Applied Science and Innovative Research 

ISSN 2474-4972 (Print) ISSN 2474-4980 (Online) 

Vol. 1, No. 2, 2017 

www.scholink.org/ojs/index.php/asir 

104 
 

Effect of an Inclined Magnetic Field on the Flow of Nanofluids 

in a Tapered Asymmetric Porous Channel with Heat Source/Sink 

and Chemical Reaction 

Ajaz Ahmad Dar1* 

1 Department of Mathematics, Annamalai University, Annamalainagar, India 

* Ajaz Ahmad Dar, E-mail: aijaz855141@gmail.com 

 

Received: September 25, 2017    Accepted: October 4, 2017    Online Published: October 13, 2017 

doi:10.22158/asir.v1n2p104       URL: http://dx.doi.org/10.22158/asir.v1n2p104 

 

Abstract 

This article deals with the effect of an inclined magnetic field with heat source/sink on the flow of 

nanofluids in a tapered asymmetric porous channel. Effect of chemical reaction has been taken into 

account. The blood is considered as an incompressible electrically conducting viscous fluid. The 

assumption of low Reynolds number and long wave length approximations has been adopted. Exact 

solutions for dimensionless axial velocity, concentration and temperature profile are obtained 

analytically. The obtained results are displayed and discussed in detail with the help of graphs for the 

variation of different emerging flow parameters. 

Keywords 

peristaltic flow, asymmetric porous channel, inclined magnetic field, heat and mass transfer, chemical 

reaction 

 

1. Introduction 

The nanofliuids have superior properties such as thermal conductivity, long-term stability, minimal 

closing in flow passages, and homogeneity due to small size and very large specific surface areas of 

nanoparticles. Hence, the nanofliuids have broad range of potential applications in pharmacological 

administration mechanisms, solar collectors, nuclear applications and peristaltic pumps for diabetic 

treatments, etc. The flow characteristics with heat and mass transfer over a stretching surface in porous 

channel have been attracted by many researchers in the recent path due to its enormous applications in 

different branches of biological and engineering fields such as wire drawing, hot rolling, metal 

extrusion, artificial fibers, etc. (Rana & Bhargava, 2002; Kothandapani & Srinivas, 2008; 

Bhattacharyya, 2013). 

Several experimental and theoretical investigations have been done by the researchers for the flow 



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analysis of nanofluids under various aspects. Nadeem et al. (2014) analyzed heat transfer of 

water-based nanofluid over an exponentially stretching sheet. Baheta and Woldeyohannes (2013) have 

studied the effect of particle size on thermal conductivity of nanofluids. Mutuku-Njane and Makinde 

(2014) analyzed the effects of MHD nanofluid flow over a permeable vertical plate with convective 

heating. The effects of slip and heat generation/absorption on MHD stagnation flow of nanofluid past a 

stretching/shrinking surface with convective boundary conditions has been discussed by Nandy and 

Mahapatra (2009). The flow and heat transfer of nanofluid past shrinking/stretching sheet with partial 

slip boundary conditions were also investigated by Mansur et al. (2014). Javed et al. (2011) 

investigated the heat transfer analysis for a hydromagnetic viscous fluid over a non-linear shrinking 

sheet. The MHD boundary-layer flow of a micropolar fluid past a wedge with constant heat flux was 

described by Anuar Ishak et al. (2013). Akbar and Nadeem (2010) reported the simulation of heat and 

chemical reactions on Reiner Rivlin fluid model for blood flow through a tapered artery with a stenosis.  

The MHD flow has got considerable attention in the last few decades due to its effect on the boundary 

layer flow control and applications in various branches of physical and engineering fields. Tripathi and 

Anwar (2012) considered a study of unsteady physiological magneto-fluid flow and heat transfer 

through a finite length channel by peristaltic pumping. Chamkha et al. (2011) analyzed the melting 

effect on unsteady hydromagnetic flow of a nanofluid past a stretching sheet. The influence of heat 

source/sink on a Maxwell fluid over a stretching surface with convective boundary condition in the 

presence of nanoparticles has been demonstrated by Ramesh and Gireesha (2014). Analytical solution 

for heat and mass transfer of MHD slip fluid in nanofluids has been derived by Nohreh abadi and 

Ghalambaz (2012). Kameshwaran et al. (2012) examined the hydromagnetic nanofluid flow due to 

stretching or shrinking sheet with viscous dissipation and chemical reaction effects. The radiation and 

inclined magnetic field effects on unsteady hydromagnetic free convection flow past an impulsively 

moving vertical plate in a porous medium has been analyzed by Sandeep and Sugunamma (2014). The 

radiation effect on the flow and heat transfer over an unsteady stretching sheet was investigated by 

El-Aziz (2009). Some other important investigations in the direction of radiation and mass transfer 

effect in various aspects were made by Rohni et al. (2012), Motsumi and Makinde (2012), Kumaran et 

al. (2011). Thermal diffusion, radiation and inclined magnetic field effects on oscillatory flow in an 

asymmetric channel in presence of heat source and chemical reaction has been studied by Ajaz and 

Elangovan (2016). The effect of inclined magnetic field and Hall current on the micropolar fluid model 

of blood flow through stenotic arteries in a porous medium has been studied by Ajaz and Elangovan 

(2016). Recently, Ajaz and Elangovan (2016a, 2016b) observed the influence of inclined magnetic field 

on the peristaltic flow of blood by considering blood as a micropolar fluid. Ajaz and Elangovan (2017) 

studied the influence of an inclined magnetic field on heat and mass transfer of the peristaltic flow of a 

couple stress fluid in an inclined channel. 

In thermal convection, the effects of temperature field as modified by heat source/sink in moving fluids 

are considerable attention in many physical problems. Krishnamurthy et al. (2015) have studied the 



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effect of viscous dissipation on hydromagnetic fluid flow and heat transfer of nanofluid over an 

exponentially stretching sheet with fluid-particle suspension. Hakeem et al. (2014) illustrated the effect 

on partial slip on hydromagnetic flow over a porous stretching sheet with non-uniform heat source/sink, 

thermal radiation and wall mass transfer past a stretching porous sheet in presence of radiation. 

Mukhopadhyay and Layek (2008) have discussed the effects of thermal radiation and variable fluid 

viscosity on the free convective flow and heat transfer past a porous stretching surface. Anjali Devi and 

Ganga (2009) have studied the viscous dissipation effects on non-linear MHD flow in a porous medium 

over a stretching porous surface. Norihan and Arifan (2011) investigated the viscous flow due to a 

permeable shrinking/stretching sheet in a nanofluid. The effect of chemical reaction and thermal 

radiation on heat and mass transfer flow of MHD micropolar fluid in a rotating frame of reference was 

investigated by Das (2011). Many researchers contributed the important investigations in this direction 

(Muthucumaraswamy et al., 2006; Pal et al., 2012; Pal & Mandal, 2014). More recently, Raju et al. 

(2015) investigated the radiation, and inclined magnetic field and cross-diffusion effects on flow over 

stretching surface. The radiation and magnetic field effects on unsteady natural convection flow of a 

nanofluid past an infinite vertical plate with heat source was studied by Mohankrishna et al. (2014). 

In view of the above facts, the aim of the present investigation is to study the effect of an inclined 

magnetic field on the flow of nanofluids in a tapered asymmetric porous channel with heat source/sink 

and chemical reaction. The highly non-linear differential equations are solved by simply using the low 

Reynolds number and high wavelength approximation approach. The expressions of velocity, 

temperature and concentration distribution are obtained which are shown and discussed with the help of 

graphs for the variations of different flow parameters. 

 

2. Mathematical Formulations 

Consider the peristaltic transport of nanofluid in an asymmetric non-uniform as shown in Figure 1. Let 

Y = -H1 and Y = H2 be the right and left wall boundaries of the tapered asymmetric channel 

respectively. The geometry of the wall surfaces is defined as 

 𝐻1
′(𝑋′, 𝑡′) = 𝑑 + 𝑚′𝑋′ + 𝑎2𝑠𝑖𝑛 [

2𝜋

𝜆
(𝑋′ − 𝑐𝑡′)]       left wall, (1) 

 𝐻2
′(𝑋′, 𝑡′) = −𝑑 − 𝑚′𝑋′ − 𝑎1𝑠𝑖𝑛 [

2𝜋

𝜆
(𝑋′ − 𝑐𝑡′) + 𝜙]      right wall, (2) 

where 𝑎1, 𝑎2 are the amplitudes of right and left walls respectively. 𝑐 is the phase speed of the wave, 

𝑑 is the half-width of the channel, 𝜆 is the wave length, the phase difference 𝜙. 

 



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Figure 1. Geometry of the Problem 

 

varies in the range 𝜙(0 ≤  𝜙 ≤ 𝜋), 𝜙 = 0 corresponds to symmetric channel with waves out of phase, 

that is both walls move towards inward or outward simultaneously and 𝑚′(<<< 1)  is the 

non-uniform parameter respectively. Moreover, 𝑎1, 𝑎2, 𝑑, and 𝜙 satisfy the following relation  

 𝑎1
2 + 𝑎2

2 + 2𝑎1𝑎2 cos 𝜙 ≤ (2𝑑)2 (3) 

The field equations governing the flow are described as  

 
𝜕𝑈′

𝜕𝑋′
+

𝜕𝑉′

𝜕𝑌′
= 0 (4) 

𝜌𝑓 [
𝜕𝑈′

𝜕𝑡′
+ 𝑈′

𝜕𝑈′

𝜕𝑋′
+ 𝑉′

𝜕𝑈′

𝜕𝑌′
] = −

𝜕𝑃′

𝜕𝑋′
+ 𝜇 (

𝜕2𝑈′

𝜕𝑋′2 +
𝜕2𝑈′

𝜕𝑌′2) 

−𝜎𝐵0
2𝑠Θ[𝑈′ cos Θ − 𝑉′ sin Θ] 

 −
𝜇

𝐾′
𝑈′ + (1 − 𝐶0)𝜌𝑓𝑔𝛼(𝑇′ − 𝑇0) + (𝜌𝑝 − 𝜌𝑓)𝑔𝛽(𝐶 − 𝐶0) (5) 

𝜌𝑓[
𝜕𝑉′

𝜕𝑡′
+ 𝑈′

𝜕𝑉′

𝜕𝑋′
+ 𝑉′

𝜕𝑉′

𝜕𝑌′
] = −

𝜕𝑃′

𝜕𝑌′
+ 𝜇(

𝜕2𝑉′

𝜕𝑋2
+

𝜕2𝑉′

𝜕𝑌′2) −
𝜇

𝐾′
𝑉′ 

 +𝜎𝐵0
2 sin Θ [𝑈′ cos Θ − 𝑉′ sin Θ] (6) 

 (𝜌𝑐′)𝑓[
𝜕𝑇′

𝜕𝑡′
+ 𝑈

𝜕𝑇′

𝜕𝑋′
+ 𝑉

𝜕𝑇′

𝜕𝑌′
] = 𝑘(

𝜕2𝑇′

𝜕𝑋′2 +
𝜕2𝑇′

𝜕𝑌′2) + 𝑄′(𝑇′ − 𝑇0) 

 +(𝜌𝑐′)𝑝𝐷𝐵(
𝑎𝑙𝐶′

𝜕𝑋′

𝜕𝑇′

𝜕𝑋′
+

𝜕𝐶′

𝜕𝑌′

𝜕𝑇′

𝜕𝑋′
) +

𝐷𝑇(𝜌𝑐′)𝑝

𝑇𝑚
[(

𝜕2𝑇′

𝜕𝑋′2 +
𝜕2𝑇′

𝜕𝑌′2)] (7) 

 [
𝜕𝐶′

𝜕𝑡′
+ 𝑈′

𝜕𝐶′

𝜕𝑋′
+ 𝑉′

𝜕𝐶′

𝜕𝑌′
] = 𝐷𝐵[

𝜕2𝐶′

𝜕𝑋′2 +
𝜕2𝐶′

𝜕𝑌′2] +
𝐷𝑇

𝑇𝑚
[
𝜕2𝑇′

𝜕𝑋′2 +
𝜕2𝑇′

𝜕𝑌′2] (8) 

where 𝑈′ and 𝑉′ are the velocity components along 𝑋, 𝑌 directions respectively. Moreover, 𝜌𝑓, 𝜌𝑝, 

𝜇 , 𝜎 , 𝐾 , Θ, 𝐵0 , 𝑔 , 𝛽 , 𝑇 , 𝑡 , 𝑘 , 𝑐𝑝 , are density of the fluid, density of the particle,viscosity, 

electrical conductivity of the fluid, porous parameter, inclination angle of magnetic field, applied 

magnetic field, acceleration due to gravity, volumetric expansion co-efficient, temperature, time, 

thermal conductivity, specific heat at constant pressure respectively. In order to derive the fluid flow, let 

us introducing the dimensionless variables and parameters as follows:  



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𝑥 =
𝑋′

𝜆
, 𝑦 =

𝑌′

𝑑
, 𝑢 =

𝑈′

𝑐
, 𝑣 =

𝑉′

𝛿𝑐
, 𝑝 =

𝑑2𝑝′

𝑐𝜆𝜇
, 𝑡 =

𝑐𝑡′

𝜆
, ℎ1 =

𝐻1
′

𝑑
, ℎ2

=
𝐻2

′

𝑑
, 𝑎 =

𝑎1

𝑑
, 

 

𝑏 =
𝑎2

𝑑
, 𝛿 =

𝑑

𝜆
, 𝑅𝑒 =

𝜌𝑓𝑐𝑑

𝜇
, 𝑃𝑟 =

𝜇𝑐𝑓

𝑘
, 𝜃 =

𝑇 − 𝑇0

𝑇1 − 𝑇0
, 𝐺𝑟 =

(1 − 𝐶𝑜)𝜌𝑓𝑔𝛼(𝑇1 − 𝑇0)𝑑2

𝑐𝜇
, 

𝐾 =
𝐾′

𝑑2
, 𝑚 =

𝜆𝑚′

𝜇
, 𝑀2 =

𝜎𝐵0
2𝑑2

𝜇
, 𝐵𝑟 =

(𝜌𝑝 − 𝜌𝑓)𝑔𝛽(𝐶1 − 𝐶0)𝑑2

𝑐𝜇
, 𝑁𝑏 =

𝜏𝐷𝐵(𝐶1 − 𝐶0)

𝜈
, 

                                            𝑁𝑡 =
𝜏𝐷𝑇(𝑇1−𝑇0)

𝜈
, Φ =

𝐶′−𝐶0

𝐶1−𝐶0
.                                (9) 

where 𝑥 and 𝑦 are the axial co-ordinate, transverse co-ordinate respectively. 𝑢 and 𝑣 denotes the 

axial velocity and transverse velocity respectively. Moreover, 𝛿, 𝜙, 𝑅𝑒, 𝑃𝑟, 𝐺𝑟, 𝑀, 𝑁𝑏, 𝑁𝑡 and 𝐵𝑟 

are wave number, phase difference, Reynolds number, Prandtl number, Grashof number, Hartmann 

number, Brownian motion parameter, thermophoresis parameter and local nanoparticle Grashof number 

respectively. By using the above non-dimensional variables and parameters from Equation (9) in 

Equations. (5)-(8) and applying the low Reynolds number and high wavelength approximation 

approach, we arrive at  

 0 =
𝜕𝑝

𝜕𝑥
+

𝜕2𝑢

𝜕𝑦2
− (

1

𝐾1
+ 𝑀2𝑐𝑜𝑠2Θ)𝑢 + 𝐺𝑟𝜃 + 𝐵𝑟𝜙 (10) 

 0 =
𝜕𝑝

𝜕𝑦
 (11) 

 0 = (
𝜕2𝜃

𝜕𝑦2
)2 + 𝑃𝑟𝑁𝑏(

𝜕𝜙

𝜕𝑦

𝜕𝜃

𝜕𝑦
) + 𝑁𝑡(

𝜕2𝜃

𝜕𝑦2
)2 + 𝑄𝜃 (12) 

 0 = (
𝜕2𝜙

𝜕𝑦2
) + (

𝑁𝑡

𝑁𝑏
)(

𝜕2𝜃

𝜕𝑦2
)2 (13) 

The corresponding dimensionless boundary conditions are  

 
𝑢 = 0,  𝜃 = 0,  Φ = 0  at    

𝑦 = ℎ1 = −1 − 𝑚𝑥 − 𝑎 sin[ 2𝜋(𝑥 − 𝑡) + 𝜙] 
(14) 

 
𝑢 = 0,    𝜃 = 1,   Φ = 1   at    

𝑦 = ℎ2 = 1 + 𝑚𝑥 + 𝑏 sin[ 2𝜋(𝑥 − 𝑡)] 
(15) 

 

3. Method of Solution 

Integrating Equation (13) twice with respect to y, and applying the boundary conditions (14), (15), the 

value of Φ becomes  

 Φ +
𝑁𝑡

𝑁𝑏
𝜃 +

𝑁𝑏 + 𝑁𝑡

(ℎ1 − ℎ2)𝑁𝑏
𝑦 −

ℎ1(𝑁𝑏 + 𝑁𝑡)

(ℎ1 − ℎ2)𝑁𝑏
= 0 (16) 

 



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Solving Equation (12) after substituting the value of Φ and applying the boundary conditions (14), 

(15), we have  

 𝜃 = 𝐴3 (−𝑒
1
2

𝐴2(𝑦−ℎ2)) sinh(𝐴4(𝑦 − ℎ1)) (17) 

𝐴2 =
𝑃𝑟(𝑁𝑏 + 𝑁𝑡)

ℎ1 − ℎ2
 

𝐴3 = Csch [
1

2
𝐴4(ℎ1 − ℎ2)] 

𝐴4 =
1

2
√(−4𝑄 + 𝐴2

2) 

Using Equation (17) in Equation (16), we have  

 Φ = 𝐴5(−𝑈) − 𝐴6(𝑦 − ℎ1) (18) 

where  

𝐴5 =
𝑁𝑡

𝑁𝑏
 

𝐴6 =
(𝑁𝑏 + 𝑁𝑡)

(ℎ1 − ℎ2)𝑁𝑏
 

Using the values of 𝜃 and Φ in Equation (10) and applying the boundary conditions, the velocity 

expressions becomes 

𝑢 = 𝑒√𝐴5(−𝑦) (𝐴5 (𝐶1 (𝑒√𝐴5(ℎ1+2𝑦) − 𝑒√𝐴5(2ℎ1+𝑦) − 𝑒√𝐴5(ℎ2+2𝑦) + 𝑒√𝐴5(2ℎ2+𝑦) + 𝐴9))) 

+𝑒√𝐴5(−𝑦) (𝐴5 (𝐴2𝐺𝑟 (𝑒√𝐴5(2ℎ1+𝑦) − 𝑒√𝐴5(2ℎ2+𝑦)) sinh(𝐴1(𝑦 − ℎ1)))) 

+𝑒√𝐴5(−𝑦) (𝐴5 (𝐴5𝐴7𝑙𝑒𝑓𝑡(− (𝑒√𝐴5(ℎ1+ℎ2) + 𝑒2√𝐴5𝑦)) + 𝐴10 (𝑒√𝐴5(ℎ2+2𝑦) − 𝑒√𝐴5(2ℎ1+ℎ2))) 

 

+𝑒√𝐴5(−𝑦) (𝐴7𝐴1
2 (𝐶1𝑒√𝐴5(ℎ1+𝑦) + 𝐶1𝑒√𝐴5(ℎ2+𝑦) − 𝐶1𝑒2√𝐴5𝑦

+ 𝐴5𝑒2√𝐴5𝑦 + 𝐴8)) /𝐴6 

(19) 

where  

𝐴7 = (𝐺𝑟 − 𝐴5𝐵𝑟)𝐴3 

𝐴8 = 𝐵𝑟𝐴6 

𝐴9 = (𝐴1(4𝐴1 − 4√𝐴1𝐴2 + 𝐴2
2 − 4𝐴4

2)(4𝐴1 + 4√𝐴1𝐴2 + 𝐴2
2 − 4𝐴4

2)) 

𝐴10 = 16𝐴1
2𝐴8ℎ1 − 8𝐴1𝐴2

2𝐴8ℎ1 + 𝐴2
4𝐴8ℎ1 − 32𝐴1𝐴4

2𝐴8ℎ1 − 8𝐴2
2𝐴4

2𝐴8ℎ1 + 16𝐴4
4𝐴8ℎ1 

 

 

 



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4. Results and Discussion 

The purpose of this section is to discuss the numerical and computational results with the help of 

graphical illustrations. The influence of different emerging physical parameters has been discussed with 

particular importance. Figure 2 illustrates the variation of axial velocity for different values of the 

Hartmann number  (𝑀) , Brownian motion parameter  (𝑁𝑏) , thermophoresis parameter (𝑁𝑡) , heat 

source/sink parameter (𝑄), non-uniform parameter (𝑚), local nanoparticle Grashof number (𝐵𝑟), and 

permeability parameter (𝐾1).  

From Figures 2(a)-(b), it is observed that the axial velocity decreases by increasing the Hartmann 

number (𝑀) and Brownian motion parameter (𝑁𝑏). From Figures 2(c)-(h), it is depicted that the 

velocity increases by increasing thermophoresis parameter (𝑁𝑡), heat source/sink parameter (𝑄), 

non-uniform parameter (𝑚), local nanoparticle Grashof number (𝐵𝑟), and permeability parameter 

(𝐾1).  

Figure 3 illustrates the variation of temperature profile (𝜃) for different values Brownian motion 

parameter (𝑁𝑏), thermophoresis parameter (𝑁𝑡) and Prandtl number 𝑃𝑟. From Figure 3(a)-(c), It is 

observed that the temperature increases by increasing the values of Brownian motion parameter (𝑁𝑏) 

and thermophoresis parameter (𝑁𝑓𝑡) and Prandtl number (𝑃𝑟). Figures 3(d)-(e), it is depicted that the 

temperature profile decreases by increasing the values heat generation and absorption parameter (𝑄).  

The variation of concentration profile for different values of physical parameters has been illustrated in 

Figure 4. From Figures 4(a)-(c), It is observed that the concentration profile decreases by increasing 

Brownian motion parameter (𝑁𝑏), thermophoresis parameter (𝑁𝑡) and Prandtl number 𝑃𝑟 . From 

Figures 4(d)-(e), we can find that the concentration profile increases by increasing the heat source/sink 

parameter (𝑄).  

 

 

                      2(a) 2(b) 



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 2(c) 2(d) 

 

 2(e) 2(f) 

 

 2(g) 2(h) 

Figures 2(a-h). Variation of M , bN , tN , Q , m , rB  and 1K  over the Axial Velocity 

( )u  with Respect to y at 0.5, 0.5, 2, 3, 0
dp

a b d x
dx

     
 

 



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                    3(a)      3(b) 

 

                     3(c)                                 3(d) 

 

                     3(e) 

Figures 3(a-e). Variation of , , ,r rN P Q  over the Temperature Profile ( )  with Respect to y 

with 0.5, 0.5, 2, 0a b d x     

 



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 4(a) 4(b) 

 

 4(c) 4(d) 

 

                     4(e) 

Figures 4(a-e). Variation of , , ,r rN P Q  over the Concentration Profile   with Respect to y 

with 0.5, 0.5, 2, 0a b d x   
 

 

4. Conclusion 

In this article, we have studied the effect of an inclined magnetic field on the flow of nanofluids in a 

tapered asymmetric porous channel with heat source/sink. The governing equations of the flow are 

solved analytically and the effects of various flow parameters on axial velocity, temperature profile and 

concentration profile are discussed finally. We conclude the main observations are enlisted below:  

a) By increasing the values of Brownian motion parameter, the axial velocity decreases.  

b) The axial velocity of a fluid increases by increasing the thermophoresis parameter, heat 

source/sink parameter, non-uniform parameter (m), local nanoparticle Grashof number and porous 



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parameter.  

c) The Grashof number and thermal radiation parameter show similar behavior on the axial velocity.  

d) The temperature of the fluid increases by increasing Brownian motion parameter, thermophoresis 

parameter and Prandtl number.  

e) Heat source and heat sink parameters show similar effects on temperature profile.  

f) The concentration profile decreases by increasing Brownian motion parameter, thermophoresis 

parameter and Prandtl number.  

 

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