Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

 68  
 

 

  

 

 

Hall and Joule's heating Influences on Peristaltic Transport of Bingham plastic Fluid 

with Variable Viscosity in an Inclined Tapered Asymmetric Channel 

 

                            Rasha Yousif Hassen                                          Hayat A. Ali 

Department of Mathematics and Application of Computer, University of Technology, 

 rashamath80@yahoo.com                           100048@uotechnology.edu.iq 

 

 

 

Abstract  

   This paper presents an investigation of peristaltic flow of Bingham plastic fluid in an 

inclined tapered asymmetric channel with variable viscosity. Taking into consideration Hall 

current, velocity, thermal slip conditions, Energy equation is modeled by taking Joule heating 

effect into consideration and by holding assumption of long wavelength and low Reynolds 

number approximation, these equations are simplified into a couple of non-linear ordinary 

differential equations that are solved by using perturbation technique. Graphical analysis has 

been involved for various flow parameters emerging in the problem. We observed two 

opposite behaviors for Hall parameter and Hartman number on velocity axial and temperature 

curves.  

Keywords: Heat Transfer, Hall Effect, Joule's Heating, Bingham plastic Fluid, Tapered 

Channel. 

 

1. Introduction 

    Peristaltic transport is a successive sinusoidal waves movement of fluids along a flexible 

channel walls. It is naturally found in human living body such as urine movement from kidney 

to bladder, food swallowing process and blood flow in the small vessels [1- 3]. Moreover, the 

peristaltic transport of non- Newtonian fluid gained much attention in various modern 

industrial and biomedical phenomena like polymer industry and artificial hearts that their 

devices designed in a manner where the fluid flows without internal moving parts [4-6]. 

Inspired by this fact and since the modern industrial fluids are characterized by their variable 

viscosity, a few researchers indicate studies regarding the peristaltic transport of fluids having 

variable viscosity. Adnan and Abdulhadi [7] analyzed the effect of an inclined magnetic field 

on peristaltic flow of Bingham plastic fluid in an inclined symmetric channel with slip 

conditions. In the same year Adnan and Abdulhadi [8] investigated the peristaltic flow of the 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Doi: 10.30526/34.1.2554 

Article history: Received 20, January, 2020, Accepted 26, February, 2020, Published in January 2021 

 

mailto:rashamath80@yahoo.com
mailto:100048@uotechnology.edu.iq
mailto:100048@uotechnology.edu.iq


  

69 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 
Bingham plastic fluid in a curved channel. Hayat et al. [9] studied the effect of soret and 

dufour on the peristaltic transport of Bingham plastic fluid considering magnetic field. While 

Lakshminarayana et al. [10] investigated the heat transfer and the effect of slip condition and 

wall properties on the peristaltic transport of Bingham fluid. Ara et al. [11] explored the 

Jeffery- Hamel flow of Bingham plastic fluid in converging channel in the presence of external 

magnetic field. However, Salih [12] illustrated the influence of varying temperature and 

concentration on (MHD) peristaltic transport of Jeffery fluid with variable viscosity through 

porous channel.  For more information see [7,13]. 

In this paper, the influence of Hall and Joule's heating on the peristaltic flow of Bingham 

plastic fluid passing through an inclined tapered asymmetric channel with variable viscosity is 

studied. A long wave number  and low Reynolds number are taken into consideration to 
simplify the problem. Perturbation technique is used to solve and find the last shape of stream 

function. 

Finally, the effects of various parameters on axial velocity, temperature, stream function and 

heat transfer coefficients are discussed graphically.     

 

2. Mathematical Modeling 

   The peristaltic transport of an incompressible Bingham plastic fluid in a tapered inclined 

channel at an angle 𝛼 which is an asymmetric channel with a total width(2𝑑) is considered. 

Characterizing the flow by the existence of a strong transverse magnetic field 𝐵 =  (0, 0, 𝛽0). 
A magnetic Reynolds number is taken small and the induced magnetic field is prescribed 

neglected. The flow is achieved by the peristaltic waves of length 𝜆 with different amplitude 

and phases moving with a constant speed 𝑐 along the channels walls. 

The geometry of the walls surfaces is described by 

𝑌1 = 𝐻1(�̅�, 𝑡) = 𝑑 + 𝑚1�̅� + 𝑎1𝐶𝑜𝑠 (
2𝜋(�̅�−𝑐𝑡)

𝜆
)                                                               (1)                                                

𝑌2 = 𝐻2(�̅�, 𝑡) = −𝑑 − 𝑚1�̅� − 𝑎2𝐶𝑜𝑠 (2𝜋(�̅� − 𝑐𝑡)/𝜆 + Ø)                                           (2)                                                

Where 𝑌1, 𝑌2  are the upper and lower wall respectively, 𝑚1  is the non-uniform 

parameter, 𝑎1, 𝑎2 are the wave amplitudes, 𝑡 is the time and (�̅�, �̅�) the rectangular coordinates 

in a fixed frame. Ø Is the phase different and Ø ∈ [0, 𝜋] such that when Ø = 0 corresponds to 

asymmetric channel with waves out of phase, and when  Ø = 𝜋 , the waves are in phase. 

Further 𝑎, 𝑏, 𝑑 𝑎𝑛𝑑 ∅ satisfy the necessary condition 

𝑎2 + 𝑏2 + 2𝑎𝑏𝑑 𝑐𝑜𝑠∅ ≤ (2𝑑)2                                                                                        (3) 

 

By applying the generalized Ohm's law [2], we include the Hall current as follows 

�⃑� = 𝐽 × �⃑⃑�                                                                                                                           (4) 

Such that 

𝐽 = 𝜎[�⃑⃑�  × �⃑⃑� −
1

𝑒𝑛
 (𝐽 × �⃑⃑�)]                                                                                               (5) 



  

70 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 
Hence,  

�⃑� = (
−𝜎𝛽0

2(𝑈−𝑚𝑉)

1+𝑚2
,
𝜎𝛽0

2(𝑉+𝑚𝑈)

1+𝑚2
, 0)                                                                                    (6) 

In which �⃑� is the magnetic force,  𝐽 assigns to the current density vector, �⃑⃑� = (𝑈, 𝑉, 0) the 

velocity field, 𝜎 the electrical conductivity, 𝑛 the number density of electron, 𝑒 the electric 

charge, 𝛽0 the magnetic field strength and (𝑚 =
𝜎𝛽0

𝑒𝑛
)  the Hall parameter. 

The fluid satisfies Bingham plastic model and its extra stress tensor is given as follows [11]: 

𝑆̅ = ( 𝜇(�̅�) +
𝜏𝑦

𝛾.
) �̅�1      For          𝜏 ≥ 𝜏𝑦

𝑆̅ = 0   For                𝜏 < 𝜏𝑦
}                                                                         (7) 

Such that 

�̅�1 = ∇�⃑⃑� + (∇�⃑⃑�)
𝑇
                                                                                                             (8) 

And 

𝛾. = √
 𝑡𝑟𝑎𝑐(�̅�1)

2

2
                                                                                                                 (9) 

𝑆̅ represents the extra stress tensor,𝛻 = (𝜕/𝜕�̅�, 𝜕/𝜕�̅�, 0) the gradient vector, 𝜏𝑦 the yield stress 

�̅�1is the first Rivlin- Erickson, and 𝜇(�̅�) is the dynamic variable viscosity.  

The fundamental equations of the flow can be written as below: 

𝑑𝑖𝑣 �̅� = 0                                                                                                                         (10)  

𝑋-component of momentum equation  

𝜌 (
𝑑�̅�

𝑑𝑡
+ �̅�

𝜕�̅�

𝜕�̅�
+ �̅�

𝜕�̅�

𝜕�̅�
) = −

𝜕�̅�

𝜕�̅�
+

𝜕�̅��̅��̅�
𝜕�̅�

+
𝜕�̅��̅��̅�
𝜕�̅�

−
𝜎𝛽2( �̅�−𝑚 �̅�)

1+𝑚2
+ 𝜌𝑔 Sin 𝛼 −

𝜇

𝜅
�̅�               (11) 

𝑌-component of momentum equation  

𝜌 (
𝑑�̅�

𝑑𝑡
+ �̅�

𝜕�̅�

𝜕�̅�
+ �̅�

𝜕�̅�

𝜕�̅�
) = −

𝜕�̅�

𝜕�̅�
+

𝜕�̅��̅��̅�
𝜕�̅�

+
𝜕�̅��̅��̅�
𝜕�̅�

−
𝜎𝛽2( �̅�+𝑚 �̅�)

1+𝑚2
+ 𝜌𝑔 Cos 𝛼 −

𝜇

𝜅
�̅�               (12) 

And 

Energy equation with Joule heating effect is 

𝜌𝑐𝑃 (
𝜕𝑇

𝜕�̅�
 + �̅�

𝜕𝑇

𝜕�̅�
+ �̅�

𝜕𝑇

𝜕�̅�
) = 𝐾 (

𝜕2𝑇

𝜕�̅�2
+

𝜕2𝑇

𝜕�̅�2
) + (𝑆�̅̅��̅� − 𝑆�̅̅��̅�) 

𝜕�̅�

𝜕�̅�
+ 𝑆�̅̅��̅�  (

𝜕�̅�

𝜕�̅�
+  

𝜕�̅�

𝜕�̅�
) +

                                                
𝜎𝛽0

2
(�̅�2+�̅�2)

1+𝑚2
                                                                             (13) 

In which 𝜎, 𝐾, 𝜅,  �̅�, 𝜇 , 𝜌, 𝑐𝑃, 𝑔 are the electrical conductivity, the thermal conductivity, the 
porosity parameter, the dynamic viscosity, the density, the specific heat, and the gravity 

respectively.  

The corresponding boundary slip conditions are 



  

71 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

�̅� ∓ 𝛾𝑆�̅�𝑦 = 0  𝑎𝑡    �̅� = 𝐻1, 𝐻2

𝑇 ∓ 𝛽1
𝜕𝑇

𝜕𝑦
= 𝑇0  𝑎𝑡    �̅� = 𝐻1, 𝐻2

}                                                                                     (14) 

 And the wall flexibility condition is 

[−𝜏
𝜕3

𝜕�̅�3
+ 𝑚2

𝜕3

𝜕�̅�𝜕𝑡2
+ 𝑑′

𝜕2

𝜕𝑡𝜕�̅�
] �̅� =

𝜕�̅�

𝜕�̅�
      𝑎𝑡    �̅� = 𝐻1, 𝐻2                                             (15) 

Where   𝑇0, 𝜏, 𝑚2, 𝑑
′, 𝛾, 𝛽1 are the temperature at the upper and lower walls, the elastic tension, 

the mass per unit area and the coefficient of viscous damping, velocity slip coefficient, and 

temperature slip coefficient respectively. 

 Dimensional analysis is used for normalizing the flow equations Eqs. (7) - (15) by using the 

following as bellows: 

 𝑥 =
�̅�

𝜆
  , 𝑦 =

�̅�

𝑑
 ,   𝑢 =

�̅�

𝑐
  , 𝑣 =

�̅�

𝑐
  , ℎ1 =

𝐻1

𝑑
, ℎ2 =

𝐻2

𝑑
 , 𝑝 =  

𝑑2�̅�

𝜆𝜇𝑐
, 𝛿 =

𝑑

𝜆
 , 𝛾∗ =

𝛾

𝑑
 , 𝛽1

∗ =
𝛽1

𝑑
, 𝑆 =

    
𝑑�̅�(�̅�)

𝜇𝑐
 , 𝑅𝑒 =

𝜌𝑐𝑑

𝜇
,   𝜃 =

𝑇−𝑇0

𝑇0
, 𝐸1 =

−𝜏𝑑3

𝜆3𝜇𝑐
 , 𝐸2 =

𝑚2𝑐𝑑
3

𝜆3𝜇𝑐
, 𝐸3 =

𝑑′𝑑3

𝜆2𝜇
, 𝑃𝑟 =

𝜇𝑐𝑃

𝑘
 , 𝐻 =

𝛽0𝑑√
𝜎

𝜇
 , 𝐸𝑐 =

𝑐2

𝑐𝑃𝑇0
 , 𝐵𝑟 = 𝐸𝑐𝑃𝑟 , 𝐹𝑟 =

𝑐

√𝑔𝑑
 , 𝑑1 =

𝑎1

𝑑
 , 𝑑2 =

𝑎2

𝑑
 , 𝑚1

∗ =
𝑚1𝜆

𝑑
, 𝐵𝑛 =

𝑑𝜏𝑦

𝜇𝑐
, 𝜇(�̅�) =

𝜇(𝑦)

𝜇
                                                                                                              (16) 

Where 𝛿  is the wave number, 𝐸1  the wall elastance parameter, 𝐸2  the mass per unit area 

parameter, 𝐸3the wall damping parameter, 𝑅𝑒 the Reynolds number, 𝑃𝑟 the Prandtl number, 𝐻 

Hartman number, 𝐸𝑐  Eckret number, 𝐵𝑟  Brinkman number,  𝐹𝑟  Froude number,  𝑚1
∗ the 

dimensionless non-uniform parameter, ℎ1 the dimensionless lower wall surface, ℎ2 upper wall 

surface,  𝑥, 𝑦  components of the dimensionless coordinates,  𝑢  axial velocity,  𝑣  transverse 

component of velocity, 𝜖 the fluid dimensionless viscosity parameter, 𝛾., 𝛽1
∗ the dimensionless 

velocity and thermal slip parameters respectively and 𝐵𝑛 

Bingham number. Note that we omitted asterisks for simplicity 

Introducing the stream function  𝜓(𝑥, 𝑦, 𝑡)  and make use of the following relation  

𝑢 = 𝜓𝑦, 𝑣 = −𝛿𝜓𝑥                                                                                                                   

Applying Eq. (17) into Eqs. (9) – (16) and making use of Eq. (17), the continuity equation (11) 

vanishs identically, other flow equations take the following form  

𝛿𝑅𝑒(𝜓𝑡𝑦 + 𝜓𝑦𝜓𝑥𝑦 − 𝛿𝜓𝑥𝜓𝑥𝑦) = −𝑃𝑥 + 𝛿
𝜕𝑆𝑥𝑥

𝜕𝑥
+

𝜕𝑆𝑥𝑦

𝜕𝑦
−

𝐻2

(1+𝑚2)
(𝜓𝑦 + 𝛿𝑚𝜓𝑥) −

𝜓𝑦

𝜅
+

                                                                 
𝑅𝑒

(𝐹𝑟)2
 𝑠𝑖𝑛𝛼                                                               (17) 

𝛿2𝑅𝑒(−𝛿2𝜓𝑡𝑥 − 𝛿
2𝜓𝑦𝜓𝑥𝑥 + 𝛿

3𝜓𝑥𝜓𝑥𝑦)

= −𝑃𝑦 + 𝛿
𝜕𝑆𝑦𝑦

𝜕𝑦
+ 𝛿2

𝜕𝑆𝑦𝑥

𝜕𝑥
−

𝐻2𝛿

(1 + 𝑚2)
(−𝛿𝜓𝑥 + 𝑚𝜓𝑦) −

𝜓𝑦

𝜅
−

𝛿𝑅𝑒

(𝐹𝑟)2
 𝑐𝑜𝑠𝛼 

                                                                                                                                          (18) 



  

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

𝑅𝑒𝑃𝑟 (𝛿
𝜕𝜃

𝜕𝑡
+ 𝛿𝜓𝑦

𝜕𝜃

𝜕𝑥
− 𝛿𝜓𝑥

𝜕𝜃

𝜕𝑦
) =

𝛿2
𝜕2𝜃

𝜕𝑥2
+

𝜕2𝜃

𝜕𝑦2
+ 𝐵𝑟𝑆𝑥𝑦(𝜓𝑦𝑦 − 𝛿

2𝜓𝑥𝑥) +                                                                
𝐻2𝐵𝑟

(1+𝑚2)
((𝜓𝑦)

2 +

𝛿2(𝜓𝑥)
2) + 𝛿𝐵𝑟(𝑆𝑥𝑥 − 𝑆𝑦𝑦)𝜓𝑥𝑦 … (19)  

Adopting the assumptions of peristaltic long wavelength and low Reynolds number, Eqs. (17)- 

(19) will be reduced into following form 

𝑃𝑥 =
𝜕𝑆𝑥𝑦

𝜕𝑦
− (

𝐻2

(1+𝑚2)
+

1

𝜅
) 𝜓𝑦 +

𝑅𝑒

(𝐹𝑟)2
 𝑠𝑖𝑛𝛼,                                                                     (20) 

𝑃𝑦 = 0,                                                                                                                             (21) 

𝜕2𝜃

𝜕𝑦2
= 𝐵𝑟𝑆𝑥𝑦𝜓𝑦𝑦 +

𝐻2𝐵𝑟

(1+𝑚2)
(𝜓𝑦)

2,                                                                                    (22) 

𝑆𝑥𝑦 = 𝑆𝑦𝑥 = 𝜇(𝑦)𝜓𝑦𝑦 + 𝐵𝑛                                                                                              (23) 

𝑆𝑥𝑥 = 0, 𝑆𝑦𝑦 = 0,                                                                                                               (24) 

And the dimensionless boundary conditions are  

𝑢 ∓ 𝛾𝑆𝑥𝑦 = 0       

𝜃 ∓ 𝛽1
𝜕𝜃

𝜕𝑦
= 0    

(𝐸1
𝜕3

𝜕𝑥3
+ 𝐸2

𝜕3

𝜕𝑥𝜕𝑡2
+ 𝐸3

𝜕2

𝜕𝑡𝜕𝑥
) 𝑦 =

𝜕𝑆𝑥𝑦

𝜕𝑦
−

𝐻2𝜓𝑦

1+𝑚2
+

𝑅𝑒

(𝐹𝑟)2
𝑠𝑖𝑛𝛼]

 
 
 
 

  At 𝑦 = ℎ1, ℎ2,                 (25) 

Where 

 ℎ1 = −1 − 𝑚1𝑥 − 𝑑1 (𝐶𝑜𝑠(𝑥 − 𝑡) + ∅) ,  ℎ2 = 1 + 𝑚1𝑥 + 𝑑2(𝐶𝑜𝑠 2𝜋(𝑥 − 𝑡))                    

Furthermore, heat transfer coefficient at lower wall is derived as 

  𝑍 =
𝜕ℎ1

𝜕𝑥
 𝜃𝑦(ℎ1)                                                                                 (26) 

Through Eqs. (20) and (21), we obtain 

𝜕2𝑆𝑥𝑦

𝜕𝑦2
− (

𝐵𝑟𝐻2

(1+𝑚2)
+

1

𝜅
) 𝜓𝑦𝑦 = 0                                                                                          (27) 

We toke the dimensionless approximate expression for 𝜇(𝑦)as 

𝜇(𝑦) = 𝑒−𝜖𝑦 = 1 − 𝜖𝑦,     where𝜖 < 1, 

𝜖 is non- dimensional viscosity parameter. 

 

3. Solution Methodology 

   By using the perturbation method for a small non- dimensional viscosity parameter 𝜖 and 

expanding the flow quantities in a power series of 𝜖  , we obtain 

 𝜓 = 𝜓0 + 𝜖 𝜓1                                                                                                                   (28) 



  

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𝜃 = 𝜃0 + 𝜖 𝜃1                                                                                                                      (29) 

Substituting Eqs. (28), (29) into Eqs. (22) - (27) and then comparing the coefficients of same 

power of 𝜖 up to the first order, we obtain the following two systems 

 

 

3.1. Zeroth order system 

 

     The general form of zeroth- order system is: - 

𝜓0𝑦𝑦𝑦𝑦 − (
𝐻2

1+𝑚2
+

1

𝜅
) 𝜓0𝑦𝑦 = 0                                                                                       (30) 

𝜃0𝑦𝑦 + 𝐵𝑟 (
𝜕2𝜓0

𝜕𝑦2
+ 𝐵𝑛) 𝜓0𝑦𝑦 +

𝐻2

1+𝑚2
𝐵𝑟𝜓0𝑦

2
= 0                                                          (31) 

Now with the respect to the boundary conditions, we have: -  

 𝜓0𝑦 ∓ 𝛾(𝜓0𝑦𝑦 + 𝐵𝑛) = 0                                                                                                 (32) 

𝜃0 ∓ 𝛽1𝜃0𝑦 = 0                                                                                                                 (33) 

𝐸1
𝜕3𝑦

𝜕𝑥3
+ 𝐸2

𝜕2𝑦

𝜕𝑥𝜕𝑡2
+ 𝐸3

𝜕2𝑦

𝜕𝑡𝜕𝑥
= 𝜓0𝑦𝑦𝑦 −

𝐻2

1+𝑚2
𝜓0𝑦 +

𝑅𝑒

(𝐹𝑟)2
𝑠𝑖𝑛𝛼                                       (34) 

𝑎𝑡 𝑦 = ℎ1, ℎ2,  

 

3.2. First order system 

     The general form of first- order system is 

𝜓1𝑦𝑦𝑦𝑦 − 𝑦𝜓0𝑦𝑦𝑦𝑦 − 2𝜓0𝑦𝑦𝑦 − (
𝐻2

1+𝑚2
+

1

𝜅
) 𝜓1𝑦𝑦 = 0                                                    (35) 

𝜕2𝜃

𝜕𝑦2
+ 𝐵𝑟(𝜓1𝑦𝑦 − 𝜓0𝑦𝑦)𝜓1𝑦𝑦 +

𝐻2

1+𝑚2
𝐵𝑟𝜓1𝑦

2
=0                                                            (36) 

With the respect to the boundary conditions  

 𝜓1𝑦 ∓ 𝛾(𝜓1𝑦𝑦 − 𝑦𝜓0𝑦𝑦) = 0                                                                                           (37) 

 𝜃1 ∓ 𝛽1𝜃1𝑦 = 0                                                                                                                 (38) 

 𝜓1𝑦𝑦𝑦 −
𝐻2

1+𝑚2
𝜓1𝑦 = 0                                                                                                    (39) 

Solving the both system using Mathematica program, we get the closed form for 𝜓, and 𝜃 

𝜓 =
1

𝑘1
2 (𝑐1𝑒

𝑘1𝑦 + 𝑐2𝑒
−𝑘1𝑦) + 𝑐3 + 𝑦𝑐4 +

𝜖

8𝑘1
3 (𝐴1 + 𝐴2 + 8𝑘1(𝐴3 + 𝑘1

2
(𝑐7 + 𝑦𝑐8)), 

𝜃 = −
𝐵𝑟

𝑘1
2𝜅

(𝐴4 + 𝐴5 + 𝐴6) + 𝑛1 + 𝑦𝑛2 −
𝜖

3840𝑘16𝜅
𝐵𝑟𝑒−2𝑘1𝑦( 𝐴7 + 𝐴8 + 𝐴9 + 𝐴10 − 𝐴11) +

𝑛3 + 𝑦𝑛4, 

Where 



  

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𝐴1 = 𝑐1𝑒
𝑘1𝑦(−3 − 2𝑘1𝑦 + 2𝑘1

2
𝑦2) , 

𝐴2 = 𝑐2𝑒
−𝑘1𝑦(3 − 2𝑘1𝑦 − 2𝑘1

2
𝑦2),  

𝐴3 = 𝑐5𝑒
𝑘1𝑦 + 𝑐6𝑒

−𝑘1𝑦, 

𝑘1 = (
𝐻2

1+𝑚2
+

1

𝜅
)
1/2

, 

𝐴4 = (𝑐1𝑐2𝑦
2 −

1

2
𝑐4

2𝑘1
2
𝑦2 +  

1

2
𝑐4

2𝑘1
4
𝑦2𝜅),  

𝐴5 =
𝑐2
2𝑒−2𝑘1𝑦(−1+2𝑘1

2𝜅)

4𝑘1
2 +

𝑐1
2𝑒2𝑘1𝑦(−1+2𝑘1

2𝜅)

4𝑘1
2 , 

𝐴6 =
𝑐2𝑒

−𝑘1𝑦(2𝑐4+𝐵𝑛𝑘1𝜅−2𝑐4𝑘1
2𝜅)

𝑘1
+

𝑐1𝑒
𝑘1𝑦(−2𝑐4+𝐵𝑛𝑘1𝜅+2𝑐4𝑘1

2𝜅)

𝑘1
, 

𝐴7 = (15𝑐2
2(−27 + 16𝑘1

5
𝑦3𝜅 + 8𝑘1

6
𝑦4𝜅 + 2𝑘1

2
(2𝑦2 + 9𝜅) − 8𝑘1

3
(𝑦3 + 3𝑦𝜅) −

4𝑘1
4
(𝑦4 + 4𝑦2𝜅)), 

𝐴8 = 5(3𝑐1
2𝑒4𝑘1𝑦(−27 − 16𝑘1

5
𝑦3𝜅 + 8𝑘1

6
𝑦4𝜅 + 2𝑘1

2
(2𝑦2 + 9𝜅) + 8𝑘1

3
(𝑦3 + 3𝑦𝜅) −

4𝑘1
4
(𝑦4 + 4𝑦2𝜅)), 

𝐴9 = 192𝑘1
2
(𝑐6

2(−1 + 2𝑘1
2
𝜅) + 4𝑐6𝑒

𝑘1𝑦𝑘1(𝑐5𝑒
𝑘1𝑦𝑘1𝑦

2 + 𝑐8(2 − 2𝑘1
2
𝜅)) +

𝑒2𝑘1𝑦(8𝑐5𝑐8𝑒
𝑘1𝑦𝑘1(−1 + 𝑘1

2
𝜅) + 2𝑐8

2𝑘1
4
𝑦2(−1 + 𝑘1

2
𝜅) + 𝑐5

2𝑒2𝑘1𝑦(−1 + 2𝑘1
2
𝜅))), 

𝐴10 = 32𝑐1𝑒
2𝑘1𝑦𝑘1(3𝑐5𝑒

2𝑘1𝑦(2 + 𝑘1𝑦 − 2𝑘1
3𝑦𝜅 + 2𝑘1

4
𝑦2𝜅 − 𝑘1

2
(𝑦2 + 3𝜅)) +

𝑘1(6𝑐8𝑒
𝑘1𝑦(3 − 6𝑘1𝑦 + 2𝑘1

2
𝑦2)(−1 + 𝑘1

2
𝜅) + 𝑐6𝑘1𝑦

2(−15 + 2𝑘1𝑦 + 𝑘1
2
(𝑦2 + 6𝜅))))), 

𝐴11 = 4𝑐2𝑘1(120𝑐6(2 − 𝑘1𝑦 + 2𝑘1
3
𝑦𝜅 + 2𝑘1

4
𝑦2𝜅 − 𝑘1

2
(𝑦2 + 3𝜅)) + 𝑒𝑘1𝑦𝑘1(−240𝑐8(3 +

6𝑘1𝑦 + 2𝑘1
2
𝑦2)(−1 + 𝑘1

2
𝜅) + 𝑒𝑘1𝑦𝑦2(40𝑐5𝑘1(−15 − 2𝑘1𝑦 + 𝑘1

2
(𝑦2 + 6𝜅)) + 𝑐1(375 −

60𝑘1
2
(𝑦2 + 4𝜅) + 4𝑘1

4
(𝑦4 + 15𝑦2𝜅))), 

Where 𝑐1, 𝑐2, 𝑐3, 𝑐4, 𝑐5, 𝑐6, 𝑐7, 𝑐8, 𝑛1, 𝑛2, 𝑛3, 𝑛4,can be found using simple calculations. 

4. Result and Discussions 

        In this section we visualize graphically the influence of different inclusive parameters on 

velocity profile, temperature distribution, heat transfer coefficient and trapping phenomenon. 

  

4.1. Velocity Profile 

    Figures 1-3 elucidate the behavior of velocity profile against the following various 

important parameters   (𝐸1, 𝐸2, 𝐸3 𝜅, 𝑚, 𝐻, 𝜙) and for fixed values of (𝜖 = 0.04, Fr = 0.8, Re =

0.2, 𝛼 =
Pi

2
, 𝛾 = 0.7, 𝑡 = 0.1, d1 = 0.4, d2 = 0.4, 𝑥 = 0.4, 𝑚1 = 0.1) . Fig.1 (a) is plotted to 

describe the effect of wall elasticity parameters on velocity profile. One can conclude a 

significant increase upon enhancement of wall rigidity and tension parameters respectively  

𝐸1, 𝐸2  whereas the velocity profile slightly rises as mass characterization parameter 

𝐸3 increase. Similar observation is seen for enhanced values of permeability parameter 𝜅 and 

Hall number 𝑚 on velocity curve i.e. 𝑢(𝑦) is increasing and these results are shown in Figure 
1(b) and Figure 2 (a). A reversed situation for the larger magnitude of both Hartman number 



  

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𝐻 and Bingham number 𝐵𝑛 are shown in Figure 2 (b) and Figure 3 (a). From Figure 3 (b), 

we notice that the velocity profile enhances for higher values of phase difference parameter 𝜙.  

 

 

1.                                                                (b) 

 
Figure 1:Velocity profile for different values of (a) Elasticity parameters 𝐸1, 𝐸2, 𝐸3 (b) permeability parameter 𝜅. 

 

           (a)                                                                  (b) 

  
 

Figure 2:  Velocity profile for different values of (a) Hall number 𝑚  (b) Hartman number 𝐻 , 

 

 

          (a)                                                                        (b) 

   
 

Figure 3:  Velocity profile for different values of (a) Bingham number 𝐵𝑛  (b) Phase difference parameter 𝜙. 
  

 

E1 0.1 , E2 0.2 , E3 0.3

E1 0.2 , E2 0.2 , E3 0.3

E1 0.1 ,E2 0.4 ,E3 0.3

E1 0.1 , E2 0.2 , E3 0.6

0.0 0.5 1.0 1.5 2.0

0

5

10

15

y

u
y 1.4

1.6

1.8

2

0.0 0.5 1.0 1.5 2.0

0

5

10

15

20

y
u

y

m 0.2

m 0.4

m 0.6

m 0.8

0.0 0.5 1.0 1.5 2.0

0

5

10

15

20

y

u
y

H 1.6

H 1.8

H 2

H 1.4

0.0 0.5 1.0 1.5 2.0

0

5

10

15

20

y

u
y

Bn 0.2

Bn 0.4

Bn 0.6

Bn 0.8

0.0 0.5 1.0 1.5 2.0

0

5

10

15

20

y

u
y

6

4

3

2

0.0 0.5 1.0 1.5 2.0

0

5

10

15

20

y

u
y



  

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4.2. Temperature Distribution 

 

    Figures 4-6 are plotted to elaborate parabolic behavior for temperature distribution against 

axial 𝑦  axis for different magnitudes of the parameters  (𝐵𝑟, 𝐹𝑟, 𝐻, 𝛽1, 𝑚, 𝐵𝑛). Figure 4(a) 

illustrates the impact of Brinkman number on  𝜃(𝑦) . It is seen that 𝐵𝑟  react directly on 

temperature profile. However, in Figure 4(b) Froude number 𝐹𝑟  record quite opposite 
behavior compared to Brinkman number. The effect of Hartman number on temperature 

profile testified in Fig. 5(a). It is evident that the rise in Lorentz force produces a resistance for 

a larger value of Hartman number and consequently reduces the temperature profile. Figures. 

5(b), 6(a), and 6(b) clarify an increment in temperature slip parameter 𝛽1 Hall number 𝑚 and 

Bingham number 𝐵𝑛 causing rise in the temperature distribution curve.   

 

         (a)                                                                          (b) 

   
Figure 4:  Temperature profile 𝜃(𝑦)for different values of (a) Brinkman number 𝐵𝑟  (b) Froude number 𝐹𝑟. 

 

 

 

 

 

 

(b) 

    
Figure 5:  Temperature profile 𝜃(𝑦)for different values of (a) Hartman number 𝐻  (b) temperature slip parameter 𝛽1. 

 

      (a)                                                                                (b) 

Br 0.3

Br 0.5

Br 0.7

Br 0.9

1.5 1.0 0.5 0.0 0.5 1.0 1.5

0

50

100

150

200

y

y Fr 0.3

Fr 0.5

Fr 0.7

Fr 0.95

1.5 1.0 0.5 0.0 0.5 1.0 1.5

20

40

60

80

100

y

y

H 1

H 1.5

H 1.7

H 2

1.5 1.0 0.5 0.0 0.5 1.0 1.5

0

20

40

60

80

100

y

y

1 0.3

1 0.5

1 0.7

1 0.9

1.5 1.0 0.5 0.0 0.5 1.0 1.5

0

20

40

60

80

y

y



  

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Figure 6:  Temperature profile 𝜃(𝑦)for different values of (a) Hall number 𝑚  (b) Bingham number 𝐵𝑛. 

 

 

 

4.3. Heat Transfer 
 

     Figures 7-8 elucide the impact of Brinkman number 𝐵𝑟 , Froude number 𝐹𝑟 , phase 

difference parameter ∅, and permability parameter 𝜅 on the coefficient of heat transfer at the 

lower wall profile. These figures show an oscillatory behavior of 𝑍(𝑥) via the flow of 
peristaltic waves along the channel wall. Figure 7(a) portrays the increasing function of heat 

transfer coffecient due to arise in  𝐵𝑟 value. However Figure 7(b) depicts an oppsite reaction 

for 𝐹𝑟. Figure 8(a) charactrizes a mixed behavior for  𝑍(𝑥) for a higher value of  ∅ . We 

deduce from Figure 8(b) that for ascending magnitude of 𝜅, the rate of heat transfer increases . 
      

         (a)                                                                          (b)  

               
Figure 7: Heat transfer coefficient Z(𝑥)for different values of (a) Brinkman number 𝐵𝑟  (b) Froude number 𝐹𝑟. 

 

        (a)                                                                  (b) 

 

m 0.2

m 0.4

m 0.6

m 0.8

1.5 1.0 0.5 0.0 0.5 1.0 1.5

0

20

40

60

y

y
Bn 0.2

Bn 0.4

Bn 0.6

Bn 0.8

1.5 1.0 0.5 0.0 0.5 1.0 1.5

0

50

100

150

200

y

y

Br 0.2

Br 0.4

Br 0.6

Br 0.8

0.0 0.5 1.0 1.5
200

100

0

100

200

300

400

500

x

Z
x

Fr 0.2

Fr 0.4

Fr 0.6

Fr 0.8

0.0 0.5 1.0 1.5

200

100

0

100

200

300

x

Z
x



  

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Figure 8: Heat transfer coefficient Z(𝑥)for different values of (a) Phase difference parameter 𝜙  (b) permeability 

parameter𝜅. 

 

 

 

4.4. Trapping phenomenon 

 

   A phenomenon in which an amount of fluid trapped in closed streamlines is called bolus. In 

this part of work, some results of the phenomenon of trapping are portrayed. Figures 9-15 

highlight the impact of wall elasticity parameters  𝐸1, 𝐸2, 𝐸3  and for 𝐻, 𝜅, 𝑚, 𝜖, 𝐹𝑟, 𝐵𝑛 values. 

Graphical results show two asymmetric regions, the first region begins from (0.2 ≤ 𝑥 ≤

0.7)while the second region  (0.7 ≤ 𝑥 ≤ 1.2). One can observe an increment in size and 
number of trapped bolus, whereas the second region witnesses a less number of generated 

bolus. The effect of wall rigidity and tension parameters respectively  𝐸1, 𝐸2  and mass 

characterization parameter 𝐸3 on trapping phenomenon are shown in Figure 9. However, it is 

important to note that both parameters 𝐸1, 𝐸2  increase the trap bolus in magnitude and number 

while a higher value of 𝐸3 parameter that reduces the size of bolus but in the right side, its 

number reduces. Figure 10 reveals that ascending value of Hartman number 𝐻 is due to 
increases in Lorentz force which resists the fluid flow as a result decreases the size of trapping 

bolus. Opposite to this result, permeability parameter 𝜅 directly acts on trapping bolus in size 

and number; see Figure 11. The variation of Hall number 𝑚 and dimensionless viscosity 

parameter  𝜖  on trapped bolus are reflected in Figures 12 and 13. One can observe the 
increasing function for them on trapped bolus size. In Figure 14, we demonstrate that a larger 

value of Froude number  𝐹𝑟  reduces both the size and circulation of bolus. Figure 15 

interprets the independence of trapping bolus of variation of Bingham parameter 𝐵𝑛. 
 

 

                                                                              (a) 

 

"
6

4

3

2

0.0 0.5 1.0 1.5

40

20

0

20

40

x

Z
x

x

0.2

0.4

0.6

0.9

0.0 0.5 1.0 1.5

4

2

0

2

4

x

Z
x



  

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      (b)                                                                          (c) 

  

Figure 9:  Streamlines for variation of parameters (a) wall rigidity  𝐸1  (b) wall tension  𝐸2   (c) mass 

characterization  𝐸3 with   𝜖 = 0.02, 𝜅 = 0.2, 𝑚 = 0.2, 𝐹𝑟 = 0.8, 𝐻 = 1, 𝑅𝑒 = 0.2, 𝛼 =
𝑃𝑖

6
, 𝛾 = 0.7, 𝜙 =

𝑃𝑖

4
, 𝑡 =

0.1, 𝑚1 =    0.5, 𝐵𝑛 = 0.5 . 
 

 

 

 

 

 

        (a)                                                           (b) 
 



  

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Figure 10 Streamlines for variation of Hartman number  𝐻  with  𝐸1 = 0.4, 𝐸2 = 0.2, 𝐸3 = 0.3 𝜖 = 0.02, 𝜅 =

0.2, 𝑚 = 0.2, 𝐹𝑟 = 0.8, 𝑑1 = 0.1, 𝑅𝑒 = 0.2, 𝛼 =
𝑃𝑖

6
, 𝛾 = 0.7, 𝜙 =

𝑃𝑖

4
, 𝑡 = 0.1, 𝑚1 = 0.5, 𝐵𝑛 = 0.5 . 

 

 

               (a)                                                                 (b) 

  
Figure 11 Streamlines for variation of permeability parameter   𝜅  with  𝐸1 = 0.4, 𝐸2 = 0.2, 𝐸3 = 0.3 𝜖 =

0.02, 𝐻 = 0.2, 𝑚 = 0.2, 𝐹𝑟 = 0.8, 𝑑1 = 0.1, 𝑅𝑒 = 0.2, 𝛼 =
𝑃𝑖

6
, 𝛾 = 0.7, 𝜙 =

𝑃𝑖

4
, 𝑡 = 0.1, 𝑚1 = 0.5, 𝐵𝑛 = 0.5 . 

 

 

 

 

 

 

 

 

 

 

        (a)                                                                 (b) 



  

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Figure 12 Streamlines for variation of Hall number  𝑚 with 𝐸1 = 0.4, 𝐸2 = 0.2, 𝐸3 = 0.3 𝜖 = 0.02, 𝐻 = 0.2, 𝜅 =

0.2, 𝐹𝑟 = 0.8, 𝑑1 = 0.1, 𝑅𝑒 = 0.2, 𝛼 =
𝑃𝑖

6
, 𝛾 = 0.7, 𝜙 =

𝑃𝑖

4
, 𝑡 = 0.1, 𝑚1 = 0.5, 𝐵𝑛 = 0.5 . 

 

        (a)                                                                         (b) 
 

   
Figure 13 Streamlines for variation of non- dimensional viscosity parameter  𝜖 with 𝐸1 = 0.4, 𝐸2 = 0.2, 𝐸3 =

0.3 𝑚 = 0.2, 𝐻 = 0.2, 𝜅 = 0.2, 𝐹𝑟 = 0.8, 𝑑1 = 0.1, 𝑅𝑒 = 0.2, 𝛼 =
𝑃𝑖

6
, 𝛾 = 0.7, 𝜙 =

𝑃𝑖

4
, 𝑡 = 0.1, 𝑚1 = 0.5, 𝐵𝑛 =

0.5 . 

 

 

 

 

 

 

 

 

 

            (a)                                                                      (b) 



  

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Figure 14 Streamlines for variation of Froude number   𝐹𝑟  with  𝐸1 = 0.4, 𝐸2 = 0.2, 𝐸3 = 0.3 𝑚 = 0.2, 𝐻 =

0.2, 𝜅 = 0.2, 𝜖 = 0.8, 𝑑1 = 0.1, 𝑅𝑒 = 0.2, 𝛼 =
𝑃𝑖

6
, 𝛾 = 0.7, 𝜙 =

𝑃𝑖

4
, 𝑡 = 0.1, 𝑚1 = 0.5, 𝐵𝑛 = 0.5 . 

 

                (a)                                                                              (b) 

  
Figure 15 Streamlines for variation of Bingham number   𝐵𝑛 with 𝐸1 = 0.4, 𝐸2 = 0.2, 𝐸3 = 0.3 𝑚 = 0.2, 𝐻 =

0.2, 𝜅 = 0.2, 𝜖 = 0.8, 𝑑1 = 0.1, 𝑅𝑒 = 0.2, 𝛼 =
𝑃𝑖

6
, 𝛾 = 0.7, 𝜙 =

𝑃𝑖

4
, 𝑡 = 0.1, 𝑚1 = 0.5, 𝐹𝑟 = 0.5 . 

 

 

 

 

 

 

 

 

 

 

 



  

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5. Conclusions 

     

   The peristaltic transport of non- Newtonian Bingham plastic fluid with variable viscosity in 

an inclined tapered asymmetric channel is performed, taking into account Hall and Joule's 

heating influences. Adopting assumptions of long wavelength and low Reynolds number the 

problem is modeled and reduced into a pair of nonlinear differential equations which are solved 

approximately by using a perturbation method. A parametric analysis is permitted through 

various graphs that made us outcome with some following important observations 

1. The velocity profile is an increasing function of 𝑚, ∅, 𝜅,  and wall elasticity parameters 

whereas it decreases with arise up of 𝐻,and 𝐵𝑛  parameters. 

2. It is observed from the figures that the Froude number is oppositely affected by the 

temperature distribution profile. 

3. Hartman number𝐻 impact on temperature profile totally revers Hall number 𝑚 effect i.e. the 
first one shows a reduction behavior while Hall number enhances it.  

4. The absolute heat transfer shows an oscillatory behavior along the length of peristaltic wave 

as well as it depicts a mixed behavior for a larger value of phase difference parameter 𝜙. 

5. The trapping phenomenon is divided into two asymmetric regions, and from the figures one 

can notice that the trapped bolus increases in size and circulation as 𝐸1, 𝐸2 increase whereas it 

decreases for a higher magnitude of 𝐸3. 

6. The trapped bolus remains unchanged in size when Bingham number 𝐵𝑛 enhances. 

 

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peristaltic transport on MHD fluid having temperature-dependent properties, Alexandria 

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radiative peristaltic flow of pseudo plastic nano fluid in a tapered asymmetric channel, J. 

Magn. Magn. Mater. 2016,408.  

3. Adnan, F.A.; Abdualhadi, A.M. Effect of a magnetic field on a peristaltic transport of 

Bingham plastic fluid in asymmetrical channel. Sci. int. (Lhore).2019, 31, 1, 29-40. 

4. Ali Nasir; Asghar Zaheer. Mixed convective heat transfer analysis for the peristaltic 

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