304 

© 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 

 
 

 
Analysis of Stream Function in a Powell-Eyring Fluid Flow with Couple-

Stress Effect and Rotation  

 

Rana Ghazi Ibraheem 1* and Liqaa Zeki Hummady 2   
1,2 Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq. 

*Corresponding Author.  

 

 Received: 8 May 2023                          Accepted: 21 August 2023                    Published: 20 April 2025 

 doi.org/10.30526/38.2.3478 

 
 Abstract   

     The effect of couple stress, slip condition, changing rotation, and other variables on the 

peristaltic flow of Powell-Eyring fluid in an inclined asymmetric channel with an inclining 

magnetic field through a porous media is investigated in this study. Constitutive equations 

obeying the Powell-Eyring fluid model are employed. In flow analysis, assumptions such as a 

low Reynolds number and a long wavelength approximation are used. The stream function and 

mechanical efficiency have closed form expressions devised. The stream function is expressed 

mathematically. Through the collection of figures, the impact of various criteria is explained and 

graphically represented Through the collection of figures, the impact has been explained of the 

Hartman number (Ha), the Darcy number (Da), the inclination of the magnetic field (β), the 

rotation (Ω), the porous medium parameter (w), the amplitude ratio (ϕ),  the slip condition (𝛽1), 

and couple stress parameter (𝛼) on stream function have been explained and graphically 

represented by using the perturbation method which is analytic method. These numerical results 

were achieved using the mathematical application MATHEMATICA. 

Keywords: Couple stress, Magnetic felid, Peristaltic flow, Porous medium, Powell- Eyring fluid. 

 

1. Introduction 

     Peristaltic pumping is a particular type of pumping that occurs when a variety of complex 

rheological fluids may be easily transferred between two locations. The term "peristaltic" refers 

to this method of pumping. The ducts through which the fluid passes undergo intermittent 

involuntary constriction and then expand. As a result, the pressure gradient rises, causing the 

fluid to move forward. After Latham's groundbreaking work(1) and because it is utilized in 

biological, engineering, and physiological systems academics have become increasingly 

interested in the different applications of peristalsis. Because it is utilized in biological, 

engineering, and physiological systems, peristaltic transport has received significant attention in 

recent years. Generally, the peristaltic wave’s circular contractions and the successive 

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0000-0002-6764-2003‎
mailto:rana.ghazi1103a@cs.uobaghdad.edu.iq‎
https://orcid.org/0000-0001-8236-4120‎
mailto:‎liqqa.hummady@sc.uobaghdad.edu.iq


IHJPAS. 2025, 38(2) 

 

305 
 

longitudinal contractions that occur during peristalsis are generated by the sinuses which 

propagate along the fluid-containing duct. This technique is the basis for several muscular tubes, 

including the gastrointestinal tract, fallopian tubes, bile ducts, ureters, esophageal tubes, and 

others. Peristalsis is used as the basis for the creation of devices such as peristaltic pumps, roller 

pumps, hose pumps, tube pumps, finger pumps, heart-lung machines, blood pump machines, and 

dialysis machines. These applications include the transportation of aggressive chemicals, high 

solid slurries, toxic (nuclear industries), and other materials. Moreover, non-Newtonian fluids 

are better than numerous industrial and physiological processes that use Newtonian fluids. 

Among the models of non-Newtonian fluids (which can exhibit various rheological effects), that 

can be accessed is Powell- Eyring fluid. Although this model is more difficult mathematically 

than models of non-Newtonian fluids, it deserves more attention because of its distinct benefits. 

Numerous researchers have been interested in the Powell-Eyring fluid's peristaltic flow 

mechanism since it was studied by Hina and Mustafa and Hayat and Alsaedi (2), Hayat and 

Naseema and Rafiq and Fua (3), Hayat and Ahmed (4), Hussain and Alvi and Latif and Asghar 

(5), and Ali and Liqaa (6). The static magnetohydrodynamic flow and heat transfer of an Eyring-

Powell fluid on an expansion plate with viscous dissipation were studied and numerically 

explained (7). The exchange of thermal energy between different system components is referred 

to as heat transfer. However, the medium's physical characteristics and the separate 

compartments' temperatures affect the speed. In recent years, research (8–11) has been conducted 

about studying the effect of heat transport on non-Newtonian fluids. In a tapered asymmetric 

channel, the issue of peristaltic transport of an incompressible non-Newtonian fluid is examined 

(12). With regard to well-established problems of the stir of semi-conductive physiological 

fluids, such as blood and blood pump machines, magnetic drug forcing, and pertinent methods 

of human digestion, the advantage of applied magnetic field (MHD) on peristaltic efficacy is 

crucial (13). It is also helpful in treating gastroparesis, chronic constipation, and morbid obesity 

as well as magnetic resonance imaging (MRI), which is used to identify brain, vascular diseases, 

and tumors. A substance that has several tiny holes scattered throughout it is referred to as a 

porous medium. In riverbeds, fluid infiltration and seepage are sustained by flows over porous 

media. Important examples of flows through a porous material are those through the ground, 

water, and oil. Oil is trapped in rock formations like limestone and sandstone, which make up 

the majority of an oil reservoir (14). Natural porous media can be found in many different forms, 

such as sand, rye bread, wood, filters, bread loaves, human lungs, and the gallbladder. Food 

processing, oxygenation, hemodialysis, tissue condition, heat convection for blood flow from 

tissues' pores, and radiation between the environment and its surface all depend on the action of 

heat transfer in the peristaltic repositioning of fluid (15–18). Research into fluid peristaltic 

transfer in the presence of an external magnetic field and rotation is necessary for many issues 

involving the flow of conductive physiological fluids, such as blood and saline water (19,20). A 

variety of values are used for the rotational parameters, the porous medium, density, amplitude 

wave, and taper of the channel, as well as a variety of values for the Hartman number and Darcy 

number, to study the effects of varying the velocity and pressure gradient. The ability to 

characterize theologically complicated fluids like liquid crystals and human blood makes the 

study of pair stress fluid extremely helpful in understanding a variety of physical issues.Couple 

stress fluid refers to a specific type of non-Newtonian fluid in which the size of the fluid's 

particles is taken into consideration. Recent researches on the couple stress fluid has been 



IHJPAS. 2025, 38(2) 

 

306 
 

conducted (21–24). This study's goal is to examine the couple-stress, slip condition, and rotation 

impacts of Powell-Eyring fluid peristaltic transport in porous media under the combined 

influence of inclined MHD. Where it was observed that increasing the values of the couple stress 

coefficient (α) leads to an increase in the size of the bolus trapped and approaching the channel 

wall, while the increases in the values of rotation (Ω) led to a decrease in the size of the bolus 

trapped and moving away from the channe walls and  when the values of the slip condition (𝛽1) 

decrease leading to the decrease in the size of the trapped bolus and moving away from the 

channel walls. 

 

2. Materials and Methods 

2.1. A mathematical formulation for asymmetric flow 

Consider the peristaltic motion of an incompressible Powell-Eyring fluid in a two-dimensional, 

asymmetric conduit with a width of (d'+d). An endless sinusoidal wave traveling along the channel walls 

at a constant forward speed (c) is what generates flow. The geometry of the wall structure is described 

as: 

h1
̅̅ ̅(x̅, t)̅ = d − a1 sin [

2π

λ
(X̅ − ct)̅]     (1) 

 h2
̅̅ ̅(x̅, t)̅ = −d′ − a2 sin [

2π

λ
(X̅ − ct)̅ + Φ]     (2) 

 In which  ℎ1
̅̅ ̅(�̅�, 𝑡̅), ℎ2

̅̅ ̅(�̅�, 𝑡)̅ are the lower and upper walls respectively, (𝑑, 𝑑′)denote the channel 

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

(Φ) varies in the range (0 ≤ Φ ≤ 𝜋), when  Φ = 0 is a symmetric channel with out-of-phase 

waves and Φ = 𝜋 waves are in phase, the rectangular coordinate system is chosen so that the 

�̅� − 𝑎𝑥𝑖𝑠 is in the direction of the wave's motion. and the �̅� − 𝑎𝑥𝑖𝑠 perpendicular to �̅�, where  𝑡̅  

is the time  

Further 𝑎1, 𝑎2 , 𝑑, 𝑑′  and Φ fulfill the following condition:   

𝑎1
2 + 𝑎2

2 + 2𝑎1𝑎2 cos Φ ≤ (𝑑 + 𝑑′)2     (3) 

The Cauchy stress tensor 𝜏̅  for a fluid that obeys the Powell- Eyring model is given as follows:- 

 τ̅ = −PI + S̅    (4) 

 
Figure 1. Coordinates for Inclined Asymmetric Channels in Cartesian Space 

 

 S̅ = [μ +
1

βγ
sinh−1 (

γ̇

c1
)] A11     (5) 

  γ̇ = √
1

2
tras(A11)2     (6) 

 A11 = ∇V̅ + (∇V̅)T     (7) 

 Where S̅ is the extra stress tensor, I is the identity tensor, ∇= (𝜕�̅�, 𝜕�̅�, 0) is the gradient vector, 



IHJPAS. 2025, 38(2) 

 

307 
 

(𝛽, 𝑐1) are the material parameters of Powell-Eyring fluid, P is the fluid pressure, and 𝜇 the 

dynamic viscosity. The term sinh−1  is approximately equivalent to 

 sinh−1 (
γ̇

c1
) =

γ̇

c1
−

γ̇3

6c1
3     , |

γ̇5

6c1
5| ≪ 1     (8) 

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

and energy, which are expressed in frame (�̅�, �̅�) as  

 
∂U̅

∂X̅
+

∂V̅

∂Y̅
= 0     (9) 

 ρ (
∂U̅

∂t̅
+ U̅

∂U̅

∂X̅
+ V̅

∂U̅

∂Y̅
) − ρΩ (ΩU̅  + 2

∂V̅

∂t̅
) =  −

∂P̅

∂X̅
+

∂S̅X̅X̅

∂X̅
+

∂S̅X̅Y̅

∂Y̅
− σβ0

2 cosβ(U̅cosβ − V̅sinβ) 

−
μ

k̅
 U̅ − μ1∇4U̅ + pg sin α∗    (10)                 

 ρ (
∂V̅

∂t̅
+ U̅

∂V̅

∂X̅
+ V̅

∂V̅

∂Y̅
) − ρΩ (ΩV̅  + 2

∂V̅

∂t̅
) =  −

∂P̅

∂Y̅
+

∂S̅X̅Y̅

∂X̅
+

∂S̅Y̅Y̅

∂Y̅
− σβ0

2 sinβ(U̅cosβ − V̅sinβ) 

−
μ

k̅
 V̅ − μ1∇4V̅ + pg cos α∗    (11) 

 ρCP (
∂

∂t̅
+ U̅

∂

∂X̅
+ V̅

∂

∂Y̅
) T̅ = k′ (

∂2

∂t̅2 +
∂2

∂X̅2 +
∂2

∂Y̅2) T̅ + μ [(
∂U̅

∂Y̅
+

∂V̅

∂X̅
)

2

+ 2 (
∂U̅

∂X̅
)

2

+ 2 (
∂V̅

∂Y̅
)

2

]      (12) 

Let ∇2= (
∂2

∂X ̅2
+

∂2

∂X̅2)    then  ∇4= (∇2)2 

Where 𝜌 is the fluid density, �̅� = [�̅�, �̅�] is the velocity vector, �̅� is the hydrodynamic pressure, 

𝑆�̅̅��̅� , 𝑆�̅̅��̅� , 𝑎𝑛𝑑  𝑆�̅̅��̅� are the elements of the extra stress tensor 𝑆̅, 𝜎  is the electrical conductivity, 𝛽0  

is the constant magnetic field,  𝛽  is the inclination of the magnetic field, Ω is the rotation 𝐶𝑃 is 

specific heat, 𝑘′ is the thermal conductivity, �̅� is a temperature, and 𝜇 for viscosity. 

Listed below are the parts of Powell-additional Eying's stress tensor, as described by Equation (5)   

S̅X̅X̅ = 2 (μ +
1

βC1
) U̅X̅ −

1

3βC1
3 [2U̅X̅

2
+ (V̅X̅ + U̅Y̅)2 + 2V̅Y

2
] U̅X̅   (13) 

S̅X̅Y̅ = 2 (μ +
1

βC1
) (V̅X̅ + U̅Y̅) −

1

6βC1
3 [2U̅X̅

2
+ (V̅X̅ + U̅Y̅)2 + 2V̅Y

2
] (V̅X̅ + U̅Y̅)   (14) 

 S̅Y̅Y̅ = 2 (μ +
1

βC1
) V̅Y̅ −

1

3βC1
3 [2U̅X̅

2
+ (V̅X̅ + U̅Y̅)2 + 2V̅Y

2
] V̅Y̅   (15) 

Natural peristaltic motion is an erratic occurrence, but it applying the transformation from the 

laboratory frame, stability can be assumed (fixed frame) (�̅�, �̅�)to wave frame (move frame) (�̅�, �̅�). 

The subsequent transformations determine the connection between pressure, velocities, and 

coordinates in a laboratory frame (�̅�, �̅�)  to wave frame (�̅�, �̅�)  

x̅ = X̅ − c, y̅ = Y ̅, u̅ = U̅ − c, v̅ = V̅,  p̅(x̅, y̅) = P̅(X̅, Y̅, t)̅   (16) 

Where  �̅� and �̅� represent the velocity factors and  �̅� represents the pressure in the wave frame. 

Now that Equation (15) has been substituted into Equation (1), (2), and (9) –(14), the resulting 

equation has been normalized using the non-dimensional variables shown below: 

𝑥 =
1

𝜆
�̅�, 𝑦 =

1

𝑑
�̅�, 𝑢 =

1

𝑐
�̅�, 𝑣 =

1

𝛿𝑐
�̅�, 𝑝 =

𝑑2

𝜆 𝜇 𝑐
�̅�, 𝑡 =

𝑐

𝜆
𝑡,̅ ℎ1 =

1

𝑑
ℎ1
̅̅ ̅, ℎ2 =

1

𝑑
ℎ2
̅̅ ̅, 𝛿 =

𝑑

𝜆
 , 𝑅𝑒 =

𝜌 𝑐 𝑑

𝜇
,   

 𝐻𝑎 = 𝑑√
𝜎

𝜇
𝛽0, 𝐷𝑎 =

�̅�

𝑑2, 𝑤 =
1

𝜇 𝛽 𝐶1
 , 𝐴 =

𝑤

6
(

𝐶

𝐶1 𝑑
)

2

,  �̅� = 𝑇 − 𝑇0, 𝜃 =
𝑇−𝑇0

𝑇1−𝑇0
 , 𝐹𝑟 =

𝑐2

𝑑𝑔
, 𝛽1 =

𝛽∗

𝑑
, 

Sxx =
λ

μ c
S̅X̅X̅,  Sxy =

d

μ c
S̅X̅Y̅, Syy =

d

μ c
S̅Y̅Y̅   (17) 

Where, (𝛿) is the wave number, (ℎ1) and (ℎ2) are non-dimensional lower and upper wall surfaces 

respectively, (Re) is the Reynolds number, (Ha) is the Hartman number, (𝛷) is the amplitude ratio, 

(w) is the non-dimensional permeability of the porous medium parameter, (Da) is the Darcy 

number, (A) is the Powell-Eyring fluid parameter, (𝑇0) and (𝑇1)  are the temperatures at the upper 

and lower walls, (Fr) is the Froude number and (𝛼∗)  inclination angle of the channel to the 



IHJPAS. 2025, 38(2) 

 

308 
 

horizontal axis. 

Following that is 

h1(x, t) = 1 − a sin(2πx),  a =
a1

d
            (18) 

 h2(x, t) = −d∗ − b sin (2πx + Φ) , b =
a2

d
 , d∗ =

d′

d
           (19) 

Where a, b, d*, and satisfy Eq.3, then 

a2 + b2 + 2ab cos Φ ≤ (1 + d∗)2    (20) 
∂u

∂x
+

∂v

∂y
= 0   (21) 

𝑅𝑒𝛿 (
𝜕𝑢

𝜕𝑡
+ 𝑢

𝜕𝑢

𝜕𝑥
+ 𝑣

𝜕𝑢

𝜕𝑦
) −

𝜌𝑑2Ω

𝜇
(Ω𝑢 + 2

𝛿𝑐

𝑑
  

𝜕𝑣

𝜕𝑡
) = −

𝜕𝑝

𝜕𝑥
+ 𝛿2  

𝜕

𝜕𝑥
  𝑆𝑥𝑥 +

𝜕

𝜕𝑦
  𝑆𝑥𝑦 

−Ha2 cos β (u cos β − δv sin β) −
1

Da
 u −

1

∝2
(δ4 ∂4

∂x4
+ 2δ2 ∂4

∂x2 ∂y2
+

∂4

∂y4
) u +

Re

Fr
sin α∗(22)Reδ3 (

∂v

∂t
+ u

∂v

∂x
+ v

∂v

∂y
) −

ρd2Ωδ

μ
(Ωδv + 2

δc

d
  

∂v

∂t
) = −

∂p

∂x
+ δ2  

∂

∂x
  Sxy + δ

∂

∂y
  Syy 

+Ha2 sin β (δu cos β − δ2v sin β) − δ2 1

Da
 v −

δ2

∝2 (δ4 ∂4

∂x4 + 2δ2 ∂4

∂x2 ∂y2 +
∂4

∂y4) v +

δ
Re

Fr
cos α∗   (23) 

Reδ (
∂θ

∂t
+ u

∂θ

∂x
+ v

∂θ

∂y
) =

1

Pr
(c2δ2 ∂2θ

∂t2 + δ2 ∂2θ

∂x2 +
∂2θ

∂y2) + Ec [(
∂u

∂y
+ δ2 ∂v

∂x
)

2

+ 2δ2 (
∂u

∂x
)

2

+

2δ2 (
∂v

∂y
)

2

]   (24) 

Sxx = 2(1 + w)
∂u

∂x
− 2A [2δ2 (

∂u

∂x
)

2

+ (
∂u

∂y
+ δ2 ∂v

∂x
)

2

+ 2δ2 (
∂v

∂y
)

2

]
∂u

∂x
   (25) 

 Sxy = (1 + w) (δ2 ∂v

∂x
+

∂u

∂y
) − A [2δ2 (

∂u

∂x
)

2

+ (
∂u

∂y
+ δ2 ∂v

∂x
)

2

+ 2δ2 (
∂v

∂y
)

2

] (δ2 ∂v

∂x
+

∂u

∂y
)   (26) 

 Syy = 2(1 + w) (δ
∂v

∂y
) − 2Aδ [2δ2 (

∂u

∂x
)

2

+ (
∂u

∂y
+ δ2 ∂v

∂x
)

2

+ 2δ2 (
∂v

∂y
)

2

]   (27) 

In previous equations, Pr is the Prandtl number, Ec is the Eckert number and θ is the dimensionless 

temperature. 

Following are the relations between the stream function (ψ) and velocity components: 

 u =
∂Ψ

∂y
, v = −

∂Ψ

∂x
   (28) 

Substituting Equation (28) into Equations (21) to (270, noting that the mass balance displayed by 

Equation (21) is similarly satisfied, produces the consequence that Equation (28) is satisfied. 

 Re δ (
∂2Ψ

∂t ∂y
+

∂3Ψ

∂x ∂y2 −
∂3Ψ

∂x ∂y2) −
ρd2Ω

μ
(Ω

∂Ψ

∂y
− 2

δc

d
  

∂2Ψ

∂t ∂x
) = −

∂p

∂x
+ δ2 ∂

∂x
 Sxx + 

∂

∂y
Sxy −

Ha2 cos β (
∂Ψ

∂y
 cos β + δ

∂Ψ

∂x
sin β) −

1 

Da

∂Ψ

∂y
−

1

∝2 (δ4 ∂4

∂x4 + 2δ2 ∂4

∂x2 ∂y2 +
∂4

∂y4)
∂Ψ

∂y
+

Re

Fr
sin α∗  

   (29) 

 Re δ3 (−
∂2Ψ

∂t ∂x
−

∂3Ψ

∂x2 ∂y
−

∂3Ψ

∂x2 ∂y
) −

ρd2Ωδ

μ
(−Ωδ

∂Ψ

∂x
− 2

δc

d
  

∂2Ψ

∂t ∂x
) = −

∂p

∂y
+ δ2 ∂

∂x
 Sxy + δ

∂

∂y
Syy +

Ha2 sin β (δ
∂Ψ

∂y
 cos β + δ2 ∂Ψ

∂x
sin β) + δ2 1 

Da

∂Ψ

∂x
+

δ2

∝2
(δ4 ∂4

∂x4
+ 2δ2 ∂4

∂x2 ∂y2
+

∂4

∂y4
)

∂Ψ

∂x
+

δ
Re

Fr
cos α∗ 9   (30) 

 

 



IHJPAS. 2025, 38(2) 

 

309 
 

 Reδ (
∂θ

∂t
+

∂Ψ

∂y
 
∂θ

∂x
−

∂Ψ

∂x

∂θ

∂y
) =

1

Pr
(c2δ2 ∂2θ

∂t2 + δ2 ∂2θ

∂x2 +
∂2θ

∂y2) + Ec [(
∂2Ψ

∂y2 + δ2 ∂2Ψ

∂x2 )
2

+

2δ2 (
∂2Ψ

∂x ∂y
)

2

+ 2δ2 (
∂2Ψ

∂x ∂y
)

2

]   (31) 

Sxx = 2(1 + w)
∂2Ψ

∂x ∂y
− 2A [2δ2 (

∂2Ψ

∂x ∂y
)

2

+ (
∂2Ψ

∂y2 − δ2 ∂2Ψ

∂x2 )
2

+ 2δ2 (
∂2Ψ

∂x ∂y
)

2

]    (32) 

  Sxy = (1 + w) (−δ2 ∂2Ψ

∂x2
+

∂2Ψ

∂y2
) − A [2δ2 (

∂2Ψ

∂x ∂y
)

2

+ (−δ2 ∂2Ψ

∂x2
+ δ2 ∂Ψ

∂y2
)

2

+

2δ2 (
∂2Ψ

∂x ∂y
)

2

] (−δ2 ∂2Ψ

∂x2 +
∂2Ψ

∂y2 )   (33) 

  Syy = −2(1 + w)δ
∂2Ψ

∂x ∂y
− 2Aδ [2δ2 (

∂2Ψ

∂x ∂y
)

2

+ (
∂2Ψ

∂y2 − δ2 ∂2Ψ

∂x2 )
2

+ 2δ2 (
∂2Ψ

∂x ∂y
)

2

] (−
∂2Ψ

∂x ∂y
)  (34) 

Now, the Equations (29- 34) become the form when (𝑅𝑒 𝑎𝑛𝑑  𝛿 ≪ 1) are present: 

 −
ρd2Ω2

μ
 
∂Ψ

∂y
= −

∂p

∂x
+

∂

∂y
Sxy − (Ha2 cos2 β +

1 

Da
)

∂Ψ

∂y
−

1

∝2

∂5Ψ

∂y5 +
Re

Fr
sin α∗   (35) 

 −
∂p

∂y
= 0   (36) 

 
∂2θ

∂y2 = −Ec. Pr (
∂2Ψ

∂y2 )
2

   (37) 

While the component of the extra stress tensor becomes the form of 

 Sxx = 2(1 + w)
∂2Ψ

∂x ∂y
− 2A (

∂2Ψ

∂y2 )
2

∂2Ψ

∂x ∂y
   (38) 

 Sxy = (1 + w) (
∂2Ψ

∂y2 ) − A (
∂2Ψ

∂y2 )
3

   (39) 

 Syy = 0   (40) 

Also, if Equation (39) is entered into Equation (35) as well as the derivative with respect to y and 

by (w+1) is taken, then the following equation is obtained: 

 
𝜕4𝛹

𝜕𝑦4 − 𝜂𝐴
𝜕2

𝜕𝑦2 (
𝜕2𝛹

𝜕𝑦2 )
3

− 𝜁
𝜕2𝛹

𝜕𝑦2 − 𝜂
1

∝2

𝜕6𝛹

𝜕𝑦6 = 0   (41) 

 Where  

 ζ =
Ha2 cos2 β+

1 

Da
−

ρd2Ω2

μ

w+1
, η =

1

w+1
  

 Ψ =
F

2
  ,

∂Ψ

∂y
= −1 , θ = 0     at     y = h1  , Ψ = −

F

2
  ,

∂Ψ

∂y
= −1 , θ = 0    at   y = h2     (42) 

∂Ψ

∂y
+ β1

∂2Ψ

∂y2
= −1   at y = h1   ,

∂Ψ

∂y
+ β1

∂2Ψ

∂y2
= −1   at y = h2                                                   (43)   

 
∂3Ψ

∂y3 = 0   at y = h1   ,
∂3Ψ

∂y3 = 0   at y = h2                                                                                   (44) 

In the wave frame, the dimensionless volume flow rate and boundary condition are as follows: 

F represents the dimensionless temporal average flow in the wave frame. Through the expression, 

it is related to the dimensionless temporal mean flow rate (Q) in the laboratory frame  

 Q = F + 1 + d∗                                                                                                                              (45) 

2.2.Effect of a couple – stress 

In this part, a relationship between the pair stress parameter (α) and the material fluid parameters 

(A) would be found. The relationship will help us in simplifying the problem’s solution.  Because 

of the use of the perturbation method in solving the stream function, and to see the impact of every 

parameter contained in the problem, the zero and first-order solutions must be found. However, 



IHJPAS. 2025, 38(2) 

 

310 
 

we just need to find the zero-order utilizing the relationship between the pair stress parameter and 

the material fluid properties. Fluid parameters from a substance with no dimensions: 

Let  A =
w

6
(

c

c1d
)

2

              

Then  

𝑑 = √
𝑤

6𝐴
(

𝑐

𝑐1
)                                                                                                                                (46) 

Since  

𝛼 = 𝑑√
𝜇

𝜇1
                                                                                                                                     (47) 

Substitute Equation (46) into Equation (47), we get   

α = √
wμ

6Aμ1
 (

c

c1
)                                                                                                                            (48) 

𝛼2 =
𝑤𝜇

6𝐴𝜇1
 (

𝑐

𝑐1
)

2

        and     
1

𝛼2
=

6𝐴𝜇1

𝑤𝜇
 (

𝑐1

𝑐
)

2

                                                                                (49) 

2.3.Solution of the problem 

By using Equation (45) with Equations (35 – 40) and boundary conditions (42 – 44) and since         

𝛿 ≤ 1, and using the approximation of a long wavelength and a low Reynolds number. For the 

appearance of the couple stress parameter in the equation, the solution is limited to the zero order 

by giving all the parameters required to solve the problem and find the results, we get the motion 

equation in terms of stream function which is  

  𝚿𝐲𝐲𝐲𝐲 − 𝛇𝚿𝐲𝐲 −
𝛈

𝛂𝟐 𝚿𝐲𝐲𝐲𝐲𝐲𝐲 = 𝟎                                                                                                (50) 

The straight forward formula for the momentum equation's solution is 

 Ψ = √2(
√2ep1yC1

p3
+

√2e−p1yC2

p3
+

√2ep2yC3

p4
+

√2e−p2yC4

p4
) + C5 + yC6 b                                       (51) 

Where  

 p1 =

√α2−√α2(α2−4ζη)

η

√2
 

 p2 =

√α2+√α2(α2−4ζη)

η

√2
 

 p3 = √α2−√α2(α2−4ζη)

η
√α2−√α4−4α2ζη

η
 

 p4 = √
α2+√α2(α2−4ζη)

η
√

α2+√α4−4α2ζη

η
 

Within the fixed frame, the axial velocity component is expressed as 

 𝑢(𝑥, 𝑦, 𝑡) = Ψy                                                                                                                         (52) 

Now, substitute equation (39) into equation (35), we get  

 
𝜕𝑝

𝜕𝑥
= (𝑤 + 1)Ψ𝑦𝑦𝑦 − (𝑤 + 1)𝜁Ψ𝑦 −

1

𝛼2 Ψ𝑦𝑦𝑦𝑦𝑦 +
𝑅𝑒

𝐹𝑟
sin 𝛼∗                                                     (53) 

 

3. Results and discussions 

       This section displays the stream function by using the MATHEMATICA software. A 

fascinating occurrence happens in peristaltic flows, where the closed stream function traps a 

quantity of fluid sometimes referred to as bolus inside the channel adjacent to walls, and moves in 



IHJPAS. 2025, 38(2) 

 

311 
 

the direction of wave propagation. The stream function is plotted for various values of the Hartman 

number (Ha), the Darcy number (Da),  the inclination of the magnetic field (β), the rotation (Ω), 

the values of the porous medium parameter (w), the amplitude ratio (ϕ), the slip condition (𝛽1),  

and the couple stress parameter (α) in the asymmetric channels illustrated in Figures (2-9). 

 Figures (2, 4 and 9) display the increases in the values of Hartman number (Ha), the inclination 

of magnetic field (β), and the couple stress parameter (α) leading to the increase in the size of the 

trapped bolus and approaching to the channel walls.  

 Figures (5 , 6) illustrate the increases in the values of the rotation (Ω)  and the values of the 

porous medium parameter (w) led to the decrease in the size of the trapped bolus and the move 

away from the channel walls. 

 Figure 3 demonstrates how the confined bolus's size has increased and its approach from the 

channel wall is caused by a drop in the values of the Darcy number (Da). 

 Figures (7 , 8) show how decreasing values of the amplitude ratio (ϕ) and the slip condition (𝛽1) 

caused the trapped bolus to shrink and move away from the channel walls. 

 

 
Figure 2. Stream function variation for different (a) Ha=7.5,  (b) Ha=8, (c ) Ha=8.5 when, 𝛽 = 2.5, Da = 10, 𝜌 =

0.7, 𝑑 = 5, 𝛺 = 1.5, 𝜇 = 3, 𝑤 = 0.01, 𝜙 = 2.5, 𝑎 = 0.4, 𝑏 = 0.6, 𝑑1 = 0.5, 𝐹° = 0.9, 𝛽1 = 4, 𝛼 = 0.6 

 
Figure 3. Stream function variation for different (a) Da=10, (b) Da=1, (c ) Da=0.5  when Ha = 7.5, 𝛽 = 2.5, 𝜌 =

0.7, 𝑑 = 5,  𝛺 = 1.5, 𝜇 = 3, 𝑤 = 0.01, 𝜙 = 2.5, 𝑎 = 0.4, 𝑏 = 0.6, 𝑑1 = 0.5, 𝐹° = 0.9, 𝛽1 = 4, 𝛼 = 0.6 

 

 

  



IHJPAS. 2025, 38(2) 

 

312 
 

   

(a) (b) (c ) 

Figure 4. Stream function variation for different (a) 𝛽 = 2.5, (b) β=2.6, (c ) β=2.7 when Ha = 7.5, Da = 10, 𝜌 = 0.7, 

 𝑑 = 5, 𝛺 = 1.5, 𝜇 = 3, 𝑤 = 0.01, 𝜙 = 2.5, 𝑎 = 0.4, 𝑏 = 0.6, 𝑑1 = 0.5, 𝐹° = 0.9, 𝛽1 = 4 , 𝛼 = 0.6 

   

(a) (b) (c ) 

Figure 5. Stream function variation for different (a) 𝛺 = 1.5, (b)  Ω=1.55, (c ) Ω=1.6 when Ha = 7.5, 𝛽 = 2.5, Da = 10, 𝜌 =

0.7, 𝑑 = 5, 𝜇 = 3, 𝑤 = 0.01, 𝜙 = 2.5, 𝑎 = 0.4, 𝑏 = 0.6, 𝑑1 = 0.5, 𝐹° = 0.9, 𝛽1 = 4, 𝛼 = 0.6 

 

   

(a) (b) (c ) 

Figure 6. Stream function variation for different (a) w=0.01, (b) w=0.41, (c )w=0.81 when Ha = 7.5, 𝛽 = 2.5, Da = 10, 

 𝜌 = 0.7, 𝑑 = 5, 𝛺 = 1.5, 𝜇 = 3, 𝜙 = 2.5, 𝑎 = 0.4, 𝑏 = 0.6, 𝑑1 = 0.5, 𝐹° = 0.9, 𝛽1 = 4 , 𝛼 = 0.6 

 



IHJPAS. 2025, 38(2) 

 

313 
 

   

(a) (b) (c ) 

Figure 7. Stream function variation for different (a) ϕ = 2.5,  (b) ϕ=2, (c ) ϕ=1.5 when Ha = 7.5, 𝛽 = 2.5, Da = 10, 

 𝜌 = 0.7, 𝑑 = 5, 𝛺 = 1.5, 𝜇 = 3, 𝑤 = 0.01, 𝑎 = 0.4, 𝑏 = 0.6, 𝑑1 = 0.5, 𝐹° = 0.9, 𝛽1 = 4, 𝛼 = 0.6 

   

(a) (b) (c ) 

Figure 8. Stream function variation for different (𝑎) 𝛽1 = 4, (b) 𝛽1 = 3.5, (c )  𝛽1 = 3 when Ha = 7.5, 𝛽 = 2.5, Da = 10, 

 𝜌 = 0.7, 𝑑 = 5, 𝛺 = 1.5, 𝜇 = 3, 𝑤 = 0.01, 𝜙 = 2.5, 𝑎 = 0.4, 𝑏 = 0.6, 𝑑1 = 0.5, 𝐹° = 0.9, 𝛼 = 0.6 

 

4. Conclusions 

     This study examines how coupling stress, slip condition, and rotation affect the peristaltic 

movement of a Powell-Eyring fluid through a porous medium that is vulnerable to inclined MHD 

and heat transfer. The asymmetric channel is formed by selecting peristaltic waves with varying 

amplitudes and phases on the non-uniform walls and a low Reynolds number. The formulas for 

stream function are produced. Multiple graphs are utilized for parameter analysis:  

When the values of the Hartman number (Ha), the inclination of the magnetic field (β), and the 

couple stress parameter (α) increase leads to an increase in the size of the trapped bolus and 

approaching the channel wall's, while the increases in the values of the rotation (Ω)  and the values 

of the porous medium parameter (w) led to the decrease in the size of the trapped bolus and move 

away from the channel walls. While, when the values of the Darcy number (Da) decrease leading 

to the increase in the size of the trapped bolus and approaching the channel walls, the opposite 

occurs when the decrease values of the amplitude ratio (ϕ) and the slip condition(𝛽1). 

 

 



IHJPAS. 2025, 38(2) 

 

314 
 

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. 

 

Funding  

     There is no funding for the article. 

 

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