Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

140 
 

 

  

 

 

 

Farah Alaa Adnan                                           Ahmad M. Abdul Hadi
 

 

 

 

 

 

 

 
 

Abstract 

     In this paper, we study the peristaltic transport of incompressible Bingham plastic fluid in a 

curved channel. The formulation of the problem is presented through, the regular perturbation 

technique for small values of 
1

𝑘
 is used to find the final expression of stream function. The 

numerical solution of pressure rise per wave length is obtained through numerical integration 

because its analytical solution is impossible. Also the trapping phenomenon is analyzed. The 

effect of the variation of the physical parameters of the problem are discussed and illustrated 

graphically. 
 

Key words: Bingham plastic fluid, a curved channel, Peristaltic transport, regular perturbation 

method. 
 

1. Introduction 

     The peristaltic flows have a fundamental function in physiology and engineering. Many 

authors have investigated such flows like Adnan et al. studied the effect of the magnetic field on 

a peristaltic flow of Bingham plastic fluid [1]. Ahmed et al. investigated the effect of a magnetic 

field on peristaltic flow of Jeffery fluid through a porous medium in a tapered asymmetric 

channel [2]. Adnan et al. investigated the effect of a magnetic field on a peristaltic transport of 

Bingham plastic fluid in a symmetrical channel [3]. And Ahmad et al. studied the peristaltic 

transport of MHD flow and heat transfer in a tapered asymmetric channel through porous 

medium and the effect of variable viscosity, velocity – non slip and temperature jump are 

discussed [4]. The flows over the curved walls are considering as engineering advantage and 

happened for example in turbo machinery and the wings of airplane. And that’s why it is 

fundamental to know the effect of curvature (which denoted by 𝑘) on the channels. The peristaltic 

motion in curved channel is an important area of research because the shape of the most of the 

physiological tubes and glandular channels is curved. It have been seen from the available studies 

Ibn Al Haitham Journal for Pure and Applied Science 

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

 

Doi:10.30526/32.3.2291  

Peristaltic Flow of the Bingham Plastic Fluid in a Curved Channel 

 

Department of Mathematics, College of Science, University of Baghdad, 

Baghdad, Iraq. 

 

f arrah.a_adnan@yahoo.com

                                            

                    

 
Article history: Received 3 April 2019, Accepted 8 May 2019, Publish September 

2019 

 

mailto:farrah.a_adnan@yahoo.com
mailto:farrah.a_adnan@yahoo.com
mailto:farrah.a_adnan@yahoo.com
mailto:farrah.a_adnan@yahoo.com


Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

141 
 

that not a lot of attention is considering in the peristaltic flow in a curved channel. Ali et al. 

studied the peristaltic flow of a non- Newtonian fluid in a curved channel [5]. Hayat et al. studied 

the peristaltic transport of viscous fluid with compliant walls in curved channel [6]. Ahmad 

studied the effect of radial magnetic field on peristaltic transport of Jeffrey fluid with variable 

viscosity in curved channel with heat and mass transfer properties [7]. Kalantari et al. 

investigated the effect of radially imposed magnetic field on the flow characteristics in a curved 

channel [8]. Norenn et al. studied the curvature effects on the MHD peristaltic flow of an 

incompressible carreau fluid in a curved channel [9]. Hina el al. studied the slip conditions effect 

on the peristaltic transport of Johnson- Segalman fluid through a curved channel [10-12]. In this 

paper, the peristaltic transport of Bingham plastic fluid in a curved channel gave been discussed. 

The concept of large wave length 𝜆 and small Ronald’s 𝑅𝑒 is used to simplify the equation. The 

nonlinear differential equation is solved analytically by the regular perturbation method for small 

values of 
1

𝑘
 where 𝑘 is the curvature of the channel. The impact of the different physical 

parameters on the velocity axial, the local shear stress, the pressure gradient, pressure rise and 

trapping phenomenon are discussed in detail with the use of the graphs.  
 

2. Problem Statement   

     Assume the peristaltic flow of an incompressible Bingham plastic fluid in a curved channel of 

thickness 2𝑎 convolute in a circle with center 0 and radius 𝑅∗ where the space inside the channel 

filled with Bingham plastic fluid. The velocity components in radial �̅�  and axial �̅� direction are 

�̅� and �̅� respectively. The flow of Bingham plastic fluid in the curved channel consequent to the 

contraction and dilation of the flexible walls of the channel where the inertial force effects are 

taken small. The walls of the channel are described mathematically as follows: 

�̅�(�̅�, 𝑡̅) = 𝑎 + 𝑏 sin [
2𝜋

𝜆
(�̅� − 𝑐𝑡)̅],                   Upper wall  

−�̅�(�̅�, 𝑡̅) = −𝑎 − 𝑏 sin [
2𝜋

𝜆
(�̅� − 𝑐𝑡̅)],              Lower wall                                                     (1) 

Where c is the speed, 𝑏 is the amplitude of wave along the upper and lower walls and 𝜆 is the 

wave length. The wave length is taken large however in more realistic situations it depends on 

time as the contraction and dilation in natural systems are account on varying 𝜆. The width of 

curved channel perpendicular to the plane is supposed to be infinite so the flow can safely be 

taken as two dimensional. The flow is assumed to be laminar. Then the velocity field which is 

consider as the following form: 

�̅� = [�̅�(�̅�, �̅�, 𝑡̅), �̅�(�̅�, �̅�, 𝑡̅), 0] 

 

 
Figure 1.  The geometry of the curved channel. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

142 
 

For an incompressible Bingham plastic fluid the conservation of mass and momentum for the 

velocity field are given as follows:  

∇. �̅� = 0  

𝜌 (
𝜕�̅�

𝜕𝑡
+ (�̅�. ∇̅)�̅�) = −∇�̅� + ∇. 𝜏̅                                                                                                    (2)           

Where 𝜌 is the density of the fluid, �̅� is the pressure and 𝜏̅ is the stress tensor. 
𝜕�̅�

𝜕�̅�
+

𝑅∗

𝑅∗+�̅�

𝜕�̅�

𝜕�̅�
+

�̅�

𝑅∗+�̅�
= 0                                                                                                                (3) 

𝜌 [
𝜕�̅�

𝜕�̅�
+ �̅�

𝜕�̅�

𝜕�̅�
+

𝑅∗�̅�

𝑅∗+�̅�
  

𝜕�̅�

𝜕�̅�
−

�̅�2

𝑅∗+�̅�
] = −

𝜕�̅�

𝜕�̅�
+

1

𝑅∗+�̅�

𝜕

𝜕�̅�
[(𝑅∗ + �̅�)𝜏�̅̅��̅�  ] + (

𝑅∗

𝑅∗+�̅�
)

𝜕�̅��̅��̅�
𝜕�̅�

−
�̅��̅��̅�

𝑅∗+�̅�
               

                                                                                                       (4) 

𝜌 [
𝜕�̅�

𝜕�̅�
+ �̅�

𝜕�̅�

𝜕�̅�
+

𝑅∗�̅�

𝑅∗+�̅�
  

𝜕�̅�

𝜕�̅�
+

�̅��̅�

𝑅∗+�̅�
] = −(

𝑅∗

𝑅∗+�̅�
)

𝜕�̅�

𝜕�̅�
+

1

(𝑅∗+�̅�)2
𝜕

𝜕�̅�
[(𝑅∗ + �̅�)2𝜏�̅̅��̅� ] + (

𝑅∗

𝑅∗+�̅�
)

𝜕�̅��̅��̅�
𝜕�̅�

   

                                                                           (5) 

With the boundary conditions are taken in the forms as follows: 

�̅� = 0  at ∓�̅�(�̅�, 𝑡̅) = ∓[𝑎 + 𝑏 sin [
2𝜋

𝜆
(�̅� − 𝑐𝑡̅)]]                                                                    (6) 

In the Fixed frame of reference(�̅�, �̅�), the flow in the channel is unsteady. However it can be 

considered as steady in the wave frame with coordinate (�̅�, �̅�) moving with speed c. the 

transformation equations that relating the two frames are given as follows: 

�̅� = �̅� − 𝑐𝑡̅, �̅� = �̅�, �̅� = �̅� − 𝑐 , �̅� = �̅�, �̅� = �̅�                                                                               (7) 

 and presenting the dimensionless quantities that used to find out the non-dimensional analysis as 

follows: 

𝑥 =
2𝜋�̅�

𝜆
, 𝜂 =

�̅�

𝑎
, 𝑢 =

𝑢

𝑐
 , 𝑣 =

�̅�

𝑐
  , 𝑝 =

2𝜋𝑎2

𝜆𝜇𝑐
 �̅�, ℎ =

�̅�

𝑎
 , 𝜏 =

𝑎�̅�

𝜇𝑐
  , 𝑅𝑒 =

𝜌𝑐𝑎

𝜇
 ,   𝛿 =

2𝜋𝑎

𝜆
 , 𝑘 =

𝑅∗

𝑎
 , 𝜑 =

𝑏

𝑎
 , �̇� =

𝜌𝑎2

𝜇0
�̅̇�                                                                                                     (8) 

In which  (𝑢, 𝑣) are the non- dimensional velocity components, 𝑝 is the non- dimensional 

pressure, 𝜏 is the non- dimensional stress tensor, 𝑅𝑒 is the Ronald’s number, 𝛿 is the wave length 

ratio (wave number), 𝑘 is the curvature parameter and 𝜑 is the amplitude ratio. 

Making use of the equation (7) and the above dimensionless quantities, the continuity and the 

motion equations will be reduced as follows: 

 (𝑘 + 𝜂)
𝜕𝑣

𝜕𝜂
+ 𝑘𝛿

𝜕𝑢

𝜕𝑥
+ 𝑣 = 0                                                                                                          (9) 

−𝛿 2𝑅𝑒
𝜕𝑣

𝜕𝑥
+ 𝑅𝑒 𝛿𝑣

𝜕𝑣

𝜕𝜂
+ 𝑅𝑒 𝛿

𝑘(𝑢+1)

𝑘+𝜂
  

𝜕𝑣

𝜕𝑥
− 𝑅𝑒 𝛿

(𝑢+1)2

𝑘+𝜂
= −

𝜕𝑝

𝜕𝜂
+  

𝛿

𝑘+𝜂

𝜕

𝜕𝜂
[(𝑘 + 𝜂)𝜏𝜂𝜂  ] +

𝛿 2  (
𝑘

𝑘+𝜂
)

𝜕𝜏𝜂𝑥

𝜕𝑥
− 𝛿

𝜏𝑥𝑥

𝑎𝑘+𝑎𝜂
                                                                                          (10) 

−𝑅𝑒 𝛿
𝑘+𝜂

𝑘
 
𝜕𝑢

𝜕𝑥
+ 𝑅𝑒 

𝑘+𝜂

𝑘
𝑣

𝜕𝑢

𝜕𝜂
+ 𝑅𝑒 𝛿 (𝑢 + 1)

𝜕𝑢

𝜕𝑥
+  𝑅𝑒

(𝑢+1)𝑣

𝑘
= −

𝜕𝑝

𝜕𝑥
+

1

𝑘
 

1

𝑘+𝜂

𝜕

𝜕𝜂
[(𝑘 + 𝜂)2𝜏𝜂𝑥 ] +

𝛿
𝜕𝜏𝑥𝑥

𝜕𝑥
                                                                                                                                          (11)                        

The boundary conditions in non-dimensional parameters, 

𝑢 + 1 = 0 , ∓[ℎ] = ∓[1 + 𝜑 sin 𝑥]                                                                                      (12) 

And the stream function [11].  

𝑢 = −
𝜕𝜓

𝜕𝜂
 , 𝑣 = 𝛿

𝑘

𝜂+𝑘

𝜕𝜓

𝜕𝑥
                                                                                                              (13) 

Applying equation (13) on the equation (12).The boundary conditions becomes   



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

143 
 

𝜕𝜓

𝜕𝜂
= 1, ∓[ℎ] = ∓[1 + 𝜑 sin 𝑥]                                                                                                   (14) 

    The general solution of the governing equations seems to be hard in most cases, therefore the 

solution can be found under the assumption of small wave number 𝛿 ≪ 1 and low Reynolds 

number approximation, thus the equations (10) and (11) can be written as follows: 
𝜕𝑝

𝜕𝜂
= 0                                                                                                                                           (15) 

𝜕𝑝

𝜕𝑥
=

1

𝑘
 

1

𝑘+𝜂

𝜕

𝜕𝜂
[(𝑘 + 𝜂)2𝜏𝜂𝑥  ]                                                                                                       (16)  

Deriving equation with respect to  𝜂, one obtain: 

𝜕

𝜕𝜂
[

1

𝑘
 

1

𝑘+𝜂

𝜕

𝜕𝜂
[(𝑘 + 𝜂)2𝜏𝜂𝑥  ]] = 0                                                                                                  (17) 

Noticing that the Cauchy stress tensor 𝜎∗ can be decomposed as follows: 

 𝜎∗ = −�̅�𝐼 + 𝜏̅ 

Where 𝜏̅ is the extra stress tensor and �̅� is the mechanical pressure. The Bingham stresses 𝜏�̅�𝑗 are 

related to the strain rates �̅�1through the constitutive equation:  

𝜏�̅�𝑗 = (2𝜇0 +
𝜏𝑦

�̅̇�
) �̅�1 for 𝜏 ≥ 𝜏𝑦 

𝜏�̅�𝑗 = 0        for 𝜏 < 𝜏𝑦. 

Where 𝜏𝑦 is the yield stress, 𝜇0is the Bingham fluid viscosity, �̅̇� = (
1

2
 𝑡𝑟𝑎𝑐�̅�1

2
)

1

2
,  and the final  

𝜏𝜂𝑥 = 2 (−  
𝜕2𝜓

𝜕𝜂2
−  

1

𝑘+𝜂
(1 −

𝜕𝜓

𝜕𝜂
)) + 𝐵𝑛                                                                                      (18)                                          

Where 
𝜌𝑎2𝜏𝑦

 𝜇0
2𝑅𝑒

=
𝑎 𝜏𝑦

𝜇0𝑐
= 𝐵𝑛 the Bingham number. Substituting the above equation (18) into 

equation (17) one obtain:  

𝜕4𝜓

𝜕𝜂4
+

2

(𝑘+𝜂)

𝜕3𝜓

𝜕𝜂3
−

1

(𝑘+𝜂)2
𝜕2𝜓

𝜕𝜂2
+ (

𝜕𝜓

𝜕𝜂
− 1)

1

(𝑘+𝜂)3
= 0                                                                   (19) 

Noticing that the dimensionless mean flows Θ in the fixed frame and F in the wave frame as: 

 Θ =
𝑄∗

𝑎𝑐
, 𝐹 =

𝑞

𝑎𝑐
    one get:   

Θ = F + 2 

In which:  

𝐹 = − ∫
𝜕𝜓

𝜕𝜂
𝑑𝜂

ℎ

−ℎ

= −(𝜓(ℎ) − 𝜓(−ℎ)) 

Then:                    

𝜓(ℎ) − 𝜓(−ℎ) = −𝐹 

𝜓(ℎ) = −
𝐹

2
  At the upper wall and 𝜓(−ℎ) =

𝐹

2
 at the lower wall. The dimensionless pressure rise 

over wave length is defined by:  

∆𝑝𝜆 = ∫
𝑑𝑝

𝑑𝑥
 𝑑𝑥

2𝜋

0
  

   

 

 

                                                                                                      



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

144 
 

3. Methods of Solution for the Problem 

     Using the perturbation for small values of  
1

𝑘
 by expanding the flow quantities in a power 

series of 
1

𝑘
 as follows:   

𝜓 = 𝜓0 +
1

𝑘
𝜓1 +  

1

𝑘2
𝜓3  

1

𝑘+𝜂
=

1

𝑘
−

𝜂

𝑘2
+

𝜂2

𝑘3
  

1

(𝑘+𝜂)2
=

1

𝑘2
−

2𝜂

𝑘3
+

3𝜂2

𝑘4
  

1

(𝑘+𝜂)3
=

1

𝑘3
−

3𝜂

𝑘4
+

6𝜂2

𝑘5
                                                                                                              (20) 

Substituting the above equation (20) into the equation (19) and then comparing the coefficients of 

the same power of  
1

𝑘
 up to the second order the following systems will be obtained:   

               

3.1 The Zeroth Order 

𝜕4𝜓0

𝜕𝜂4
= 0                                                      (21)  

 Along with its boundary conditions: 

 𝜓0 = −
𝐹

2
,

𝜕𝜓0

𝜕𝜂
= 1  at the upper wall  

 𝜓0 =
𝐹

2
,   

𝜕𝜓0

𝜕𝜂
= 1  at the lower wall                                                                                           (22) 

 

3.2 The First Order 
𝜕4𝜓1

𝜕𝜂4
+ 2

𝜕3𝜓0

𝜕𝜂3
= 0                                                                     (23) 

Along with its boundary conditions: 

 𝜓1 = 0,
𝜕𝜓1

𝜕𝜂
= 0  at the upper wall 

𝜓1 = 0,
𝜕𝜓1

𝜕𝜂
= 0  at the lower wall                                                      (24) 

 

3.3 The Second Order 
𝜕4𝜓2

𝜕𝜂4
+ 2

𝜕3𝜓1

𝜕𝜂3
− 2𝜂

𝜕3𝜓0

𝜕𝜂3
−

𝜕3𝜓0

𝜕𝜂3
= 0                                                      (25) 

Along with its boundary condition:  

𝜓1 = 0,
𝜕𝜓1

𝜕𝜂
= 0  at the upper wall  

  𝜓1 = 0,
𝜕𝜓1

𝜕𝜂
= 0  at the lower wall                                           (26) 

Solving the above systems of the zeroth, first and second order by mathematic program, the final 
calculation for the stream function would be as follows: 

𝜓 =

𝑎1 + 𝑎2𝜂 + 𝑎3𝜂
2 + 𝑎4𝜂

3 +
𝑎5+𝑎6𝜂+𝑎7𝜂

2+𝑎8𝜂
3−

a4𝜂4

2

𝑘
+

𝑎9+𝑎10𝜂+𝑎11𝜂
2+𝑎11𝜂

3+
1

12
((𝑎3−6𝑎8)𝜂

4+
21𝑎4𝜂

5

5
)

𝑘2
           

                                                                                                                     (27) 

 And, 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

145 
 

𝑎1 = −
𝐹ℎ1

3−3𝐹ℎ1
2ℎ2  −2ℎ1

3ℎ2  −3𝐹ℎ1ℎ2  
2+𝐹ℎ2  

3+2ℎ1ℎ2  
3

2(−ℎ1+ℎ2  )
3

,   

𝑎2 = −
−ℎ1

3−6𝐹ℎ1ℎ2  −3ℎ1
2ℎ2  +3ℎ1ℎ2  

2+ℎ2  
3

(ℎ1−ℎ2  )
3

, 𝑎3 = −
3(𝐹+ℎ1−ℎ2  )(ℎ1+ℎ2  )

(ℎ1−ℎ2  )
3

,  

𝑎4 =
2(𝐹+h1−ℎ2  )

(ℎ1−ℎ2  )
3

, 𝑎5 = −
1

2
𝑎4ℎ1

2
ℎ2  

2, 𝑎6 = 𝑎4ℎ1
2

ℎ2   + 𝑎4ℎ1ℎ2  
2,  

𝑎7 =
1

2
(−𝑎4ℎ1

2
− 4𝑎4ℎ1ℎ2 − 𝑎4ℎ2

2
), 𝑎8 = 𝑎4ℎ1 + 𝑎4ℎ2  , 

𝑎9 =
1

60
(5𝑎3ℎ1

2
ℎ2  

2 − 30𝑎8ℎ1
2

ℎ2
2

+ 42𝑎4ℎ1
3

ℎ2
2

+ 42𝑎4ℎ1
2

ℎ2
3

),  

𝑎10 = −
1

60
ℎ1  (10𝑎3ℎ1ℎ2 − 60𝑎8ℎ1ℎ2 + 84𝑎4ℎ1

2
ℎ2 + 10𝑎3ℎ2

2
− 60𝑎8ℎ2

2
+ 147𝑎4ℎ1ℎ2

2
+

84𝑎4ℎ2  
3), 𝑎11 =

1

60
(5𝑎3ℎ1

2
− 30𝑎8ℎ1

2
+ 42𝑎4ℎ1

3
+ 20𝑎3ℎ1ℎ2   − 120𝑎8ℎ1ℎ2   +

168𝑎4ℎ1
2

ℎ2 + 5𝑎3ℎ2
2

− 30𝑎8ℎ2
2

+ 168𝑎4ℎ1ℎ2
2

+ 42𝑎4ℎ2
3

), 𝑎12 =
1

60
(−10𝑎3ℎ1 +

60𝑎8ℎ1 − 63𝑎4ℎ1
2

− 10𝑎3ℎ2 + 60𝑎8ℎ2 − 84𝑎4ℎ1ℎ2 − 63𝑎4ℎ2  
2)     

                                                                                                                                                     (28) 

4. Results and Discussion 

       This section presented the graphical description of different parameters on the out coming of 

flow quantities. The variation of the axial velocity, local shear stress, the pressure gradient, the 

pressure rise and the stream lines are illustrated and analyzed graphically. 
 

4.1 The Axial Velocity 

      The outcome in terms of different parameters of the axial velocity  𝑢 (𝜂) have been plotted as 

a function of 𝜂 at 𝑥 =
𝜋

2
 the wider part of the channel. It is clear from the figurers that the axial 

velocity attains parabolic in nature other than some points of reflection on velocity curves which 

opposite the situation from increase to decrease and vice versa including the following figures.      

Figure 2. Describes the mixed behavior of small values of the curvature of the channel on the 

axial velocity profile. It is clear that the axial velocity profile is not symmetric at the central line 

of the channel 𝜂 = 0, and the maximal of the profile shift to the end of the channel. However in 

Figure 3. For large values of  𝑘 the axial velocity profile becomes symmetric at the central line 

of the channel 𝜂 = 0. It has been noted that the axial velocity stops and fixed for large values of  

𝑘. Figure 4. Studied the effect of 𝜙 on the axial velocity, the graph that an increase in 𝜙  increase 

the axial velocity profile near the walls and decreases the velocity profile at the central part of the 

channel. It also shows that the axial velocity profile is not symmetric at the central line of the 

channel 𝜂 = 0 and the axial velocity profile attains its maximal at the beginning of the channel. 

Figure 5. Portrays that for ascending values of the flow rate the axial velocity profile increases in 

the central part of the channel and decreases at the wall of the channel.  

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

146 
 

 
Figure 2.The velocity profile for fixed values of 

𝜙 = 0.75, F = 1.5. 

 
Figure 3.The velocity profile for fixed values of 

𝜙 = 0.75, F = 1.5. 

 
Figure 4.The velocity profile for fixed values of 

𝑘 = 5, F = 1.5. 

 
Figure 5.The velocity profile for fixed values of 

𝑘 = 5, 𝜙 = 0.75. 

4.2 The Local Shear Stress 

     In this subsection, the discussions of the influence of the different parameters on the local 

shear stress were investigated. Figure 6. Discusses the variation of 𝑘 on the magnitude of shear 

stress. It shows that the shear stress increases in the whole channel specially at the central part of 

the channel with an increase of 𝑘. However the shear stress stops increasing and fixed for large 

values of 𝑘 as it is shown in Figure 7. Figure 8. Exhibits the effect of an increase of 𝜙 on the 

shear stress. It is shown that when 𝜙 increases the magnitude of shear stress decreases in (𝑥 < 0 

and 𝑥 > 3) and for 0 ≤ 𝑥 ≤ 3 the magnitude of shear stress increases. Figure 9. Deduced that 

for large values of the flow rate 𝐹 the shear stress is decreasing function in the whole channel. 

Figure 10. displays the impact of Bingham fluid number on the shear stress. It is noticed that the 

shear stress is an increasing function to the increase of Bingham fluid number 𝐵𝑛 .  

 

  
Figure 7. The shear stress for fixed values[ 𝜂 =

0.5, 𝜙 = 0.5, F = 1.5, Bn = 5]. 
Figure 6. The shear stress for fixed values[ 𝜂 =

0.5, 𝜙 = 0.5, F = 1.5, Bn = 5]. 

k 2

k 5

k 10

2 1 0 1 2

2

1

0

1

u

k 100

k 200

k 300

2 1 0 1 2

1.5

1.0

0.5

0.0

0.5

1.0

u

0.5

0.75

1

2 1 0 1 2

1.5

1.0

0.5

0.0

0.5

1.0

u

F 1

F 1.5

F 2

2 1 0 1 2

4

3

2

1

0

1

2

u

k 100

k 150

k 200

1 0 1 2 3 4

6

4

2

0

2

x

x

k 5

k 10

k 15

1 0 1 2 3 4

6

4

2

0

2

x

x



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

147 
 

  
Figure 9.The shear stress for fixed values[ 𝜂 =

0.5, 𝑘 = 5, 𝜙 = 0.5, Bn = 5]. 
Figure 8. The shear stress for fixed values[ 𝜂 =

0.5, 𝑘 = 5, F = 1.5, Bn = 5]. 

 
Figure 10. The shear stress for fixed values[ 𝜂 = 0.5, 𝑘 = 5, 𝜙 = 0.5, F = 1.5]. 

 

4.3 The Pressure Gradient (
𝒅𝒑

𝒅𝒙
) 

       In this section the effect of the variation of different parameters on the pressure gradient is 

recorded in the following plots. Figure 11. Demonstrates that an increase in the value of 𝑘 

increases the pressure gradient curves .the interesting feature the pressure gradient stops 

increasing and fixed for large values of   𝑘. Figure 12. Portrays that for ascending values of flow 

rate the pressure gradient is decreasing function. Figure 13. Illustrates the effect of amplitude 

ratio parameter (𝜙) on the pressure gradient curves, when 𝜙 increase the amounts of pressure 

gradient decreases at 3 < 𝑥 ≤ 5 and increase at the central part of the channel. Figure 14. 

Describes the effect of increasing the value of Bingham number 𝐵𝑛 on the pressure gradient. It 

shows that The pressure gradient (
𝑑𝑝

𝑑𝑥
) is decreasing function for the increase of 𝐵𝑛. 

  
Figure 11. The pressure gradient for fixed values [𝜂 =

0.5, F = 1, 𝜙 = 0.5, Bn = 5]. 

 
Figure 12. The pressure gradient for fixed values [𝜂 =

0.5, 𝑘 = 5, 𝜙 = 0.5, Bn = 5]. 

1 0 1 2 3 4

30

20

10

0

x

0.5

0.75

1

1 0 1 2 3 4
140

120

100

80

60

40

20

0

x

x

Bn 5

Bn 15

Bn 25

1 0 1 2 3 4

10

0

10

20

x

k 3

k 4

k 5

2 1 0 1 2 3 4 5

0.20

0.15

0.10

0.05

0.00

x

d
P

d
x

0.5

1

1.5

2 1 0 1 2 3 4 5
0.05

0.04

0.03

0.02

0.01

0.00

x

d
P

d
x



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

148 
 

 
Figure 13.The pressure gradient for fixed values [𝜂 =

0.5, 𝑘 = 5, F = 1, Bn = 5]. 

 
Figure 14.The pressure gradient for fixed values [𝜂 =

0.5, 𝑘 = 5, F = 1, 𝜙 = 0.5].

4.4 The Pressure Rise 

     The relationship between the non-dimensional average pressure rise per wave length and 

dimensionless mean flow rate Θ. The following Figurers 15-17. Shows the variation of  Δp 

against mean flow rate Θ where a linear relationship between Δp and Θ as it is shown in the 

figurers. The whole region is considered to five parts  

1. Peristaltic pumping region where Δp > 0, Θ > 0. 

2. Augmented pumping (co- pumping region) where  Δp < 0, Θ > 0 and no flow is 

recognize in this region. 

3. Retrograde pumping region where Δp > 0, Θ < 0 and no flow is recognize in this region. 

4. The co- pumping region where Δp < 0, Θ < 0. 

5. Free pumping region where  Δp = 0. 

     Figure 15. discusses the variation of 𝑘 on the pressure rise per wave length. It is observed that 

the pumping rate increases in the co- pumping region where Δp < 0, Θ < 0 while it is decreasing 

in the peristaltic pumping region where Δp > 0, Θ > 0. Figure 16. Plots the effect of amplitude 

ratio parameter 𝜙 on pressure rise per wave length, it has been noticed that an increase in 𝜙 

increase the pumping rate in the peristaltic pumping region(Δp > 0, Θ > 0), and decrease the 

pumping rate in the co- pumping region (Δp < 0, Θ < 0). Figure 17. Shows the effect of an 

increase of Bingham number parameter Bn on the pressure rise per wave length, it shows that the 

pumping rate decreases in both peristaltic pumping region (Δp > 0, Θ > 0) and the co- pumping 

region( Δp < 0, Θ < 0). 
 

 
Figure 15. The pressure rise for fixed   values [𝜙 =

0.5, Bn = 5]. 
 

 
Figure 16.The pressure rise for fixed values [𝑘 =

5, Bn = 5]. 

0.5

0.75

1

2 1 0 1 2 3 4 5

0.5

0.4

0.3

0.2

0.1

0.0

x

d
P

d
x

Bn 5

Bn 25

Bn 50

2 1 0 1 2 3 4 5

0.05

0.04

0.03

0.02

0.01

0.00

x

d
P

d
x

k 50

k 75

k 100

10 5 0 5 10

0.00010

0.00005

0.00000

0.00005

0.00010

p

0.5

1

1.5

10 5 0 5 10

2

0

2

4

p



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

149 
 

                                       Figure 17.The pressure rise for fixed values [𝑘 = 5, 𝜙 = 0.5]. 

4.5 The Stream Lines 

      Trapping if formation of circular bolus by internally splitting of stream function. The 

movement of bolus forward through peristaltic wave with the same speed as the wave itself. The 

bolus or circulation region is symmetric about the center line; it is split in to two boluses 

appearing in the upper and lower halves of the channel for large values of 𝑘 where the shape and 

size of the boluses in the two halves is a symmetric. However for small values of 𝑘 the bolus 

region is not symmetric about the center line 𝜂 = 0, in fact it is split into two boluses appearing 

in the upper and lower halves of the channel and the shape and size of the boluses in the two 

halves is not symmetric. The effect of 𝑘 the curvature of the channel on the trapped bolus is 

shown in Figure 18. one can see for small values of 𝑘 the bolus decreases in size and number. It 

is interesting that the size and circulation of the two boluses stops for large values of 𝑘. Figure 

19. deduced that for large values of 𝜙 and for large values of 𝑘 the size and the number of 

trapped boluses increased where the size and the shape of the bolus is symmetric. Figure 20. 

shows that the size and number of the stream lines increases with an increase of  𝜙 for small 

values of 𝑘 in which the two boluses are not symmetric around the center line 𝜂 = 0. The impact 

of flow rate on the stream line in case of large values of 𝑘 is shown in Figure 21. it shows that 

for large magnitude of the flow rate the size of the trapping bolus increases. And the impact of 

flow rate on the stream line in case of small values of 𝑘 is shown in Figure 22. It shows the size 

of the trapping bolus increases it is not symmetric around the center line 𝜂 = 0. 

 

 

 
 

 

Figure 18. Stream lines for variation of 𝑘 and fixed {𝜙 = 1, 𝑘 = 5, F = 1} and 𝑘 = 2, 𝑘 = 5, 𝑘 = 50, 𝑘 = 100. 
 

 

Bn 5

Bn 25

Bn 50

10 5 0 5 10

1.0

0.5

0.0

0.5

1.0

p

A

  
B

  

C

  
D  



  

150 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

 

 
Figure 19. Stream lines for variation of 𝜙 and fixed {𝑘 = 100, 𝐹 = 1} , 𝜙 = 1, 𝜙 = 1.5 , 𝜙 = 2 . 

 

 

 
Figure 20. Stream lines for variation of 𝜙 and fixed {𝑘 = 5, 𝐹 = 1} and 𝜙 = 1, 𝜙 = 1.5 , 𝜙 = 2 . 

 
Figure 21. Stream lines for variation of 𝐹 and fixed {𝜙 = 1, 𝑘 = 100} and 𝐹 = 1, 𝐹 = 2, 𝐹 = 3. 

 

 
Figure 22.Stream lines for variation of 𝐹 and fixed {𝜙 = 1, 𝑘 = 5} and 𝐹 = 1, 𝐹 = 2, 𝐹 = 3. 

5. Conclusion  

    In this paper the peristaltic transport of Bingham plastic fluid in a curved channel is examined.  

Important findings of the analysis have been written as follows: 

A

  

C

  

A

  

B

  

C

  

A

  

B

  

C  

A

  

B

  
C

  



  

151 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

 

1. The axial velocity is an increasing function near the walls under the effect of the parameter 

𝜙  and a decreasing function at the central part of the channel. However the situation is 

reversed for the increase of flow rate.  

2. A mixed behavior of local shear stress is noticed due to an increase of 𝜙. Shear stress is an 

increasing function to the increase of 𝐵𝑛 and small values of 𝑘, however the situation is 

opposite for the increase of 𝐹. 

3. The magnitude of pressure gradient decreases with an increase of 𝜙 and 𝐵𝑛. However the 

response is completely reverse with an increase of 𝑘. 

4. In the pumping regions, the pumping rate increased in the peristaltic pumping region via 

increasing of  𝜙 and 𝐵𝑛, while the pumping rate increased in the  co-pumping region with 

an increase of  𝑘 and 𝐵𝑛. 

5. The size of the trapped bolus increases with an increase of 𝜙 and 𝐹 but the size and shape 

of the trapped bolus is not symmetric in case of small values of 𝑘 and symmetric in case of 

large values of 𝑘. 

References  

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

Bingham plastic fluid. Jour. of adv. Research and control.2018, 10, 10, 2007-2024. 

2. Ahmad, T.S.; Abdulhadi, A.M. Effect of a magnetic field on peristaltic flow of Jeffery 

fluid through a porous medium in a tapered asymmetric channel. Global journal of 

mathematics.2017, 9, 2, 6889-6893. 

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

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

4. Ahmad, T.S.; Abdulhadi, A.M. Peristaltic transport of MHD flow and heat transfer in a 

tapered asymmetric channel through porous medium: effect of variable viscosity, velocity 

– non slip and temperature jump. International journal of Advance scientific and technical 

research.2017, 7, 2, 53-69. 

5. Ali, N.; Sajid, M.; Abbas, Z. non- Newtonian fluid induced by peristaltic waves in a 

curved channel. European journal of mechanic B/fluids.2010, 29, 387-394. 

6. Hayat, T.; Javed, M.; Hendi, A. Peristaltic transport of viscous fluid in curved channel 

with compliant walls. International journal of heat and mass transfer.2011, 54, 1615-

1621.  

7. Ahmad, T.S.; Abdualhadi, A.M. Effect of radial magnetic field on peristaltic transport of 
Jeffrey fluid in curved channel with heat and mass transfer. Journal of physics. conference 

iop.2018, 1003, 12053, 1-28. 

8. Kalantari, A.; Riasiand, A.; Sadeghy, K. Peristaltic flow of Giesekus fluid through curved 
channels: an approximate solution. Journal of the society of rholugy japan.2014, 42, 1, 9-

17.  

9. Norenn, S.; Hayat, T.; Alsaedi, A. Flow of MHD carreau in a curved channel. Applied 
Bionies and Bionechanics.2013, 10, 29-39.  

10. Hina, S.; Mustafa, M.; Hayat, T. Peristaltic transport of Johnson - Segalman fluid in a 
curved channel with slip condition. Plos one.2014, 9, 12, 1-25. 



  

152 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

 

11. Sato, H.; Kawai, T.; Fujita, T.; Okabe, M. Two dimensional peristaltic flows in curved 
channels. Trans. The Japan soc. Mech. Eng.2000, 66, 679-688. 

12. Tawfiq, L. N. M.; Jabber. A.K. Mathematical Modeling of Groundwater Flow. Global 

Journal of Engineering science and Researches. 2016, 3, 10, 15-22. 

 

http://scholar.google.com/scholar?cluster=11364619293857166181&hl=en&oi=scholarr