384 

©2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), University 

of Baghdad. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 

International License 

 

Effect of Couple-Stress on Peristaltic Transport of Sutterby Fluid in an 

Asymmetric Channel 

 

Asmaa A. Mohammed 1*  and Liqaa Zeki Hummady 2  
1 Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq. 

1,2 Department of Mathematics, College of Science for Women, University of Baghdad, Baghdad, Iraq. 

*Corresponding Author. 

 

 Received:6 May 2023                                Accepted: 14 August 2023                     Published:20 January 2025 

 doi.org/10.30526/38.1.3462 

 
Abstract 

     In this paper, the effect of a couple–stress and other variables on the peristaltic flow of Sutterby 

fluid in an inclined asymmetric channel containing a porous medium with heat transfer is examined. 

In the presence of rotation and couple–stress, mathematical modeling is developed using constitutive 

equations based on the model of Sutterby fluid. In flow analysis, assumptions such as long wavelength 

approximation and low Reynolds number are used. The resulting nonlinear equation was numerically 

solved using the perturbation method. The effects of various physical parameters such as the couple-

stress parameter, the Grashof number, the Hartmann number, the Reynold number, the Froude number, 

the Hall parameter, the Darcy number, the magnetic field, the Sutterby fluid parameter, and heat 

transfer analysis on the stream function are analyzed graphically. Utilizing MATHEMATICA 

software, numerical results were computed. 

Keywords: Peristaltic flow, Heat transform, Inclined channel, Porous medium, Couple-stress, and 

Sutterby fluid. 

 

1. Introduction 

     Peristaltic pumping is a special type of pumping when it is simple to transport a variety of complex 

rheological fluids from one location to another. This pumping principle is referred to as peristaltic 

(1). Some examples of such physiological processes are the passage of food, chyme, and urine. 

Peristalsis is the driving force behind everything from worm movement to the transfer of noxious and 

clean fluids to the operation of finger pumps and the heart-lung machine. Damping, dispensability, 

and tension in the vasculature play a critical part in physiological processes involving peristalsis, such 

as blood flow (2). Studies of peristalsis were first introduced by (3,4). Since then, researchers have 

made numerous attempts to dissect the peristaltic movement of fluids and its implications in the 

medical and business worlds. In biological systems and industrial fluid transport, heat transfer is a 

fundamental principle. One of the most essential roles of the cardiovascular system is maintaining 

the body's temperature. Air that enters the lungs must also be tempered to the body's temperature. 

This is accomplished through the use of all blood vessels. There are three methods of heat 

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
https://doi.org/10.30526/38.1.3462
https://orcid.org/0000-0002-2723-0055
mailto:asmaaaa_math@csw.uobaghdad.edu.iq
https://orcid.org/0000-0001-8236-4120
mailto:liqqa.hummady@sc.uobaghdad.edu.iq


IHJPAS. 2025, 38 (1) 

 

385 

 

 

 

transmission; however, convection is the most relevant for fluid circulation in the human body. 

Human and animal bodies use convection heat transfer to release heat generated by metabolic 

processes into the environment (1). In recent years, the effects of changing viscosity, heat transfer, 

and mass transfer on magnetohydrodynamic (MHD) peristaltic flow in an asymmetric tapering 

inclined channel with porous material were Examined (5). For a high magnetic field like in MHD  

flows, hall current has significant effects. This phenomenon is widely used in a variety of fields, 

including the design of power generators, Hall accelerators, refrigeration coils, electric transformers, 

and spacecraft propulsion systems. The peristaltic transport in the presence of Hall current has been 

the subject of several published works. The effect of Hall current, viscosity variation, and porous 

medium on the peristaltic transport of viscoelastic fluid through irregular microchannels was studied 

(6). The effect of magnetohydrodynamic (MHD) on a viscous fluid generalized burgers' fluid with a 

gradient constant pressure and an exponentially accelerating plate, where the no-slip hypothesis 

between the burgers' fluid and the exponential plate is no longer applicable, were studied (7). (8) 

studied a couple stress of peristaltic transport of Sutterby micropolar nanofluid within a symmetric 

channel with a powerful magnetic field and Hall currents effect. The influence of couple stress as 

well as rotation on the peristaltic flow of a Powell-Eyring in an inclined, tapered, and asymmetrical 

channel investigates by (9). (10) have examined the effect of MHD on a peristaltic flow of Newtonian 

fluid with a couple stress through porous media, where the assumption of no slippage between the 

wall and the fluid is no longer applicable. 

Since (11–13) examined the mechanism of peristaltic transport, it has attracted the interest of 

numerous researchers. Viscous liquids are less prevalent in industrial and physiological processes 

than non-Newtonian fluids. Shampoo, ketchup, lubricants, paints, and blood are all examples of non-

Newtonian substances found in nature. Among that, Sutterby liquid (14) is one of the materials that 

characterize ionic high polymer solutions. Convection and Hall current was used to simulate the MHD 

peristaltic transport of a Sutterby nanofluid (15). Waveform motion of non-Newtonian fluids through 

porous channels is discussed (16,17), where the effects of rotation and an inclined MHD are 

considered. Magnetohydrodynamic (MHD) for Williamson fluid with variable temperatures and 

variable concentrations in a slanted channel with variable viscosity has been investigated (18). The 

effects of radiation and convection in a Sutterby fluid are discussed (19). In Ramesh (20), 

electroosmotic peristaltic transport of Sutterby nanofluids is investigated. The peristaltic flow of a 

Sutterby liquid in an inclined channel was investigated (21).  

In this paper, the study will look at the effects of rotation on heat transfer for peristaltic transport in 

an inclined asymmetric channel with a porous medium. This will be done by using different values 

of the parameters of rotation, amplitude wave, and channel taper, as well as different values of the 

Grashof number, the Hartmann number, the Reynold number, the Froude number, the Hall parameter, 

the Darcy number, the magnetic field, the Sutterby fluid parameter, and heat transfer analysis, based 

on the changes in stream function. 

 

2. Materials and Methods 

2.1. A mathematical formulation for asymmetric flow 

Consider the peristaltic transport of an incompressible Sutterby fluid through a two-dimensional 

asymmetric conduit that has a width of (𝑑′ + 𝑑). whereas motion is constant within a coordinate 

system pumped at wave speed (c) in the wave framer (�̅�,  �̅�).  

The geometry of a wall's structure is described as:  



IHJPAS. 2025, 38 (1) 

 

386 

 

 

 

ℎ1
̅̅ ̅(�̅�, 𝑡̅) = 𝑑 −  𝑎1 sin [

2𝜋

𝜆
(�̅� − 𝑐𝑡̅)]                                                                                             (1) 

ℎ2
̅̅ ̅(�̅�, 𝑡̅) = −𝑑′ − 𝑎2 sin [

2𝜋

𝜆
(�̅� − 𝑐𝑡)̅ + Φ]  (2) 

In which  ℎ1
̅̅ ̅(�̅�, 𝑡̅), ℎ2

̅̅ ̅(�̅�, 𝑡̅) are the lower and upper wall respectively, (𝑑, 𝑑′) indicates the channel 

width, (𝑎1, 𝑎2) are the wave's amplitudes, (𝜆) represents the wavelength, (𝑐) is the speed of a wave, 

(Φ) varies in the range (0 ≤ Φ ≤ 𝜋), when the value of Φ = 0 the channel is symmetric with waves 

out of phase and Φ = 𝜋 waves are in phase the rectangular coordinates has been designed in such a 

method that �̅� − 𝑎𝑥𝑖𝑠 is along the path that waves use for propagation and �̅� − 𝑎𝑥𝑖𝑠 perpendicular 

to �̅�, 𝑡̅  represents the time. 

Further 𝑎1, 𝑎2, 𝑑, 𝑑′  and Φ satisfy the following condition  

𝑎1
2 + 𝑎2

2 + 2a1a2 cos Φ ≤ (𝑑 + 𝑑′)2   (3) 

2.2 Basic equation 

The additional stress tensor for the Sutterby model is determined by (20): 

S̅ =
μ

2
[

sinh−1(nγ̇)

nγ̇
]

𝑚∗

A1  (4) 

γ̇ = √
1

2
tras(A1)2  (5) 

A1 = ∇V̅ + (∇V̅)T  (6) 

Where 𝑆̅ denotes the stress of the extra tensor, n, and 𝑚∗ represents the material constants of the 

Sutterby fluid, 𝛻 = (𝜕�̅�, 𝜕�̅�, 0) is the gradient vector, 𝜇 represents the dynamic viscosity and 

A1 represents the first Rivilin–Ericksen tensor. The phrase sinh−1  is approximately equivalent to 

sinh−1 (
γ̇

𝑛
) =

γ̇

n
−

γ̇3

6n3     , |
γ̇5

6n5| ≪ 1      (7) 

The constituents of the extra stress tensor of Sutterby defined by Equation ((4) are listed below: 

𝑆�̅̅��̅� =
𝜇

2
[1 −

𝑚𝑛2

6
(2�̅��̅�

2
+ (�̅��̅� + �̅��̅�)2 + 2�̅�𝑌

2
)]2�̅��̅�  (8) 

𝑆�̅̅��̅� =
𝜇

2
[1 −

𝑚𝑛2

6
(2�̅��̅�

2
+ (�̅��̅� + �̅��̅�)2 + 2�̅�𝑌

2
)] (�̅��̅� +  �̅��̅�)  (9) 

𝑆�̅̅��̅� =
𝜇

2
[1 −

𝑚𝑛2

6
(2�̅��̅�

2
+ (�̅��̅� + �̅��̅�)2 + 2�̅�𝑌

2
)] 2�̅��̅�  (10) 

2.3. The governing equations 

The flow is controlled by three coupled nonlinear partial differentials of continuity, momentum, and 

energy, the governing equations in frame (�̅�, �̅�) can be written as follows:  
𝜕�̅�

𝜕�̅�
+

𝜕�̅�

𝜕�̅�
= 0    (11) 

𝜌 (
𝜕�̅�

𝜕�̅�
+ �̅�

𝜕�̅�

𝜕�̅�
+ �̅�

𝜕�̅�

𝜕�̅�
) − 𝜌Ω (Ω�̅�  + 2

𝜕�̅�

𝜕�̅�
) =  −

𝜕�̅�

𝜕�̅�
+

𝜕�̅��̅��̅�

𝜕�̅�
+

𝜕�̅��̅��̅�

𝜕�̅�
−

𝜎𝐵0
2  

(1+𝑚2)
(�̅� −

𝑚�̅�)+g𝜌𝛽𝑇(𝑇 − 𝑇0) −
𝜇

𝑘0
�̅� − 𝜇1∇4�̅� + 𝜌𝑔𝑠𝑖𝑛(𝛼1)  

(12) 

𝜌 (
𝜕�̅�

𝜕�̅�
+ �̅�

𝜕�̅�

𝜕�̅�
+ �̅�

𝜕�̅�

𝜕�̅�
) − 𝜌Ω (Ω�̅� − 2

𝜕�̅�

𝜕�̅�
) =  −

𝜕�̅�

𝜕�̅�
+

𝜕�̅��̅��̅�

𝜕�̅�
+

𝜕�̅��̅��̅�

𝜕�̅�
−

𝜎𝐵0
2  

(1+𝑚2)
(�̅� + 𝑚�̅�) −

𝜇

𝑘0
�̅� − 𝜇1∇4�̅�+ 𝜌𝑔𝑐𝑜𝑠(𝛼1)  

(13) 

𝜌𝐶𝑃 (
𝜕

𝜕�̅�
+ �̅�

𝜕

𝜕�̅�
+ �̅�

𝜕

𝜕�̅�
) �̅� = 𝜅 (

𝜕2

𝜕�̅�2 +
𝜕2

𝜕�̅�2 +
𝜕2

𝜕�̅�2) �̅� + 𝜑0  (14) 

 Where 𝜌 is the fluid density, (�̅�,  �̅�) the velocity components, �̅� represents the hydrodynamic  

pressure, 𝑆�̅̅��̅� , 𝑆�̅̅��̅�, and  𝑆�̅̅��̅� are the constituents of the extra stress tensor 𝑆̅. 𝜎  is the electrical 



IHJPAS. 2025, 38 (1) 

 

387 

 

 

 

conductivity, 𝜑0 is the steady heat addition/absorption, 𝐵0 is an applied magnetic field, 𝛽𝑇 is the 

thermal expansion coefficient, g is the gravitational acceleration, 𝜇1 is a constant link to the couple 

stress,  and Ω represents the rotation. The specific heat is denoted by 𝐶𝑃, 𝛼1 is the channel's angle of 

an inclination concerning the horizontal axis, 𝑘0 material constant, the thermal conductivity by 𝑘, and 

the temperature by �̅�. And  

∇2=
𝜕2

𝜕�̅�2 +
𝜕2

𝜕�̅�2 , ∇4=
𝜕4

𝜕�̅�4 + 2
𝜕4

𝜕�̅�2𝜕�̅�2 +
𝜕4

𝜕�̅�4   

Peristaltic movement in reality is an unsteady behavior, but it can be considered to be steady via the 

change from the experimental frame (fixed frame) (�̅�,  �̅�)  to the wave frame (move frame) (�̅�, �̅�). 

The following transformations establish the link between coordinates, velocities, and pressure in 

laboratory frame (�̅�,  �̅�)  to wave frame (�̅�, �̅�): 

�̅� = �̅� − 𝑐𝑡̅  , �̅� = 𝑦 ̅  , �̅� = �̅� − 𝑐 , �̅� = �̅�  , �̅�(�̅�, �̅�, 𝑡̅) =  �̅�(�̅�, �̅�)   (15) 

Where �̅� and �̅� represent the components of velocity, and  �̅� denotes the pressure in the wave frame. 

Now, Equations (15) will be substituted into Equations (1),(2) and (8)-(14) and then normalize the 

equation that is produced by doing so by utilizing the non-dimensional quantities that are listed as 

follows: 

𝑥 =
1

𝜆
�̅�, 𝑦 =

1

𝑑
�̅�, 𝑢 =

1

𝑐
�̅�, 𝑣 =

1

𝑐
�̅�, 𝑎 =

𝑎1

𝑑
, 𝑏 =

𝑎2

𝑑
 , 𝑑1 =

𝑑′

𝑑
, 𝑝 =

𝑑2

𝜆 𝜇 𝑐
�̅�, 𝑡 =

𝑐

𝜆
𝑡̅, ℎ1 =

1

𝑑
ℎ1
̅̅ ̅,  ℎ2 =

1

𝑑
ℎ2
̅̅ ̅, 𝛿 =

𝑑

𝜆
, 𝑅𝑒 =

𝜌 𝑐 𝑑

𝜇
, �̅� = 𝑇 − 𝑇0, 𝜃 =

𝑇−𝑇0

𝑇1−𝑇0
,  𝑆𝑖𝑗 =

𝑑

𝜇 𝑐
𝑆�̅� ̅𝐽̅, 𝐺𝑟 =

𝑔𝛽𝑇(𝑇−𝑇0)𝑑2

𝜇𝑐
, 𝑃𝑟 =

𝜇𝑐𝑝

𝑘
, 𝐹𝑟 =

𝑐2

𝑔𝑑
, 𝐷𝑎 =

𝑘0

𝑑2 , 𝛼 = 𝑑√
𝜇

 𝜇1
     

(16) 

Where, (𝛿) represents the wave number, (ℎ1) 𝑎𝑛𝑑 (ℎ2) is the nondimensional upper and lower wall 

surface respectively, (Re) represents the Reynolds number, (Pr) represents the Prandtl number, (Gr) 

represents the Grashof number, (Fr) represents the Froude number, (M) represents the Hartman 

number, (Da) represents Darcy number, (𝛷) represents the face difference,(A) represents the 

parameter of Sutterby liquid, (𝛼) represents the couple stress parameter, and (𝑇0) 𝑎𝑛𝑑 (𝑇1)  are the 

wall temperatures at the top and bottom, respectively. Then, in view of Equations (16), (1), (2), and 

(8)-(14) take the form : 

ℎ1(𝑥)  =  1 + 𝑎 𝑠𝑖𝑛 𝑥  (17) 

ℎ2(𝑥) = −𝑑1 − 𝑏  𝑠𝑖𝑛 (𝑥 + Ф)  (18) 

𝛿
𝜕𝑢

𝜕𝑥
+

𝜕𝑣

𝜕𝑦
= 0  (19) 

𝑅𝑒 (𝛿
𝜕𝑢

𝜕𝑡
+ 𝛿𝑢

𝜕𝑢

𝜕𝑥
+ 𝑣

𝜕𝑢

𝜕𝑦
) −

𝜌𝑑2

𝜇
𝛺 (𝛺𝑢 + 2𝛿

𝜕𝑣

𝜕𝑡
) =  −

𝜕𝑝

𝜕𝑥
+ 𝛿

𝜕𝑆𝑥𝑥

𝜕𝑥
+

𝜕Sxy

𝜕𝑦
−

𝜎𝐵0
2  

(1+𝑚2)
(𝑢 −

𝑚𝑣) +  𝐺𝑟 𝜃 −
1

𝐷𝑎
𝑢 −

1

𝛼2
(𝛿4 𝜕4𝑢

𝜕𝑥4
+ 2𝛿2 𝜕4𝑢

𝜕𝑥2𝜕𝑦2
+

𝜕4𝑢

𝜕𝑦4
) +

𝑅𝑒

𝐹𝑟
sin(𝛼1)  

 

(20) 

𝑅𝑒𝛿 (𝛿
𝜕𝑣

𝜕𝑡
+ 𝛿𝑢

𝜕𝑣

𝜕𝑥
+ 𝑣

𝜕𝑣

𝜕𝑦
) − 𝑅𝑒

𝑑

𝑐
𝛺 (𝛺𝛿𝑣 − 2𝛿2 𝜕𝑢

𝜕𝑡
) =  −

𝜕𝑝

𝜕𝑦
+ 𝛿2 𝜕𝑆𝑥𝑦

𝜕𝑥
+ 𝛿

𝜕𝑆𝑦𝑦

𝜕𝑦
−

𝜎𝐵0
2  

(1+𝑚2)

𝑑2

𝜇
𝛿(𝑣 + 𝑚𝑢) −

1

𝐷𝑎
𝛿𝑣 −

1

𝛼2 (𝛿5 𝜕4𝑣

𝜕𝑥4 + 2𝛿3 𝜕4𝑣

𝜕𝑥2𝜕𝑦2 + 𝛿
𝜕4𝑣

𝜕𝑦4) +
𝑅𝑒

𝐹𝑟
cos(𝛼1)  

 

(21) 

𝑅𝑒𝑃𝑟𝛿 (
𝜕

𝜕𝑡
+ 𝑢

𝜕

𝜕𝑥
+ 𝑣

𝜕

𝜕𝑦
) 𝜃 = (𝛿2 𝑐2𝜕2

𝜕𝑡2 + 𝛿2 𝜕2

𝜕𝑥2 +
𝜕2

𝜕𝑦2) 𝜃 + 𝐵  (22) 

Introduction to fluid flow (𝜓) through a relationship:  



IHJPAS. 2025, 38 (1) 

 

388 

 

 

 

𝑢 = 𝜓𝑦 , 𝑣 = −𝛿𝜓𝑥  (23) 

 Substituted Equations (23) into Equations (19)-(22) respectively, 

 

𝛿
𝜕𝜓𝑦

𝜕𝑥
− 𝛿

𝜕𝜓𝑥

𝜕𝑦
= 0  (24) 

𝑅𝑒 (𝛿
𝜕𝜓𝑦

𝜕𝑡
+ 𝛿𝜓𝑦

𝜕𝜓𝑦

𝜕𝑥
− 𝛿𝜓𝑥

𝜕𝜓𝑦

𝜕𝑦
) −

𝜌d2

𝜇
𝛺 (𝛺𝜓𝑦 − 2𝛿2 𝜕𝜓𝑥

𝜕𝑡
) =  −

𝜕𝑝

𝜕𝑥
+ 𝛿

𝜕𝑆𝑥𝑥

𝜕𝑥
+

𝜕𝑆𝑥𝑦

𝜕𝑦
−

𝜎𝐵0
2  

(1+𝑚2)
(𝜓𝑦 + 𝑚𝛿𝜓𝑥) +  Gr 𝜃 −

1

𝐷𝑎
𝜓𝑦 −

1

𝛼2 (𝛿4 𝜕4𝜓𝑦

𝜕𝑥4 + 2𝛿2 𝜕4𝜓𝑦

𝜕𝑥2𝜕𝑦2 +
𝜕4𝜓𝑦

𝜕𝑦4 ) +
𝑅𝑒

𝐹𝑟
sin(𝛼1)  

  

(25) 

 𝑅𝑒𝛿 (−𝛿2 𝜕𝜓𝑥

𝜕𝑡
− 𝛿2𝜓𝑦

𝜕𝜓𝑥

𝜕𝑥
− 𝛿2𝜓𝑥

𝜕𝜓𝑥

𝜕𝑦
) − 𝑅𝑒

𝑑

𝑐
𝛺 (−𝛺𝛿2𝜓𝑥 − 2𝛿2 𝜕𝜓𝑦

𝜕𝑡
) =  −

𝜕𝑝

𝜕𝑦
+

𝛿2 𝜕𝑆𝑥𝑦

𝜕𝑥
+ 𝛿

𝜕𝑆𝑦𝑦

𝜕𝑦
+

𝜎𝐵0
2  

(1+𝑚2)

𝑑2

𝜇
𝛿(−𝛿𝜓𝑥 + 𝑚𝜓𝑦) +

1

𝐷𝑎
𝛿𝜓𝑥 −

1

𝛼2 (−𝛿6 𝜕4𝜓𝑥

𝜕𝑥4 − 2𝛿4 𝜕4𝜓𝑥

𝜕𝑥2𝜕𝑦2 −

𝛿2 𝜕4𝜓𝑥

𝜕𝑦4 ) +
𝑅𝑒

𝐹𝑟
cos (𝛼1) 

 

      

(26) 

𝑅𝑒𝑃𝑟𝛿 (
𝜕

𝜕𝑡
+ 𝜓𝑦

𝜕

𝜕𝑥
− 𝛿𝜓𝑥

𝜕

𝜕𝑦
) 𝜃 = (𝛿2 𝑐2𝜕2

𝜕𝑡2 + 𝛿2 𝜕2

𝜕𝑥2 +
𝜕2

𝜕𝑦2) 𝜃 + 𝐵     (27) 

When (𝛿 <<1), the Equations (25)-(27) become as follows: 

−
𝜌d2

𝜇
𝛺2𝜓𝑦 = −

𝜕𝑝

𝜕𝑥
+

𝜕𝑆𝑥𝑦

𝜕𝑦
− (

𝑀2 

1 + 𝑚2
+

1

𝐷𝑎
)𝜓𝑦 +  𝐺𝑟 𝜃 −

1

𝛼2

𝜕5𝜓

𝜕𝑦5
+

𝑅𝑒

𝐹𝑟
sin (𝛼1) (28) 

−
𝜕𝑝

𝜕𝑦
= 0   (29) 

 
𝜕2𝜃

𝜕𝑦2 + B = 0   (30) 

While an additional stress tensor component takes the following form: 

𝑠𝑥𝑦 =
1

2

𝜕2𝜓

𝜕𝑦2 − 𝐴 (
𝜕2𝜓

𝜕𝑦2)
3

, 𝑠𝑥𝑥 = 0, 𝑠𝑦𝑦 = 0   (31) 

Where 𝑀 = √
𝜎  

μ
𝐵0𝑑 the Hartman number, 𝐴 =

𝒎𝒃𝟐𝒄𝟐

𝟔𝒅𝟐
 the Sutterby liquid parameter and 𝐵 =

𝑑2𝜑0

𝑘(𝑇1−𝑇0)
 

the constant heat radiation 

If Equation (31) is substituted into Equation (28) and the derivative concerning y is taken, 

the following equation is obtained 

:
2

𝛼2

𝜕6𝜓

𝜕𝑦6 −
𝜕4𝜓

𝜕𝑦4 [1 − 3𝐴 (
𝜕2𝜓

𝜕𝑦2)
2

] + 6𝐴
𝜕2𝜓

𝜕𝑦2 (
𝜕3𝜓

𝜕𝑦3)
2

− 2 (
𝜌d2

𝜇
Ω2 −

𝑀2

𝑚2+1
−

1

𝐷𝑎
)

𝜕2𝜓

𝜕𝑦2 −

2𝐺𝑟
𝜕𝜃

𝜕𝑦
= 0  

 

(32) 

𝜕2𝜃

𝜕𝑦2 + 𝐵 = 0   (33) 

In wave frames, the dimensionless boundary conditions are: 



IHJPAS. 2025, 38 (1) 

 

389 

 

 

 

𝜕𝜓

𝜕𝑦
+ 𝛽

𝜕2𝜓

𝜕𝑦2
= −1, 𝜓 =

𝐹

2
,

𝜕3𝜓

𝜕𝑦3
= 0      𝑎𝑡 𝑦 = ℎ1   (34) 

𝜕𝜓

𝜕𝑦
− 𝛽

𝜕2𝜓

𝜕𝑦2 = −1, 𝜓 = −
𝐹

2
,

𝜕3𝜓

𝜕𝑦3 = 0    𝑎𝑡 𝑦 = ℎ2  (35) 

𝜃 = 0  𝑎𝑡 𝑦 = ℎ1 , 𝜃 = 1   𝑎𝑡 𝑦 = ℎ2  (36) 

Where F is just the flow rate, which is dimensionless in time in the frame of the wave. It is associated 

with the form that has no dimensions' temporal flow rate Q1 in the experimental frame via the 

expression: 

𝑄1 = 𝐹 + 1 + 𝑑   (37) 

as 𝑎, 𝑏, Ф, and d achieve Equation (3): 

𝑎2 + 𝑏2 + 2𝑎𝑏𝑐𝑜𝑠(Φ) ≤ (1 + 𝑑1)2  (38) 

Initially, the nonlinear equation Equation (33) is solved by integrating and substituting the boundary 

conditions Equation (36), and then the solution to Equation (37) is obtained : 

𝜃 = −
−2h1+h1

2h2𝐵−h1h2
2𝐵

2(ℎ1−h2)
−

(2−h1
2𝐵+h2

2𝐵)𝑦

2(h1−h2)
−

𝐵𝑦2

2
  (39) 

By differentiating Equation (39) to y and substituting it into Equation (32), obtaining the following 

nonlinear equation: 

2

𝛼2

𝜕6𝜓

𝜕𝑦6 −
𝜕4𝜓

𝜕𝑦4 [1 − 3𝐴 (
𝜕2𝜓

𝜕𝑦2)
2

] + 6𝐴
𝜕2𝜓

𝜕𝑦2 (
𝜕3𝜓

𝜕𝑦3)
2

− 2 (
𝜌d2

𝜇
Ω2 −

𝑀2

𝑚2+1
−

1

𝐷𝑎
)

𝜕2𝜓

𝜕𝑦2 +

2𝐺𝑟 (−
(2−h1

2𝐵+h2
2𝐵)

2(h1−h2)
−   By) = 0     

(40) 

2.4. Solution of the problem 

It is not possible to that construct a solution in closed form for every one of the arbitrary parameters 

involved in Equation (40), as it is highly non-linear and convoluted. Therefore, the perturbation 

approach is used to get the answer. The solution was expanded to include perturbation (22) : 

𝜓 = 𝜓0 + 𝐴𝜓1 + 𝑜(𝐴2)   (41) 

And by substituting the expressions Equation (41) into Equation (40), along with the boundary 

conditions Equation (34) and Equation (35) and equating the coefficients of similar powers of A, The 

following system of equations is obtained: 

2.4.1. Zeroth order system 

When such terms of order (A) in a zero-order system are negligible, the result is 

2

𝛼2

𝜕6𝜓0

∂y6 −
𝜕4𝜓0

∂y4 − 𝜁
𝜕2𝜓0

∂y2 + 𝛾𝑦 − 𝜂 = 0   (42) 

Where 𝜁 = 2(
𝜌𝑑2

𝜇
Ω2 −

1

𝐷𝑎
−

𝑀2

𝑚2+1
), 𝛾 = 2𝐺𝑟𝐵, and 𝜂 = 𝐺𝑟[𝐵(ℎ1 + ℎ2) −

2

ℎ1−ℎ2
] 



IHJPAS. 2025, 38 (1) 

 

390 

 

 

 

Such that 

𝜕𝜓0

𝜕𝑦
+ 𝛽1

𝜕2𝜓0

𝜕𝑦2
= −1, 𝜓0 =

𝐹0

2
,

𝜕3𝜓0

𝜕𝑦3
= 0      𝑎𝑡 𝑦 = ℎ1  (43) 

and  

𝜕𝜓0

𝜕𝑦
− 𝛽1

𝜕2𝜓0

𝜕𝑦2 = −1, 𝜓0 = −
𝐹0

2
,

𝜕3𝜓0

𝜕𝑦3 = 0    𝑎𝑡 𝑦 = ℎ2  (44) 

2.4.2. First order system 

2

𝛼2  
𝜕6𝜓1

∂y6 −
𝜕4𝜓1

∂y4 − 𝜁
𝜕2𝜓1

∂y2 + 3
∂4𝜓0

𝜕𝑦4 (
𝜕2𝜓0

𝜕𝑦2 )
2

+ 6
𝜕2𝜓0

𝜕𝑦2 (
𝜕3𝜓0

𝜕𝑦3 )
2

+ 𝛾𝑦 − 𝜂 = 0  (45) 

𝜕𝜓1

𝜕𝑦
+ 𝛽1

𝜕2𝜓1

𝜕𝑦2
= −1, 𝜓1 =

𝐹1

2
,

𝜕3𝜓1

𝜕𝑦3
= 0      𝑎𝑡 𝑦 = ℎ1   (46) 

and  

𝜕𝜓1

𝜕𝑦
− 𝛽1

𝜕2𝜓1

𝜕𝑦2
= −1, 𝜓1 = −

𝐹1

2
,

𝜕3𝜓1

𝜕𝑦3
= 0    𝑎𝑡 𝑦 = ℎ2  (47) 

Solving the relevant zeroth-order and first-order systems yields the final stream function equation. 

𝜓 = 𝜓0 + A𝜓1  

 
(48) 

3. Results 

        This section consists of one subsection. Using MATHEMATICA, the velocity distribution is 

depicted in the first, and the pressure gradient is presented in the second. 

Trapping Phenomena is another fascinating phenomenon of peristaltic motion. Essentially, it is the 

production of an internally circulating fluid gap utilizing a closed streamline. This captured gap 

propelled the head along peristaltic waves. (Figure 1-Figure 9) Describe the effect of the parameters 

𝜴, 𝑴, 𝑮𝒓, 𝒎, 𝑨, 𝜶, 𝑩, 𝑫𝒂, 𝒂𝒏𝒅 𝝓 on stream function.  

 

        
Figure 1. Distribution of streamlines for (a)𝛀=1, (b)𝛀=1.033, (c)𝛀=1.066, 𝑴 = 𝟎. 𝟗𝟗, 𝐆𝐫 = 𝟎. 𝟓, 𝒎 = 𝟎. 𝟎𝟓, 𝑨 = 𝟑, 𝜶 =

𝟐. 𝟓, 𝑩 = 𝟏, 𝐃𝐚 = 𝟔, 𝝓 = 𝟐. 𝟒, 𝝆 = 𝟏, 𝒅 = 𝟏, 𝝁 = 𝟏, 𝒂 = 𝟎. 𝟖, 𝒃 = 𝟎. 𝟖, 𝒅𝟏 = 𝟎. 𝟎𝟎𝟏,  𝑭𝟎 = 𝟎. 𝟎𝟏, 𝜷𝟏 = 𝟎. 𝟎𝟓                                 

(1)  In 2 dimensions  (2)  In 3 dimensions. 

upper 

gap 

lower 

gap 

upper 

gap lower 

gap 

upper 

gap 
lower 

gap 



IHJPAS. 2025, 38 (1) 

 

391 

 

 

 

 

 
Figure 2. Distribution of streamlines for 𝛺=1, (a)𝑀=0.92,(b)𝑀 = 0.95, (𝑐)𝑀 = 0.99, Gr = 0.5, 𝑚 = 0.05, 𝐴 = 3, 𝛼 =
2.5, 𝐵 = 1, Da = 6, 𝜙 = 2.4, 𝜌 = 1, 𝑑 = 1, 𝜇 = 1, 𝑎 = 0.8, 𝑏 = 0.8, 𝑑1 = 0.001,  𝐹0 = 0.01, 𝛽1 = 0.05                                      

(1)  In 2 dimensions  (2)  In 3 dimensions 

 

 
Figure 3. Distribution of streamlines for 𝛺=1, 𝑀 = 0.99, (𝑎)𝐺𝑟 = 1, (𝑏)𝐺𝑟 = 5, (𝑐)Gr = 10, 𝑚 = 0.05, 𝐴 = 3, 𝛼 =
2.5, 𝐵 = 1, Da = 6, 𝜙 = 2.4, 𝜌 = 1, 𝑑 = 1, 𝜇 = 1, 𝑎 = 0.8, 𝑏 = 0.8, 𝑑1 = 0.001, 𝐹0 = 0.01, 𝛽1 = 0.05                                       

(1) In 2 dimensions  (2)  In 3 dimensions.. 

upper 

gap lower 

gap 

upper 

gap lower 

gap 

upper 

gap lower 

gap 

upper 

gap 
lower 

gap 

upper 

gap lower 

gap 

upper 

gap 
lower 

gap 



IHJPAS. 2025, 38 (1) 

 

392 

 

 

 

 

 
Figure 4. Distribution of streamlines for 𝛺=1, 𝑀 = 0.99, 𝐺𝑟 = 0.5, (𝑎)𝑚 = 0.05, (𝑏)𝑚 = 0.2, (𝑐)𝑚 = 0.4, 𝐴 = 3, 𝛼 =
2.5, 𝐵 = 1, 𝐷𝑎 = 6, 𝜙 = 2.4, 𝜌 = 1, 𝑑 = 1, 𝜇 = 1, 𝑎 = 0.8, 𝑏 = 0.8, 𝑑1 = 0.001,  𝐹0 = 0.01, 𝛽1 = 0.05                                   

(1) In 2 dimensions (2)  In 3 dimensions 

 

 
Figure 5. Distribution of streamlines for 𝛺=1, 𝑀 = 0.99, Gr = 0.5, 𝑚 = 0.05, (𝑎)𝐴 = 1, (𝑏)𝐴 = 5, (𝑐)𝐴 = 9, 𝛼 =
2.5, 𝐵 = 1, Da = 6, 𝜙 = 2.4, 𝜌 = 1, 𝑑 = 1, 𝜇 = 1, 𝑎 = 0.8, 𝑏 = 0.8, 𝑑1 = 0.001, 𝐹0 = 0.01, 𝛽1 = 0.05                                                      

(1) In 2 dimensions  (2)  In 3 dimensions 

 

upper 

gap lower 

gap 

upper 

gap lower 

gap 

upper 

gap lower 

gap 

upper 

gap lower 

gap 

upper 

gap 
lower 

gap 

upper 

gap lower 

gap 



IHJPAS. 2025, 38 (1) 

 

393 

 

 

 

 

 
Figure 6. Distribution of streamlines for 𝛺=1, 𝑀 = 0.99, 𝐺𝑟 = 0.5, 𝑚 = 0.05, 𝐴 = 3, (𝑎)𝛼 = 2.5, (𝑏)𝛼 = 2.75, (𝑐)𝛼 =
3, 𝐵 = 1, 𝐷𝑎 = 6, 𝜙 = 2.4, 𝜌 = 1, 𝑑 = 1, 𝜇 = 1, 𝑎 = 0.8, 𝑏 = 0.8, 𝑑1 = 0.001,  𝐹0 = 0.01, 𝛽1 = 0.05                                        

(1) In 2 dimensions (2)  In 3 dimensions 

     

 
Figure 7. Distribution of streamlines for Ω=1, M = 0.99, Gr = 0.5, m = 0.05, A = 3, α = 2.5, (a)B = 1, (b)B =
2, (c)B = 3, Da = 6, ϕ = 2.4, ρ = 1, d = 1, μ = 1, a = 0.8, b = 0.8, d1 = 0.001,  F0 = 0.01, β1 = 0.05                      
(1) In 2 dimensions  (2) In 3 dimensions. 

 

upper 

gap lower 

gap 

upper 

gap lower 

gap 

upper 

gap 
lower 

gap 

upper 

gap lower 

gap 

upper 

gap lower 

gap 

upper 

gap lower 

gap 



IHJPAS. 2025, 38 (1) 

 

394 

 

 

 

    

 
Figure 8.  Distribution of streamlines for 𝛺=1, 𝑀 = 0.99, 𝐺𝑟 = 0.5, 𝑚 = 0.05, 𝐴 = 3, 𝛼 = 2.5, 𝐵 = 1, (a)𝐷𝑎 =
3, (𝑏)𝐷𝑎 = 6, (𝑐)𝐷𝑎 = 9, 𝜙 = 2.4, 𝜌 = 1, 𝑑 = 1, 𝜇 = 1, 𝑎 = 0.8, 𝑏 = 0.8, 𝑑1 = 0.001,  𝐹0 = 0.01, 𝛽1 = 0.05                    

(1) In 2 dimensions (2)  In 3 dimensions. 

 

 
Figure 9.  Distribution of streamlines for 𝛺=1, 𝑀 = 0.99, 𝐺𝑟 = 0.5, 𝑚 = 0.05, 𝐴 = 3, 𝛼 = 2.5, 𝐵 = 1, 𝐷𝑎 = 6, (𝑎)𝜙 =
2.1, (𝑏)𝜙 = 2.4, (𝑐)𝜙 = 2.7, 𝜌 = 1, 𝑑 = 1, 𝜇 = 1, 𝑎 = 0.8, 𝑏 = 0.8, 𝑑1 = 0.001,  𝐹0 = 0.01, 𝛽1 = 0.05                              

(1) In 2 dimensions (2)  In 3 dimensions. 
 

4. Discussions 

        As show in result, the upper gap size and the lower gap size do not affect by an increase in the 

rotation (Ω). The upper gap size and the lower gap size do not affect by increasing the Hartmann 

number (M). With increasing the Thermal Grashof number (Gr), the size of the upper gap size and 

lower gap decreases. The size of the upper gap and lower gap have no effect by increasing Hall 

parameter (m). The size of the upper and lower gaps decreases with the increase of the fluid parameter 

(A). The upper gap size and the lower gap size don't change with increasing the couple stress 

upper 

gap lower 

gap 

upper 

gap lower 

gap 

upper 

gap lower 

gap 

upper 

gap lower 

gap 

upper 

gap lower 

gap 

upper 

gap lower 

gap 



IHJPAS. 2025, 38 (1) 

 

395 

 

 

 

parameter (𝛼). The upper gap and lower gap slightly decreases with increasing the constant heat 

radiation (𝐵). As the Darcy number (𝐷𝑎) goes up, the size of the upper gap and lower gap don't 

change. The size of the upper gap and lower gap decreases with the increase of the wavelength (𝜙). 

 

5. Conclusions  

       In this article, the influence of heat transfer and rotation on a Sutterby fluid in an asymmetric 

channel was investigated. In this investigation, a lot of attention has been paid to the analysis of things 

like stream function based on a simple analytical solution. The key findings of the current research 

are summarized below: 

❖ As (Ω), (M), (m), and (Da) goes up, the upper gap size and the lower gap size have no effect.  

❖ As (𝑮𝒓), (𝑨), (𝜶), and (𝝓) increase, The size of the upper gap and lower gap slightly decreases. 

❖ As (𝑩) increases, The size of the upper gap and lower gap slightly decreases. 

 

Acknowledgment 

      We hereby confirm that all the Figures and Tables in the manuscript are ours. Furthermore, any 

figures and images, that are not ours, have been included with the necessary permission for 

republication, which is attached to the manuscript. 

 

Conflict of Interest  

      The authors declare that they have no conflicts of interest. 

 

Ethical Clearance 

      The project was approved by the local ethical committee in University of Baghdad. 

 

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