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The hemodynamics of variable liquid properties on the MHD peristaltic mechanism of Jeffrey fluid with heat and mass transfer
Alexandria Engineering Journal (2020) 59, 693–706
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
The hemodynamics of variable liquid properties on the MHD peristaltic mechanism of Jeffrey fluid with heat and mass transfer B.B. Divya a, G. Manjunatha a,*, C. Rajashekhar a, Hanumesh Vaidya b, K.V. Prasad b a Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Karnataka 576104, India b Department of Mathematics, Vijayanagara Srikrishnadevaraya University, Ballari, Karnataka 583105, India
Received 11 April 2019; revised 4 November 2019; accepted 23 January 2020
KEYWORDS Darcy number; Biot number; Variable viscosity; Variable thermal conductivity; Concentration slip
Abstract The current work intends to look into the effects of variable liquid properties on the magnetohydrodynamics of peristaltic flow exhibited by Jeffrey fluid through a compliant-walled channel. In order to make realistic approximations for the flow characteristics of blood, the channel is considered to be inclined and porous. Furthermore, convective boundary conditions and concentration slip have been employed in the analysis. The mathematical formulation is established on the grounds of low Reynolds number and long wavelength approximations. Perturbation solution is obtained for the resulting non-linear differential equations of momentum and energy for small values of variable viscosity and variable thermal conductivity, whereas exact solution is found for the concentration field. The impact of various parameters included in the study is displayed graphically. A rise in the parameter for variable viscosity is found to accelerate the fluid flow, hence resulting in an increased bolus size. For variable thermal conductivity, a similar influence on the heat transfer was observed. The behaviour of the skin-friction coefficient, Nusselt and Sherwood numbers have also been plotted for the pertinent parameters. Ó 2020 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Peristalsis is the mechanism through which fluids are transported within a distensible tube. The fluid transport is * Corresponding author. E-mail address: [email protected] (G. Manjunatha). Peer review under responsibility of Faculty of Engineering, Alexandria University.
facilitated by means of a progressive wave of expansion and contraction, travelling along the tube wall. Peristalsis is a critical component of various biological processes such as the motion of food stuff through the oesophagus, transport of chyme through the gastrointestinal tract and many other such processes needed for the function of biological systems. One such process is the flow of urine through the ureter, which was studied by Latham [1] resulting in the first investigation of the peristaltic mechanism. Since then, multiple investiga-
https://doi.org/10.1016/j.aej.2020.01.038 1110-0168 Ó 2020 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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Nomenclature g B0 Br Bi Dm Da x w p0 y u w Ec E5 M E2 Tm Nu Pr Re Sc Sh Sr Cp k
acceleration due to gravity applied magnetic field Brinkman number Biot number coefficient of mass diffusivity Darcy number dimensionless axial distance dimensionless axial velocity dimensionless pressure dimensionless radial distance dimensionless radial velocity dimensionless streamline Eckert number elastic parameter of the wall Hartmann number mass characterising parameter mean temperature of the fluid Nusselt number Prandtl number Reynolds number Schmidt number Sherwood number Soret number specific heat at constant pressure thermal conductivity of the fluid
tions have been conducted to study the mechanism by which peristaltic transport of fluids takes place. While these investigations have considered both Newtonian and non-Newtonian fluids, the latter is of greater importance from a biological standpoint, as most physiological fluids exhibit nonNewtonian behaviour. Various non-Newtonian models for the peristaltic transport of fluids have been investigated, including the Casson model [2] and the Rabinowitsch fluid model [3]. Among the many non-Newtonian models, the Jeffrey model was found to be more significant in explaining the flow of blood through arteries. Hayat et al. [4] carried out one of the first studies on the Jeffrey model. Combinations of the Jeffrey fluid model and other non-Newtonian fluids have also been investigated. Sreenadh et al. [5] considered a twofluid model consisting of the power-law fluid in contact with a Jeffrey fluid flowing through an inclined channel. Over the past few years, researchers have also explored the peristaltic transport of Jeffrey fluid considering different geometrical configurations [6–8]. For biological systems, the effect of heat transfer in peristaltic transport also needs to be considered, as processes such as oxygenation of blood and hemodialysis involve the transfer of heat. Additionally, the human lungs, stones in the gallbladder, blood vessels of small radius etc. act as natural porous media, which makes it necessary to give due importance to porous medium considerations. It has also been observed that the walls of the capillaries have a flattened endothelial layer of cells around them, which are porous in nature. Furthermore, when blood clots in the lumen of artery, the resulting clogged region forms a porous medium. Some of the initial investigations considering both the aspects of heat transfer and porous
KT t a1 E3 E4 E1 b1 c
thermal diffusion ratio time undeformed radius of the tube wall damping parameter wall rigidity parameter wall tension parameter wave amplitude wave speed
Greek Symbols amplitude ratio a angle of inclination of the channel c coefficient of thermal conductivity a1 coefficient of variable viscosity / concentration of the fluid b1 concentration slip parameter h dimensionless temperature r electrical conductivity of the fluid q fluid density k1 Jeffrey parameter kðhÞ variable thermal conductivity b velocity slip parameter lðyÞ viscosity varying with channel width d wave number k wave length
media were conducted by Vajravelu et al. [9] and Srinivas and Gayathri [10]. Additionally, variable fluid viscosity needs to be considered in order to provide accurate results for the flow of blood through small arteries, which exhibit variation in thickness. One of the first investigations on this front was carried out by Srivastava et al. [11]. The simultaneous effect of heat transfer and variable viscosity in peristalsis was investigated by Nadeem et al. [12] ,which showed that increasing values of variable viscosity lead to a decrease in the axial velocity. Other considerations include the influence of mass transfer, which explains functions such as diffusion of nutrients from the blood to neighbouring tissues, and the compliant properties of the channel walls. Asghar et al. [13] studied the simultaneous effects of heat and mass transfer of a non-Newtonian Sisko fluid through a curved channel filled with porous medium. Their studies revealed that higher values of permeability parameter lowered the fluid velocity. Farooq et al. [14] studied the chemical reaction and thermophoretic deposition in the heat and mass transfer of a hyperbolic tangent fluid through a curved channel. They imposed the slip effects for velocity, temperature as well as concentration. In the study, the temperature and velocity profiles indicated a contrasting behaviour of the respective slip parameters. Among heat transfer mechanisms, convection is the most prevalent in blood flow. Hence, investigations have been conducted by considering the convective boundary conditions for fluids in different geometries and assumptions [15–17]. In addition, blood flowing through arteries can be best approximated by fluids that show variable thermal conductivity [18,19]. As the erythrocyte (red blood cell) is biomagnetic in nature, the presence of a magnetic field has been shown to influence
The hemodynamics of variable liquid properties on the MHD
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peristalsis in the arterial walls. Specifically, blood flow with a shear rate below 100 s1 has been shown to represent a mathematical model for MHD (magnetohydrodynamic) flow. Peristaltic transport of a fluid with MHD considerations in a channel helps in explaining problems concerning the movement of physiological fluids, operation of blood pump machines, research on the working of a peristaltic MHD compressor, etc. Also, magnetic fields are known to have safe and healing effects on the human body. Magnetic fields of suitable intensity can be used to control excessive bleeding during surgeries, transport drugs and treat cancerous/tumor cells. Thus, the effect of magnetic field on peristalsis of non-Newtonian fluids has been explored both in biological and industrial applications [20,21]. The analysis of MHD two-phase flow for a dusty electrically conducting fluid, carried out by Chamkha [22], suggested that increased values of the Hartmann number lead to a reduction in the volumetric flow rate and the skin-friction coefficient of both the phases. Effects of heat and mass transfer on peristaltic transport in the presence of a magnetic field have also been investigated [23–26]. Studies with magnetic field considerations for the Jeffrey fluid model have been conducted by Hayat et al. [27], which demonstrate the effects of Hartmann number and Jeffrey fluid parameter on pressure rise needed to produce zero flow rate. Further, non-linear convective flow of Jeffrey nanofluid was analyzed by Waqas et al. [28] in which they also considered the aspects of thermal radiation and heat generation. Sinha et al. [29] extended upon the work on heat transfer by examining the impact of velocity slip as well as temperature jump along with variable viscosity. Additionally, the effects of heat and mass transfer on the peristaltic motion for a magnetohydrodynamic fluid were also investigated with porous medium considerations along with the channel wall properties [30,31]. The hydromagnetic flow of two immiscible liquids was investigated by Chamkha [32] through both porous as well as non-porous channels. This extensive parameteric study revealed that the higher values of magnetic parameter and inverse Darcy number result in a reduction of the fluid velocity. Similar observations were made by Umavathi et al. [33] for a porous channel with viscous and Darcy dissipation effects. Srinivas et al. [34] examined the simultaneous effects of velocity slip and heat transfer on the MHD flow of a fluid through porous channel and compliant wall considerations. They observed that the permeability parameters strengthen the slip at the walls, whereas the Hartmann number weakens it. In the recent years, many studies have been done which include magnetic effects and transfer of heat and mass [35–38]. The present paper aims to build upon the work of the aforementioned researchers to examine the peristaltic mechanism for a magnetohydrodynamic Jeffrey fluid in the presence of heat and mass transfer. The authors have tried to build a mathematical model to give realistic manifestations for the flow of blood. For this purpose, the channel of peristaltic transport is considered to be inclined, with porous and compliant walls. No work has been reported yet, considering the combination of the two variable properties (viscosity as well as thermal conductivity) of an electrically conducting blood which is exposed to a radial magnetic field. The current work is one of the first attempts to include these complex rheological properties of blood. Further, convective boundary conditions have been considered for heat transfer and slip conditions are imposed for mass transfer. The results of the current investigation finds
applications in the peristaltic transport of blood in coronary arteries of smaller diameter, where the formation of blood clots or growth of tumors due to excessive cell division result in the lumen of the blood vessel acting as a porous media. 2. Development of the problem The two-dimensional flow of an incompressible nonNewtonian fluid is induced by the peristaltic waves travelling along the length of the channel. The wavelength of these wave trains is considered to be k which propagates with a speed c. The cartesian coordinates ðx0 ; y0 Þ are chosen where the propagation of the fluid takes place in the x0 direction and y0 is transverse to it. For simplicity, the channel is assumed to be axisymmetric. The channel is also considered to be inclined at an angle a with the horizontal surface, with its walls being flexible due to the effects of wall properties. Further, the flow is exposed to an external magnetic field of strength B0 which is applied in the direction perpendicular to the fluid flow (see Fig. 1). The electric field is taken to be zero. Also, the magnetic Reynolds number is assumed to be very small so that the induced magnetic field is negligible in comparison to the external magnetic field. The variable viscosity and variable thermal conductivity (variable liquid properties) of the non-Newtonian Jeffrey model is considered in the analysis. The porous conditions, convective conditions and concentration slip are taken into account. Geometry of the surface of the channel walls is represented as below: 2p 0 y ¼ Hðx0 ; t0 Þ ¼ a1 þ b1 sin ðx ct0 Þ ; ð1Þ k where a1 is the undeformed radius of the tube, b1 is the wave amplitude and t is time. Considering all the above flow properties, the equations governing the fluid flow in the laboratory frame are @w0 @v0 þ ¼ 0; @x0 @y0
ð2Þ
0 0 0 @w @p0 @s0x0 x0 @s0x0 y0 0 @w 0 @w ¼ þ w þ v þ þ q @t0 @x0 @y0 @x0 @x0 @y0 þ qg sin a rB20 w0 ;
ð3Þ
0 0 0 @v @p0 @s0x0 y0 @s0y0 y0 0 @v 0 @v q ¼ þ w þ v þ þ qg cos a; @t0 @x0 @y0 @y0 @x0 @y0 ð4Þ
Fig. 1
Geometry of the channel wall.
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Fig. 2 Velocity profiles for (a) M, (b) a1 , (c) Da, (d) b, (e) a and (f) k1 with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:02; k1 ¼ 0:2; a ¼ p4 ; a1 ¼ 0:02; ¼ 0:6; F ¼ 2; M ¼ 1. Table 1 Values of axial velocity for t ¼ 0:1; b ¼ 0:2; Da ¼ 0:02; k1 ¼ 0:2; a ¼ p4 ; a1 ¼ 0:02; ¼ 0:6; F1 ¼ 2; M ¼ 1 with varying value of the elastic parameters. Axial velocity (w) E2
E1
E3
E4
E5
y
0.09
0.1
0.11
0.04
0.045
0.05
0.4
0.45
0.5
0.001
0.0015
0.002
0.01
0.03
0.05
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1.0
0.4756 0.4735 0.4670 0.4563 0.4413 0.4223 0.3993 0.3726 0.3422 0.3085 0.2716 0.2319 0.1896 0.1451 0.0987 0.0507 0.0015 0.0485 0.0989 0.1494 1.3778
2.4333 2.4283 2.4133 2.3886 2.3541 2.3102 2.2571 2.1953 2.1252 2.0473 1.9621 1.8704 1.7728 1.6700 1.5629 1.4521 1.3385 1.2231 1.1066 0.9900 0.8741
4.3909 4.3831 4.3597 4.3208 4.2668 4.1980 4.1149 4.0180 3.9082 3.7861 3.6527 3.5090 3.3561 3.1950 3.0271 2.8535 2.6755 2.4946 2.3121 2.1294 1.9479
2.4333 2.4283 2.4133 2.3886 2.3541 2.3102 2.2571 2.1953 2.1252 2.0473 1.9621 1.8704 1.7728 1.6700 1.5629 1.4521 1.3385 1.2231 1.1066 0.9900 0.8741
3.4121 3.4057 3.3865 3.3547 3.3104 3.2541 3.1860 3.1067 3.0167 2.9167 2.8074 2.6897 2.5644 2.4325 2.2950 2.1528 2.0070 1.8589 1.7094 1.5597 1.4110
4.3909 4.3831 4.3597 4.3208 4.2668 4.1980 4.1149 4.0180 3.9082 3.7861 3.6527 3.5090 3.3561 3.1950 3.0271 2.8535 2.6755 2.4946 2.3121 2.1294 1.9479
2.4333 2.4283 2.4133 2.3886 2.3541 2.3102 2.2571 2.1953 2.1252 2.0473 1.9621 1.8704 1.7728 1.6700 1.5629 1.4521 1.3385 1.2231 1.1066 0.9900 0.8741
1.3014 1.2981 1.2880 1.2714 1.2482 1.2187 1.1830 1.1415 1.0943 1.0420 0.9847 0.9231 0.8575 0.7884 0.7163 0.6418 0.5655 0.4879 0.4096 0.3312 0.2533
0.1696 0.1679 0.1627 0.1542 0.1423 0.1272 0.1089 0.0876 0.0635 0.0366 0.0073 0.0243 0.0579 0.0933 0.1302 0.1684 0.2075 0.2473 0.2874 0.3276 0.3675
10.1616 10.1455 10.0971 10.0169 9.9053 9.7630 9.5912 9.3911 9.1641 8.9118 8.6362 8.3393 8.0232 7.6904 7.3433 6.9845 6.6169 6.2430 5.8659 5.4883 5.1131
6.2974 6.2869 6.2552 6.2027 6.1297 6.0366 5.9242 5.7932 5.6446 5.4795 5.2992 5.1048 4.8980 4.6802 4.4531 4.2183 3.9777 3.7331 3.4862 3.2391 2.9936
2.4333 2.4283 2.4133 2.3886 2.3541 2.3102 2.2571 2.1953 2.1252 2.0473 1.9621 1.8704 1.7728 1.6700 1.5629 1.4521 1.3385 1.2231 1.1066 0.9900 0.8741
2.4333 2.4283 2.4133 2.3886 2.3541 2.3102 2.2571 2.1953 2.1252 2.0473 1.9621 1.8704 1.7728 1.6700 1.5629 1.4521 1.3385 1.2231 1.1066 0.9900 0.8741
2.3341 2.3292 2.3147 2.2907 2.2572 2.2145 2.1630 2.1029 2.0348 1.9592 1.8765 1.7874 1.6926 1.5928 1.4887 1.3811 1.2708 1.1587 1.0455 0.9323 0.8197
2.2349 2.2302 2.2161 2.1928 2.1603 2.1189 2.0689 2.0106 1.9445 1.8711 1.7909 1.7044 1.6124 1.5155 1.4145 1.3101 1.2031 1.0942 0.9845 0.8745 0.7653
0 2 0 0 0 @T @ T @ 2 T0 @w0 0 @T 0 @T ¼ k þ s0x0 x0 0 þ qCp þ w þ v þ 0 0 0 02 02 @t @x @y @x @y @x 0 @v @w0 @v0 0 0 þ sy0 y0 0 ; þ ð5Þ sx0 y0 @x0 @y0 @y
2 0 0 0 @C0 @ C @ 2 C0 Dm KT @ 2 T0 @ 2 T0 0 @C 0 @C þ : þ w þ v ¼ D þ þ m @t0 @x0 @y0 @x02 @y02 Tm @x02 @y02 ð6Þ
The hemodynamics of variable liquid properties on the MHD The fluid flow is unsteady in the laboratory frame. To overcome this, it is necessary to introduce a wave frame ð x; yÞ moving away from the laboratory frame of reference ðx0 ; y0 Þ with a constant velocity c. The transformations corresponding to this are ¼ w0 c; v ¼ v0 and pð xÞ ¼ p0 ðx0 ; t0 Þ: x ¼ x0 ct0 ; y ¼ y0 ; w 0
ð7Þ
0
and v are the axial and transverse velocities in Here, w ; v ; w the laboratory and wave frame of reference respectively. Also, T0 is the fluid temperature, and C0 is the concentration of the fluid. The above dimensional parameters are rendered dimensionless using the following x¼
ct0 H a1 a1 x y w v ; y ¼ ; w ¼ ; v ¼ ; t ¼ ; h ¼ ; sij ¼ s0ij ; d ¼ ; a1 k a1 c cd k lc k
qDm KT T0 lCp ;M ¼ ; Pr ¼ Tm lC0 k
rffiffiffi r C0 C0 B0 a1 ; / ¼ ; l C0
ð8Þ
where the above non-dimensional parameters are mentioned in the nomenclature. On making the transformation from laboratory frame to wave frame with Eq. (7) and using the dimensionless quantities given by Eq. (8), the Eqs. (3)–(6) take the following form @w @p @sxx @sxy sin a þ M2 ðw þ 1Þ þ Red w ¼ þd ; ð9Þ @x @y @x @x F @v @v @p @sxy @syy cos a d3 Re w þ d þv ¼ þ d2 ; @x @y @x @y @y F
Fig. 3
@h ¼ Red w @x þ v @h @y
ð10Þ
Velocity profiles for k1 as obtained by Bhatti et al. [39].
@2 h @y2
2
@ h þ d2 @x 2 i @v @w sxy d @x þ @y ;
1 Pr
h @v þ Ec d sxx @w þ þ syy @y @x
ð11Þ
@/ @/ 1 @2/ @2/ @2h @2h ¼ Red w þ d2 2 þ Sr d2 2 þ 2 ; ð12Þ þv 2 @x @y Sc @y @x @x @y
where the constitutive equation for Jeffrey fluid is 2dlðyÞ dk2 c @ v @ @w 1þ w þ ; sxx ¼ 1 þ k1 a1 @x d @y @x sxy ¼
ð13Þ
lðyÞ dk2 c @ v @ @w @v @w 1þ w þ þd ; ð14Þ 1 þ k1 a1 @x d @y @y @x @x
syy ¼
qca1 cl T0 T0 a2 c2 l ;F ¼ Re ¼ ;h ¼ ; p ¼ 1 p0 ; Ec ¼ ; Sc ¼ ; 2 qga1 qDm l T0 ckl Cp T0
Sr ¼
697
2dlðyÞ dk2 c @ v @ @w 1þ w þ : 1 þ k1 a1 @x d @y @x
ð15Þ
The variable viscosity lðyÞ and variable thermal conductivity kðhÞ is considered to be varying exponentially with y and h as lðyÞ ¼ ea1 y ¼1 a1 y þ Oða21 Þ;
ð16Þ
kðhÞ ¼ ech ¼1 þ ch þ Oðc2 Þ;
ð17Þ
where a1 and c are the coefficients of variable viscosity and variable thermal conductivity respectively. On assuming long wavelength(d 1) and low Reynolds number, the final set of equations representing our problem is given by @p ¼0; @y
ð18Þ
@p @ lðyÞ @w sin a ¼ M2 ðw þ 1Þ þ ; @x @y 1 þ k1 @y F
ð19Þ
2 @ @h lðyÞ @w kðhÞ ¼Br ; @y @y 1 þ k1 @y
ð20Þ
@2/ @2h ¼ ScSr 2 : 2 @y @y
ð21Þ
The corresponding boundary conditions in dimensional form are pffiffiffiffiffiffiffi Da @w @w ¼ 0 at y ¼ 0; w ¼ 1 at y ¼ h; b @y @y
the
non-
ð22Þ
Fig. 4 Skin-friction coefficient along the channel length for (a) a1 , (b) Da and (c) k1 with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:2; b ¼ 0:2; Da ¼ 0:04; k1 ¼ 0:2; a ¼ p4 ; a1 ¼ 0:02; ¼ 0:2; F ¼ 2; M ¼ 1.
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@h @h ¼ 0 at y ¼ 0; þ Bih ¼ 0 at y ¼ h; @y hy
ð23Þ
@/ @/ ¼ 0 at y ¼ 0; b1 þ / ¼ 0 at y ¼ h; @y @y
ð24Þ
@3 @3 @2 @5 @ E1 3 þ E2 þ E þ E þ E 3 4 5 @x @x@t2 @x@t @x5 @x @ lðyÞ @w sin a M2 ðw þ 1Þ þ at y ¼ h: h¼ @y 1 þ k1 @y F
ð25Þ
3. Solution methodology Due to the non-linearity in the Eqs. (19) and (20), we cannot find an exact solution for the problem. Hence, we adopt the method of perturbation to expand the velocity field w and tem-
perature profile h about small values of variable viscosity a1 and variable thermal conductivity c. However, closed form solution is obtained for concentration from Eq. (21). The perturbation series for w and h are taken as w ¼w0 þ a1 w1 þ Oða21 Þ;
ð26Þ
h ¼h0 þ ch1 þ Oðc2 Þ:
ð27Þ
Substituting the above in Eqs. (19), (20), (22) and (23), we obtain the expressions for velocity and temperature profiles from perturbation technique (up to first order) as, w¼
C1 a1 y C1 M1 y2 cosðM1 yÞ þ C3 a1 sinðM1 yÞ C1 þ C2 a1 þ 4 4
1 ð28Þ 2 ðP fÞð1 þ k1 Þ þ M21 ; M1
Fig. 5 Temperature profiles for (a) c, (b) a1 , (c) M, (d) Bi, (e) Br, (f) a, (g) Da, (h) k1 and (i) elastic parameters with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:08; k1 ¼ 0:2; a ¼ p4 ; a1 ¼ 0:02; M ¼ 3; Bi ¼ 5; Br ¼ 2; c ¼ 0:2; ¼ 0:6; F1 ¼ 2.
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Fig. 6 Nusselt number along the channel length for (a) c, (b) Bi, (c) Br, (d) Da, (e) k1 and (f) elastic parameters with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:08; k1 ¼ 2; a ¼ p4 ; a1 ¼ 0:02; M ¼ 3; Bi ¼ 5; Br ¼ 2; c ¼ 0:2; ¼ 0:2; F ¼ 2.
Br cC2 y2 l1 ðyÞ þ C4 y þ C5 þ cC6 C4 C5 cy 4 1 þ k1 2 2 l1 ðyÞBr c cBr þ ðC4 l2 ðyÞy þ C5 l1 ðyÞÞ; 1 þ k1 2 1 þ k1
h¼
ð29Þ
Br / ¼ ScSr l1 ðyÞ þ C4 y þ C5 þ cC6 C4 C5 cy 1 þ k1 # 2 cC24 y2 l1 ðyÞBr c cBr ðC4 l2 ðyÞy þ C5 l1 ðyÞÞ þ C7 y þ C8 ; ð30Þ þ 2 1 þ k1 2 1 þ k1
where M1 ¼ f¼ P¼
pffiffiffiffiffiffiffiffiffiffiffiffiffi M 1 þ k1 ; sin a ; F 8p3 E2p3
@h @h ; @x @y y¼h
ð35Þ
@h @/ : Sh ¼ @x @y y¼h
ð36Þ
Nu ¼
E5 sinð2pðx tÞÞ ðE1 þ E2 4p2 E4 4p 2 Þ cosð2pðx tÞÞ :
The expressions for l1 ðyÞ; l2 ðyÞ and Ci ; Aj ði ¼ 1; 2; . . . 8; j ¼ 1; 2; . . . ; 9Þ are as Appendix. We have the stream function w given by @w @w ;v ¼ ; w¼ @y @x and
4. Discussion of the graphical solutions the constants given in the
w ¼ 0 at y ¼ 0:
ð31Þ
ð32Þ
Using the above, we obtain the expression for w as w¼
The key dimensionless parameters of interest in the field of bioengineering are the skin-friction coefficient ðCf Þ, Nusselt number ðNuÞ and Sherwood number ðShÞ. These quantities are defined as below: @h @w Cf ¼ ; ð34Þ @x @y y¼h
sinðM1 yÞðC1 þ C2 a1 Þ C3 a1 cosðM1 yÞ C1 a1 þ M1 M1 4M21 C1 M1 a1 ðy2 M21 4M31 y cosðM1 yÞ þ 2yM1 sinðM1 yÞ þ 2 cosðM1 yÞÞ 2 M1 ðyM1 sinðM1 yÞ þ cosðM1 yÞÞ
½ðP fÞð1 þ k1 Þ þ M21 :
ð33Þ
The present section aims to analyse the graphical results of the influence of various parameters involved in the investigation. Discussions are carried out on the behaviour of velocity profiles, temperature and concentration fields, streamlines, skin-friction coefficient, Nusselt number and Sherwood number. 4.1. Flow characteristics Fig. 2 exhibits the behaviour of velocity profile for variations in M; a1 ; Da; b; a; k1 and wall properties (E1 ; E2 ; E3 ; E4 and E5 ). It is evident from these figures that the axial velocity has a parabolic trajectory, with the velocity being maximum in the central region of the channel. It is seen from Fig. 2(a) that Hartmann number M decelerates the fluid flow. This behaviour is expected, as transverse magnetic field resists the flow. Particularly in blood, transverse magnetic field results in the
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Fig. 7 Concentration profiles for (a) c, (b) a1 , (c) M, (d) Bi, (e) k1 (f) elastic parameters, (g) Sc, (h) Sr and (i) b1 with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:08; k1 ¼ 0:2; a ¼ p4 ; a1 ¼ 0:02; M ¼ 3; Bi ¼ 5; Br ¼ 2; Sc ¼ 0:2; Sr ¼ 0:3; b1 ¼ 0:2; ¼ 0:6; c ¼ 0:2; F1 ¼ 2.
formation of rouleaux leading to a decrease in velocity. The increasing effect of variable viscosity on axial velocity near the channel walls is sketched in Fig. 2(b). The consideration of porous walls is significant in the study of blood flow through arteries. As the porosity of the walls increase, the flow resistance decreases, thus resulting in an increase in the fluid velocity near the channel walls (see Fig. 2(c)). The role of b is found to be opposite in comparison to Da, as observed in Fig. 2(d). Most of the blood vessels are not horizontal and are inclined at various angles. Thus it becomes important in a physical context to consider the inclination of channel in biological systems. As the channel becomes more inclined away from the horizontal surface, the reduction in gravitational forces increase the velocity of the fluid, which is clearly seen in Fig. 2(e). Results for horizontal channel can be derived from
the present model by substituting a as zero. Similar results are shown in Fig. 2(f) for increasing values of the Jeffrey parameter k1 . The corresponding results for a Newtonian fluid can be obtained by taking k1 as zero. Thus, Jeffrey model helps us attain the results for both Newtonian and non-Newtonian behaviour of blood. Table 1 gives the dependence of axial velocity on wall properties. The parameters E1 and E2 account for the elastic nature of the walls, while E3 ; E4 and E5 are damping/resistive forces. Accordingly, it can be seen clearly that fluid flows rapidly with E1 and E2 and the flow is retarded for increasing values of E3 ; E4 and E5 . Fig. 3 depicts the velocity profile for varying k1 obtained by Bhatti et al. [39] in their study on MHD peristalsis of a Jeffrey fluid. For Da ¼ 0; a ¼ 0 and a1 ¼ 0, results of the current work are in excellent agreement with those obtained by Bhatti et al. [39].
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Fig. 8 Sherwood number along the channel length for (a) c, (b) Br, (c) Da, (d) elastic parameters, (e) Sc and (f) Sr with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:08; k1 ¼ 2; a ¼ p4 ; a1 ¼ 0:02; M ¼ 3; Bi ¼ 5; Br ¼ 2; Sc ¼ 0:2; Sr ¼ 0:3; b1 ¼ 0:2; ¼ 0:2; c ¼ 0:2; F ¼ 2.
Fig. 9 Streamlines for (a) M ¼ 3 and (b) M ¼ 3:1 with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:02; k1 ¼ 2; a ¼ p4 ; a1 ¼ 0:02; ¼ 0:6; F1 ¼ 2.
The influence of a1 ; Da and k1 on skin-friction coefficient Cf can be seen in Fig. 4. The oscillatory behaviour of the graphs is due to the sinusoidal movement of peristaltic waves along the channel walls. Fig. 4(a) and (b) show an increase in the magnitude of Cf for increasing values of a1 and Da, whereas Fig. 4(c) shows a contrasting behaviour for k1 . 4.2. Heat transfer characteristics The fluctuations in temperature profile of the fluid during heat transfer as a result of variations in
c; a1 ; M; Bi; Br; a; Da; k1 ; E1 ; E2 ; E3 ; E4 and E5 are scrutinized in Fig. 5. The temperature profiles are parabolic with maximum temperature occuring near the channel center. This behaviour can be explained by the effect of viscous dissipation which increases the temperature in the centre of the channel. This phenomenon occurs due to the viscosity of the fluid, leading to the conversion of kinetic energy of the fluid to internal thermal energy. It is known that the thermal conductivity of a fluid indicates the ability of the fluid to retain or liberate heat to its surroundings. Thus, when the thermal conductivity of the fluid bound inside the channel is larger than the wall
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Fig. 10 Streamlines for (a) a1 ¼ 0:01 and (b) a1 ¼ 0:03 with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:02; k1 ¼ 2; a ¼ p4 ; M ¼ 3; ¼ 0:6; F1 ¼ 2.
Fig. 11 Streamlines for (a) k1 ¼ 2 and (b) k1 ¼ 2:2 with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; M ¼ 3; k1 ¼ 2; a ¼ p4 ; a1 ¼ 0:02; ¼ 0:6; F1 ¼ 2.
temperature, thermal reading of the fluid rises (see Fig. 5(a)). Fig. 5(b) depicts the increasing effect of variable viscosity on temperature profile. This nature is justified by the fact that with an increase in the variable viscosity of the fluid, its heat transfer capacity decreases. The decreasing effect of M on temperature is seen in Fig. 5(c). The variation of Bi is illustrated in Fig. 5(d). Biot number decreases the thermal conductivity which in turn reduces the fluid temperature. This decreasing behaviour can be observed in the figure. An opposite behaviour of Br on temperature profile is seen in Fig. 5(e). This is attributed to an intensification in the flow resistance due to an increase in Br, which causes the viscous dissipation effects to inflate the internal thermal energy. Contrasting effects of a and Da are seen in Fig. 5(f) and (g), wherein the temperature is found to rise with the channel inclination and drop with the increasing porosity of the channel walls. The temperature of the fluid is enhanced with the Jeffrey parameter k1 (see Fig. 5(h)). From Fig. 5(i), the temperature profile intensifies for increasing values of E1 and E2 , whereas an opposite tendency is noticed for E3 ; E4 and E5 . The plots shown in Fig. 6 show the variation of Nusselt number Nu with c; Bi; Br; Da; k1 and the wall properties. Nu
measures the temperature gradient at the channel walls. The dimensionless quantity of Nusselt number is the ratio of convective to conductive heat transfer. The parameters c; Bi and Br have an increasing effect on the absolute value of Nusselt number, as seen in Fig. 6(a)–(c). On the other hand, Fig. 6(d) and (e) shows that higher values of Da and k1 result in a decrease in the magnitude of Nu. From Fig. 6(f), it can be observed that larger values of E1 ; E2 and E3 lead to an increase in the magnitude of Nu, whereas the absolute value of Nu is reduced by increasing E4 and E5 . 4.3. Mass transfer characteristics Fig. 7 depicts the concentration profiles for different values of c; a1 ; M; Bi; k1 ; Sc; Sr; b1 ; E1 ; E2 ; E3 ; E4 and E5 . Figs. 7(a)-(f) reveal an opposite behaviour for the respective parameters as compared to the temperature profiles. As mass and heat are known to be inversely proportional, this behaviour is expected in a physical context. Additionally, it can be inferred from the profiles that the particulate matter within the fluid is more concentrated at the periphery compared to the central region of
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Fig. 12 Streamlines for (a) E1 ¼ 0:09; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01, (b) E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01, (c) E1 ¼ 0:1; E2 ¼ 0:045; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01, (d) E1 ¼ 0:1; E2 ¼ 0:045; E3 ¼ 0:45; E4 ¼ 0:002; E5 ¼ 0:01, (e) E1 ¼ 0:1; E2 ¼ 0:045; E3 ¼ 0:45; E4 ¼ 0:0022; E5 ¼ 0:01 and (f) E1 ¼ 0:1; E2 ¼ 0:045; E3 ¼ 0:45; E4 ¼ 0:0022; E5 ¼ 0:04 with t ¼ 0:1; b ¼ 0:2; M ¼ 3; k1 ¼ 2; a ¼ p4 ; a1 ¼ 0:02; ¼ 0:6; F1 ¼ 2.
the channel. From a biological standpoint, this behaviour is in order to facilitate the diffusion of vital nutrients from blood and other fluids to the adjacent cells and tissues. Fig. 7(g)-(i) demonstrate a decreasing trend for increasing values of Sc; Sr and b1 . Fig. 8 shows the variation of Sherwood number with c; Br; Da, wall properties ðE1 E5 Þ; Sc and Sr. Sherwood number measures the concentration gradient at the walls of the channel. It is a dimensionless quantity, which gives the
ratio of convective mass transfer to rate of diffusive mass transport. The Sherwood number sees an increase in magnitude for increasing values of c and Br (see Fig. 8(a) and (b)), while the higher values of Da decreases the absolute value of Sh (see Fig. 8(c)). Fig. 8(d) reveals a rise in the magnitude of Sh as E1 ; E2 and E3 increase, and a drop in Sh for higher values of E4 and E5 . Increasing values of Sc and Sr bear an incraesing effect on the absolute value of Sh, as indicated in Fig. 8(e) and (f).
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4.4. Trapping phenomenon
Declaration of Competing Interest
Trapping is the most important and widely discussed topic in the peristaltic mechanism. When the streamlines get closed, it stimulates the formation of a bolus of fluid which circulates internally and moves forward along with the peristaltic waves. This phenomenon may be responsible for the thrombus formation which eventually blocks the flow of blood through narrow vessels. This naturally occurring episode of trapping was first noticed by Shapiro [40] in his studies on peristalsis. The variation in the size of bolus formed by trapping is shown in Figs. 9– 12. The volume of trapped bolus is found to increase for M and a1 as seen in Figs. 9 and 10. The volume of the trapped bolus is found to decrease with increasing Jeffrey parameter (k1 ) (see Fig. 11). Fig. 12 elucidates the effects of the various parameters accounting the wall properties on the size of the trapped bolus.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment The authors sincerely thank the valuable and constructive comments of the referees for the improvement of this research article.
Appendix A. l1 ðyÞ ¼
5. Summary and conclusions
1 ½C21 ð525sinð2yM1 Þa31 þ 210cosð2yM1 ÞM1 a21 ð3 þ 4ya1 Þ 26880M31
4y6 M71 a21 ð7 þ 5ya1 Þ þ 630sinð2yM1 ÞM21 a1 ð4 þ ya1 þ y2 a21 Þþ
The impact of variable liquid properties (variable viscosity and variable thermal conductivity) are examined on the MHD peristaltic flow through an inclined channel with porous walls. The flow of blood is modelled by the Jeffrey fluid. The influence of critical parameters on velocity, heat and mass transfer are sketched using MATLAB, analysed and discussed in detail. The streamlines are also plotted to observe the trapping phenomena. The results reported in the work have potential applications in industries as well as medical science. Along with the consideration of variable liquid properties in a MHD peristaltic flow, physical conditions such as an asymmetric/tapered asymmetric artery with porous medium can be considered in the future works. The current work serves a fair theoretical estimate for further work in this directon. The major outcomes of the present article can be summarised as: a1 accelerates the fluid and Da decelerates it in the central part of the channel. The magnetic parameter M causes a drop in the velocity and temperature, but they are enhanced by the increasing inclination of the channel. A rise in temperature is observed for increasing values of c and Br, and decreasing values of Bi and k1 . The wall property parameters E1 and E2 play a vital role in the growth of velocity and temperature profiles, while increasing values of E3 ; E4 and E5 result in a fall in velocity and temperature. The behaviour of temperature and concentration profiles are opposite to each other. The increase in the values of parameters Sc; Sr and b1 is responsible for reduction in the concentration of particles in the fluid. a1 leads to depressed values of the magnitude of C f whereas c has an increasing effect on the absolute value of Nu. The magnitude of Sh sees an increase for higher values of Sc and Sr. The size of the bolus trapped during peristalsis is seen to increase with M and a1 , whereas the Jeffrey parameter (k1 ) reduces the size of the trapped bolus.
210y2 sinð2yM1 ÞM41 a1 ð8 þ 6ya1 þ y2 a21 Þ 140M31 ð24cosð2yM1 Þþ 12ycosð2yM1 Þa1 6y2 cosðyM1 Þ2 a21 þ y3 ð1 þ 3cosð2yM1 ÞÞa31 Þþ 14y2 M51 ð480 þ 80ya1 5y2 ð17 þ 3cosð2yM1 ÞÞa21 þ 3y3 ð7 þ 5 cosð2yM1 ÞÞa31 ÞÞ 1120M21 a21 ðC23 ð3sinð2yM1 Þa1 þ 2y2 M31 ð3 þ ya1 Þ 3cosð2yM1 ÞM1 ð1 þ ya1 ÞÞ þ C22 ð3sinð2yM1 Þa1 þ 2y2 M31 ð3þ ya1 Þ þ 3cosð2yM1 ÞM1 ð1 þ ya1 ÞÞ þ 6C2 C3 ðcosð2yM1 Þa1 þ sinð2yM1 Þ M1 ð1 þ ya1 ÞÞÞ þ 56C1 M1 a1 ðC3 ð15sinð2yM1 Þa21 15cosð2yM1 ÞM1 a1 ð7 þ ya1 Þ þ 2y4 M51 a1 ð5 þ 3ya1 Þ 30sinð2yM1 ÞM21 ð4 þ 3ya1 Þ 10 y2 M31 a1 ð3ð2 þ cosð2yM1 ÞÞ þ yð2 þ 3cosð2yM1 ÞÞa1 ÞÞ þ 5C2 ð3cosð2yM1 Þ a21 þ 6y2 sinð2yM1 ÞM31 a1 ð1 þ ya1 Þ þ 3sinð2yM1 ÞM1 a1 ð7 þ ya1 Þ 6 cosð2yM1 ÞM21 ð4 þ 3ya1 Þ 2y2 M41 ð24 þ 2ya1 þ 3y2 a21 ÞÞÞ; l2 ðyÞ ¼
1 ½C21 ð4095cosð2yM1 Þa31 þ 420y4 sinð2yM1 ÞM51 a21 ð1 þ ya1 Þ 107520M41
2y7 M81 a21 ð8 þ 5ya1 Þ þ 210sinð2yM1 ÞM1 a21 ð5 þ 29ya1 Þ þ 70y2 M41 a1 ð4 þ ya1 Þð12cosð2yM1 Þ þ yð2 þ 9cosð2yM1 ÞÞa1 Þ 210cosð2yM1 Þ M21 a1 ð40 22ya1 þ 21y2 a21 Þ 420sinð2yM1 ÞM31 ð16 þ 16ya1 14y2 a21 þ5y3 a31 Þ 28y3 M61 ð320 40ya1 þ 34y2 a21 þ 7y3 a31 ÞÞ 1120M21 a21 ð6C2 C3 ð3sinð2yM1 Þa1 þ 2cosð2yM1 ÞM1 ð1 þ ya1 ÞÞ þ C23 ð9cosð2yM1 Þa1 þ 2y3 M41 ð4 þ ya1 Þ 6sinð2yM1 ÞM1 ð1 þ ya1 ÞÞ þ C22 ð9cosð2yM1 Þa1 þ 2y3 M41 ð4 þ ya1 Þ þ 6sinð2yM1 ÞM1 ð1 þ ya1 ÞÞÞ þ 112C1 M21 a1 ðC3 ð10 y3 M31 a1 ð4 þ ya1 Þ þ 2y5 M51 a1 ð2 þ ya1 Þ 30y2 sinð2yM1 ÞM21 a1 ð1þ ya1 Þ þ 15sinð2yM1 Þa1 ð11 þ 2ya1 Þ 15cosð2yM1 ÞM1 ð8 8ya1 þ 3y2 a21 ÞÞ þC2 ð30y2 cosð2yM1 ÞM21 a1 ð1 þ ya1 Þ þ 15cosð2yM1 Þa1 ð11 þ 2ya1 Þþ 15sinð2yM1 ÞM1 ð8 8ya1 þ 3y2 a21 Þ 2y3 M41 ð80 þ 5ya1 þ 6y2 a21 ÞÞÞ;
The hemodynamics of variable liquid properties on the MHD C1 ¼ C2 ¼
1 þ M12 ½ðP fÞð1 þ k1 Þ þ M21 1 pffiffiffiffi ; cosðM1 hÞ bDa M1 sinðM1 hÞ
705 A3 ¼
1 ½C2 ð4095cosð2hM1 Þa31 þ 420h4 sinð2hM1 ÞM51 a21 ð1 þ ha1 Þ 107520M41 1 2h7 M81 a21 ð8 þ 5ha1 Þ þ 210sinð2hM1 ÞM1 a21 ð5 þ 29ha1 Þþ
1
pffiffiffiffi ½1 C3 sinðM1 hÞ cosðM1 hÞ bDa M1 sinðM1 hÞ
70h2 M41 a1 ð4 þ ha1 Þð12cosð2hM1 Þ þ hð2 þ 9cosð2hM1 ÞÞa1 Þ
pffiffiffiffiffiffiffi Da C1 h C1 h cosðM1 hÞ þ M1 sinðM1 hÞ ½C3 M1 cosðM1 hÞ 4 4 b C1 þ ðcosðM1 hÞ M21 h2 cosðM1 hÞ 3hM1 sinðM1 hÞÞ ; 4 2
210cosð2hM1 ÞM21 a1 ð40 22ha1 þ 21h2 a21 Þ 420sinð2hM1 ÞM31 ð16 þ 16ha1 14h2 a21 þ 5h3 a31 Þ 28h3 M61 ð320 40ha1 þ 34h2 a21 þ 7h3 a31 ÞÞ 1120M21 a21 ð6C2 C3 ð3sinð2hM1 Þa1 þ 2cosð2hM1 ÞM1 ð1 þ h a1 ÞÞ þ C23 ð9cosð2hM1 Þa1 þ 2h3 M41 ð4 þ ha1 Þ 6sinð2hM1 ÞM1 ð1 þ ha1 ÞÞ þ C22 ð9cosð2hM1 Þa1 þ 2h3 M41 ð4 þ ha1 Þ þ 6sinð2hM1 Þ
C3 ¼
C1 ; 4M1
M1 ð1 þ ha1 ÞÞÞ þ 112C1 M21 a1 ðC3 ð10h3 M31 a1 ð4 þ ha1 Þ þ 2h5 M51
C4 ¼
Br ½240a21 M21 ð4C2 C3 M1 þ C22 a1 C23 a1 Þþ 1920M21 ð1 þ kÞ
a1 ð11 þ 2ha1 Þ 15cosð2hM1 ÞM1 ð8 8ha1 þ 3h2 a21 ÞÞ þ C2 ð30h2
C21 ð240M21 a1 15a31 Þ þ 20C1 M1 a1 ð24C2 M1 a1 þ C3 ð48M21 þ 3a21 ÞÞ; Br A1 1 þh ; þ A2 C4 C5 ¼ 1 þ k1 Bi Bi 2 Br A1 A2 A22 h h2 1 þ þh C6 ¼ þ þ C24 þ C4 C5 1 þ k1 Bi 2 Bi Bi 2 Br C4 A2 h C4 A3 C5 A1 þ þ þ C4 A3 h þ C5 A2 ; 1 þ k1 Bi Bi Bi Br A4 þ C4 A6 ; C7 ¼ScSr 1 þ k1 Br A1 þ C4 A7 þ ScSrðA8 þ cA9 Þ C7 h; C8 ¼b1 ScSr 1 þ k1 A1 ¼
1 ½C2 ð15cosð2hM1 Þa31 30h4 sinð2hM1 ÞM51 a21 ð1 þ ha1 Þ 1920M21 1
a1 ð2 þ ha1 Þ 30h2 sinð2hM1 ÞM21 a1 ð1 þ ha1 Þ þ 15sinð2hM1 Þ cosð2hM1 ÞM21 a1 ð1 þ ha1 Þ þ 15cosð2hM1 Þa1 ð11 þ 2ha1 Þ þ 15 sinð2hM1 ÞM1 ð8 8ha1 þ 3h2 a21 Þ 2h3 M41 ð80 þ 5ha1 þ 6h2 a21 ÞÞÞ; 1 þ k1 ; A4 ¼C4 Br 1 A5 ¼ ½1120M21 a21 ð3C22 M1 þ 3C23 M1 þ 6C2 C3 a1 Þþ 26880M31 C21 ð3360M31 þ 630M1 a21 Þ þ 56C1 M1 a1 ð105C3 M1 a1 þ 5C2 ð24M21 þ 3a21 ÞÞ; 2 Br Br A6 ¼ A4 A5 ðC4 A5 þ C5 A4 Þ þ C4 C5 ; 1 þ k1 1 þ k1 2 Br Br A1 A2 ðC4 A2 þ C4 A1 h þ C5 A1 Þ þ C24 h þ C4 C5 ; A7 ¼ 1 þ k1 1 þ k1 Br A8 ¼ A2 þ C4 h þ C5 ; 1 þ k1 2 1 A2 Br Br C2 h2 þ ðC4 A3 h þ C5 A2 Þ 4 C4 C5 h þ C6 : A9 ¼ 2 1 þ k1 1 þ k1 2
15sinð2hM1 ÞM1 a21 ð3 þ 2ha1 Þ 2h5 M61 a21 ð6 þ 5ha1 Þ þ 30M21 a1 ð8cosð2hM1 Þ þ hð2 þ 5cosð2hM1 ÞÞa1 h2 a21 Þ þ 30sinð2hM1 ÞM31
References
2
ð16 þ 7h2 a21 þ 4h3 a31 Þ þ 5hM41 ð192 þ 96hsinðhM1 Þ a1 þ 4h2 ð17 þ 6cosð2hM1 ÞÞa21 42h3 sinðhM1 Þ2 a31 ÞÞ þ 240M21 a21 ð2C3 C4 ðsinð2hM1 Þa1 þ 2cosð2hM1 ÞM1 ð1 þ ha1 ÞÞ þ C23 ðcosð2hM1 Þa1 2hM21 ð2 þ ha1 Þ þ 2sinð2hM1 ÞM1 ð1 þ ha1 ÞÞ C24 ðcosð2hM1 Þa1 þ 2hM21 ð2 þ ha1 Þ þ 2sinð2hM1 ÞM1 ð1 þ ha1 ÞÞÞ þ 20C1 M1 a1 ð3C3 ðsinð2hM1 Þa21 þ 2cosð2hM1 ÞM1 a1 ð4 þ ha1 Þ þ 2sinð2hM1 ÞM21 2
ð8 þ 4ha1 þ 3h2 a21 Þ þ 4hM31 ð8 2hcosðhM1 Þ a1 þ h2 ð2þ cosð2hM1 ÞÞa21 ÞÞ þ C4 ð3cosð2hM1 Þa21 þ 12h2 sinð2hM1 ÞM31 a1 ð1þ ha1 Þ þ 6sinð2hM1 ÞM1 a1 ð4 þ ha1 Þ þ 2h3 M41 a1 ð4 þ 3ha1 Þ 6M21 2
ð8cosð2hM1 Þ 8hsinðhM1 Þ a1 þ h2 ð2 þ 3cosð2hM1 ÞÞa21 ÞÞÞ; A2 ¼
1 ½C2 ð525sinð2hM1 Þa31 þ 210cosð2hM1 ÞM1 a21 ð3 þ 4ha1 Þ 26880M31 1 4h6 M71 a21 ð7 þ 5ha1 Þ þ 630sinð2hM1 ÞM21 a1 ð4 þ ha1 þ h2 a21 Þþ 210h2 sinð2hM1 ÞM41 a1 ð8 þ 6ha1 þ h2 a21 Þ 140M31 ð24cosð2hM1 Þ þ 12hcosð2hM1 Þa1 6h2 cosðhM1 Þ2 a21 þ h3 ð1 þ 3cosð2hM1 ÞÞa31 Þþ 14h2 M51 ð480 þ 80ha1 5h2 ð17 þ 3cosð2hM1 ÞÞa21 þ 3h3 ð7 þ 5 cosð2hM1 ÞÞa31 ÞÞ 1120M21 a21 ðC23 ð3sinð2hM1 Þa1 þ 2h2 M31 ð3 þ ha1 Þ 3cosð2hM1 ÞM1 ð1 þ ha1 ÞÞ þ C22 ð3sinð2hM1 Þa1 þ 2h2 M31 ð3þ ha1 Þ þ 3cosð2hM1 ÞM1 ð1 þ ha1 ÞÞ þ 6C2 C3 ðcosð2hM1 Þa1 þ sinð2hM1 ÞM1 ð1 þ ha1 ÞÞÞ þ 56C1 M1 a1 ðC3 ð15sinð2hM1 Þa21 15 cosð2hM1 ÞM1 a1 ð7 þ ha1 Þ þ 2h4 M51 a1 ð5 þ 3ha1 Þ 30sinð2hM1 Þ M21 ð4 þ 3ha1 Þ 10h2 M31 a1 ð3ð2 þ cosð2hM1 ÞÞ þ hð2 þ 3 cosð2hM1 ÞÞa1 ÞÞ þ 5C2 ð3cosð2hM1 Þa21 þ 6h2 sinð2hM1 ÞM31 a1 ð1þ ha1 Þ þ 3sinð2hM1 ÞM1 a1 ð7 þ ha1 Þ 6cosð2hM1 ÞM21 ð4 þ 3ha1 Þ 2h2 M41 ð24 þ 2ha1 þ 3h2 a21 ÞÞÞ;
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706 [11] L.M. Srivastava, V.P. Srivastava, S.N. Sinha, Peristaltic transport of a physiological fluid, Biorheology 20 (2) (1983) 153–166. [12] S. Nadeem, T. Hayat, Noreen Sher Akbar, M.Y. Malik, On the influence of heat transfer in peristalsis with variable viscosity, Int. J. Heat Mass Transf. 52 (21–22) (2009) 4722–4730. [13] Zeeshan Asghar, Nasir Ali, Raheel Ahmed, Muhammad Waqas, Waqar Azeem Khan, A mathematical framework for peristaltic flow analysis of non-newtonian sisko fluid in an undulating porous curved channel with heat and mass transfer effects, Comput. Methods Programs Biomed. 182 (2019) 105040. [14] S. Farooq, M. Ijaz Khan, T. Hayat, M. Waqas, A. Alsaedi, Theoretical investigation of peristalsis transport in flow of hyperbolic tangent fluid with slip effects and chemical reaction, J. Mol. Liq. 285 (2019) 314–322. [15] Abderrahim Wakif, Zoubair Boulahia, Farhad Ali, Mohamed R. Eid, Rachid Sehaqui, Numerical analysis of the unsteady natural convection mhd couette nanofluid flow in the presence of thermal radiation using single and two-phase nanofluid models for cu-water nanofluids, Int. J. Appl. Comput. Math. 4 (3) (2018) 81. [16] Fateh Mebarek-Oudina, Convective heat transfer of titania nanofluids of different base fluids in cylindrical annulus with discrete heat source. Heat Transfer—Asian, Research 48 (1) (2019) 135–147. [17] Ali J Chamkha, Double-diffusive convection in a porous enclosure with cooperating temperature and concentration gradients and heat generation or absorption effects, Numerical Heat Transfer: Part A: Applications 41 (1) (2002) 65–87. [18] K.V. Prasad, K. Vajravelu, Hanumesh Vaidya, Neelufer Z Basha, V. Umesh, Thermal and species concentration of mhd casson fluid at a vertical sheet in the presence variable fluid properties, Ain Shams Eng. J. (2017). [19] K.V. Prasad, K. Vajravelu, Hanumesh Vaidya, M.M. Rashidi, Z. Basha Neelufer, Flow and heat transfer of a casson liquid over a vertical stretching surface: Optimal solution, Am. J. Heat Mass Transfer 5 (1) (2018) 1–22. [20] Abderrahim Wakif, Zoubair Boulahia, Rachid Sehaqui, Numerical analysis of the onset of longitudinal convective rolls in a porous medium saturated by an electrically conducting nanofluid in the presence of an external magnetic field, Res. Phys. 7 (2017) 2134–2152. [21] Mohammad Rahimi-Gorji, O. Pourmehran, M. Gorji-Bandpy, D.D. Ganji, Unsteady squeezing nanofluid simulation and investigation of its effect on important heat transfer parameters in presence of magnetic field, J. Taiwan Inst. Chem. Eng. 67 (2016) 467–475. [22] Ali Chamkha, Hydromagnetic two-phase flow in a channel, 1995. [23] Safia Akram, S. Nadeem, Anwar Hussain, Effects of heat and mass transfer on peristaltic flow of a bingham fluid in the presence of inclined magnetic field and channel with different wave forms, J. Magn. Magn. Mater. 362 (2014) 184–192. [24] Ajaz Ahmad Dar, K. Elangovan, Influence of an inclined magnetic field on heat and mass transfer of the peristaltic flow of a couple stress fluid in an inclined channel. World, J. Eng. 14 (1) (2017) 7–18.
B.B. Divya et al. [25] Muhammad Ijaz Khan, Muhammad Waqas, Tasawar Hayat, Ahmed Alsaedi, Soret and dufour effects in stretching flow of jeffrey fluid subject to newtonian heat and mass conditions, Res. Phys. 7 (2017) 4183–4188. [26] Ali Chamkha, Mhd flow of a micropolar fluid past a stretched permeable surface with heat generation or absorption, 2009. [27] T. Hayat, Maryam Shafique, A. Tanveer, A. Alsaedi, Hall and ion slip effects on peristaltic flow of jeffrey nanofluid with joule heating, J. Magn. Magn. Mater. 407 (2016) 51–59. [28] M. Waqas, S.A. Shehzad, T. Hayat, M. Ijaz Khan, A. Alsaedi, Simulation of magnetohydrodynamics and radiative heat transport in convectively heated stratified flow of jeffrey nanofluid, J. Phys. Chem. Solids 133 (2019) 45–51. [29] A. Sinha, G.C. Shit, N.K. Ranjit, Peristaltic transport of mhd flow and heat transfer in an asymmetric channel: Effects of variable viscosity, velocity-slip and temperature jump, Alexandria Eng. J. 54 (3) (2015) 691–704. [30] T. Hayat, M. Umar Qureshi, Q. Hussain, Effect of heat transfer on the peristaltic flow of an electrically conducting fluid in a porous space, Appl. Math. Model. 33 (4) (2009) 1862–1873. [31] M. Gnaneswara Reddy, Heat and mass transfer on magnetohydrodynamic peristaltic flow in a porous medium with partial slip, Alexandria Eng. J. 55 (2) (2016) 1225–1234. [32] Ali J Chamkha, Flow of two-immiscible fluids in porous and nonporous channels, J. Fluids Eng. 122 (1) (2000) 117–124. [33] Jawali C. Umavathi, J. Prathap Kumar, Ali J. Chamkha, Ioan Pop, Mixed convection in a vertical porous channel, Transp. Porous Media 61 (3) (2005) 315–335. [34] S. Srinivas, R. Gayathri, M. Kothandapani, The influence of slip conditions, wall properties and heat transfer on mhd peristaltic transport, Comput. Phys. Commun. 180 (11) (2009) 2115–2122. [35] O. Pourmehran, M. Rahimi-Gorji, D.D. Ganji, Heat transfer and flow analysis of nanofluid flow induced by a stretching sheet in the presence of an external magnetic field, J. Taiwan Inst. Chem. Eng. 65 (2016) 162–171. [36] Fateh Mebarek-Oudina, Oluwole Daniel Makinde, Numerical simulation of oscillatory mhd natural convection in cylindrical annulus: Prandtl number effect, in: Defect and Diffusion Forum, vol. 387, Trans Tech Publ, 2018, pp. 417–427. [37] Muhammad Waqas, Muhammad Farooq, Muhammad Ijaz Khan, Ahmed Alsaedi, Tasawar Hayat, Tabassum Yasmeen, Magnetohydrodynamic (mhd) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition, Int. J. Heat Mass Transf. 102 (2016) 766–772. [38] H.S. Takhar, A.J. Chamkha, G. Nath, Unsteady flow and heat transfer on a semi-infinite flat plate with an aligned magnetic field, Int. J. Eng. Sci. 37 (13) (1999) 1723–1736. [39] M.M. Bhatti, R. Ellahi, A. Zeeshan, Study of variable magnetic field on the peristaltic flow of jeffrey fluid in a non-uniform rectangular duct having compliant walls, J. Mol. Liq. 222 (2016) 101–108. [40] Ascher H. Shapiro, Michel Yves Jaffrin, Steven Louis Weinberg, Peristaltic pumping with long wavelengths at low reynolds number, J. Fluid Mech. 37 (4) (1969) 799–825.
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
The hemodynamics of variable liquid properties on the MHD peristaltic mechanism of Jeffrey fluid with heat and mass transfer B.B. Divya a, G. Manjunatha a,*, C. Rajashekhar a, Hanumesh Vaidya b, K.V. Prasad b a Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Karnataka 576104, India b Department of Mathematics, Vijayanagara Srikrishnadevaraya University, Ballari, Karnataka 583105, India
Received 11 April 2019; revised 4 November 2019; accepted 23 January 2020
KEYWORDS Darcy number; Biot number; Variable viscosity; Variable thermal conductivity; Concentration slip
Abstract The current work intends to look into the effects of variable liquid properties on the magnetohydrodynamics of peristaltic flow exhibited by Jeffrey fluid through a compliant-walled channel. In order to make realistic approximations for the flow characteristics of blood, the channel is considered to be inclined and porous. Furthermore, convective boundary conditions and concentration slip have been employed in the analysis. The mathematical formulation is established on the grounds of low Reynolds number and long wavelength approximations. Perturbation solution is obtained for the resulting non-linear differential equations of momentum and energy for small values of variable viscosity and variable thermal conductivity, whereas exact solution is found for the concentration field. The impact of various parameters included in the study is displayed graphically. A rise in the parameter for variable viscosity is found to accelerate the fluid flow, hence resulting in an increased bolus size. For variable thermal conductivity, a similar influence on the heat transfer was observed. The behaviour of the skin-friction coefficient, Nusselt and Sherwood numbers have also been plotted for the pertinent parameters. Ó 2020 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Peristalsis is the mechanism through which fluids are transported within a distensible tube. The fluid transport is * Corresponding author. E-mail address: [email protected] (G. Manjunatha). Peer review under responsibility of Faculty of Engineering, Alexandria University.
facilitated by means of a progressive wave of expansion and contraction, travelling along the tube wall. Peristalsis is a critical component of various biological processes such as the motion of food stuff through the oesophagus, transport of chyme through the gastrointestinal tract and many other such processes needed for the function of biological systems. One such process is the flow of urine through the ureter, which was studied by Latham [1] resulting in the first investigation of the peristaltic mechanism. Since then, multiple investiga-
https://doi.org/10.1016/j.aej.2020.01.038 1110-0168 Ó 2020 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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Nomenclature g B0 Br Bi Dm Da x w p0 y u w Ec E5 M E2 Tm Nu Pr Re Sc Sh Sr Cp k
acceleration due to gravity applied magnetic field Brinkman number Biot number coefficient of mass diffusivity Darcy number dimensionless axial distance dimensionless axial velocity dimensionless pressure dimensionless radial distance dimensionless radial velocity dimensionless streamline Eckert number elastic parameter of the wall Hartmann number mass characterising parameter mean temperature of the fluid Nusselt number Prandtl number Reynolds number Schmidt number Sherwood number Soret number specific heat at constant pressure thermal conductivity of the fluid
tions have been conducted to study the mechanism by which peristaltic transport of fluids takes place. While these investigations have considered both Newtonian and non-Newtonian fluids, the latter is of greater importance from a biological standpoint, as most physiological fluids exhibit nonNewtonian behaviour. Various non-Newtonian models for the peristaltic transport of fluids have been investigated, including the Casson model [2] and the Rabinowitsch fluid model [3]. Among the many non-Newtonian models, the Jeffrey model was found to be more significant in explaining the flow of blood through arteries. Hayat et al. [4] carried out one of the first studies on the Jeffrey model. Combinations of the Jeffrey fluid model and other non-Newtonian fluids have also been investigated. Sreenadh et al. [5] considered a twofluid model consisting of the power-law fluid in contact with a Jeffrey fluid flowing through an inclined channel. Over the past few years, researchers have also explored the peristaltic transport of Jeffrey fluid considering different geometrical configurations [6–8]. For biological systems, the effect of heat transfer in peristaltic transport also needs to be considered, as processes such as oxygenation of blood and hemodialysis involve the transfer of heat. Additionally, the human lungs, stones in the gallbladder, blood vessels of small radius etc. act as natural porous media, which makes it necessary to give due importance to porous medium considerations. It has also been observed that the walls of the capillaries have a flattened endothelial layer of cells around them, which are porous in nature. Furthermore, when blood clots in the lumen of artery, the resulting clogged region forms a porous medium. Some of the initial investigations considering both the aspects of heat transfer and porous
KT t a1 E3 E4 E1 b1 c
thermal diffusion ratio time undeformed radius of the tube wall damping parameter wall rigidity parameter wall tension parameter wave amplitude wave speed
Greek Symbols amplitude ratio a angle of inclination of the channel c coefficient of thermal conductivity a1 coefficient of variable viscosity / concentration of the fluid b1 concentration slip parameter h dimensionless temperature r electrical conductivity of the fluid q fluid density k1 Jeffrey parameter kðhÞ variable thermal conductivity b velocity slip parameter lðyÞ viscosity varying with channel width d wave number k wave length
media were conducted by Vajravelu et al. [9] and Srinivas and Gayathri [10]. Additionally, variable fluid viscosity needs to be considered in order to provide accurate results for the flow of blood through small arteries, which exhibit variation in thickness. One of the first investigations on this front was carried out by Srivastava et al. [11]. The simultaneous effect of heat transfer and variable viscosity in peristalsis was investigated by Nadeem et al. [12] ,which showed that increasing values of variable viscosity lead to a decrease in the axial velocity. Other considerations include the influence of mass transfer, which explains functions such as diffusion of nutrients from the blood to neighbouring tissues, and the compliant properties of the channel walls. Asghar et al. [13] studied the simultaneous effects of heat and mass transfer of a non-Newtonian Sisko fluid through a curved channel filled with porous medium. Their studies revealed that higher values of permeability parameter lowered the fluid velocity. Farooq et al. [14] studied the chemical reaction and thermophoretic deposition in the heat and mass transfer of a hyperbolic tangent fluid through a curved channel. They imposed the slip effects for velocity, temperature as well as concentration. In the study, the temperature and velocity profiles indicated a contrasting behaviour of the respective slip parameters. Among heat transfer mechanisms, convection is the most prevalent in blood flow. Hence, investigations have been conducted by considering the convective boundary conditions for fluids in different geometries and assumptions [15–17]. In addition, blood flowing through arteries can be best approximated by fluids that show variable thermal conductivity [18,19]. As the erythrocyte (red blood cell) is biomagnetic in nature, the presence of a magnetic field has been shown to influence
The hemodynamics of variable liquid properties on the MHD
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peristalsis in the arterial walls. Specifically, blood flow with a shear rate below 100 s1 has been shown to represent a mathematical model for MHD (magnetohydrodynamic) flow. Peristaltic transport of a fluid with MHD considerations in a channel helps in explaining problems concerning the movement of physiological fluids, operation of blood pump machines, research on the working of a peristaltic MHD compressor, etc. Also, magnetic fields are known to have safe and healing effects on the human body. Magnetic fields of suitable intensity can be used to control excessive bleeding during surgeries, transport drugs and treat cancerous/tumor cells. Thus, the effect of magnetic field on peristalsis of non-Newtonian fluids has been explored both in biological and industrial applications [20,21]. The analysis of MHD two-phase flow for a dusty electrically conducting fluid, carried out by Chamkha [22], suggested that increased values of the Hartmann number lead to a reduction in the volumetric flow rate and the skin-friction coefficient of both the phases. Effects of heat and mass transfer on peristaltic transport in the presence of a magnetic field have also been investigated [23–26]. Studies with magnetic field considerations for the Jeffrey fluid model have been conducted by Hayat et al. [27], which demonstrate the effects of Hartmann number and Jeffrey fluid parameter on pressure rise needed to produce zero flow rate. Further, non-linear convective flow of Jeffrey nanofluid was analyzed by Waqas et al. [28] in which they also considered the aspects of thermal radiation and heat generation. Sinha et al. [29] extended upon the work on heat transfer by examining the impact of velocity slip as well as temperature jump along with variable viscosity. Additionally, the effects of heat and mass transfer on the peristaltic motion for a magnetohydrodynamic fluid were also investigated with porous medium considerations along with the channel wall properties [30,31]. The hydromagnetic flow of two immiscible liquids was investigated by Chamkha [32] through both porous as well as non-porous channels. This extensive parameteric study revealed that the higher values of magnetic parameter and inverse Darcy number result in a reduction of the fluid velocity. Similar observations were made by Umavathi et al. [33] for a porous channel with viscous and Darcy dissipation effects. Srinivas et al. [34] examined the simultaneous effects of velocity slip and heat transfer on the MHD flow of a fluid through porous channel and compliant wall considerations. They observed that the permeability parameters strengthen the slip at the walls, whereas the Hartmann number weakens it. In the recent years, many studies have been done which include magnetic effects and transfer of heat and mass [35–38]. The present paper aims to build upon the work of the aforementioned researchers to examine the peristaltic mechanism for a magnetohydrodynamic Jeffrey fluid in the presence of heat and mass transfer. The authors have tried to build a mathematical model to give realistic manifestations for the flow of blood. For this purpose, the channel of peristaltic transport is considered to be inclined, with porous and compliant walls. No work has been reported yet, considering the combination of the two variable properties (viscosity as well as thermal conductivity) of an electrically conducting blood which is exposed to a radial magnetic field. The current work is one of the first attempts to include these complex rheological properties of blood. Further, convective boundary conditions have been considered for heat transfer and slip conditions are imposed for mass transfer. The results of the current investigation finds
applications in the peristaltic transport of blood in coronary arteries of smaller diameter, where the formation of blood clots or growth of tumors due to excessive cell division result in the lumen of the blood vessel acting as a porous media. 2. Development of the problem The two-dimensional flow of an incompressible nonNewtonian fluid is induced by the peristaltic waves travelling along the length of the channel. The wavelength of these wave trains is considered to be k which propagates with a speed c. The cartesian coordinates ðx0 ; y0 Þ are chosen where the propagation of the fluid takes place in the x0 direction and y0 is transverse to it. For simplicity, the channel is assumed to be axisymmetric. The channel is also considered to be inclined at an angle a with the horizontal surface, with its walls being flexible due to the effects of wall properties. Further, the flow is exposed to an external magnetic field of strength B0 which is applied in the direction perpendicular to the fluid flow (see Fig. 1). The electric field is taken to be zero. Also, the magnetic Reynolds number is assumed to be very small so that the induced magnetic field is negligible in comparison to the external magnetic field. The variable viscosity and variable thermal conductivity (variable liquid properties) of the non-Newtonian Jeffrey model is considered in the analysis. The porous conditions, convective conditions and concentration slip are taken into account. Geometry of the surface of the channel walls is represented as below: 2p 0 y ¼ Hðx0 ; t0 Þ ¼ a1 þ b1 sin ðx ct0 Þ ; ð1Þ k where a1 is the undeformed radius of the tube, b1 is the wave amplitude and t is time. Considering all the above flow properties, the equations governing the fluid flow in the laboratory frame are @w0 @v0 þ ¼ 0; @x0 @y0
ð2Þ
0 0 0 @w @p0 @s0x0 x0 @s0x0 y0 0 @w 0 @w ¼ þ w þ v þ þ q @t0 @x0 @y0 @x0 @x0 @y0 þ qg sin a rB20 w0 ;
ð3Þ
0 0 0 @v @p0 @s0x0 y0 @s0y0 y0 0 @v 0 @v q ¼ þ w þ v þ þ qg cos a; @t0 @x0 @y0 @y0 @x0 @y0 ð4Þ
Fig. 1
Geometry of the channel wall.
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Fig. 2 Velocity profiles for (a) M, (b) a1 , (c) Da, (d) b, (e) a and (f) k1 with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:02; k1 ¼ 0:2; a ¼ p4 ; a1 ¼ 0:02; ¼ 0:6; F ¼ 2; M ¼ 1. Table 1 Values of axial velocity for t ¼ 0:1; b ¼ 0:2; Da ¼ 0:02; k1 ¼ 0:2; a ¼ p4 ; a1 ¼ 0:02; ¼ 0:6; F1 ¼ 2; M ¼ 1 with varying value of the elastic parameters. Axial velocity (w) E2
E1
E3
E4
E5
y
0.09
0.1
0.11
0.04
0.045
0.05
0.4
0.45
0.5
0.001
0.0015
0.002
0.01
0.03
0.05
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1.0
0.4756 0.4735 0.4670 0.4563 0.4413 0.4223 0.3993 0.3726 0.3422 0.3085 0.2716 0.2319 0.1896 0.1451 0.0987 0.0507 0.0015 0.0485 0.0989 0.1494 1.3778
2.4333 2.4283 2.4133 2.3886 2.3541 2.3102 2.2571 2.1953 2.1252 2.0473 1.9621 1.8704 1.7728 1.6700 1.5629 1.4521 1.3385 1.2231 1.1066 0.9900 0.8741
4.3909 4.3831 4.3597 4.3208 4.2668 4.1980 4.1149 4.0180 3.9082 3.7861 3.6527 3.5090 3.3561 3.1950 3.0271 2.8535 2.6755 2.4946 2.3121 2.1294 1.9479
2.4333 2.4283 2.4133 2.3886 2.3541 2.3102 2.2571 2.1953 2.1252 2.0473 1.9621 1.8704 1.7728 1.6700 1.5629 1.4521 1.3385 1.2231 1.1066 0.9900 0.8741
3.4121 3.4057 3.3865 3.3547 3.3104 3.2541 3.1860 3.1067 3.0167 2.9167 2.8074 2.6897 2.5644 2.4325 2.2950 2.1528 2.0070 1.8589 1.7094 1.5597 1.4110
4.3909 4.3831 4.3597 4.3208 4.2668 4.1980 4.1149 4.0180 3.9082 3.7861 3.6527 3.5090 3.3561 3.1950 3.0271 2.8535 2.6755 2.4946 2.3121 2.1294 1.9479
2.4333 2.4283 2.4133 2.3886 2.3541 2.3102 2.2571 2.1953 2.1252 2.0473 1.9621 1.8704 1.7728 1.6700 1.5629 1.4521 1.3385 1.2231 1.1066 0.9900 0.8741
1.3014 1.2981 1.2880 1.2714 1.2482 1.2187 1.1830 1.1415 1.0943 1.0420 0.9847 0.9231 0.8575 0.7884 0.7163 0.6418 0.5655 0.4879 0.4096 0.3312 0.2533
0.1696 0.1679 0.1627 0.1542 0.1423 0.1272 0.1089 0.0876 0.0635 0.0366 0.0073 0.0243 0.0579 0.0933 0.1302 0.1684 0.2075 0.2473 0.2874 0.3276 0.3675
10.1616 10.1455 10.0971 10.0169 9.9053 9.7630 9.5912 9.3911 9.1641 8.9118 8.6362 8.3393 8.0232 7.6904 7.3433 6.9845 6.6169 6.2430 5.8659 5.4883 5.1131
6.2974 6.2869 6.2552 6.2027 6.1297 6.0366 5.9242 5.7932 5.6446 5.4795 5.2992 5.1048 4.8980 4.6802 4.4531 4.2183 3.9777 3.7331 3.4862 3.2391 2.9936
2.4333 2.4283 2.4133 2.3886 2.3541 2.3102 2.2571 2.1953 2.1252 2.0473 1.9621 1.8704 1.7728 1.6700 1.5629 1.4521 1.3385 1.2231 1.1066 0.9900 0.8741
2.4333 2.4283 2.4133 2.3886 2.3541 2.3102 2.2571 2.1953 2.1252 2.0473 1.9621 1.8704 1.7728 1.6700 1.5629 1.4521 1.3385 1.2231 1.1066 0.9900 0.8741
2.3341 2.3292 2.3147 2.2907 2.2572 2.2145 2.1630 2.1029 2.0348 1.9592 1.8765 1.7874 1.6926 1.5928 1.4887 1.3811 1.2708 1.1587 1.0455 0.9323 0.8197
2.2349 2.2302 2.2161 2.1928 2.1603 2.1189 2.0689 2.0106 1.9445 1.8711 1.7909 1.7044 1.6124 1.5155 1.4145 1.3101 1.2031 1.0942 0.9845 0.8745 0.7653
0 2 0 0 0 @T @ T @ 2 T0 @w0 0 @T 0 @T ¼ k þ s0x0 x0 0 þ qCp þ w þ v þ 0 0 0 02 02 @t @x @y @x @y @x 0 @v @w0 @v0 0 0 þ sy0 y0 0 ; þ ð5Þ sx0 y0 @x0 @y0 @y
2 0 0 0 @C0 @ C @ 2 C0 Dm KT @ 2 T0 @ 2 T0 0 @C 0 @C þ : þ w þ v ¼ D þ þ m @t0 @x0 @y0 @x02 @y02 Tm @x02 @y02 ð6Þ
The hemodynamics of variable liquid properties on the MHD The fluid flow is unsteady in the laboratory frame. To overcome this, it is necessary to introduce a wave frame ð x; yÞ moving away from the laboratory frame of reference ðx0 ; y0 Þ with a constant velocity c. The transformations corresponding to this are ¼ w0 c; v ¼ v0 and pð xÞ ¼ p0 ðx0 ; t0 Þ: x ¼ x0 ct0 ; y ¼ y0 ; w 0
ð7Þ
0
and v are the axial and transverse velocities in Here, w ; v ; w the laboratory and wave frame of reference respectively. Also, T0 is the fluid temperature, and C0 is the concentration of the fluid. The above dimensional parameters are rendered dimensionless using the following x¼
ct0 H a1 a1 x y w v ; y ¼ ; w ¼ ; v ¼ ; t ¼ ; h ¼ ; sij ¼ s0ij ; d ¼ ; a1 k a1 c cd k lc k
qDm KT T0 lCp ;M ¼ ; Pr ¼ Tm lC0 k
rffiffiffi r C0 C0 B0 a1 ; / ¼ ; l C0
ð8Þ
where the above non-dimensional parameters are mentioned in the nomenclature. On making the transformation from laboratory frame to wave frame with Eq. (7) and using the dimensionless quantities given by Eq. (8), the Eqs. (3)–(6) take the following form @w @p @sxx @sxy sin a þ M2 ðw þ 1Þ þ Red w ¼ þd ; ð9Þ @x @y @x @x F @v @v @p @sxy @syy cos a d3 Re w þ d þv ¼ þ d2 ; @x @y @x @y @y F
Fig. 3
@h ¼ Red w @x þ v @h @y
ð10Þ
Velocity profiles for k1 as obtained by Bhatti et al. [39].
@2 h @y2
2
@ h þ d2 @x 2 i @v @w sxy d @x þ @y ;
1 Pr
h @v þ Ec d sxx @w þ þ syy @y @x
ð11Þ
@/ @/ 1 @2/ @2/ @2h @2h ¼ Red w þ d2 2 þ Sr d2 2 þ 2 ; ð12Þ þv 2 @x @y Sc @y @x @x @y
where the constitutive equation for Jeffrey fluid is 2dlðyÞ dk2 c @ v @ @w 1þ w þ ; sxx ¼ 1 þ k1 a1 @x d @y @x sxy ¼
ð13Þ
lðyÞ dk2 c @ v @ @w @v @w 1þ w þ þd ; ð14Þ 1 þ k1 a1 @x d @y @y @x @x
syy ¼
qca1 cl T0 T0 a2 c2 l ;F ¼ Re ¼ ;h ¼ ; p ¼ 1 p0 ; Ec ¼ ; Sc ¼ ; 2 qga1 qDm l T0 ckl Cp T0
Sr ¼
697
2dlðyÞ dk2 c @ v @ @w 1þ w þ : 1 þ k1 a1 @x d @y @x
ð15Þ
The variable viscosity lðyÞ and variable thermal conductivity kðhÞ is considered to be varying exponentially with y and h as lðyÞ ¼ ea1 y ¼1 a1 y þ Oða21 Þ;
ð16Þ
kðhÞ ¼ ech ¼1 þ ch þ Oðc2 Þ;
ð17Þ
where a1 and c are the coefficients of variable viscosity and variable thermal conductivity respectively. On assuming long wavelength(d 1) and low Reynolds number, the final set of equations representing our problem is given by @p ¼0; @y
ð18Þ
@p @ lðyÞ @w sin a ¼ M2 ðw þ 1Þ þ ; @x @y 1 þ k1 @y F
ð19Þ
2 @ @h lðyÞ @w kðhÞ ¼Br ; @y @y 1 þ k1 @y
ð20Þ
@2/ @2h ¼ ScSr 2 : 2 @y @y
ð21Þ
The corresponding boundary conditions in dimensional form are pffiffiffiffiffiffiffi Da @w @w ¼ 0 at y ¼ 0; w ¼ 1 at y ¼ h; b @y @y
the
non-
ð22Þ
Fig. 4 Skin-friction coefficient along the channel length for (a) a1 , (b) Da and (c) k1 with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:2; b ¼ 0:2; Da ¼ 0:04; k1 ¼ 0:2; a ¼ p4 ; a1 ¼ 0:02; ¼ 0:2; F ¼ 2; M ¼ 1.
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@h @h ¼ 0 at y ¼ 0; þ Bih ¼ 0 at y ¼ h; @y hy
ð23Þ
@/ @/ ¼ 0 at y ¼ 0; b1 þ / ¼ 0 at y ¼ h; @y @y
ð24Þ
@3 @3 @2 @5 @ E1 3 þ E2 þ E þ E þ E 3 4 5 @x @x@t2 @x@t @x5 @x @ lðyÞ @w sin a M2 ðw þ 1Þ þ at y ¼ h: h¼ @y 1 þ k1 @y F
ð25Þ
3. Solution methodology Due to the non-linearity in the Eqs. (19) and (20), we cannot find an exact solution for the problem. Hence, we adopt the method of perturbation to expand the velocity field w and tem-
perature profile h about small values of variable viscosity a1 and variable thermal conductivity c. However, closed form solution is obtained for concentration from Eq. (21). The perturbation series for w and h are taken as w ¼w0 þ a1 w1 þ Oða21 Þ;
ð26Þ
h ¼h0 þ ch1 þ Oðc2 Þ:
ð27Þ
Substituting the above in Eqs. (19), (20), (22) and (23), we obtain the expressions for velocity and temperature profiles from perturbation technique (up to first order) as, w¼
C1 a1 y C1 M1 y2 cosðM1 yÞ þ C3 a1 sinðM1 yÞ C1 þ C2 a1 þ 4 4
1 ð28Þ 2 ðP fÞð1 þ k1 Þ þ M21 ; M1
Fig. 5 Temperature profiles for (a) c, (b) a1 , (c) M, (d) Bi, (e) Br, (f) a, (g) Da, (h) k1 and (i) elastic parameters with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:08; k1 ¼ 0:2; a ¼ p4 ; a1 ¼ 0:02; M ¼ 3; Bi ¼ 5; Br ¼ 2; c ¼ 0:2; ¼ 0:6; F1 ¼ 2.
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Fig. 6 Nusselt number along the channel length for (a) c, (b) Bi, (c) Br, (d) Da, (e) k1 and (f) elastic parameters with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:08; k1 ¼ 2; a ¼ p4 ; a1 ¼ 0:02; M ¼ 3; Bi ¼ 5; Br ¼ 2; c ¼ 0:2; ¼ 0:2; F ¼ 2.
Br cC2 y2 l1 ðyÞ þ C4 y þ C5 þ cC6 C4 C5 cy 4 1 þ k1 2 2 l1 ðyÞBr c cBr þ ðC4 l2 ðyÞy þ C5 l1 ðyÞÞ; 1 þ k1 2 1 þ k1
h¼
ð29Þ
Br / ¼ ScSr l1 ðyÞ þ C4 y þ C5 þ cC6 C4 C5 cy 1 þ k1 # 2 cC24 y2 l1 ðyÞBr c cBr ðC4 l2 ðyÞy þ C5 l1 ðyÞÞ þ C7 y þ C8 ; ð30Þ þ 2 1 þ k1 2 1 þ k1
where M1 ¼ f¼ P¼
pffiffiffiffiffiffiffiffiffiffiffiffiffi M 1 þ k1 ; sin a ; F 8p3 E2p3
@h @h ; @x @y y¼h
ð35Þ
@h @/ : Sh ¼ @x @y y¼h
ð36Þ
Nu ¼
E5 sinð2pðx tÞÞ ðE1 þ E2 4p2 E4 4p 2 Þ cosð2pðx tÞÞ :
The expressions for l1 ðyÞ; l2 ðyÞ and Ci ; Aj ði ¼ 1; 2; . . . 8; j ¼ 1; 2; . . . ; 9Þ are as Appendix. We have the stream function w given by @w @w ;v ¼ ; w¼ @y @x and
4. Discussion of the graphical solutions the constants given in the
w ¼ 0 at y ¼ 0:
ð31Þ
ð32Þ
Using the above, we obtain the expression for w as w¼
The key dimensionless parameters of interest in the field of bioengineering are the skin-friction coefficient ðCf Þ, Nusselt number ðNuÞ and Sherwood number ðShÞ. These quantities are defined as below: @h @w Cf ¼ ; ð34Þ @x @y y¼h
sinðM1 yÞðC1 þ C2 a1 Þ C3 a1 cosðM1 yÞ C1 a1 þ M1 M1 4M21 C1 M1 a1 ðy2 M21 4M31 y cosðM1 yÞ þ 2yM1 sinðM1 yÞ þ 2 cosðM1 yÞÞ 2 M1 ðyM1 sinðM1 yÞ þ cosðM1 yÞÞ
½ðP fÞð1 þ k1 Þ þ M21 :
ð33Þ
The present section aims to analyse the graphical results of the influence of various parameters involved in the investigation. Discussions are carried out on the behaviour of velocity profiles, temperature and concentration fields, streamlines, skin-friction coefficient, Nusselt number and Sherwood number. 4.1. Flow characteristics Fig. 2 exhibits the behaviour of velocity profile for variations in M; a1 ; Da; b; a; k1 and wall properties (E1 ; E2 ; E3 ; E4 and E5 ). It is evident from these figures that the axial velocity has a parabolic trajectory, with the velocity being maximum in the central region of the channel. It is seen from Fig. 2(a) that Hartmann number M decelerates the fluid flow. This behaviour is expected, as transverse magnetic field resists the flow. Particularly in blood, transverse magnetic field results in the
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Fig. 7 Concentration profiles for (a) c, (b) a1 , (c) M, (d) Bi, (e) k1 (f) elastic parameters, (g) Sc, (h) Sr and (i) b1 with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:08; k1 ¼ 0:2; a ¼ p4 ; a1 ¼ 0:02; M ¼ 3; Bi ¼ 5; Br ¼ 2; Sc ¼ 0:2; Sr ¼ 0:3; b1 ¼ 0:2; ¼ 0:6; c ¼ 0:2; F1 ¼ 2.
formation of rouleaux leading to a decrease in velocity. The increasing effect of variable viscosity on axial velocity near the channel walls is sketched in Fig. 2(b). The consideration of porous walls is significant in the study of blood flow through arteries. As the porosity of the walls increase, the flow resistance decreases, thus resulting in an increase in the fluid velocity near the channel walls (see Fig. 2(c)). The role of b is found to be opposite in comparison to Da, as observed in Fig. 2(d). Most of the blood vessels are not horizontal and are inclined at various angles. Thus it becomes important in a physical context to consider the inclination of channel in biological systems. As the channel becomes more inclined away from the horizontal surface, the reduction in gravitational forces increase the velocity of the fluid, which is clearly seen in Fig. 2(e). Results for horizontal channel can be derived from
the present model by substituting a as zero. Similar results are shown in Fig. 2(f) for increasing values of the Jeffrey parameter k1 . The corresponding results for a Newtonian fluid can be obtained by taking k1 as zero. Thus, Jeffrey model helps us attain the results for both Newtonian and non-Newtonian behaviour of blood. Table 1 gives the dependence of axial velocity on wall properties. The parameters E1 and E2 account for the elastic nature of the walls, while E3 ; E4 and E5 are damping/resistive forces. Accordingly, it can be seen clearly that fluid flows rapidly with E1 and E2 and the flow is retarded for increasing values of E3 ; E4 and E5 . Fig. 3 depicts the velocity profile for varying k1 obtained by Bhatti et al. [39] in their study on MHD peristalsis of a Jeffrey fluid. For Da ¼ 0; a ¼ 0 and a1 ¼ 0, results of the current work are in excellent agreement with those obtained by Bhatti et al. [39].
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Fig. 8 Sherwood number along the channel length for (a) c, (b) Br, (c) Da, (d) elastic parameters, (e) Sc and (f) Sr with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:08; k1 ¼ 2; a ¼ p4 ; a1 ¼ 0:02; M ¼ 3; Bi ¼ 5; Br ¼ 2; Sc ¼ 0:2; Sr ¼ 0:3; b1 ¼ 0:2; ¼ 0:2; c ¼ 0:2; F ¼ 2.
Fig. 9 Streamlines for (a) M ¼ 3 and (b) M ¼ 3:1 with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:02; k1 ¼ 2; a ¼ p4 ; a1 ¼ 0:02; ¼ 0:6; F1 ¼ 2.
The influence of a1 ; Da and k1 on skin-friction coefficient Cf can be seen in Fig. 4. The oscillatory behaviour of the graphs is due to the sinusoidal movement of peristaltic waves along the channel walls. Fig. 4(a) and (b) show an increase in the magnitude of Cf for increasing values of a1 and Da, whereas Fig. 4(c) shows a contrasting behaviour for k1 . 4.2. Heat transfer characteristics The fluctuations in temperature profile of the fluid during heat transfer as a result of variations in
c; a1 ; M; Bi; Br; a; Da; k1 ; E1 ; E2 ; E3 ; E4 and E5 are scrutinized in Fig. 5. The temperature profiles are parabolic with maximum temperature occuring near the channel center. This behaviour can be explained by the effect of viscous dissipation which increases the temperature in the centre of the channel. This phenomenon occurs due to the viscosity of the fluid, leading to the conversion of kinetic energy of the fluid to internal thermal energy. It is known that the thermal conductivity of a fluid indicates the ability of the fluid to retain or liberate heat to its surroundings. Thus, when the thermal conductivity of the fluid bound inside the channel is larger than the wall
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Fig. 10 Streamlines for (a) a1 ¼ 0:01 and (b) a1 ¼ 0:03 with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; Da ¼ 0:02; k1 ¼ 2; a ¼ p4 ; M ¼ 3; ¼ 0:6; F1 ¼ 2.
Fig. 11 Streamlines for (a) k1 ¼ 2 and (b) k1 ¼ 2:2 with E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01; t ¼ 0:1; b ¼ 0:2; M ¼ 3; k1 ¼ 2; a ¼ p4 ; a1 ¼ 0:02; ¼ 0:6; F1 ¼ 2.
temperature, thermal reading of the fluid rises (see Fig. 5(a)). Fig. 5(b) depicts the increasing effect of variable viscosity on temperature profile. This nature is justified by the fact that with an increase in the variable viscosity of the fluid, its heat transfer capacity decreases. The decreasing effect of M on temperature is seen in Fig. 5(c). The variation of Bi is illustrated in Fig. 5(d). Biot number decreases the thermal conductivity which in turn reduces the fluid temperature. This decreasing behaviour can be observed in the figure. An opposite behaviour of Br on temperature profile is seen in Fig. 5(e). This is attributed to an intensification in the flow resistance due to an increase in Br, which causes the viscous dissipation effects to inflate the internal thermal energy. Contrasting effects of a and Da are seen in Fig. 5(f) and (g), wherein the temperature is found to rise with the channel inclination and drop with the increasing porosity of the channel walls. The temperature of the fluid is enhanced with the Jeffrey parameter k1 (see Fig. 5(h)). From Fig. 5(i), the temperature profile intensifies for increasing values of E1 and E2 , whereas an opposite tendency is noticed for E3 ; E4 and E5 . The plots shown in Fig. 6 show the variation of Nusselt number Nu with c; Bi; Br; Da; k1 and the wall properties. Nu
measures the temperature gradient at the channel walls. The dimensionless quantity of Nusselt number is the ratio of convective to conductive heat transfer. The parameters c; Bi and Br have an increasing effect on the absolute value of Nusselt number, as seen in Fig. 6(a)–(c). On the other hand, Fig. 6(d) and (e) shows that higher values of Da and k1 result in a decrease in the magnitude of Nu. From Fig. 6(f), it can be observed that larger values of E1 ; E2 and E3 lead to an increase in the magnitude of Nu, whereas the absolute value of Nu is reduced by increasing E4 and E5 . 4.3. Mass transfer characteristics Fig. 7 depicts the concentration profiles for different values of c; a1 ; M; Bi; k1 ; Sc; Sr; b1 ; E1 ; E2 ; E3 ; E4 and E5 . Figs. 7(a)-(f) reveal an opposite behaviour for the respective parameters as compared to the temperature profiles. As mass and heat are known to be inversely proportional, this behaviour is expected in a physical context. Additionally, it can be inferred from the profiles that the particulate matter within the fluid is more concentrated at the periphery compared to the central region of
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703
Fig. 12 Streamlines for (a) E1 ¼ 0:09; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01, (b) E1 ¼ 0:1; E2 ¼ 0:04; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01, (c) E1 ¼ 0:1; E2 ¼ 0:045; E3 ¼ 0:4; E4 ¼ 0:002; E5 ¼ 0:01, (d) E1 ¼ 0:1; E2 ¼ 0:045; E3 ¼ 0:45; E4 ¼ 0:002; E5 ¼ 0:01, (e) E1 ¼ 0:1; E2 ¼ 0:045; E3 ¼ 0:45; E4 ¼ 0:0022; E5 ¼ 0:01 and (f) E1 ¼ 0:1; E2 ¼ 0:045; E3 ¼ 0:45; E4 ¼ 0:0022; E5 ¼ 0:04 with t ¼ 0:1; b ¼ 0:2; M ¼ 3; k1 ¼ 2; a ¼ p4 ; a1 ¼ 0:02; ¼ 0:6; F1 ¼ 2.
the channel. From a biological standpoint, this behaviour is in order to facilitate the diffusion of vital nutrients from blood and other fluids to the adjacent cells and tissues. Fig. 7(g)-(i) demonstrate a decreasing trend for increasing values of Sc; Sr and b1 . Fig. 8 shows the variation of Sherwood number with c; Br; Da, wall properties ðE1 E5 Þ; Sc and Sr. Sherwood number measures the concentration gradient at the walls of the channel. It is a dimensionless quantity, which gives the
ratio of convective mass transfer to rate of diffusive mass transport. The Sherwood number sees an increase in magnitude for increasing values of c and Br (see Fig. 8(a) and (b)), while the higher values of Da decreases the absolute value of Sh (see Fig. 8(c)). Fig. 8(d) reveals a rise in the magnitude of Sh as E1 ; E2 and E3 increase, and a drop in Sh for higher values of E4 and E5 . Increasing values of Sc and Sr bear an incraesing effect on the absolute value of Sh, as indicated in Fig. 8(e) and (f).
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4.4. Trapping phenomenon
Declaration of Competing Interest
Trapping is the most important and widely discussed topic in the peristaltic mechanism. When the streamlines get closed, it stimulates the formation of a bolus of fluid which circulates internally and moves forward along with the peristaltic waves. This phenomenon may be responsible for the thrombus formation which eventually blocks the flow of blood through narrow vessels. This naturally occurring episode of trapping was first noticed by Shapiro [40] in his studies on peristalsis. The variation in the size of bolus formed by trapping is shown in Figs. 9– 12. The volume of trapped bolus is found to increase for M and a1 as seen in Figs. 9 and 10. The volume of the trapped bolus is found to decrease with increasing Jeffrey parameter (k1 ) (see Fig. 11). Fig. 12 elucidates the effects of the various parameters accounting the wall properties on the size of the trapped bolus.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment The authors sincerely thank the valuable and constructive comments of the referees for the improvement of this research article.
Appendix A. l1 ðyÞ ¼
5. Summary and conclusions
1 ½C21 ð525sinð2yM1 Þa31 þ 210cosð2yM1 ÞM1 a21 ð3 þ 4ya1 Þ 26880M31
4y6 M71 a21 ð7 þ 5ya1 Þ þ 630sinð2yM1 ÞM21 a1 ð4 þ ya1 þ y2 a21 Þþ
The impact of variable liquid properties (variable viscosity and variable thermal conductivity) are examined on the MHD peristaltic flow through an inclined channel with porous walls. The flow of blood is modelled by the Jeffrey fluid. The influence of critical parameters on velocity, heat and mass transfer are sketched using MATLAB, analysed and discussed in detail. The streamlines are also plotted to observe the trapping phenomena. The results reported in the work have potential applications in industries as well as medical science. Along with the consideration of variable liquid properties in a MHD peristaltic flow, physical conditions such as an asymmetric/tapered asymmetric artery with porous medium can be considered in the future works. The current work serves a fair theoretical estimate for further work in this directon. The major outcomes of the present article can be summarised as: a1 accelerates the fluid and Da decelerates it in the central part of the channel. The magnetic parameter M causes a drop in the velocity and temperature, but they are enhanced by the increasing inclination of the channel. A rise in temperature is observed for increasing values of c and Br, and decreasing values of Bi and k1 . The wall property parameters E1 and E2 play a vital role in the growth of velocity and temperature profiles, while increasing values of E3 ; E4 and E5 result in a fall in velocity and temperature. The behaviour of temperature and concentration profiles are opposite to each other. The increase in the values of parameters Sc; Sr and b1 is responsible for reduction in the concentration of particles in the fluid. a1 leads to depressed values of the magnitude of C f whereas c has an increasing effect on the absolute value of Nu. The magnitude of Sh sees an increase for higher values of Sc and Sr. The size of the bolus trapped during peristalsis is seen to increase with M and a1 , whereas the Jeffrey parameter (k1 ) reduces the size of the trapped bolus.
210y2 sinð2yM1 ÞM41 a1 ð8 þ 6ya1 þ y2 a21 Þ 140M31 ð24cosð2yM1 Þþ 12ycosð2yM1 Þa1 6y2 cosðyM1 Þ2 a21 þ y3 ð1 þ 3cosð2yM1 ÞÞa31 Þþ 14y2 M51 ð480 þ 80ya1 5y2 ð17 þ 3cosð2yM1 ÞÞa21 þ 3y3 ð7 þ 5 cosð2yM1 ÞÞa31 ÞÞ 1120M21 a21 ðC23 ð3sinð2yM1 Þa1 þ 2y2 M31 ð3 þ ya1 Þ 3cosð2yM1 ÞM1 ð1 þ ya1 ÞÞ þ C22 ð3sinð2yM1 Þa1 þ 2y2 M31 ð3þ ya1 Þ þ 3cosð2yM1 ÞM1 ð1 þ ya1 ÞÞ þ 6C2 C3 ðcosð2yM1 Þa1 þ sinð2yM1 Þ M1 ð1 þ ya1 ÞÞÞ þ 56C1 M1 a1 ðC3 ð15sinð2yM1 Þa21 15cosð2yM1 ÞM1 a1 ð7 þ ya1 Þ þ 2y4 M51 a1 ð5 þ 3ya1 Þ 30sinð2yM1 ÞM21 ð4 þ 3ya1 Þ 10 y2 M31 a1 ð3ð2 þ cosð2yM1 ÞÞ þ yð2 þ 3cosð2yM1 ÞÞa1 ÞÞ þ 5C2 ð3cosð2yM1 Þ a21 þ 6y2 sinð2yM1 ÞM31 a1 ð1 þ ya1 Þ þ 3sinð2yM1 ÞM1 a1 ð7 þ ya1 Þ 6 cosð2yM1 ÞM21 ð4 þ 3ya1 Þ 2y2 M41 ð24 þ 2ya1 þ 3y2 a21 ÞÞÞ; l2 ðyÞ ¼
1 ½C21 ð4095cosð2yM1 Þa31 þ 420y4 sinð2yM1 ÞM51 a21 ð1 þ ya1 Þ 107520M41
2y7 M81 a21 ð8 þ 5ya1 Þ þ 210sinð2yM1 ÞM1 a21 ð5 þ 29ya1 Þ þ 70y2 M41 a1 ð4 þ ya1 Þð12cosð2yM1 Þ þ yð2 þ 9cosð2yM1 ÞÞa1 Þ 210cosð2yM1 Þ M21 a1 ð40 22ya1 þ 21y2 a21 Þ 420sinð2yM1 ÞM31 ð16 þ 16ya1 14y2 a21 þ5y3 a31 Þ 28y3 M61 ð320 40ya1 þ 34y2 a21 þ 7y3 a31 ÞÞ 1120M21 a21 ð6C2 C3 ð3sinð2yM1 Þa1 þ 2cosð2yM1 ÞM1 ð1 þ ya1 ÞÞ þ C23 ð9cosð2yM1 Þa1 þ 2y3 M41 ð4 þ ya1 Þ 6sinð2yM1 ÞM1 ð1 þ ya1 ÞÞ þ C22 ð9cosð2yM1 Þa1 þ 2y3 M41 ð4 þ ya1 Þ þ 6sinð2yM1 ÞM1 ð1 þ ya1 ÞÞÞ þ 112C1 M21 a1 ðC3 ð10 y3 M31 a1 ð4 þ ya1 Þ þ 2y5 M51 a1 ð2 þ ya1 Þ 30y2 sinð2yM1 ÞM21 a1 ð1þ ya1 Þ þ 15sinð2yM1 Þa1 ð11 þ 2ya1 Þ 15cosð2yM1 ÞM1 ð8 8ya1 þ 3y2 a21 ÞÞ þC2 ð30y2 cosð2yM1 ÞM21 a1 ð1 þ ya1 Þ þ 15cosð2yM1 Þa1 ð11 þ 2ya1 Þþ 15sinð2yM1 ÞM1 ð8 8ya1 þ 3y2 a21 Þ 2y3 M41 ð80 þ 5ya1 þ 6y2 a21 ÞÞÞ;
The hemodynamics of variable liquid properties on the MHD C1 ¼ C2 ¼
1 þ M12 ½ðP fÞð1 þ k1 Þ þ M21 1 pffiffiffiffi ; cosðM1 hÞ bDa M1 sinðM1 hÞ
705 A3 ¼
1 ½C2 ð4095cosð2hM1 Þa31 þ 420h4 sinð2hM1 ÞM51 a21 ð1 þ ha1 Þ 107520M41 1 2h7 M81 a21 ð8 þ 5ha1 Þ þ 210sinð2hM1 ÞM1 a21 ð5 þ 29ha1 Þþ
1
pffiffiffiffi ½1 C3 sinðM1 hÞ cosðM1 hÞ bDa M1 sinðM1 hÞ
70h2 M41 a1 ð4 þ ha1 Þð12cosð2hM1 Þ þ hð2 þ 9cosð2hM1 ÞÞa1 Þ
pffiffiffiffiffiffiffi Da C1 h C1 h cosðM1 hÞ þ M1 sinðM1 hÞ ½C3 M1 cosðM1 hÞ 4 4 b C1 þ ðcosðM1 hÞ M21 h2 cosðM1 hÞ 3hM1 sinðM1 hÞÞ ; 4 2
210cosð2hM1 ÞM21 a1 ð40 22ha1 þ 21h2 a21 Þ 420sinð2hM1 ÞM31 ð16 þ 16ha1 14h2 a21 þ 5h3 a31 Þ 28h3 M61 ð320 40ha1 þ 34h2 a21 þ 7h3 a31 ÞÞ 1120M21 a21 ð6C2 C3 ð3sinð2hM1 Þa1 þ 2cosð2hM1 ÞM1 ð1 þ h a1 ÞÞ þ C23 ð9cosð2hM1 Þa1 þ 2h3 M41 ð4 þ ha1 Þ 6sinð2hM1 ÞM1 ð1 þ ha1 ÞÞ þ C22 ð9cosð2hM1 Þa1 þ 2h3 M41 ð4 þ ha1 Þ þ 6sinð2hM1 Þ
C3 ¼
C1 ; 4M1
M1 ð1 þ ha1 ÞÞÞ þ 112C1 M21 a1 ðC3 ð10h3 M31 a1 ð4 þ ha1 Þ þ 2h5 M51
C4 ¼
Br ½240a21 M21 ð4C2 C3 M1 þ C22 a1 C23 a1 Þþ 1920M21 ð1 þ kÞ
a1 ð11 þ 2ha1 Þ 15cosð2hM1 ÞM1 ð8 8ha1 þ 3h2 a21 ÞÞ þ C2 ð30h2
C21 ð240M21 a1 15a31 Þ þ 20C1 M1 a1 ð24C2 M1 a1 þ C3 ð48M21 þ 3a21 ÞÞ; Br A1 1 þh ; þ A2 C4 C5 ¼ 1 þ k1 Bi Bi 2 Br A1 A2 A22 h h2 1 þ þh C6 ¼ þ þ C24 þ C4 C5 1 þ k1 Bi 2 Bi Bi 2 Br C4 A2 h C4 A3 C5 A1 þ þ þ C4 A3 h þ C5 A2 ; 1 þ k1 Bi Bi Bi Br A4 þ C4 A6 ; C7 ¼ScSr 1 þ k1 Br A1 þ C4 A7 þ ScSrðA8 þ cA9 Þ C7 h; C8 ¼b1 ScSr 1 þ k1 A1 ¼
1 ½C2 ð15cosð2hM1 Þa31 30h4 sinð2hM1 ÞM51 a21 ð1 þ ha1 Þ 1920M21 1
a1 ð2 þ ha1 Þ 30h2 sinð2hM1 ÞM21 a1 ð1 þ ha1 Þ þ 15sinð2hM1 Þ cosð2hM1 ÞM21 a1 ð1 þ ha1 Þ þ 15cosð2hM1 Þa1 ð11 þ 2ha1 Þ þ 15 sinð2hM1 ÞM1 ð8 8ha1 þ 3h2 a21 Þ 2h3 M41 ð80 þ 5ha1 þ 6h2 a21 ÞÞÞ; 1 þ k1 ; A4 ¼C4 Br 1 A5 ¼ ½1120M21 a21 ð3C22 M1 þ 3C23 M1 þ 6C2 C3 a1 Þþ 26880M31 C21 ð3360M31 þ 630M1 a21 Þ þ 56C1 M1 a1 ð105C3 M1 a1 þ 5C2 ð24M21 þ 3a21 ÞÞ; 2 Br Br A6 ¼ A4 A5 ðC4 A5 þ C5 A4 Þ þ C4 C5 ; 1 þ k1 1 þ k1 2 Br Br A1 A2 ðC4 A2 þ C4 A1 h þ C5 A1 Þ þ C24 h þ C4 C5 ; A7 ¼ 1 þ k1 1 þ k1 Br A8 ¼ A2 þ C4 h þ C5 ; 1 þ k1 2 1 A2 Br Br C2 h2 þ ðC4 A3 h þ C5 A2 Þ 4 C4 C5 h þ C6 : A9 ¼ 2 1 þ k1 1 þ k1 2
15sinð2hM1 ÞM1 a21 ð3 þ 2ha1 Þ 2h5 M61 a21 ð6 þ 5ha1 Þ þ 30M21 a1 ð8cosð2hM1 Þ þ hð2 þ 5cosð2hM1 ÞÞa1 h2 a21 Þ þ 30sinð2hM1 ÞM31
References
2
ð16 þ 7h2 a21 þ 4h3 a31 Þ þ 5hM41 ð192 þ 96hsinðhM1 Þ a1 þ 4h2 ð17 þ 6cosð2hM1 ÞÞa21 42h3 sinðhM1 Þ2 a31 ÞÞ þ 240M21 a21 ð2C3 C4 ðsinð2hM1 Þa1 þ 2cosð2hM1 ÞM1 ð1 þ ha1 ÞÞ þ C23 ðcosð2hM1 Þa1 2hM21 ð2 þ ha1 Þ þ 2sinð2hM1 ÞM1 ð1 þ ha1 ÞÞ C24 ðcosð2hM1 Þa1 þ 2hM21 ð2 þ ha1 Þ þ 2sinð2hM1 ÞM1 ð1 þ ha1 ÞÞÞ þ 20C1 M1 a1 ð3C3 ðsinð2hM1 Þa21 þ 2cosð2hM1 ÞM1 a1 ð4 þ ha1 Þ þ 2sinð2hM1 ÞM21 2
ð8 þ 4ha1 þ 3h2 a21 Þ þ 4hM31 ð8 2hcosðhM1 Þ a1 þ h2 ð2þ cosð2hM1 ÞÞa21 ÞÞ þ C4 ð3cosð2hM1 Þa21 þ 12h2 sinð2hM1 ÞM31 a1 ð1þ ha1 Þ þ 6sinð2hM1 ÞM1 a1 ð4 þ ha1 Þ þ 2h3 M41 a1 ð4 þ 3ha1 Þ 6M21 2
ð8cosð2hM1 Þ 8hsinðhM1 Þ a1 þ h2 ð2 þ 3cosð2hM1 ÞÞa21 ÞÞÞ; A2 ¼
1 ½C2 ð525sinð2hM1 Þa31 þ 210cosð2hM1 ÞM1 a21 ð3 þ 4ha1 Þ 26880M31 1 4h6 M71 a21 ð7 þ 5ha1 Þ þ 630sinð2hM1 ÞM21 a1 ð4 þ ha1 þ h2 a21 Þþ 210h2 sinð2hM1 ÞM41 a1 ð8 þ 6ha1 þ h2 a21 Þ 140M31 ð24cosð2hM1 Þ þ 12hcosð2hM1 Þa1 6h2 cosðhM1 Þ2 a21 þ h3 ð1 þ 3cosð2hM1 ÞÞa31 Þþ 14h2 M51 ð480 þ 80ha1 5h2 ð17 þ 3cosð2hM1 ÞÞa21 þ 3h3 ð7 þ 5 cosð2hM1 ÞÞa31 ÞÞ 1120M21 a21 ðC23 ð3sinð2hM1 Þa1 þ 2h2 M31 ð3 þ ha1 Þ 3cosð2hM1 ÞM1 ð1 þ ha1 ÞÞ þ C22 ð3sinð2hM1 Þa1 þ 2h2 M31 ð3þ ha1 Þ þ 3cosð2hM1 ÞM1 ð1 þ ha1 ÞÞ þ 6C2 C3 ðcosð2hM1 Þa1 þ sinð2hM1 ÞM1 ð1 þ ha1 ÞÞÞ þ 56C1 M1 a1 ðC3 ð15sinð2hM1 Þa21 15 cosð2hM1 ÞM1 a1 ð7 þ ha1 Þ þ 2h4 M51 a1 ð5 þ 3ha1 Þ 30sinð2hM1 Þ M21 ð4 þ 3ha1 Þ 10h2 M31 a1 ð3ð2 þ cosð2hM1 ÞÞ þ hð2 þ 3 cosð2hM1 ÞÞa1 ÞÞ þ 5C2 ð3cosð2hM1 Þa21 þ 6h2 sinð2hM1 ÞM31 a1 ð1þ ha1 Þ þ 3sinð2hM1 ÞM1 a1 ð7 þ ha1 Þ 6cosð2hM1 ÞM21 ð4 þ 3ha1 Þ 2h2 M41 ð24 þ 2ha1 þ 3h2 a21 ÞÞÞ;
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