IHJPAS. 36(2)2023 

436 
This work is licensed under a Creative Commons Attribution 4.0 International License 

 

   

 

 

 

 

 

 

 

 

 

 
 

 

Abstract 

 In this paper, the impact of magnetic force, rotation, and nonlinear heat radiation on the peristaltic 

flow of a hybrid bio -nanofluids through a symmetric channel are investigated. Under the assumption 

of a low Reynolds number and a long wavelength, the exact solution of the expression for stream 

function, velocity, heat transfer coefficient, induced magnetic field, magnetic force, and temperature 

are obtained by using the Adomian decomposition method. The findings show that the magnetic 

force contours improve when the magnitude of the Hartmann number M is high and decreases when 

rotation increases. Lastly, the effects of essential parameters that appear in the problem are analyzed 

through a graph. Plotting all figures is done using the MATHEMATICA  software. 

Keywords  Adomain decomposition technique, Peristaltic transport, Magnetic force, Symmetric 

channel, Rotation frame. 

1. Introduction  

Recently, nanotechnology has gained a lot of attention. The performance of this field in several 

applications, such as photocatalysts, heat exchangers, engineering, and biomedicine such as 

destroying tumor cells and cancer diagnosis was studied in the modern era. Many nanoparticles 

such as copper, gold, and silver particles are used in proteins and nucleic acids. Because these 

nanoparticles have highly biocompatible, magnetic, chemical, mechanical, and thermal properties. 

Due to their superior quenching efficiency when compared to other nanoparticles, nanoparticles 

are widely used in medicinal applications for the treatment of malignant tumors[1-3]. 

Peristaltic is a significant mechanism produced by the propagation of waves all along the walls of 

a tube or a channel. This mechanism is well known to physiologists as one of the principle 

mechanisms for fluid transportation. Peristaltic mechanisms are utilized in a variety of industrial 

applications and biomedical, this mechanism is helpful in many different systems, including the 

motion of ovulation in the fallopian channel, the swallowing of food down to the stomach, the flow 

doi.org/10.30526/36.2.3066 

Article history: Received 1 October 2022, Accepted 12 December 2022, Published in April 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Rotation and Magnetic Force Effects on Peristaltic Transport of Non -Newtonian 

Fluid in a Symmetric Channel  

 
Amaal Mohi Nassief  

Department of Mathematics, College of 

Science, University of Baghdad, 

Baghdad, Iraq 

laama82@yahoo.com 

Ahmed M. Abdulhadi 

Department of Mathematics, College of 

Science, University of Baghdad, 

Baghdad, Iraq 

ahm6161@yahoo.com 

https://creativecommons.org/licenses/by/4.0/
mailto:laama82@yahoo.com
mailto:ahm6161@yahoo.com


IHJPAS. 36(2)2023 

437 
 

of tiny blood vessels, blood pump in the heart-lung machine, and the transport of urine from the 

renal to the bladder[4]. There are some academic research has investigated the peristaltic flow 

process of nanofluids, and some of these researches, including . [5] discussed the peristaltic 

transport affects the magnetic field and thermal properties of copper-water nanofluids. [6] 

discussed Zinc Oxide nanoparticles moving via tapering arteries while suspended in blood and 

subject to magnetic effects. [7] examined the impacts of various nanoparticle types on the 

peristaltic transport of nanofluid. [8] studied the impact of the apply magnetic field on heat 

radiation and the magnetic force on gold and copper nanoparticles in peristaltic flow. [9] discussed 

the effect of nanoparticles in the motion of blood in a vertical channel. [10] discussed the peristaltic 

of couple stress nanofluid organized by the presence the electrical field and magnetic field into 

micro channel. 

The rotation phenomenon has several uses in cosmic and geophysical processes .It can be used to 

understand when galaxies arise and the oceans circulate. The orientation of nanoparticles in fluids 

is explained by rotational diffusion.The peristalsis of magnetic field fluid in the presence of 

rotation is significant in some flow situations involving the motion of physiological fluids, such as 

saline water and blood. The magnetic field and rotation are beneficial for the movement of bio 

fluids through the intestines, ureters, and arterioles.  Numerous scholars have been interested in 

the impact of rotation and the peristaltic transport mechanism since they were examined by [11-

17]. 

This study intends to investigate how both magnetic force and rotation affect the peristaltic 

transport of hybrid bio-nanofluids through a symmetric channel. The precise solutions for 

magnetic force, stream lines, heat transfer coefficient, temperature, and velocity have been 

obtained utilizing the Adomian decomposition technique. Graphs are used to illustrate physical 

characteristics that affect the flow. 

2. Adomain Decomposition Method (ADM) 

This method can be utilized to solve linear and non-linear differential equations as well as integral 

equations, and it produces better results than other methods. George Adomian first proposed this 

method, and it has since been used to solve a variety of problems[18 and19]. To illustrate a general 

overview of the Adomain Decomposition Method, let's consider a form equation [20-23]. 

𝐿𝑢 + 𝑅𝑢 + 𝑁𝑢 = 𝑔                                                                                          (1) 

Where L is invertible linear operator, N represent the nonlinear terms, R is reminder of the linear 

operator. Applying the L−1 on both sides in equation (1), yield: 

𝑢 = L−1(𝑔)− L−1(𝑅𝑢)− L−1(𝑁𝑢)                                                                (2) 

Now L−1 represent the n-fold integration for nth order L,by using the given conditions all are 

assumed to be prescribed ,the ADM define the solution: 

𝑢 = ∑ 𝑢𝑚
∞
𝑚=0 .  

Where the  𝑢0,𝑢1,𝑢3,…  recursively determined by using the relation : 

  𝑢0 = 𝑓(𝑥), 



IHJPAS. 36(2)2023 

438 
 

  𝑢𝑚+1 = L
−1(𝑅  𝑢𝑚) − L

−1(𝑁  𝑢𝑚) ,𝑚 ≥ 0                                                                                   (3) 

The last term of the Eq. (3) can be computed by substituting : 

𝑁𝑢 = ∑ 𝐴𝑚(𝑢0

∞

𝑚=0

,𝑢1,…,𝑢𝑚) . 

Where 𝐴𝑚 represent generated Adomain polynomials for the specified nonlinearity, they depend 

only on the 𝑢0 ,…,𝑢𝑚 components and form rapidly converge series, the 𝐴𝑚 are given: 

𝐴0 = 𝑓(𝑢0), 

𝐴1 = 𝑢1𝑓
(1)(𝑢0), 

𝐴2 = 𝑢2𝑓
(1)(𝑢0)+

𝑢1
2

2!
𝑓(2)(𝑢0). 

And can be found from the formula : 

𝐴𝑚 =
1

𝑚!

𝑑𝑚

𝑑𝜆𝑚
[𝑁(∑𝜆𝑖

∞

𝑖=0

𝑢𝑖)]

𝜆=0

. 

3. Problem Formulation  

Let's consider the peristaltic flow of an electrically conducted hybrid bio-nanofluid in the presence 

of rotation and magnetic field, blood is considered in this research as a basic non-Newtonian fluid, 

while copper and gold are considered nanoparticles in a 2-D channel of width 2𝑑1. A  

(X̅, Y̅)  Cartesian coordinate system. We selected X̅  in the direction of wave propagation and Y̅   

transverse to it, a constant magnetic field of strength 𝐻0 acting in the transverse direction results 

in an induced magnetic field  𝐻(�̅�𝑥(X̅, Y̅, t)̅, �̅�𝑦(X̅, Y̅, t)̅ + 𝐻0,0).  H
∗refers to the total magnetic 

field composed of (�̅�𝑥(X̅, Y̅, t)̅, �̅�𝑦(X̅, Y̅, t)̅ + 𝐻0,0). The channel wall's geometry is depicted by: 

ℎ = 𝜉(X̅, t)̅ = 𝑑1 + 𝑑2 𝑆𝑖𝑛 (
2𝜋

𝜆
(X̅− Ct)̅)                                                                       (4) 

The flow of the fluid is induced by the infinite sinusoidal wave that travels along the channel walls 

at a wave speed of C and a wavelength of 𝜆. Where 𝑑2  is the wave amplitude, 𝑑1 is the half channel 

width, and t ̅is the time. The governing for the problem are:  

𝛻.𝐸 = 0    ,𝐽 = 𝛻 × 𝐻 ,  𝛻.𝐻 = 0 ,   𝜎[𝐸 + 𝜇𝑒(𝑉 ×𝐻
∗)] =  𝐽, 

𝛻 ×𝐸 = −𝜇𝑒
𝜕𝐻

𝜕𝑡
                                                                                                                                      (5)                                                                                                                                                 

Where V is velocity vector, 𝛔 is electrical conductivity, J is current density; E is electric field ,and 

𝛍𝐞 is magnetic permeability. The governing equations for an incompressible, unsteady, hydro-

magnetic, viscous bio- nanofluid (Au-Cu Nanoparticles) include momentum, induction, and 

heat  [24and 25]:  



IHJPAS. 36(2)2023 

439 
 

𝛻.𝑉 = 0                                                                                                                                                     (6) 

𝜌𝑛𝑓
𝜕𝑉

𝜕𝑡
+ 𝜌𝑛𝑓(Ω̅× (Ω̅ ×𝑉) + 2Ω̅ ×

𝜕𝑉

𝜕𝑡
= −𝛻𝑃 + 𝜇𝑛𝑓 𝛻

2𝑉 + 𝜇𝑒(𝐻
∗ .𝛻)𝐻∗ − 𝜇𝑒

𝛻𝐻∗2 

2
    (7) 

𝜕𝐻∗

𝜕𝑡
= 𝛻 × (𝑉 × 𝐻∗) +

1

𝜁
𝛻2𝐻∗                                                                                                           (8) 

( 𝜌𝑐𝑝)𝑛𝑓
𝐷𝑇

𝐷𝑡
= 𝑘𝑛𝑓𝛻

2𝑇 +𝜇𝑛𝑓(𝛻𝑉 + (𝛻𝑉)
𝑇)2 −

𝜕𝑞

𝜕𝑌
+ 𝑄0                                                            (9) 

Where P is fluid pressure, 𝜁 = 𝜇𝑒𝜎𝑛𝑓, represent the magntic diffusivity,  𝜎𝑛𝑓 is the electrical 

conductivity,T represent the temperature distribution, 𝑞 = −
4𝜎𝑓

∗

3𝑘𝑓
∗

𝜕𝑇4

𝜕𝑌
 is the radiative heat flux, 𝑘𝑓

∗ 

and  𝜎𝑓
∗ are mean absorption coeffient and Stefan Boltzmann constants, Q0 is the heat source, 

experimental formulation for the physical characteristics of hybrid nanofluid[25] are shown in 

Table 1, which are given by: 

Table 1 Physical properties of hybrid nanofluid . 

property Nanofluids 

Density 𝜌𝑛𝑓 = ((1 − ∅1)𝜌𝑓 + ∅1𝜌1)(1 −∅2) + ∅2𝜌2 

Heat capacity ( 𝜌𝑐𝑝)𝑛𝑓 = (( 𝜌𝑐𝑝)𝑓
(1 −∅1) + ( 𝜌𝑐𝑝)1

∅1)(1 − ∅2) + ( 𝜌𝑐𝑝)2
∅2      

Dynamic viscosity 
𝜇𝑛𝑓 =

𝜇𝑓
(1 − ∅1)

2.5(1 − ∅2)
2.5

 

Thermal conductivity 

 

knf = (
k2 +(m − 1)k3 − (k3 − k2)∅2(m −1)

k3(m −1) +k2 + ∅2(k3 −k2)
)k3    

k3 = (
k1 +kf (m − 1) − (kf − k1)(m − 1)∅1

k1 + ∅1(kf − k1) + kf (m − 1)
) kf     

Ectrical conductivity 

 

σnf = σ3 (
  σ2(1 + 2∅2) + 2(1 − ∅2) σ3
 σ3(2 +∅2) + σ2(1 − ∅2)

)          

σ3 = σf (
  σ1(1 + 2∅1) + 2σf(1 − ∅1) 

 σ1(1− ∅1) + σf(2 + ∅1)
) 

 Where ∅1  is the volume fraction of gold nanoparticles, ∅2  is the volume fraction of copper 

nanoparticles, and m represents the shape of a factor of the indicated nanoparticles, respectively. 

The physical characteristics of nanoparticles are classified in the Table 2. 

 

 

 

 

 



IHJPAS. 36(2)2023 

440 
 

Table 2. Based fluid and nanoparticles properties[25]. 

Properties Based fluid Nanoparticle 

(Gold Au) 

Nanoparticle 

(Copper Cu) 

𝜌 density 𝜌𝑓 = 1050 𝜌1 = 19300 ρ2 = 8933 

𝐶p heat capcity 𝐶𝑝𝑓 = 3617 𝐶𝑝1 = 129 𝐶𝑝2 = 385 

K thermal conductivity 𝑘𝑓=0.52 k1 = 318 𝑘2 = 401 

𝜎 electrical conductivity 𝜎𝑓 = 1.33 σ1 = 4.1∗ (10
7) σ2 = 5.96∗ (10

7) 

The velocity �⃑⃑�  of 2-D flows 𝐕⃑⃑  ⃑ is defined as [U̅(X̅, Y̅, t)̅, V̅(X̅, Y̅, t),0], where �̅� denotes the velocity 

component in coordinate X ̅and �̅�  denotes  the velocity component in coordinate Y̅. 

Choose a wave frame(𝑥 ̅̅̅̅ , �̅�) that moves at a speed of C away from the fixed frame by the following 

transformation given below. 

 �̅� = �̅� − 𝐶𝑡 ̅,       �̅� = �̅�   , �̅� = �̅� − 𝐶   , �̅� = �̅�                                                                            (10) 

The governing equations were simplified to take the following form: 

𝜕�̅�

𝜕�̅�
+
𝜕�̅�

𝜕�̅�
= 0                                                                                                                                        (11) 

𝜌𝑛𝑓 (�̅�
𝜕�̅�

𝜕�̅�
+ �̅�

𝜕�̅�

𝜕�̅�
)− 𝜌𝑛𝑓Ω(Ω(�̅�)+ 2

𝜕�̅�

𝜕𝑡̅
) = −

𝜕�̅�

𝜕�̅�
+ 𝜇𝑛𝑓 (

𝜕2

𝜕�̅�2
+
𝜕2

𝜕�̅�2
)�̅� −

𝜇𝑒
2

𝜕�̅�∗2 

𝜕�̅�
+𝜇𝑒    

(�̅�𝑥
𝜕 �̅�𝑥
𝜕�̅�

+   (�̅�𝑦 + 𝐻0)
𝜕 �̅�𝑥
𝜕�̅�

)                                                                                                       (12)

 

𝜌𝑛𝑓 (�̅�
𝜕𝑣

𝜕�̅�
+ �̅�

𝜕�̅�

𝜕�̅�
)− 𝜌𝑛𝑓Ω(Ω�̅� + 2

𝜕�̅�

𝜕𝑡̅
) = −

𝜕�̅�

𝜕�̅�
+ 𝜇𝑛𝑓 (

𝜕2

𝜕�̅�2
+
𝜕2

𝜕�̅�2
)𝑣 ̅ −

𝜇𝑒
2

𝜕�̅�∗2 

𝜕�̅�
+𝜇𝑒   

(�̅�𝑥
𝜕 �̅�𝑦

𝜕�̅�
+ (�̅�𝑦 + 𝐻0)

𝜕 

𝜕�̅�
�̅�𝑦)                                                                                                        (13)

 

    

1

𝜇𝑒

𝜕�̅�

𝜕�̅�
=
𝜕�̅�

𝜕�̅�
(�̅�𝑦 + 𝐻0) −

𝜕�̅�

𝜕�̅�
�̅�𝑥 +

1

𝜁
𝛻2𝐻𝑥

∗                                                                                   (14)  

( 𝜌𝑐𝑝)𝑛𝑓 (�̅�
𝜕

𝜕�̅�
+ �̅�

𝜕

𝜕�̅�
)�̅�

= 𝑘𝑛𝑓 (
𝜕2�̅�

𝜕�̅�2
+
𝜕2 �̅�

𝜕�̅�2
)+ 𝜇𝑛𝑓(4(

𝜕�̅�

𝜕�̅�
)
2

+ (
𝜕�̅�

𝜕�̅�
+
𝜕�̅�

𝜕�̅�
)2 −

𝜕𝑞𝑟
𝜕�̅�

+ 𝑄0                (15) 

 

 

 

 



IHJPAS. 36(2)2023 

441 
 

Using the non-dimensional quantities listed below: 

𝐻𝑦 = 
�̅�𝑦

𝐻0
 𝑝 =

  𝑑1
2�̅�

𝐶𝜆𝜇𝑓
     𝑦 =

�̅�

𝑑1
 𝑥 =

�̅�

𝜆
 

𝐻𝑥 = 
�̅�𝑥
𝐻0

 𝑅𝑒 =
𝜌𝑓𝐶𝑑1

𝜇𝑓
 𝑣 =

�̅� 

𝐶
 𝑢 =

�̅�

𝐶
 

𝜃 =
𝑇 − 𝑇𝑢
𝑇𝑙 − 𝑇𝑢

   
𝑅𝑚 = 𝜎𝑓𝜇𝑒𝐶𝑑1 

𝐸 = −
�̅�

𝐶𝐻0 𝜇𝑒
 𝛿 =

𝑑1
𝜆

 

𝑃𝑟 =
𝜇𝑓(𝑐𝑝)𝑓
𝑘𝑓

 𝑆
2 =

𝑀2

𝑅𝑒𝑅𝑚
 𝑅𝑑 =

4𝜎𝑓
∗(𝑇𝑙 − 𝑇𝑢)

3

3𝑘𝑓
∗𝑘𝑓

 𝑡 = 𝐶
𝑡̅

𝜆
 

𝐸𝑐 =
𝐶2

(𝑐𝑝)𝑓
(𝑇𝑙 −𝑇𝑢)

 𝑀 = 𝐻0𝑑1√
𝜎𝑓

𝜇𝑓
 𝛽 =

𝑄0𝑑1
2

𝑘𝑓(𝑇𝑙 − 𝑇𝑢)
 𝜙 =

  �̅�

𝐻0𝑑1
 

𝐴4 =
𝑘𝑛𝑓

𝑘𝑓
 

𝐴5 =
(𝜌𝑐𝑝)𝑛𝑓

(𝜌𝑐𝑝)𝑓

 
𝐴2 =

𝜇𝑛𝑓

𝜇𝑓
 𝐴1 =

𝜌𝑛𝑓

𝜌𝑓
 

In the preceding expressions 𝑇𝑢 is temperature at upper wall ,𝑇𝑙 is temperature at lower wall , �̅� is 

stream function, 𝐸 is strength of the electric field, 𝛿 is wavenumber, 𝑅𝑒 𝑖𝑠 Reynolds number, Ω is 

rotation, 𝑅𝑚 is magnetic Reynolds number, 𝐸𝑐 isEckert number, 𝑃𝑟 is Prandtl 

number, �̅� 𝑖𝑠 magnetic force function, M is Hartmann number, 𝜃 is temperature in the non-

dimensional form , 𝛽 is internal heat generation, and S is Strommer number respectively, the non-

dimensional form of the peristaltic wave can be expressed by : 

 ℎ =
h̅

𝑑1
= 1 + 𝜖 𝑆𝑖𝑛 (2𝜋𝑥), where amplitude ratio  𝜖 =

𝑑2

𝑑1
 . 

 Introduction of dimensionless magnetic force function 𝜙 and stream function 𝜓 by using the 

relations : 

𝑢 =
𝜕𝜓

𝜕𝑦
 ,𝑣 = −𝛿

𝜕𝜓

𝜕𝑥
 , 𝐻𝑥 =

𝜕𝜑

𝜕𝑦
    , 𝐻𝑦 = −𝛿

𝜕𝜑

𝜕𝑥
                                                                    (16) 

Substituting Eq.(16) into Eqs. (12 -15), giving us the equations: 

𝐴1𝑅𝑒 𝛿(𝜓𝑦
𝜕

𝜕𝑥
− 𝜓𝑥

𝜕

𝜕𝑦
)𝜓𝑦 −

𝜌𝑛𝑓 𝑑1
2

𝜇𝑓
Ω2(𝜓𝑦)

= −( 𝑃𝑚)𝑥 + 𝐴2𝛻
2𝜓𝑦 + 𝑆

2𝑅𝑒(𝛿𝜙𝑦𝜙𝑥𝑦 − 𝛿𝜙𝑥𝜙𝑦𝑦 + 𝜙𝑦𝑦)                             (17) 

𝐴1𝑅𝑒𝛿
3 (𝜓𝑦

𝜕

𝜕𝑥
−𝜓𝑥

𝜕

𝜕𝑦
)𝜓𝑥 −

𝜌𝑛𝑓 𝑑1
2

𝜇𝑓
Ω2𝛿2𝜓𝑥

= ( 𝑃𝑚)𝑦 + 𝐴2 𝛿
2𝛻2𝜓𝑥 + 𝑆

2𝑅𝑒𝛿2( 𝛿𝜙𝑦𝜙𝑥𝑥 − 𝛿𝜙𝑥𝜙𝑥𝑦 + 𝜙𝑥𝑦)                     (18) 

𝐸 = (𝜓𝑦 −𝛿(𝜓𝑦𝜙𝑥 − 𝜓𝑥𝜙𝑦) +
1 

𝐴3𝑅𝑚.
𝛻2𝜙                                                                                     (19) 



IHJPAS. 36(2)2023 

442 
 

𝐴5𝑅𝑒 𝑃𝑟𝛿(𝜓𝑦𝜃𝑥 − 𝜓𝑥𝜃𝑦)

= 𝐴4𝛻
2𝜃 + 𝐴2 𝐸𝑐  𝑃𝑟 (4𝛿

2𝜓𝑥𝑦
2 + (𝜓𝑦𝑦 − 𝛿

2𝜓𝑥𝑥)
2
) + 𝑅𝑑(𝜃

4)𝑦𝑦 + 𝛽           (20) 

𝑊ℎ𝑒𝑟𝑒 𝑃𝑚 is the sum magnetic and ordinary pressure, which is the total pressure. The 

corresponding stream function, temperature function, and magnetic force function boundaries for 

non-conductive elastic walls in the wave frame are as follows: 

𝜓 = 0,    𝜙𝑦 = 0,  𝜓𝑦𝑦 = 0,𝑎𝑡   𝑦 = 0,                                                                                                       

 𝜙 = 0,𝜓𝑦 = −1,𝜃 = 0,𝜓 =
𝑞

2
,𝑎𝑡   𝑦 = ℎ,                                                                                              

𝜃 = 1    𝑎𝑡    𝑦 = −ℎ .                                                                                                                            (21)

  

Where q represent mean of flow rate. Using the long wave length approximation and neglect the 

wavenumber along the low Reynolds number, one can find from Equations (17)- (20) That:  

( 𝑃𝑚)𝑥 =
𝜌𝑛𝑓 𝑑1

2

𝜇𝑓
Ω2𝜓𝑦 + 𝐴2𝜓𝑦𝑦𝑦 + 𝑆

2𝑅𝑒𝜙𝑦𝑦                                                                               (22) 

( 𝑃𝑚)𝑦 = 0                                                                                                                                             (23)  

𝐸 = 𝜓𝑦 +
1 

𝐴3𝑅𝑚.
𝜙𝑦𝑦                                                                                                                            (24) 

𝜃𝑦𝑦 =
1 

𝐴4
(−𝐴2 𝐸𝑐  𝑃𝑟(𝜓𝑦𝑦)

2
−𝑅𝑑(𝜃

4)𝑦𝑦 − 𝛽 )                                                                          (25) 

By using cross derivation to eliminate the pressure from Eq.(22) and Eq.(23), giving us the equation: 

𝜌𝑛𝑓 𝑑1
2

𝜇𝑓
Ω2𝜓𝑦𝑦 + 𝐴2𝜓𝑦𝑦𝑦𝑦 + 𝑆

2𝑅𝑒𝜙𝑦𝑦𝑦 = 0                                                                                   (26) 

Join Eq.s (24) and (26), become :  

𝜓𝑦𝑦𝑦𝑦 =
−1

𝐴2
(𝑀2𝐴3𝜓𝑦𝑦 + 𝐾𝜓𝑦𝑦)                                                                                                     (27) 

Where  𝐾 =
𝜌𝑛𝑓 𝑑1

2

𝜇𝑓
Ω2. 

4. Solution of the Problem 

In this section, the Adomain decomposition method solution will be determined for the stream 

function, temperature, and magnetic force equation. In the operator 𝐿𝑖𝑦(∗) =
𝜕𝑚(∗)

𝜕𝑦𝑚
 apply Eqs. 

(27,24,and 25) in accordance with the Adomain decomposition method: 

𝐿𝑦𝑦𝑦𝑦𝜓 =
−1

𝐴2
(𝑀2𝐴3𝜓𝑦𝑦 + 𝐾𝜓𝑦𝑦)                                                                                                 (28)  

𝐿𝑦𝑦𝜙 = 𝐸𝐴3𝑅𝑚 − 𝜓𝑦                                                                                                                          (29) 



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𝐿𝑦𝑦𝜃 =
1 

𝐴4
(−𝐴2 𝐸𝑐  𝑃𝑟(𝜓𝑦𝑦)

2
− 𝑅𝑑(𝜃

4)𝑦𝑦 − 𝛽 )                                                                      (30) 

Applying the inverse operator  𝐿𝑖𝑦
−1(∗) = ∫ (∗)𝑑𝑦

𝑦

0⏟    
𝑚−𝑡𝑖𝑚𝑒𝑠

, 𝑖 = 1,2,3,… ,allows us to write  Eqs(28-30): 

𝜓 =  𝐿𝑦𝑦𝑦𝑦
−1 (

−1

𝐴2
(𝑀2𝐴3𝜓𝑦𝑦 + 𝐾𝜓𝑦𝑦) )                                                                                            (31) 

𝜙 =  𝐿𝑦𝑦
−1(𝐸𝐴3𝑅𝑚 − 𝜓𝑦 )                                                                                                                    (32) 

𝜃 =  𝐿𝑦𝑦
−1(

1 

𝐴4
(−𝐴2 𝐸𝑐  𝑃𝑟(𝜓𝑦𝑦)

2
− 𝑅𝑑(𝜃

4)𝑦𝑦 − 𝛽 )                                                                   (33) 

Decompose the stream function, temperature, and magnetic force equations using boundary 

equation (21). 

ψ = ∑ ψm,

∞

m=0

 ϕ = ∑  ϕm,

∞

m=0

    θ = ∑ θm                                                                                  (34 ) 

∞

m=0

 

By definition of   𝐿𝑖𝑦
−1,giving us 

𝜓0 =
1

2
𝑦2𝐶1 +

1

6
𝑦3𝐶2 + 𝐶3 + 𝑦𝐶4 

𝜙0 = −
𝐴3
2
ERm+

𝐴3
2
𝐸Rm𝑦2 

𝜃0 = −
−ℎ2𝛽 − 𝐴4

2𝐴4
+ 𝑦(−

1

2ℎ
) −

𝑦2𝛽

2𝐴4
                                                                                          (35) 

𝜓𝑛+1 =  𝐿𝑦𝑦𝑦𝑦
−1 (

−1

𝐴2
(𝑀2𝐴3𝜓𝑛𝑦𝑦 + 𝐾𝜓𝑛𝑦𝑦) )                                              

𝜙𝑛+1 =  𝐿𝑦𝑦
−1(𝜓𝑛𝑦 )                                                                                                                                    

𝜃𝑛+1 =  𝐿𝑦𝑦
−1(

1 

𝐴4
(−𝐴2 𝐸𝑐  𝑃𝑟(𝜓𝑛𝑦𝑦)

2
− 𝑅𝑑(𝜃𝑛

4)𝑦𝑦 )                                                                  (36)  

Due to length colocation ,they have computed up to the second term only : 

𝜓1 = 𝐶5 + 𝑦𝐶6 + 𝑦
2𝐶7 +𝑦

3𝐶8 +
𝐾𝑦5

40ℎ2𝐴2
+
𝐾𝑞𝑦5

80ℎ3𝐴2
−
𝑀2𝑦5𝐴3
40ℎ2𝐴2

−
𝑀2𝑞𝑦5𝐴3
80ℎ3𝐴2

 , 

𝜙1 = 𝑠1 + 𝑦𝑠2 +
Rm(−

1
2
ℎ2(2ℎ + 3𝑞)𝑦2 +

1
4
(2ℎ + 𝑞)𝑦4)𝐴3

4ℎ3
 , 



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444 
 

𝜃1 = 𝑐1 +𝑦𝑐2 −
1

16ℎ6𝐴4
5
𝑦2(ℎ6Rd(−4ℎ6 + 6ℎ4𝑦2 − 4ℎ2𝑦4 + 𝑦6)𝛽4

− 4ℎ5Rd(3ℎ5 − 3ℎ4𝑦 − 3ℎ3𝑦2 + 3ℎ2𝑦3 + ℎ𝑦4 − 𝑦5)𝛽3𝐴4
− 6ℎ4Rd(ℎ4 − 4ℎ3𝑦 + ℎ2𝑦2 + 2ℎ𝑦3 − 𝑦4)𝛽2𝐴4

2

+ 4ℎ3Rd(2ℎ3 + 2ℎ2𝑦 − 3ℎ𝑦2 + 𝑦3)𝛽𝐴4
3

+ 𝐴4
4(ℎ2Rd(6ℎ2 − 4ℎ𝑦 + 𝑦2)+ 3Ec(2ℎ + 𝑞)2𝑦2𝐴2𝑝𝑟)), 

𝜓2 = 𝐶9 + 𝑦𝐶10 + 𝑦
2𝐶11 + 𝑦

3𝐶12

−

(2ℎ + 𝑞)(𝐾 − 𝑀2𝐴3)(−6𝑦
5𝐴2 +

1
35
𝑦5(−21ℎ2 + 5𝑦2)(𝐾 − 𝑀2𝐴3))

480ℎ3𝐴2
2

 , 

𝜙2
= 𝑠3 + 𝑦𝑠4

+
Rm𝐴3(−10ℎ

2(2ℎ + 3𝑞)𝑦2𝐴2 + 5(2ℎ + 𝑞)𝑦
4𝐴2 +

1
6
(2ℎ + 𝑞)(ℎ2 − 𝑦2)3(𝐾 − 𝑀2𝐴3))

80ℎ3𝐴2
, 

𝜃2 = 𝑐3 + 𝑦𝑐4 −
1

102400ℎ24𝐴2𝐴4
21
(𝑐5 − 25ℎ

54Rd5𝑦2𝛽16𝐴2 +
375

2
ℎ52Rd5𝑦4𝛽16𝐴2

− 875ℎ50Rd5𝑦6𝛽16𝐴2 +
11375

4
ℎ48Rd5𝑦8𝛽16𝐴2 − 6825ℎ

46Rd5𝑦10𝛽16𝐴2

+
25025

2
ℎ44Rd5𝑦12𝛽16𝐴2 − 17875ℎ

42Rd5𝑦14𝛽16𝐴2

+
160875

8
ℎ40Rd5𝑦16𝛽16𝐴2 − 17875ℎ

38Rd5𝑦18𝛽16𝐴2

+
25025

2
ℎ36Rd5𝑦20𝛽16𝐴2 − 6825ℎ

34Rd5𝑦22𝛽16𝐴2 +
11375

4
ℎ32Rd5𝑦24𝛽16𝐴2

− 875ℎ30Rd5𝑦26𝛽16𝐴2 +
375

2
ℎ28Rd5𝑦28𝛽16𝐴2 

−25ℎ26Rd5𝑦30𝛽16𝐴2 +
25

16
ℎ24Rd5𝑦32𝛽16𝐴2 − 375ℎ

52Rd5𝑦2𝛽15𝐴2𝐴4

+ 375ℎ51Rd5𝑦3𝛽15𝐴2𝐴4 + 2625ℎ
50Rd5𝑦4𝛽15𝐴2𝐴4 − 2625ℎ

49Rd5𝑦5𝛽15𝐴2𝐴4
− 11375ℎ48Rd5𝑦6𝛽15𝐴2𝐴4 + 11375ℎ

47Rd5𝑦7𝛽15𝐴2𝐴4
+ 34125ℎ46Rd5𝑦8𝛽15𝐴2𝐴4 − 34125ℎ

45Rd5𝑦9𝛽15𝐴2𝐴4
− 75075ℎ44Rd5𝑦10𝛽15𝐴2𝐴4 +75075ℎ

43Rd5𝑦11𝛽15𝐴2𝐴4
+ 125125ℎ42Rd5𝑦12𝛽15𝐴2𝐴4 − 125125ℎ

41Rd5𝑦13𝛽15𝐴2𝐴4
− 160875ℎ40Rd5𝑦14𝛽15𝐴2𝐴4 + 160875ℎ

39Rd5𝑦15𝛽15𝐴2𝐴4). 

The c1,…, c5, 𝐶1, …,𝐶12, and s1,..,s4 are large constants coefficients can be determined by using 

boundary condition Eq.(21) and MATHMATICA software. So on. Then the approximation system 

solution takes the following form: 

𝜓 = 𝜓0 + 𝜓1 + 𝜓2 + ⋯, 

𝜙 = 𝜙0 + 𝜙1 + 𝜙2 + ⋯, 



IHJPAS. 36(2)2023 

445 
 

𝜃 = 𝜃0 +𝜃1 + 𝜃2 + ⋯. 

The equation for the coefficient of heat transfer at the wall is: 

𝑍 = 𝜉𝑥𝜃𝑦                                                                                                                                                 (37) 

So,the electric field E is obtain from : 

𝐸 = 𝜓𝑦 +
1 

𝐴3𝑅𝑚.
𝜙𝑦𝑦 . 

 5. Discussion and Graphs of the Results 

The purpose of this section is to examine the graphic results of a variety of important parameter 

utilized in the specified modeling. In particular, streamlines, velocity, induced magnetic field, 

magnetic force contours, heat transfer coefficient, and temperature distribution are depicted in the 

figures below. We examined numerous cases, including the Hartmann number M, the magnetic 

Reynolds number Rm,and rotation Ω. The numerical variables were chosen based on previous 

literature[26-28], and flow trapping is discussed graphically. All figures are plotted using from the 

MATHEMATICA  program. 

5.1 Velocity Profile  

Figure1 depicts the velocity changes with respect to the axial y for various Hartmann number M, 

rotation, and dynamic viscosity values µ. The effect of Hartmann number M on velocity is shown 

in Fig.1a. As M increases, the velocity u near the channel's middle rises, whereas the opposite 

occurs near the peristaltic wall. This fact is related to the Lorentz force, which occurs when an 

external magnetic field is used and, in turn, leads the fluid motion to slow down. It demonstrates 

that the Lorentz force is much stronger near the wall than in the channel's middle. Display the 

effect of dynamic viscosity on velocity in Fig.1b. It is noticed that when µ increases, the velocity 

along the channel walls slowly decreases, whereas it increases at the channel center. Fig.1c 

displays the rotational effect. It can be seen that as rotation increases, the velocity u decreases near 

the channel's middle, whereas the reverse behavior can be seen near the peristaltic wall. 

 

Figure1a. Impact of  M on the velocity u  



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446 
 

  

Figure 1c. Impact of Ω on the velocity u Figure 1b. Impact of  µ on the velocity u 

5.2 Induced Magnetic Field Profile 

This subsection describes an examination of induced magnetic field  𝐻𝑥. Graphics have been 

drawn to show the development of the induced magnetic field profile in Figure 2. 

To study the impacts of magnetic Reynolds numbers Rm, Hartmann number M, and rotation Ω on 

the induced magnetic field 𝐻𝑥 with respect to axial y. We are carried out from Figure 2a-2d.The 

effects of magnetic Reynolds number Rm, and magnetic field M on the induced magnetic field 

profile can be observed in Figs.2a and 2b. It is noticed that the relationship between 𝐻𝑥 and y is 

inversely proportional to each other , with the rise in Rm and M in the region 𝑦 ≤  0 ,the induced 

magnetic field decreases and while in the region 𝑦 ≥ 0, the induced magnetic field 𝐻𝑥 rises with 

the increase in Rm and M. The impact of rotation Ω on the induced magnetic field  𝐻𝑥 is seen in 

Fig.2c. At 𝑦 ≥ 0, the induced magnetic field 𝐻𝑥 increases as rotation Ω increases, but it decreases 

at 𝑦 ≤  0. 

  

Figure 2b. Impact of M on  the induced magnetic field profile Figure 2a. Impact of Rm on  the induced magnetic field 

profile 



IHJPAS. 36(2)2023 

447 
 

5.3 Temperature Profile 

The temperature profile  behavior for the Hartmann number M, heat radiation Rd, rotation Ω, heat 

absorption β, Eckert number Ec, and Prandtl number pr are examined in Figure3. 

The impact of Hartmann number M on the temperature profile 𝜃 are examined by Fig.3a. If a rise 

in temperature is observed with increase values of Hartmann numbers M. The temperature profile 

𝜃 enhances by increasing of Prandtl number Pr in Fig.3b, this is because heat generation form 

friction brought by shear in the flow is more pronounced when the fluid is highly viscous or 

moving quickly. Same behavior is observed for Eckert number that can be shown from Fig.3c As 

the rotation parameter values increase, the temperature profile is seen to decrease Fig. 3d. 

Influence of heat absorption β is depicted in Fig. 3e the temperature profile 𝜃 decreases in the 

center of the channel and merges for near the wall.Fig.3f shows the impact of the radiation Rd on 

the temperature profile 𝜃, initially, the  𝜃 decreases and then merges near the wall. 

 
 

Figure 3b. Impact of Pr on the temperature profile Figure 3a. Impact of M on the temperature profile 

 
Figure 2c. Impact of M on  the induced magnetic field profile 



IHJPAS. 36(2)2023 

448 
 

  

Figure 3d. Impact of Ω on the temperature profile Figure 3c. Impact of Ec on the temperature profile 

  

Figure 3e. Impact of Rd on the temperature profile Figure 3d. Impact of β on the temperature profile 

 

5.4 Heat Transfer Coefficient  

The behaviors of Prandtl number pr, heat absorption β, Eckert number Ec, rotation Ω, and heat 

radiation Rd on heat transfer coefficient 𝑍 have been noticed through Fig.4. 

Fig.4a shows that when rotation increases, the heat transfer coefficient 𝑍 between the fluid and the 

wall of the channel decreases. In Fig.4b depicts the behavior of the heat transfer coefficient for 

various values of heat radiation Rd. An increase in the heat transfer is observed for rising values 

of the heat radiation Rd. The impact of Eckert number Ec and Prandtl number Pr on heat transfer 

coefficient, it is noticed from Fig.4c and 4d that heat transfer coefficient enhances for higher values 

of Ec and Pr. 



IHJPAS. 36(2)2023 

449 
 

  

Figure 4b. Impact of Rd on  the heat transfer coefficient Figure 4a. Impact of Ω on the heat transfer coefficient 

  

Figure 4d. Impact of Ec on  the heat transfer coefficient Figure 4c. Impact of pr on the heat transfer coefficient 

5.5 Trapping Phenomenon  

Typically, streamlines have the same shape as a boundary wall in the wave frame. However, under 

certain situations, some stream lines split and enclose a bolus, which moves as a whole with the 

waves. "Trapping" is the term for this phenomenon. In order to investigate the impacts of the 

trapping phenomenon at various values of Hartmann number M and rotation Figs.5 and 6 were 

drawn. It can be shown from Fig.5 that as Hartmann number M increases, the trapped bolus rises 

in size. The impact of rotation Ω on trapping can be noticed from Fig .6, it is shown that when the 

value of Ω increases, the size of the trapped bolus diminishes. 

Figs.7,8,and 9 show the magnetic force for various values of essential parameters. The contours of 

magnetic force are parallelly to the flow field. Fig.7 illustrates that rotation affects the magnetic 

force contours; it is observed that as rotation increases, the magnetic force lines diminish and the 

size of the bolus changes. Figs.8 and 9 illustrate how magnetic Reynolds number Rm and 

Hartmann number M influence the magnetic force contours. As the values of M and Rm are 

raised,the magnetic force contours move themselves forward and the magnetic force gradually 

increases, as can be seen in these figures. 



IHJPAS. 36(2)2023 

450 
 

 
  

Figure 5.  Impact of Hartmann number on the stream lines 

 
  

Figure 6. Impact of rotation on the stream lines 

   

Figure 7. Impact of rotation on the magnetic force contours 



IHJPAS. 36(2)2023 

451 
 

  
 

Figure 8. Impact of Reynolds number on the magnetic force contours 

   

Figure 9. Impact of Hartmann number on the magnetic force contours 

6. Conclusion 

The impact of magnetic force ,rotation,and nonlinear heat radiation on hybrid bio-nanofluids 

peristaltic flow in a symmetric channel under the influence of a magnetic field are discussed. 

The governing equations representing momentum, Maxwell, and heat equations are considered. 

The exact expressions for velocity, heat transfer coefficient, stream lines, temperature, induced 

magnetic field, and magnetic force are obtained by using the Adomain decomposition method. 

The major results can be summarized as follows: 

• The behavior of the velocity distribution u near the center part of the channel increases with 

a rises in  Hartmann number M, while rotation parameters oppose the velocity. 

• Induced magnetic field exhibits dual conduct in the two zones 𝑦 ≤ 0, 𝑦 ≥ 0 against all the 

important parameters. 

•  The temperature profile in the Prandtl number Pr and the Eckert number Ec is improved.  

• The temperature profile decline when higher values of rotation.    

• The heat transfer coefficient raises with rising values of the radiation parameter, Prandtl 

number, and Hartmann number M, and decreases with rotation. 



IHJPAS. 36(2)2023 

452 
 

• The trapped bolus rises in size at increasing of Hartmann number M and decline at 

increasing of rotation. 

• The contours of magnetic force improve because of an increase in the magnitude of 

Hartmann number M. 

• The contours of magnetic force decrease because of an increase in the magnitude of the 

rotation Ω. 

Finally, we conclude that the findings of this paper should be beneficial for researchers aiming 

to develop fluid mechanics as well as those in biomedical engineering and technology. 

 

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