 2.0 1.5

Results of the present study when 0.3,Re 30.0

0.5 0.0

Results of Kothandapani and Srinivas 2008

0.5

Results of the present study when 0,Re 0

1.0 1.5

1.0

0.5

0.0

0.5

1.0

1.5

y

Fig. 2. Comparison of axial velocity distribution between the results of the present study and the study of Kothandapani and Srinivas.40

Figures 3–5 represent the variation of axial velocity u across the channel height for different values of the Hartmann number Ha, wave number and slip parameter . Figure 3 shows that the axial velocity decreases in the central region of the channel with increasing Hartmann number Ha, while the axial velocity increases in the vicinity of the channel wall in order to maintain constant flow rate. The reason behind this fact is the Lorentz force that arises due to the application of an external magnetic field, which in turn produces decelerating effect of the fluid motion. It confirms that the Lorentz force is very strong in the central region of the channel rather than near the wall. Figure 4 illustrates that the axial velocity increases in the central region with increasing wave number and it has decreasing effect in the vicinity of the

2.0 1.5 1.0 Ha 0.0 u


u

1.0

Ha 1.0

0.5

Ha 2.0 0.0

Ha 4.0 Ha 6.0

0.5 1.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

y

Fig. 3. Variation of axial velocity u for different values of Ha with Re ¼ 30; ¼ 0:1; ¼ 0:0; ¼ 0:0. 1750007-13


2

1 u

0.1 0.5

0

0.7 0.9

1.5

1.0

0.5

0.0

0.5

1.0

1.5

y

Fig. 4. Variation of axial velocity u for different values of with Ha ¼ 1:0; Re ¼ 30; ¼ 0:0; ¼ 0:0.

channel walls. It may be noted from this figure that the wave number has significant impact on the axial velocity, particularly near the vessel wall. Figure 5 shows that the axial velocity decreases in the central region with the increase of the slip parameter , whereas the velocity increases near the channel walls. This fact is influenced by the presence of velocity slip at the walls. We observe that the effects of magnetic field and slip velocity have similar impact on the axial velocity. 4.2. Trapping phenomena It is known that the phenomenon of trapping is the formation of an internally circulating bolus of the fluid is a region of closed streamlines that move with the speed in the wave frame. The bolus forms in a region where the flow rate is high 2.0

1.5

1.0 0.0 u


1

0.5

0.2

0.0

0.4 0.6

0.5

1.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

y

Fig. 5. Variation of axial velocity u for different values of with Ha ¼ 1:0; Re ¼ 30; ¼ 0:0001; ¼ 0:0. 1750007-14



(cf. Shapiro et al.5). Owing to the trapping phenomenon, there exist stagnation points, where both the velocity components of the fluid vanish in the wave frame. To see the effect of various non-dimensional parameters on the streamlines, we presented here some important results through Figs. 6–16. The distribution of streamlines for different values of the Hartmann number Ha have shown in Figs. 6– 8. We observe from these figures that the formation of trapped boluses decreases in size with the increase of Hartmann number Ha. As the magnetic field strength increases, the circulating bolus split up into small trapped boluses and transported in the forward direction with wave speed. Figures 9–11 show the variation of streamline patterns for Reynolds number 10, 50 and 70, respectively. From these figures, we observe that as the Reynolds number increases, more streamlines formed closed region of space and transported in the forward direction. The streamlines in the central region become laminar type when both the effects of Hartmann number and Reynolds number come into play. However, from Figs. 12–14, one can show that the more streamlines become trapped in the central region as well as in the vicinity of the channel walls in the presence of wave number. As the amplitude of the wave increases, more amount of trapped fluid passes through the channel. Thus, it may be pointed out that the wave number has significant impact on streamlines to form more large occlusions. Therefore, the consideration of wave number cannot be overcome while studying the peristaltic phenomena. Figures 15 and 16 illustrate the variation of streamlines with different inclination angle of the channel walls. As the inclination angle increases, the size of the trapped boluses also increases and is

1.50.988 1.144

1.04 1.24

1.04

0.988 1.248

1.0

0.988 1.248 0.936

1.04 1.196 0.728 0.364

0.0 0 0.572 0.5 0.884

1.3 1.0 1.248 1.092 1.5 0.988

1.196 0.936

1.196

1.092 0.5

0.98

0.988

0.78 0.884 0.8321.144

1.092

0.78 0.832 1.144 0.884

1.092

0.832 0.884 0.988 0.78 0.676 0.416 0.468 0.624 0.52 0.312 0.26 0.208 0.104 0.052 0.052 0.104 0.156 0.208 0.26 0.312 0.364 0.416 0.624 0.468 0.676 0.52 0.728 0.78 0.832 0.988 1.04 0.936 1.144 0.884 1.196 0.832 0.884 1.248 0.832 1.248 1.09 0.78 0.78 0.936 0.936 1.144 1.3 1.196 1.092 1.04 1.3 1.04 0.988 0.988 1.196 1.14 0.936

0.572 0.156

1.0

Fig. 6. Streamline pattern

0.98

0.988 0.5

0.0

0.5

1.0

for Ha ¼ 0:0 with ¼ 0:0; Re ¼ 30; ¼ 0:1; ¼ 0:0; ¼ 0:0. 1750007-15


1.50.988 1.144

0.988 0.936

1.04

1.196

0.78 0.884 0.832

0.50.936

1.196

1.092

1.196 0.936

1.248

1.144 0.78 0.832 1.144 0.884

1.04

1.092

0.78 0.988 0.728 0.884 0.676 0.416 0.572 0.52 0.364 0.208 0.312 0.26 0.156 0.052 0.0 0.104 0 0.052 0.156 0.208 0.26 0.364 0.416 0.624 0.468 0.676 0.52 0.57 0.728 0.78 0.832 0.884 0.988 1.04 0.5 0.936 1.092 0.884 1.04 0.832 1.248 0.884 0.832 1.248 1.09 0.78 0.78 1.196 1.248 0.936 0.936 0.988 1.144 1.092 1.3 1.3 1.0 1.04 1.3 1.144 0.988 1.196 1.19 1.144 1.5 0.988 0.98 0.988 0.832 0.468 0.104 0.312

0.624


1.04 1.248

1.092

0.988 1.248

1.0

0.98

0.988

1.0

0.5

0.0

1.5 0.98

0.98 0.98

0.98 1.176

1.029

1.029

1.176

0.98

1.0 1.127

1.029

0.931

0.931

1.127

1.078

0.98 0.5 0.735

1.0

for Ha ¼ 2:0 with ¼ 0:0; Re ¼ 30; ¼ 0:1; ¼ 0:0; ¼ 0:0.


.176

0.5

1.078

0.882 0.784 1.127 0.833

0.882 0.784 0.833

0.93 1.078

0.784

0.686

0.882 0.833 0.588 0.539 0.343 0.294 0.49 0.392 0.245 0.196 0.147 0.098 0.049 0.0 0 0.049 0.147 0.09 0.196 0.245 0.294 0.392 0.343 0.539 0.637 0.441 0.686 0.49 0.784 0.735 0.58 0.98 0.833 1.029 0.5 1.078 0.882 0.882 0.833 0.931 0.931 1.176 0.833 0.784 0.784 0.882 1.176 1.225 1.078 0.931 1.12 1.225 0.98 0.98 1.029 1.127 1.0 1.225 1.22 1.029 0.441

0.637

1.176

1.127 1.5 0.98 1.0


1.07 0.9

0.98 0.5

0.0

0.5

1.0

for Ha ¼ 6:0 with ¼ 0:0; Re ¼ 30; ¼ 0:1; ¼ 0:0; ¼ 0:0.

1750007-16



1.50.992 1.1160.961

0.992 0.961 1.0851.20 1.085 0.992 1.0 1.116 1.209 0.992 0.961 0.961 1.178 1.054 1.147 1.147 1.054 1.0231.178 1.085 1.0231.147 0.961 0.961 0.992 1.023 1.116 1.054 0.5 0.837 0.93 0.961 0.899 0.775 0.868 0.682 0.589 0.651 0.806 0.744 0.713 0.527 0.62 0.558 0.403 0.496 0.465 0.341 0.434 0.372 0.31 0.155 0.279 0.124 0.248 0.217 0.186 0.093 0.062 0.031 0.0 0 0.031 0.155 0.062 0.093 0.124 0.186 0.217 0.341 0.279 0.24 0.31 0.372 0.496 0.403 0.527 0.465 0.434 0.589 0.558 0.713 0.62 0.775 0.65 0.806 0.744 0.682 0.868 0.837 0.992 0.5 0.93 1.023 1.023 0.89 1.054 1.085 0.961 0.961 0.961 1.116 1.209 1.209 0.992 0.961 1.178 1.11 1.023 0.992 1.178 1.054 0.961 1.2 1.147 1.178 1.0 0.992 1.085 1.147 1.24 1.054 1.24 0.961 0.992 1.20 0.961 1.1470.961 0.9611.08 1.116 1.5 0.992 0.99 0.992 1.0

0.992 0.961 0.992 1.178 0.992 1.209 0.961

0.5


0.5

1.0

for Re ¼ 10:0 with ¼ 0:0; Ha ¼ 1:0; ¼ 0:1; ¼ 0:1; ¼ 0:0.

1.50.992 .1470.961

0.99 0.992

0.992 1.085

1.178 0.992 1.0

0.0

0.961 1.178 0.992

0.992

1.116 0.93

1.116 1.178 1.023

1.147 0.93

1.147 1.023 1.085 1.116 1.085 1.054 1.054 0.961 1.054 0.961 0.899 0.50.806 1.023 0.961 0.775 0.992 0.868 0.93 0.682 0.837 0.744 0.558 0.589 0.713 0.651 0.62 0.403 0.465 0.341 0.496 0.279 0.434 0.527 0.372 0.31 0.186 0.248 0.217 0.155 0.124 0.093 0.062 0.031 0.0 0.031 0 0.062 0.093 0.124 0.155 0.248 0.372 0.186 0.341 0.21 0.31 0.279 0.434 0.403 0.527 0.589 0.465 0.62 0.496 0.682 0.651 0.744 0.713 0.55 0.837 0.93 0.775 0.806 0.992 0.899 0.961 0.86 0.5 1.116 0.961 0.992 0.992 1.023 1.085 1.023 1.178 0.961 1.116 1.08 1.24 0.93 1.178 0.961 1.271 0.93 0.992 1.209 1.023 1.209 1.147 1.0 1.271 1.147 1.178 1.2 1.054 1.24 1.054 1.054 0.961 1.209 1.14 1.116 1.085 0.992 1.5 0.992 0.992 0.99 1.0


0.5

0.0

0.5

1.0

for Re ¼ 50:0 with ¼ 0:0; Ha ¼ 1:0; ¼ 0:1; ¼ 0:1; ¼ 0:0.

1750007-17


1.50.992 0.96 1.152

0.992 0.992 1.152 1.12

0.992 1.056

0.992 .2481.184

0.96 1.184 0.992

0.992 1.216 1.0241.216 1.24 1.088 1.152 0.896 0.896 1.184 1.088 1.024 1.12 1.248 1.12 0.928 0.928 1.056 1.056 0.96 0.96 0.96 1.088 0.992 0.86 0.5 0.832 1.024 0.928 0.8 0.672 0.896 0.736 0.60 0.576 0.768 0.704 0.64 0.544 0.448 0.512 0.384 0.32 0.48 0.416 0.352 0.288 0.224 0.256 0.192 0.16 0.128 0 0.0 0.064 0.032 0.096 0.064 0.032 0.096 0.192 0.352 0.16 0.224 0.256 0.12 0.288 0.32 0.512 0.448 0.38 0.544 0.48 0.416 0.704 0.64 0.672 0.576 0.608 0.8 0.768 0.736 0.928 0.864 0.896 0.83 0.5 1.12 0.992 0.96 1.152 1.024 0.96 0.96 1.0881.248 1.12 0.928 1.152 0.992 1.280.992 1.216 0.896 0.896 1.216 1.0881.216 1.152 0.96 1.184 1.0 1.28 0.928 1.12 1.24 1.056 1.248 1.056 1.024 1.184 0.992 1.18 0.992 0.96 1.088 1.05 1.024 1.5 0.992 0.99 0.992


1.0 1.216

1.0

0.5


0.0

1.0

for Re ¼ 70:0 with ¼ 0:0; Ha ¼ 1:0; ¼ 0:1; ¼ 0:1; ¼ 0:0.

1.50.988

0.98

0.988

1.144

1.04

0.988 1.248

1.0

0.5

1.196

1.04 1.24 0.988

0.936

1.04

1.196 0.936

1.248

1.092 1.196 0.884 1.092 0.78 0.78 0.8321.144 0.832 1.144 0.884 1.092 0.5 0.988 0.832 0.884 0.728 0.936 0.78 0.676 0.468 0.624 0.416 0.572 0.52 0.364 0.312 0.26 0.208 0.156 0.104 0.052 0.0 0 0.052 0.104 0.156 0.208 0.26 0.312 0.364 0.416 0.624 0.468 0.676 0.52 0.728 0.572 0.78 0.832 0.988 0.884 1.04 0.936 0.5 1.144 0.884 0.884 1.248 0.832 1.248 1.09 0.78 0.78 1.196 0.832 0.936 0.936 1.196 1.3 1.144 1.3 1.092 1.04 1.0 0.988 1.3 1.04 0.988 1.248 1.196 1.14 1.092 1.5 0.988 0.98 0.988 1.0


0.5

0.0

0.5

1.0

for ¼ 0:1 with ¼ 0:0; Ha ¼ 1:0; Re ¼ 30; ¼ 0:0; ¼ 0:0.

1750007-18


1.5 1.2

0.96

1.36

1.36

1.0

0.56 0.0 0 0.72


1.44 1.04 1.76

1.04 1.441.28 0.88 1.12

0.5 0.96

0.5 1.36 1.0 1.44 1.28 1.5 1.04

1.04

0.96

1.2

1.121.76 1.52

0.88 1.6 1.36 1.92 1.84 1.6 1.28 0.88 1.68 0.80.88 0.8 1.12 1.441.92 1.841.52 1.68 0.8 0.64 1.28 1.2 0.88 0.72 0.4 0.32 0.24 0.48 0.16 0.08 0.08 0.16 0.2 0.32 0.48 0.64 0.4 0.8 0.56 1.28 0.88 1.36 0.96 1.68 1.76 1.04 1.92 1.12 2 1.52 0.88 1.44 0.8 0.88 0.8 1.12 1.12 1.2 1.84 1.76 2 1.52 1.6 0.88 0.88 1.92 1.44 1.2 1.84 1.2 1.68 1.04 1.6 0.96 1.36 1.28 0.96

1.0


0.5

0.0

0.5

1.0

for ¼ 0:5 with ¼ 0:0; Ha ¼ 1:0; Re ¼ 30; ¼ 0:0; ¼ 0:0.

1.5 1.02 1.02

1.02 1.36 1.872.72 3.4 0.85 0.510.85 1.19 3.74 0.34 2.21 3.06 0.68 0.68 1.53 1.53 0.85 2.55 3.57 0.17 0.17 2.04 0.5 1.19 3.23 1.02 3.91 2.89 2.38 1.19 1.7 0.17 0.68 0.0 0.17 0.17 1.36 0.34 2.04 0.68 2.55 0.17 3.4 1.02 1.02 3.064.08 0.5 4.42 1.19 0 3.74 0.85 1.7 2.72 1.53 2.21 3.23 0 0.68 0.68 4.25 0.85 3.91 1.19 1.19 3.57 0.85 0.85 1.87 1.0 1.7 2.89 2.38

1.53 2.38 3.06 2.04 1.19 3.4

1.36 1.0 1.7

1.36

1.53 1.02

2.72 3.74 0.85 1.7 3.23 2.21 1.19 2.89 3.91 3.57 1.02 2.55 1.87 1.360.85 0 1.02 0.51 2.89 1.36 1.87 3.572.38 1.19 3.91 4.25 0.85 3.23 2.724.42 3.74 1.7 2.21 1.19 4.08 3.06 3.4 2.55 2.04 1.53 1.02

1.5 1.0


0.5

0.0

0.5

1.0

for ¼ 0:9 with ¼ 0:0; Ha ¼ 1:0; Re ¼ 30; ¼ 0:0; ¼ 0:0.

1750007-19



1.50.992 1.1160.961

0.992 0.992 1.17 1.116

0.992 0.961 1.178 0.992

1.054 1.178 0.992 0.992 1.0 1.023 1.085 1.209 0.93 0.93 1.209 1.147 1.147 1.023 1.147 1.2091.023 1.085 1.116 0.961 0.992 0.961 1.085 0.961 1.054 1.054 0.5 0.806 0.899 0.713 0.868 0.93 0.744 0.837 0.775 0.558 0.62 0.651 0.589 0.465 0.682 0.527 0.496 0.372 0.434 0.403 0.248 0.341 0.31 0.217 0.279 0.186 0.124 0.155 0.093 0.062 0.031 0.0 0 0.031 0.093 0.124 0.155 0.062 0.248 0.217 0.372 0.279 0.341 0.18 0.403 0.31 0.465 0.43 0.558 0.62 0.527 0.496 0.651 0.713 0.589 0.682 0.837 0.744 0.806 0.775 0.868 0.899 0.5 0.961 1.054 1.023 1.054 1.054 0.9 0.992 0.961 0.961 0.992 1.209 1.116 1.209 0.93 1.178 1.11 0.992 1.178 1.023 0.93 1.023 1.2 1.147 1.178 1.0 0.961 1.085 1.147 1.24 1.085 1.24 0.961 0.992 1.20 1.147 1.116 1.08 0.992 1.5 0.992 0.992 0.99 1.0


0.5

0.0

0.5

1.0

for ¼ 0:0 with ¼ 0:0; Ha ¼ 1:0; Re ¼ 30; ¼ 0:1; ¼ 0:1.

0.99 2

1.245 1.411 1.079 1.577

1.079 1

1.328

1.245 1.162

0.9960.996 0.913 0.996 0.664 1.162 1.079 1.494 1.411 0.498 1.328 0.415 1.411 1.577 1.245 0.747 0.083 0.249 0 0 0.2491.577 1.411 0.581 0.166 1.411 1.162 1.245 0.332 0.498 1.494 1.328 0.913 1.079 0.747 1.162 0.996 0.996 0.996 1

1.494 1.16

0.83 0.581 0.16

0.332 0.083

0.415 0.664

1.162

1.494 1.079 1.411

1.245 1.079

0.8

1.6

1.245 1.57

2

1.32 0.99 1.0


0.5

0.0

0.5

1.0

for ¼ 5:0 with ¼ 0:0; Ha ¼ 1:0; Re ¼ 30; ¼ 0:1; ¼ 0:1.

1750007-20


transported in the downstream direction. Thus, in the case of diverging channel, the trapped boluses as a whole can easily pass to its downward direction. 4.3. Wall shear stress The WSS plays an important role in the dynamics of blood flow through arteries. In cardiovascular research, the low WSS regions are identified because of the localization of atherosclerotic plaques at the endothelium of the vessel wall. It is known that the oscillatory WSS with low magnitude around ð 5 dyne=cm2 Þ increases

Ha 0.0

25

Ha 1.0 20

Ha 2.0 Ha 4.0

15 τw

Ha 6.0 10 5 0 5

1.5

1.0

0.5

0.0

0.5

1.0

1.5

x

Fig. 17. Variation of wall shear stress w along the length of the channel for different values of the Hartmann number Ha when ¼ 0:0; ¼ 0:0; Re ¼ 30; ¼ 0:1; ¼ 0:1.

50

0 τw


30

50 0.1 0.5 0.7 0.9

100

1.5

1.0

0.5

0.0

0.5

1.0

1.5

x

Fig. 18. Variation of wall shear stress w along the length of the channel for different values of the wave number when ¼ 5:0; ¼ 0:0; Ha ¼ 2:0; Re ¼ 30; ¼ 0:1. 1750007-21



endothelial levels. Figure 17 depicts the variation of WSS for different magnetic field strength that is for different Hartmann number Ha. It shows that the WSS is high in the wider part of the channel whereas, low wall stress is found in the narrowing part of the channel. The magnitude of the WSS increases with the increase of the Hartmann number. The distribution of wall shear stress for different wave number is shown in Fig. 18. It reveals that the WSS also oscillates as the channel walls propagate along the length of the channel with wave speed. The magnitude of the WSS increases rapidly as the wave number increases. Therefore, while studying biofluid dynamics particularly the blood flow problems, more attention should be given to the elastic response of the artery induced by the peristaltic motion. 5. Concluding Remarks In this paper, we have theoretically studied the blood flow through small blood vessels induced by peristaltic waves in the presence of slip velocity and externally applied magnetic field. The problem is solved analytically via ADM with an aim to analyze the non-linear phenomena. The present analysis pays due attention to see the effects of magnetic field, slip velocity as well as the wavelength and Reynolds number on the peristaltic transport of a physiological fluid. The main findings of the present study are summarized as follows: .

The axial velocity at the central region has a decelerating effect with the increasing values of the Hartmann number and the velocity slip parameter. However, the reversal trend is observed in the case of increasing wave number. . The role of slip velocity, magnetic field and Reynolds number have similar impact on the streamline patterns. . The wave number has also a significant role in transporting fluid mass in the form of bolus. . The magnitude of the WSS effectively increases with increasing wave number and thereby decreasing wavelength. Finally, we conclude that the low Reynolds number and long wavelength should not be overestimated during the study of blood flow problems induced by peristaltic wave. Therefore, our theoretical investigation bears the potential to useful in the biomedical engineering and technology. Acknowledgments The authors wish to convey their sincere thanks to all the esteemed reviewers for their comments and suggestions which led to improvement of the present version of the manuscript. Author G. C. Shit sincerely acknowledges SERB, Department of Science and Technology (DST), Govt. of India, New Delhi for the financial support of this investigation through the research project Grant No. SB/EMEQ-081/2013.

1750007-22


References 1. 2. 3. 4. 5. 6. 7.


8. 9.

10. 11. 12.

13. 14. 15. 16.

17.

18. 19. 20. 21. 22. 23. 24.

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25. Jekic M, Ding Y, Dzwonczyk R, Burns P, Raman SV, Simonetti OP, Magnetic field threshold for accurate electrocardiography in the MRI environment, Magn Reson Med 64:1586–1591, 2010. 26. Hina S, Mustafa M, Hayat T, Alsaedi A, Peristaltic flow of couple-stress fluid with heat and mass transfer: An application in biomedicine, J Mech Med Biol 15:1550042, 2015. 27. Hayat T, Javed M, Ali N, MHD Peristaltic transport of a Jeffery fluid in a channel with compliant walls and porous space, Transp Porous Media 74:259–274, 2008. 28. Zaman A, Ali N, Beg OA, Unsteady magnetohydrodynamic blood flow in a poroussaturated overlapping stenotic artery: Numerical modeling, J Mech Med Biol 16:1650049, 2015. 29. Shit GC, Roy M, Ng EYK, Effect of induced magnetic field on peristaltic flow of a micropolar fluid in an asymmetric channel, Int J Numer Methods Biomed Eng 26:1380– 1403, 2010. 30. Shit GC, Roy M, Effect of induced magnetic field on blood flow through a constricted channel: An analytical approach, J Mech Med Biol 16:1650030, 2016. 31. Misra JC, Sinha A, Shit GC, Mathematical modeling of a blood flow in a porous vessel having double stenoses in the presence of an external magnetic field, Int J Biomath 4:207–225, 2011. 32. Adomian G, Application of the decomposition method to the Navier–Stokes equations, J Math Anal Appl 119:340–360, 1986. 33. Adomian G, Nonlinear transport in moving fluids, Appl Math Lett 6:35–38, 1993. 34. Bayramoglu H, Bildik N, The solution of two dimensional nonlinear differential equation by the Adomian decomposition method, Appl Math Comput 163:519–524, 2005. 35. Adomian G, Rach R, Analytic solution of nonlinear boundary value problems in several dimensions, J Math Anal Appl 174:118–137, 1993. 36. Bayramoglu H, Bildik N, The solution of two dimensional nonlinear differential equation by the Adomian decomposition method, Appl Math Comput 163:519–524, 2005. 37. Beg OA, Tripathi D, Sochi T, Gupta PK, Adomian decomposition method (ADM) simulation of magneto-bio-tribological squeeze flim with magnetic induction effects, J Mech Med Biol 15:1550072, 2015. 38. Cherruault Y, Adomian G, Decomposition method: A new proof of convergence, Math Comput Model 18:103–106, 1993. 39. Shapiro AH, Latham TW, On peristaltic pumping, Proc Ann Conf Eng Med Biol San Francisco, Cal 8:147, 1966. 40. Kothandapani M, Srinivas S, Nonlinear peristaltic transport of a Newtonian fluid in an inclined asymmetric channel through a porus medium, Phys Lett A 372:1265–1276, 2008.

Appendix A The expressions that appear in Sec. 3 are listed as follows: 2 5 2 c1 3 h1 c1 ¼ þ h1 c2 ðh1 þ Þ ; ; c2 ¼ 5 ; c3 ¼ 1 c1 2 3 1 4 4 2 3 2 F c h c h c4 ¼ 0 1 1 2 1 c3 h1 ; 2 6 2 2 2 h1 h2 þ h1 h2 ; 2 ¼ ½ðh1 þ Þ ðh2 Þ; 1 ¼ 2 2

1750007-24


2 2 h 31 h 32 h h 1 h 22 ðh1 h2 Þ 2 h2 ; 4 ¼ ðh1 h2 Þðh2 Þ ; 6 2 2 5 ¼ ½F0 þ ðh1 h2 Þ;

3 ¼

c1 c 000 9h 81 c c 00 8h 71 c c 000 7h 6 c c 000 6h 5 c c 000 8h 71 1 þ 1 2 þ 1 3 1þ 1 4 1þ 2 1 12 3024 4 1680 2 840 2 360 6 1680 6 6 5 000 6h 5 000 4 000 000 000 5h 4 c c c c c2 c 000 7h c c 5h c c 7h c c 6h 2 1þ 2 3 1þ 2 4 1þ 3 1 1þ 3 2 1þ 3 3 1 þ 2 840 360 120 6 840 2 360 120 0 00 3 8 7 6 0 00 0 c 00 c c3 c 000 4h c c 9h c c 8h 7h 1 1 1 4 þ 1 2 þ 1 3 1 þ Re3 1 1 24 6 3024 6 1680 6 840 c 02 c 001 8h 71 c 02 c 002 7h 61 c 02 c 003 6h 51 þ þ þ 4 1680 2 840 2 360 6 5 0 00 0 00 0 00 c 3 c 1 7h 1 c 3 c 2 6h 1 c 3 c 3 5h 41 c 04 c 001 6h 51 c 04 c 002 5h 41 c 04 c 003 4h 31 þ þ þ þ þ þ 2 840 360 120 2 360 120 24 0 6 5 3 0 4 0 c1 c 1 7h 1 c1 c 2 6h 1 c1 c 3 5h 1 5h 41 4h 31 0 4h 1 2 þ þ þ c1 c 4 þ c2 Re þ Ha c1 6 840 2 360 120 24 120 24 0000 6 5 3 3 0000 4 4 c 7h 1 c 2 6h 5h 1 4h 1 5h 4h þ þ 1 þ c 0000 þ c 0000 2 1 2 c 001 1 þ c 002 1 3 4 6 840 2 360 120 24 120 24 0 6 5 5 4 0 4 c 1 c1 7h 1 6h 1 5h 1 c 2 c1 6h 1 5h 1 4h 31 0 0 0 0 þ c 1 c2 þ c 1 c3 þ þ c 2 c2 þ c 2 c3 þ Re ; 2 840 360 120 2 360 120 24 000 5 6 c c 72h 71 c1 c 002 56h 61 c1 c 000 c c 000 30h 41 c2 c 000 3 42h 1 1 56h 1 3 þ þ þ 1 4 þ N2 ¼ Re 1 1 12 3024 4 1680 2 840 2 360 6 1680 5 3 5 3 000 000 4 000 000 000 c c 42h 1 c2 c 3 30h 1 c2 c 4 20h 1 c3 c 1 42h 1 c3 c 2 30h 41 c3 c 000 3 20h 1 þ þ þ þ þ þ 2 2 2 840 360 120 6 840 2 360 120 0 00 7 6 5 6 0 00 000 2 0 00 0 00 c c 12h 1 c c 72h 1 c 1 c 2 56h 1 c 1 c 3 42h 1 c 2 c 1 56h 1 þ þ þ þ 3 4 Re3 1 1 24 6 3024 6 1680 6 840 4 1680 5 5 3 0 00 0 00 4 0 00 0 00 4 0 00 c c 42h 1 c 2 c 3 30h 1 c 3 c 1 42h 1 c 3 c 2 30h 1 c 3 c 3 20h 1 þ þ þ þ þ 2 2 2 840 2 360 2 840 360 120 3 0 00 4 0 00 0 00 2 c c 30h 1 c 4 c 2 20h 1 c 4 c 3 12h 1 þ þ þ 4 1 2 360 120 24 0 5 0 4 2 c1 c 1 42h 1 c1 c 2 30h 1 c1 c 03 20h 31 0 12h 1 þ þ þ c1 c 4 Re 6 840 2 360 120 24 3 2 20h 1 12h 1 þ c2 þ Ha2 c1 120 24 0000 5 3 0000 2 2 c 1 42h 1 c 2 30h 41 20h 31 0000 20h 1 0000 12h 1 00 12h 1 2 00 þ þ þ c3 þ c4 þ c2 2 c1 6 840 2 360 120 24 120 24 0 5 3 3 4 0 4 c 1 c1 42h 1 30h 1 20h 1 c 2 c1 30h 1 20h 1 12h 21 0 0 0 0 þ c 1 c2 þ c 1 c3 þ þ c 2 c2 þ c 2 c3 þ Re ; 2 840 360 120 2 360 120 24


N1 ¼ Re

3

1750007-25


N3 ¼ ½N1 þ N2 ; 000 c c 000 7h 6 c c 000 6h 5 c c 000 8h 72 c c 9h 82 c c 00 8h 72 þ 1 2 þ 1 3 2þ 1 4 2þ 2 1 N4 ¼ Re3 1 1 12 3024 4 1680 2 840 2 360 6 1680 6 c2 c 000 c c 000 6h 5 c c 000 5h 4 c c 000 7h 6 2 7h 2 þ 2 3 2þ 2 4 2þ 3 1 2 2 840 360 120 6 840 5 3 c c 000 6h c c 000 5h 4 c c 000 4h þ 3 2 2þ 3 3 2þ 3 4 2 2 360 120 24 0 00 8 0 00 c 0 c 00 7h 6 c 0 c 00 8h 72 c c 9h 2 c c 8h 72 c 0 c 00 7h 6 c 0 c 00 6h 5 þ 1 2 þ 1 3 2þ 2 1 þ 2 2 2þ 2 3 2 Re3 1 1 6 3024 6 1680 6 840 4 1680 2 840 2 360 6 5 5 3 0 00 0 00 0 00 4 0 00 0 00 4 0 00 c c 7h c c 6h c c 5h c c 6h c c 5h c c 4h þ 3 1 2þ 3 2 2þ 3 3 2þ 4 1 2þ 4 2 2þ 4 3 2 2 840 360 120 2 360 120 24 0 6 5 3 0 0 4 c1 c 1 7h 2 c1 c 2 6h 2 c1 c 3 5h 2 5h 42 4h 32 0 4h 2 2 þ þ þ c1 c 4 þ c2 Re þ Ha c1 6 840 2 360 120 24 120 24 0000 6 5 3 3 0000 4 4 c 7h 2 c 2 6h 5h 2 4h 2 5h 4h þ þ 2 þ c 0000 þ c 0000 2 1 2 c 001 2 þ c 002 2 3 4 6 840 2 360 120 24 120 24 0 6 5 5 4 0 4 c 1 c1 7h 2 6h 2 5h 2 c 2 c1 6h 2 5h 2 4h 32 0 0 0 0 þ c 1 c2 þ c 1 c3 þ þ c 2 c2 þ c 2 c3 þ Re ; 2 840 360 120 2 360 120 24


þ

N5 ¼

Re3

7 5 6 c1 c 000 c c 00 56h 62 c1 c 000 c c 000 30h 42 c2 c 000 3 42h 2 1 72h 2 1 56h 2 þ 1 2 þ þ 1 4 þ 12 3024 4 1680 2 840 2 360 6 1680

5 3 4 c2 c 000 c c 000 30h 42 c2 c 000 c c 000 42h 52 c3 c 000 c c 000 20h 32 2 42h 2 2 30h 2 4 20h 2 þ 2 3 þ þ 3 1 þ þ 3 3 2 840 360 120 6 840 2 360 120 0 00 7 6 5 000 2 0 00 0 00 c c 12h 2 c c 72h 2 c 1 c 2 56h 2 c 1 c 3 42h 2 þ 3 4 þ þ Re3 1 1 24 6 3024 6 1680 6 840

þ

c 02 c 001 56h 62 c 02 c 002 42h 52 c 02 c 003 30h 42 c 03 c 001 42h 52 c 03 c 002 30h 42 þ þ þ þ 4 1680 2 840 2 360 2 840 360 3 3 0 00 0 00 4 0 00 0 00 2 c c 20h 2 c 4 c 1 30h 2 c 4 c 2 20h 2 c 4 c 3 12h 2 þ 3 3 þ þ þ 120 2 360 120 24 0 c c 42h 52 c1 c 02 30h 42 c1 c 03 20h 32 12h 22 þ þ þ c1 c 04 Re 1 1 6 840 2 360 120 24 3 2 20h 2 12h 2 þ c2 þ Ha2 c1 120 24 0000 3 2 2 c 42h 52 c 0000 30h 42 20h 32 0000 12h 2 00 12h 2 00 20h 2 þ 2 þ þ c 0000 þ c þ c 2 1 2 c 3 2 4 1 6 840 2 360 120 24 120 24 0 5 3 3 4 0 4 c 1 c1 42h 2 30h 2 20h 2 c 2 c1 30h 2 20h 2 12h 22 0 0 0 0 þ c 1 c2 þ c 1 c3 þ þ c 2 c2 þ c 2 c3 þ Re ; 2 840 360 120 2 360 120 24 þ

N6 ¼ ½N4 N5 ; 1750007-26


N7 ¼

Re3

c1 c 000 h9 c c 00 h 8 c c 000 h 7 c c 000 h 6 c c 000 h 8 1 1 þ 1 2 1 þ 1 3 1 þ 1 4 1 þ 2 1 1 12 3024 4 1680 2 840 2 360 6 1680

c2 c 000 h7 c c 000 h 6 c c 000 h 5 c c 000 h 7 c c 000 h 6 c c 000 h 5 c c 000 h 4 2 1 þ 2 3 1þ 2 4 1þ 3 1 1 þ 3 2 1 þ 3 3 1þ 3 4 1 2 840 360 120 6 840 2 360 120 24 0 00 9 8 7 8 7 0 00 0 00 0 00 0 00 0 c c h c c h c c h c c h c c h c c 00 h 6 Re3 1 1 1 þ 1 2 1 þ 1 3 1 þ 2 1 1 þ 2 2 1 þ 2 3 1 6 3024 6 1680 6 840 4 1680 2 840 2 360 7 6 0 00 0 00 6 0 00 5 0 00 0 00 5 0 00 4 c c h c c h c c h c c h c c h c c h þ 3 1 1 þ 3 2 1þ 3 3 1þ 4 1 1 þ 4 2 1þ 4 3 1 2 840 360 120 2 360 120 24 0 7 6 0 5 0 4 c1 c 1 h 1 c1 c 2 h 1 c1 c 3 h 1 h 51 h 41 0 h1 2 þ þ þ c1 c 4 þ c2 Re þ Ha c1 6 840 2 360 120 24 120 24 0000 7 6 5 5 0000 4 4 c h1 c h h1 h1 h h þ 2 þ 1 þ c 0000 þ c 0000 2 1 2 c 001 1 þ c 002 1 3 4 6 840 2 360 120 24 120 24 0 7 6 5 6 5 0 c 1 c1 h 1 h1 h1 c 2 c1 h 1 h1 h 41 0 0 0 0 þ c 1 c2 þ c 1 c3 þ þ c 2 c2 þ c 2 c3 þ Re ; 2 840 360 120 2 360 120 24 F N8 ¼ 1 N 7 ; 2


þ

N9 ¼

Re3

c1 c 000 h9 c c 00 h 8 c c 000 h 7 c c 000 h 6 c c 000 h 8 1 2 þ 1 2 2 þ 1 3 2 þ 1 4 2 þ 2 1 2 12 3024 4 1680 2 840 2 360 6 1680

6 5 5 4 c2 c 000 h 72 c2 c 000 c2 c 000 c3 c 000 h 72 c3 c 000 h 62 c3 c 000 c3 c 000 2 3 h2 1 2 3 h2 4 h2 4 h2 þ þ þ þ þ þ þ 2 840 360 120 6 840 2 360 120 24 0 00 9 8 7 8 7 0 00 0 00 0 00 0 00 0 c c h c c 00 h 6 c c h c c h c c h c c h Re3 1 1 2 þ 1 2 2 þ 1 3 2 þ 2 1 2 þ 2 2 2 þ 2 3 2 6 3024 6 1680 6 840 4 1680 2 840 2 360 7 6 5 0 00 0 00 6 0 00 5 0 00 4 0 00 0 00 c c h c c h c c h c c h c c h c c h þ 3 1 2 þ 3 2 2þ 3 3 2þ 4 1 2 þ 4 2 2þ 4 3 2 2 840 360 120 2 360 120 24 0 7 6 5 0 0 4 cc h cc h cc h h h5 h4 Re 1 1 2 þ 1 2 2 þ 1 3 2 þ c1 c 04 2 þ Ha2 c1 2 þ c2 2 6 840 2 360 120 24 120 24 0000 7 6 5 5 0000 4 4 c h2 c h h2 h2 h h þ 2 þ 2 þ c 0000 þ c 0000 2 1 2 c 001 2 þ c 002 2 3 4 6 840 2 360 120 24 120 24 0 7 6 5 6 5 0 c c h h h c c h h h4 þ Re 1 1 2 þ c 01 c2 2 þ c 01 c3 2 þ 2 1 2 þ c 02 c2 2 þ c 02 c3 2 ; 2 840 360 120 2 360 120 24

N10 N13

F1 þ N9 ; N11 ¼ ½N8 N10 ; N12 ¼ ½N3 N6 ; ¼ 2 2 2 h1 h2 þ h1 h2 ; ¼ 2 2

N14 ¼ ½ðh1 þ Þ ðh2 Þ; N15 ¼ ½N11 N3 ðh1 h2 Þ;

1750007-27


h 21 þ h1 ; 2 2 h 1 h 22 N17 ¼ ðh1 h2 Þðh1 þ Þ ; 2 2 N12 N17 N15 N14 N15 N16 m1 m1 ¼ ; m2 ¼ ; N13 N17 N16 N14 N17 2 3 h1 h 2 m1 h 22 m2 þ h1 ðh1 þ Þm2 ; m4 ¼ N10 þ þ h2 m3 : m 3 ¼ N3 2 6 2 h 31 h 32 6 6

ðh1 h2 Þ


N16 ¼

1750007-28