IHJPAS. 36(2)2023 

314 
This work is licensed under a Creative Commons Attribution 4.0 International License 

 

 

   

 

 

 

 

 

 

 

 

 
 

 

 

Abstract  

        The purpose of this research is to investigate the effects of rotation on heat transfer using 

inclination magnetohydrodynamics for a couple-stress fluid in a non-uniform canal. When the 

Reynolds number is low and the wavelength is long, math formulas are used to describe the stream 

function, as well as the gradient of pressure, temperature, pressure rise and axial velocity per 

wavelength, which have been calculated analytically. The many parameters in the current model 

are assigned a definite set of values. It has been noticed that both the pressure rise and the pressure 

gradient decrease with the rise of the rotation and couple stress, while they increase with an 

increase in viscosity and Hartmann number. The explanation of parameters is shown graphically 

by a series of figures using "MATHEMATICA".  

                                                                                                   

Keywords: Non-uniform channel, rotation, inclined MHD, Couple stress, Porous medium 

 

1. Introduction 

         Using a couple stress fluids as a model for describing biologically complicated fluids, such 

as colloidal fluids, polymeric suspensions, animal and human blood, and lubrication, is extremely 

beneficial in understanding a wide range of physical issues. One of the numerous models created 

to describe the reaction explanation of non-Newtonian fluids is the Couple stress fluid model. Non-

Newtonian fluids, such as couple stress fluids, are those in which particle size is taken into account 

and studied by [1-4]. Shit and Ranjit [5] studied the fluid's peristalsis in non-uniform and 

asymmetric channels when an external magnetic field is provided. Murad and Abdulhadi [6] 

Analysis of mixed convective heat transfer for peristaltic transport of viscoplastic fluid: 

perturbation and numerical studies. Abdulla and Hummady [7] considered the influence of sliding 

speed on channel walls and the effect of nonlinear particle size. To make things even more 

complicated, non-Newtonian fluids can't be studied with non-slip boundary conditions because it's 

doi.org/10.30526/36.2.2987 

Article history: Received 23 August 2022, Accepted 9 October 2022, Published in April 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

The Effect of Rotation on the Heat Transfer of a Couple Stress Fluid in A Non-

uniform Inclined Asymmetrical Channel with Inclined MHD 

 

 
Hanaa Abdulhussein 

Department of Mathematics, 

College of Science, University of 

Baghdad, Baghdad, Iraq. 

hshatty@uowasit.edu.iq 
 

 

 

 

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:hshatty@uowasit.edu.iq
mailto:ahm6161@yahoo.com


IHJPAS. 36(2)2023 

315 
 

easy to see how the walls slide. In technology, boundary slip condition fluids can be used to polish 

artificial hearts. See [8-13] for a list of studies that use this condition. 

      Researchers have recently looked into how a magnetic field and rotation affect how fluid 

moves through an asymmetric channel, Abd-Alla and Abo-Dahab [14]. The effects of rotation and 

MHD on an asymmetric channel through a porous medium where a Jeffrey fluid flows nonlinearly 

have been looked at by Abdulhadi and Al-Hadad [15]. In porous media with non-symmetric canals, 

rotating waveform motion in two-dimensional channels of non-Newtonian fluid was explored by 

Alshareef [16]. For viscoplastic fluid and variable viscosity on peristalsis, they discussed the 

influence of rotation of the mixture on convection heat transfer analysis by [17, 18].            

        This study's objective is to examine the effects of rotation on heat transfer with inclination 

magnetohydrodynamics in a non-uniform channel containing a couple stress fluid. The equation 

for the governing equation is examined using low Reynolds approximations and long-wavelength 

assumptions, respectively. Graphs are used to show the impact of various factors on fluid flow.    

 

2. mathematical Formulation  

     Consider that the effect of a magnetic field and rotation on heat transfer can be explained by a 

porous medium with an inclined asymmetry of the couple stress fluid.   

      Let's say 𝑌∗̅ =ℎ1̃ (X ̃ , t ̃ )  and 𝑌∗̅ =ℎ2̃ (X  ̃, t ̃ ) here is a representation of the channel's top and 

bottom walls. 

ℎ1̃ (X ̃ , t ̃ ) =𝑑1 + ( �̃� − 𝑐�̃�) tan �̃� + 𝑎1
∗ cos [

2𝜋

𝜆
(�̃� − 𝑐�̃�)]                                               (1) 

 ℎ2̃ (X ̃ , t ̃ )   =−𝑑2 − ( �̃� − 𝑐�̃�) tan �̃� − 𝑎2
∗ cos [

2𝜋

𝜆
(�̃� − 𝑐�̃�) + 𝜙°]                                (2) 

         Where 𝑎1
∗𝑎𝑛𝑑 𝑎2

∗  are the amplitudes of waves λ   is  wavelength, �̃�  is the time    c is wave  

speed , (0 ≤ 𝜙° ≤ 𝜋) the phase difference 𝜙°between the walls of the channel the rectangular 

Cartesian coordinates are used �̃�and  �̃�. The channel's axis is measured by �̃�, and the transverse 

axis is measured by �̃�,   which is perpendicular to �̃�. The constant heights of the upper and lower 

walls of the channel from the central line are denoted by 𝑑1𝑎𝑛𝑑 𝑑2, respectively. It's worth noting 

that 𝜙° = 0 corresponds to an asymmetry with out-of-phase waves, whereas 𝜙° = 𝜋 relates to 

waves that are in phase. Furthermore, 𝑑1 , 𝑑2, 𝑎2
∗ , 𝑎1

∗and 𝜙° satisfy the condition; 

 𝑎1
∗2 + 𝑎2

∗2 + 2𝑎1
∗𝑎2

∗𝑐𝑜𝑠𝜙° ≤ (𝑑1 + 𝑑2)
2.                                                                                         

The governing equations.  

∂Ũ 

∂X̃
 +  

∂Ṽ

∂Ỹ
= 0                                                                                                                                   (3)   

𝜌 (
𝜕Ũ

𝜕�̃�
+ Ũ

𝜕Ũ

𝜕�̃�
+ Ṽ

𝜕Ũ

𝜕�̃�
) − 𝜌�̀� (�̀��̃� + 2

𝜕Ṽ

𝜕�̃�
) = −

𝜕P̃

𝜕X̃
+ 𝜇∇2�̃� − 𝜂∇4�̃� − �̃�𝐵∗

2 𝑐𝑜𝑠 Φ (Ũ 𝑐𝑜𝑠 Φ −

Ṽ 𝑠𝑖𝑛 Φ) −
𝜇

𝑘°
Ũ + 𝜌g�̇� (T − T°)𝑠𝑖𝑛�̃� +  𝜌𝑔𝑠𝑖𝑛α̃                                                             (4)  

𝜌 (
𝜕Ṽ

𝜕�̃�
+ Ũ

∂Ṽ

∂X̃
+ Ṽ

∂Ṽ

∂Ỹ
) − 𝜌�̀� (�̀��̃� − 2

𝜕Ũ

𝜕�̃�
) = −

𝜕P̃

𝜕Ỹ
+ 𝜇∇2Ṽ − 𝜂∇4Ṽ + �̃�𝐵∗

2 𝑠𝑖𝑛 Φ (Ũ 𝑐𝑜𝑠 Φ −

Ṽ 𝑠𝑖𝑛 Φ) −
𝜇

 𝑘°
Ṽ −  𝜌𝑔𝑐𝑜𝑠�̃�                                                                                                           (5) 

𝜌𝐶𝑝 (
𝜕T

𝜕�̃�
+ Ũ

𝜕T

𝜕X̃
+ Ṽ

𝜕T

𝜕�̃�
) = K̃ (

∂2T

∂�̃�2
+

∂2T

∂�̃�2
) + 𝜇 [2 {(

∂Ũ

∂�̃�
)
2

+ (
∂Ṽ

∂Ỹ
)
2

} + (
∂Ũ

∂�̃�
+

∂Ṽ

∂X̃
)
2

] +

𝜂 [(
∂2Ũ

∂X̃2
+

∂2Ũ

∂�̃�2
)
2

+ (
∂2Ṽ

∂�̃�2
+

∂2Ṽ

∂�̃�2
)
2

] + �̃�𝐵∗
2(Ũ 𝑐𝑜𝑠 Φ − Ṽ 𝑠𝑖𝑛 Φ)

2
                                          (6)  



IHJPAS. 36(2)2023 

316 
 

      Where V′⃑⃑  ⃑=(�̃�, Ṽ, 0)  be the velocity vector, Ὼ  is rotation, �̇� the Coefficient of thermal 
expansion , �̃� is the fluid pressure , 𝜌   is density of fluid , Φ  is the inclination of the  magnetic 
field angle, μ the dynamic viscosity , η is a constant  linked to the couple stress, 𝐶𝑝 is the specific 

heat at constant pressure. impact B∗⃑⃑⃑⃑ =(𝐵∗ 𝑠𝑖𝑛 Φ , 𝐵∗ 𝑐𝑜𝑠 Φ ,0) the magnetic field vector, 𝑘°  is the 
permeability parameter ,T  is the temperature , 𝑔 is the acceleration by gravity , �̃� is the fluids 
electrical conductivity ,K̃ is the thermal conductivity . The induced electric field is not taken into 
consideration at all, because assuming a low magnetic Reynolds number                                                                                                                                           
     The fixed frame's(�̃� , �̃�) flow field and the wave frame's(�̀� , ỳ) wave field are also considered 
the motions of an unsteady and steady-state . It's important to think about the relationship between 

the wave (�̀� , ỳ)and the fixed frames (�̃� , �̃�) with a velocity of c as they move apart from one 
another  as a result of the transformations below .                                                                                                        

v̀ = �̃�  ,  ỳ= Ỹ , T = T̃  ,  x̀ = �̃� − 𝑐�̃�,   ù  = �̃� − 𝑐   ,  �̀� = �̃�                                              (7) 

      In which (ù, v̀) and (Ũ, Ṽ)are the waves' and laboratories' velocities, respectively. the 

governing equations (3) ,(4) ,(5), and (6) be expressed in Wave frame as using the 

transformations mentioned above. 

∂ù 

∂x̀
 +  

𝜕v̀

𝜕ỳ
= 0                                                                                                                           (8) 

𝜌 ((ù + 𝑐)
∂ù 

∂x̀
+ v̀

𝜕ù

𝜕ỳ
) − 𝜌�̀� (𝛺 ̀ (�̀� + 𝑐) + 2

𝜕v̀

𝜕�̃�
)

= −
𝜕�̀�

𝜕x̀
+ 𝜇∇2ù − 𝜂∇4ù − �̃�𝐵∗

2 𝑐𝑜𝑠 Φ ((ù + 𝑐) 𝑐𝑜𝑠 Φ − v̀ 𝑠𝑖𝑛 Φ) −
𝜇

𝑘°
(ù + 𝑐)  

+ 𝜌𝑔�̇� (T̃ − T°)𝑠𝑖𝑛�̃� +   𝜌𝑔𝑠𝑖𝑛�̃�                                               (9) 

𝜌 ((ù + 𝑐)
𝜕v̀

𝜕x̀
+ v̀

𝜕v̀

𝜕ỳ
) − 𝜌�̀� (𝛺 ̀ v̀ − 2

𝜕ù

𝜕�̃�
) = −

𝜕�̀�

𝜕ỳ
+ 𝜇∇2v̀ − 𝜂∇4v̀ + �̃�𝐵∗

2 𝑠𝑖𝑛 Φ ((ù + 𝑐) 𝑐𝑜𝑠 Φ −

v̀ 𝑠𝑖𝑛 Φ) −
𝜇

𝑘°
v̀   −

 𝜌𝑔𝑐𝑜𝑠�̃�                                                                                                                                              (10)   

𝜌𝐶𝑝 ((ù + 𝑐)
∂T̃

∂x̀
+ v̀

∂T̃

∂ỳ
) = �̃� (

∂2T̃

∂x̀2
+

∂2T̃

∂ỳ2
) + 𝜇 [2 {(

∂ù

∂x̀
)
2

+ (
∂v̀

∂ỳ
)
2

} + (
∂ù

∂ỳ
+

∂v̀

∂x̀
)
2

] +

𝜂 [(
∂2ù

∂x̀2
+

∂2ù

∂y2
)
2

+ (
∂2v̀

∂x̀2
+

∂2v̀

∂ỳ2
)
2

] + �̃�𝐵∗
2((ù + 𝑐) 𝑐𝑜𝑠 Φ − v̀ 𝑠𝑖𝑛 Φ)

2
                               (11) 

         To reduce the number of additional parameters, we shall define the following non-

dimensional quantities: 

𝑥 =
x̀

𝜆
 , 𝑦 =

ỳ

𝑑1
 , ℎ1

∗(𝑥) =
 ℎ1̃ (�̅� )

𝑑1
 ,  ℎ2

∗ (𝑥) =
 ℎ2̃ (�̅� )

𝑑1
 ,𝜃° =

T̃−T0

T1−T0
  , u∗ =

ù

𝑐
 , v∗ = 

𝜆v̀

𝑐𝑑1
  , 

 t∗ = 
𝑐�̃�

𝜆
 ,𝑅𝑒 =

𝑐𝜌𝑑1

𝜇
  , 𝛿 =  

𝑑1

𝜆
  , ℋ=Β∗𝑑1√

�̃�

𝜇
   ,  𝑃𝑟 =

𝜇 𝐶𝑝

�̃�
  , 𝛾 = 𝑑1√

𝜇

𝜂
   , 𝐷𝑎 =

𝑘𝜊 

 𝑑1
2, 𝐹𝑟 =

𝑐2

𝑔𝑑1
     (12) 

,   
𝑑1

2 �̀�(�̀�) 

𝜆𝜇𝑐
 = 𝑝 , 𝐵𝑟=𝑃𝑟. 𝐸𝑐  ,    𝐺𝑟 =

𝜌 𝑔(T1−T0) �̇� 𝑑1

𝜇 𝑐
 ,  𝐸𝑐 =

𝑐2

 (T1−T0) 𝐶𝑝
. 

     Where ℋ  is Hartmann number , 𝑅𝑒 is Reynolds number, 𝛿 is Wave number , 𝛾  is Couple 

stress parameter , 𝑃𝑟  is Prandtl number , 𝐷𝑎 is Darcy number  ,  𝐹𝑟 is Froude number, 𝐵𝑟 is 

Brinkman number ,  𝐸𝑐  is Eckert number ,and   𝐺𝑟 is Grashof number . 



IHJPAS. 36(2)2023 

317 
 

     According to equations (1) and (2), the dimensionless shape of the peristaltic channel walls 

may be shown in ℎ1
∗(𝑥) and ℎ2

∗(𝑥) 

ℎ1
∗(𝑥) = 1 + 𝑘𝑥 + 𝑎 cos(2𝜋𝑥)                                                                                                      (13) 

ℎ2
∗ (𝑥) = −𝑑 − 𝑘𝑥 − 𝑏 cos(2𝜋𝑥 + 𝜙°)                                                                                         (14) 

Where 𝑎 =
𝑎1

∗

𝑑1
 , 𝑏 =

𝑎2
∗

𝑑1
 , 𝑑 =

𝑑2

𝑑1
, 𝑘 = (

𝜆 tan �̃�

𝑑1
) is referred to as the channel's Non-uniform 

parameter and 𝜙° the  relation  𝑎
2 + 𝑏2 + 2𝑎𝑏𝑐𝑜𝑠𝜙° ≤ (1 + 𝑑)

2 

       Where(𝜓° )  stream function of velocity components  u
∗ and v∗ that is dimensionless u∗ =

𝜕𝜓° 

𝜕𝑦
 and v∗ = −

𝜕𝜓° 

𝜕𝑥
  , respectively, and satisfy the continuity equation (8). 

    In terms of stream function 𝜓° , the dimensionless variables are specified in equations. (9), 

(10), and (11) were translated into the following equations. 

𝑅𝑒. 𝛿 [(
𝜕𝜓°

𝜕𝑦
.

𝜕

𝜕𝑥
−

𝜕𝜓°

𝜕𝑥
.

𝜕

𝜕𝑦
)

𝜕𝜓°

𝜕𝑦
+

𝜕2𝜓°

𝜕𝑥𝜕𝑦
] −

𝜌Ὼ2𝑑1
2

𝜇
(
𝜕𝜓°

𝜕𝑦
+ 1) + 2Ὼ𝑅𝑒𝛿

2 𝜕
2𝜓°

𝜕𝑥𝜕 𝑡∗
= −

𝜕𝑝

𝜕𝑥
+ 𝛿2

𝜕3𝜓°

𝜕𝑥2𝜕𝑦
+

𝜕3𝜓°

𝜕𝑦3
−

1

𝛾2
(𝛿4

𝜕5𝜓°

𝜕𝑥4𝜕𝑦
+ 2𝛿2

𝜕5𝜓°

𝜕𝑥2𝜕𝑦3
+

𝜕5𝜓°

𝜕𝑦5
) − ℋ2 𝑐𝑜𝑠 Φ [(

𝜕𝜓°

𝜕𝑦
+ 1) 𝑐𝑜𝑠 Φ +

𝜕𝜓°

𝜕𝑥
 𝛿 𝑠𝑖𝑛 Φ] −

1

𝐷𝑎
(
𝜕𝜓°

𝜕𝑦
+ 1) +

𝑅𝑒

𝐹𝑟
sin α̃ +

G𝑟 θ sinα̃                                                                                                                                              (15)   

𝑅𝑒. 𝛿
3 [(−

𝜕𝜓°

𝜕𝑦
.

𝜕

𝜕𝑥
+

𝜕𝜓°

𝜕𝑥
.

𝜕

𝜕𝑦
)

𝜕𝜓°

𝜕𝑥
−

𝜕2𝜓°

𝜕𝑥2
] +

𝜌Ὼ2𝑑1
2

𝜇
𝛿2

𝜕𝜓°

𝜕𝑥
+ 2Ὼ𝑅𝑒𝛿

2 𝜕
2𝜓°

𝜕𝑦𝜕 𝑡∗
= −

𝜕𝑝

𝜕𝑦
−

𝛿2 (𝛿2
𝜕3𝜓°

𝜕𝑥3
+

𝜕3𝜓°

𝜕𝑦2𝜕𝑥
) +

𝛿2

𝛾2
(𝛿4

𝜕5𝜓°

𝜕𝑥5
+ 2𝛿2

𝜕5𝜓°

𝜕𝑥3𝜕𝑦2
+

𝜕5𝜓°

𝜕𝑦4𝜕𝑥
) + ℋ2 𝛿 𝑠𝑖𝑛 Φ [(

𝜕𝜓°

𝜕𝑦
+ 1) 𝑐𝑜𝑠 Φ +

𝜕𝜓°

𝜕𝑥
 𝛿 𝑠𝑖𝑛 Φ] +

𝛿2

𝐷𝑎
.
𝜕𝜓°

𝜕𝑥
−

𝑅𝑒

𝐹𝑟
 𝛿 𝑐𝑜𝑠 �̃�                                                            (16) 

            

 𝑅𝑒. 𝛿. 𝑃𝑟 (
𝜕𝜓°

𝜕𝑦
.
𝜕𝜃°

𝜕𝑥
+

𝜕𝜃°

𝜕𝑥
−

𝜕𝜓°

𝜕𝑥
.
𝜕𝜃°

𝜕𝑦
) = (𝛿2

𝜕2𝜃°

𝜕𝑥2
+

𝜕2𝜃°

𝜕𝑦2
) + 𝐵𝑟 [4𝛿

2 (
𝜕2𝜓°

𝜕𝑥𝜕𝑦
)
2

+ (
𝜕2𝜓°

𝜕𝑦2
− 𝛿

𝜕2𝜓°

𝜕𝑥2
)
2

+

1

𝛾2
[(𝛿2

𝜕3𝜓°

𝜕𝑥2𝜕𝑦
+

𝜕3𝜓°

𝜕𝑦3
)
2

+ (𝛿2
𝜕3𝜓°

𝜕𝑥3
+

𝜕3𝜓°

𝜕𝑥𝜕𝑦2
)
2

]] + 𝐵𝑟 ℋ
2 [(

𝜕𝜓°

𝜕𝑦
+ 1) 𝑐𝑜𝑠 Φ +

𝜕𝜓°

𝜕𝑥
 𝛿 𝑠𝑖𝑛 Φ]

2

                                                                                                                  

(17) 

        Cross differentiation from dimensionless equations is used to remove the pressure term. 

(17), (18), and (19) may be formulated in the context of stream function differential equation and 

temperature under  low Reynolds approximations and long-wavelength assumptions(δ<< 1) .  

𝜕6𝜓°
𝜕𝑦6

− 𝛾2
𝜕4𝜓°
𝜕𝑦4

+ 𝛾2
𝜕2𝜓°
𝜕𝑦2

[ℋ2 𝑐𝑜𝑠2 Φ +
1

𝐷𝑎
−

𝜌�̀�2𝑑1
2

𝜇
] = 0                                           (18) 

𝜕2θ°
𝜕𝑦2

+ 𝐵𝑟 [(
𝜕2𝜓°
𝜕𝑦2

)

2

+
1

𝛾2
(
𝜕3𝜓°
𝜕𝑦3

)

2

]  + 𝐵𝑟 ℋ
2 𝑐𝑜𝑠2 Φ (

𝜕𝜓°
𝜕𝑦

)
2

= 0                                 (19)   

 

 

 

 

 

 



IHJPAS. 36(2)2023 

318 
 

The dimensionless boundary conditions in the wave frame are [5]: 

𝜕𝜓°

𝜕𝑦
+ 𝛽

𝜕2𝜓°

𝜕𝑦2
= −1   𝑜𝑛 𝑦 = ℎ1

∗ 

𝜕𝜓°
𝜕𝑦

− 𝛽
𝜕2𝜓°
𝜕𝑦2

= −1   𝑜𝑛 𝑦 = ℎ2
∗  

𝜓° =
𝐹

2
 ,     𝜃° = 0  𝑜𝑛  𝑦 = ℎ1

∗                                                                                                (20) 

𝜓° = −
𝐹

2
,       θ° = 1  𝑜𝑛  𝑦 = ℎ2

∗           

𝜕3𝜓°
𝜕𝑦3

= 0 𝑜𝑛 𝑦 = ℎ1
∗𝑎𝑛𝑑 𝑦 = ℎ2

∗  

As a result of solving equations (18) and (19), the associated boundary conditions (20) are 

satisfied. 

𝜓° =
2ⅇ−𝑦𝑎1𝑡1

𝑎3
+

2ⅇ𝑦𝑎1𝑡2
𝑎3

+
2ⅇ−𝑦𝑎2𝑡3

𝑎4
+

2ⅇ𝑦𝑎2𝑡4
𝑎4

+ 𝑡5 + 𝑦𝑡6                                            (21) 

θ° = 𝑟1 + 𝑦𝑟2 +
1

2𝛾2𝑎3
3𝑎4

3 (−
1

2𝑎1
ⅇ−2𝑦𝑎1+3𝑦(𝑎1+𝑎2)−𝑦(𝑎1+3𝑎2)(ℋ2 + 2ℋ2𝐶𝑜𝑠[2𝛷] +

ℋ2𝐶𝑜𝑠[4Φ] + 2𝑖ℋ2𝑆𝑖𝑛[2Φ] + 𝑖ℋ2𝑆𝑖𝑛[4Φ] + 4𝐶𝑜𝑠[2Φ]𝑎1
2 + 4𝑖𝑆𝑖𝑛[2Φ]𝑎1

2). ..       (22)  

It's possible to write the velocity as:  

W =  −
2𝑒−𝑦𝑎1𝑎1𝑡1

𝑎3
+

2𝑒𝑦𝑎1𝑎1𝑡2

𝑎3
−

2𝑒−𝑦𝑎2𝑎2𝑡3

𝑎4
+

2𝑒𝑦𝑎2𝑎2𝑡4

𝑎4
+ 𝑡6                                                  (23) 

The axial pressure gradient can be calculated once we've identified the stream function.        

𝜕𝑝

𝜕𝑥
=

𝜕3𝜓°

𝜕𝑦3
−

1

𝛾2
.
𝜕5𝜓°

𝜕𝑦5
− ℋ2 𝑐𝑜𝑠2 Φ [(

𝜕𝜓°

𝜕𝑦
+ 1)] −

1

𝐷𝑎
(
𝜕𝜓°

𝜕𝑦
+ 1) +

𝑅𝑒

𝐹𝑟
sin α̃ + G𝑟θ° sinα̃ +

𝜌Ὼ2𝑑1
2

𝜇
(
𝜕𝜓°

𝜕𝑦
+ 1)                                                                                                                                  (24) 

𝜕𝑝

𝜕𝑦
= 0                                                                                                                                  (25) 

In non-dimensional form, the pressure rise per wavelength Δ𝑝∗ is defined as 

Δ𝑝∗ = ∫
𝜕𝑝

𝜕𝑥

1

0

dx                                                                                                                                  (26) 

3.Volumetric flow rate 

   In the laboratory frame, the volumetric flow rate is equal to 

�̃� =  ∫ Ũ 
ℎ1̃

ℎ2̃
  (X̃ , Ỹ , t̃ )d Ỹ                                                                                                    (27) 

whereℎ1̃ andℎ2̃  are functions of �̃� and  �̃� .   

In the wave frame, the volumetric flow rate is calculated: 

𝑞° =  ∫ ù
ℎ1̃

ℎ2̃
(x̀ , ỳ)d �̀�      (28)     

The relationship between 𝑄 and 𝑞  can  is calculated as follows: 

�̃� = 𝑞°  + 𝑐(ℎ1̃ − ℎ2̃)                                                                                                                        (29) 

The time-mean flow over a span of time 𝑇∗ fixed in place �̅�   as 

𝑄° =  
1

𝑇∗
∫ �̃�𝑑𝑡                                                                                                                               (30)

𝑇∗

0

 

Using equation  (29) in equation (30) the flow rate   𝑄° has the From  

𝑄° =  
1

𝑇∗
∫ 𝑞°𝑑𝑡 +

𝑇∗

0

 𝑐(ℎ1̃ − ℎ2̃) 



IHJPAS. 36(2)2023 

319 
 

= 𝑞° + 𝑐𝑑1 + 𝑐𝑑2 + 2𝑐𝑥𝜆 tan �̃� + 𝑐𝑎1 cos(2𝜋𝑥) + 𝑐𝑎2 cos(2𝜋𝑥 + 𝜙°)                           (31) 

The non-dimensional of equation (28) is provided by 

𝜔 = 𝐹 + 1 + 𝑑 + 2𝑘𝑥 + 𝑎 cos(2𝜋𝑥) + 𝑏 cos(2𝜋𝑥 + 𝜙°)                                                   (32) 

where 𝜔 =
𝑄°

cd1
and F =

𝑞°

cd1
 has the expression in the form 

𝐹 = ∫
𝜕𝜓°
𝜕𝑦

ℎ1
∗

ℎ2
∗

dy = 𝜓°(ℎ1
∗) − 𝜓°(ℎ2

∗)                                                                                           (33) 

4. Results and Discussion 

      This section focuses on " velocity"W, "temperature"   θ°, " gradient of pressure " δP
∗, "pressure 

rise"   Δ𝑝∗ ,  and "stream function"  𝜓° . To get the numerical values corresponding to the above-

mentioned analytical formulas, we utilized "MATHEMATICA " program.  

 4.1.  The distribution of velocity 

          Now, we'll look at the effect of axial velocity (W) variation across the canal on several 

parameters. Note that the velocity is parabolic. In the center, the axial velocity rises with increasing 

rotation parameter " �̀�", density "ρ", the width of the channel "𝑑1", Darcy number " 𝐷𝑎" and the 

magnetic field inclination angle  "Φ", whereas the axial velocity reduces near the channel wall's 

edge, as seen in Figure 1 (a-e). According to Figure 1(f) "𝜙°", is shown to increase axial velocity 

in the middle of the canal while axial velocity lowers at one wall boundary and rises at the other. 

As can be seen in Figures 1(g-k), as " ℋ ", "γ", "β", viscosity "μ", and " k"  increase, the axial 
velocity decreases in the channel's middle while increasing near the channel's wall border. 

 

  

(a)                                                                      (b) 

 
(c )                                                                                     (d)  

 

 

(e)                                                                                         (f) 
 



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320 
 

 
(g)                                                                          (h) 

  
(i)                                                                              (j) 

 

        (k) 

Figure 1.Variation the  axial velocity "W" with different parameters  {𝐷𝑎 =0.5 , ℋ = 3  , 𝛾 = 2   ,  𝑎 = 0.6,  𝜇 =

0.1 , �̀� = 1 , 𝑏 =0.7 , 𝜙° =
𝜋

6
 , 𝑑 = 1 ,  𝛽 =0.1 , Φ =

𝜋

4
 ,   𝑘 =0.2 ,   𝑥 = 1  ,  𝜌 =0.5  ,   𝐹 = 1  , 𝑑1 =  0.2} 

 

4.2.   Pumping characteristics 

       In this subsection, we will analyze the pressure through the channel. 

4.2.1. pressure gradient" 𝛅𝐏∗  " 

          Each of the figures in this section illustrates the pressure gradient and the axial axis of 

fluctuation along the canal at one wavelength 𝑥 ∈[0,1].  Figures like this show that flow is 

restricted in the narrowing portion of channel 𝑥 ∈[0.2,0.6].  As a result, a larger pressure gradient 

is needed to achieve a normal flow. Because of the smaller pressure gradient in the larger area of 

the channel 𝑥 ∈ [0,0.2]∪ [0.6,1] fluid may pass readily. Figure 2(a-b) shows that when viscosity 

"𝜇" and the Hartmann number" ℋ"  raises, the pressure gradient also increases in size. This graphic 

shows that greater pressure is required to move some volume of fluid through the narrower region 

of the channel. When looking at channel parameters such as the magnetic field inclination angle 

"Φ" , the inclination" 𝛼 ̃" of the channel, rotation parameter  " �̀�"  ,density "𝜌" ,width of the channel 

"𝑑1" , Grashof numbers "G𝑟"   ,slip parameter"𝛽", Darcy number  " 𝐷𝑎"  , Reynolds 

number"𝑅𝑒",couple stress "𝛾" , temperature  "θ° ", the phase difference " 𝜙°" and Non-uniformity 

"𝑘" as depicted in Figures 2( c –o), the pattern is the opposite. Because of the inclination, less 

pressure is needed to move liquids. 

 

 

 

 



IHJPAS. 36(2)2023 

321 
 

4.2.2. pressure rise"  𝚫𝐩∗ " 

          Figures 3(a-o) show the relationship between pressure rise and volumetric flow rate.  The 

connection between pressure rise and the rate of volumetric flow for each wavelength is seen to 

be linear. The whole region is considered into five parts (1) the peristaltic pumping region 

where(∆𝑝∗ > 0, 𝐹 > 0) (2) when (∆𝑝∗ > 0  , 𝐹 < 0), then it is a retrograde pumping region (3) 

augmented pumping (co-pumping) region where (∆𝑝∗ < 0 , 𝐹 > 0) (4) There is a co-pumping 

region where (∆𝑝∗ < 0 , 𝐹 < 0) (5) (∆𝑝∗ = 0)corresponds to the free pumping region.                                                 

       Figure3 (a-b) shows the impact of  viscosity "𝜇" and Hartmann number " ℋ "on ∆𝑝∗ on It has 

been noticed in the retrograde pumping region. The pumping rate increases as Hartmann number 

"ℋ"increases. Co-pumping reduces when in this region and increases in " ℋ "and "𝜇" . Volumetric 

flow rate and pressure increase ∆𝑝∗ are shown to be linearly related in Figures 3 (c - f ) for a variety 

of Grashof numbers "G𝑟" , "𝑅𝑒", " �̃�" and  "θ° ". We observed that a rise results in an increase in 

retrograde pumping rate and augmented pumping (co-pumping) region an increase in the pressure 

rise. Figures 3 (g - m ) Graph shows that with an increase in rotation parameter  " �̀�"  ,density "𝜌" 

,width of the channel "d1",  " 𝐷𝑎", "𝛽", "𝛾" and" Φ "  . the pumping rate decreases in the region of 

retrograde pumping while in the augmented pumping region found to increase. Figure 3 (n)shows 

an impact of channel non-uniform parameter "𝑘"on ∆𝑝∗. It is observed that in a retrograde pumping 

region and a free pumping region (∆𝑝∗ = 0), the pumping rate decreases with an increase in "𝑘". 

Figure 3 (o) depicts the effect of "𝜙°" on ∆𝑝∗. It is observed that in a retrograde pumping region 

and a free pumping region (∆𝑝∗ = 0), the pumping rate increases with an increase in"𝜙°" .  

 
)a)                                                                                   (b( 

 
)c)                                                                                   (d) 

 

)e)                                                                                        (f) 



IHJPAS. 36(2)2023 

322 
 

  

                 (g)                                                                                (h(   

 
          (i)                                                                                  (j) 

 
             (k)                                                                                 (l) 

 

   
            (m)                                                                         (n) 

 
(o) 

Figure 2. Variation the  pressure gradient" δP∗  with different parameters   {𝐷𝑎 =0.5 , ℋ = 3  , 𝛾 = 2   ,  𝑎 = 0.6,  

𝜇 = 0.1 , �̀� = 1 , 𝑏 =0.7 , 𝜙° =
𝜋

6
 , 𝑑 = 1 ,  𝛽 =0.1 , Φ =

𝜋

4
 ,   𝑘 =0.2 ,   𝑦 = 1  ,  𝜌 =0.5  ,   𝐹 =  0.1 , 𝑑1 =  0.2  

, 𝐹𝑟 = 0.5  , �̃� =
𝜋

6
 , 𝐺𝑟 = 1 , Θ° = 1} 

 



IHJPAS. 36(2)2023 

323 
 

 
                                         (a)                                                                                      (b)           

  
                                    (c)                                                                                      (d)  

 
                                           (e)                                                                                      (f) 

 
 

                                           (g)                                                                                      (h) 

  
                                    (i)                                                                                (j) 



IHJPAS. 36(2)2023 

324 
 

 
   

                                        (k)                                                                                     (l) 

 

                                         (m)                                                                                    (n)  

            
(o) 

Figure 3 . Variation the  pressure rise  𝛥𝑝∗   with different parameters    {𝐷𝑎 =0.5 , ℋ = 3  , 𝛾 = 2   ,  𝑎 = 0.6,  𝜇 =

0.1 , �̀� = 1 , 𝑏 =0.7 , 𝜙° =
𝜋

6
 , 𝑑 = 1 ,  𝛽 =0.1 , Φ =

𝜋

4
 ,   𝑘 =0.2 ,   𝑦 = 1  ,  𝜌 =0.5  ,   𝐹 =  0.1 , 𝑑1 =  0.2  , 𝐹𝑟 = 

0.5  , �̃� =
𝜋

6
 , 𝐺𝑟 = 1 , Θ° = 1} 

4.3. temperature profile  "𝛉° " 

         An examination of the fluid temperature profile for a fixed value of x = 1 yields a parabolic 

temperature profile, with a higher graph in the middle. Fluids that have a high viscosity are more 

likely to convert kinetic energy into internal energy, which causes them to get hotter. There is no 

doubt about that. Its flow and movement resistance diminish with increasing temperature, a 

phenomenon caused by the tiny distances between molecules and the cohesive forces that exist 

between them. Figures 4 (a-g) show that the temperature increases values " �̀�"  , "𝜌" , "d1"  " 𝐷𝑎" 

," H " , "𝑘"and "𝐵𝑟"increase, so does the temperature in the channel's center and temperature 

parabolic. The increase in temperature is accompanied by  "𝐵𝑟". 𝐵𝑟𝑖𝑛𝑘𝑚𝑎𝑛 number  "𝐵𝑟" of raised 

flow resistances given by shear. As parameter values "𝜇"  , " "Φ " "𝛽",  "𝛾"and "𝜙°" increase, they 

show that the temperature decreases in Figures 4 (h-l ). 

 
                                              (a)                                                                                (b) 



IHJPAS. 36(2)2023 

325 
 

 
                                                  (c)                                                                                  (d) 

 
                                               (e)                                                                                      (f) 

 
                                          (g)                                                                                      (h) 

 
                                                   (i)                                                                                     (j) 

 
                                                 (k)                                                                                  (l) 

Figure 4. Variation the temperature "θ° "  with different parameters    {𝐷𝑎 =0.5 , ℋ = 3  , 𝛾 = 2   ,  𝑎 = 0.6,  𝜇 =

0.1 , �̀� = 1 , 𝑏 =0.7 , 𝜙° =
𝜋

6
 , 𝑑 = 1 ,  𝛽 =0.1 , Φ =

𝜋

4
 ,   𝑘 =0.2 ,   𝑥 = 1  ,  𝜌 =0.5  ,   𝐹 =  0.1 , 𝑑1 =  0.2  , 𝐵𝑟 = 

1} 

4.4.  The trapping phenomena 

         In order to explain the trapping phenomena, the formation of a circulating bolus of fluid, 

which is a closed streamline region, at the speed of the wave. There will be points in the wave 

frame where the fluid's velocity is zero due to the trapping phenomenon. The volumetric flow rate 

via a line linking any two places is computed by taking into account the difference in stream 

function values at the two sites in question, which is why studying streamline patterns is so 



IHJPAS. 36(2)2023 

326 
 

important. Figures 5–15 indicate that bolus formation occurs from both sides of the center line in 

the extended region. Bigger and larger volumes of the trapped bolus can be extracted from the 

system by raising the strength of the magnetic field in Figures ( 5-9) (a, b, and c), as "Φ " , " �̀�", 

"𝜌" , "d1" and " 𝐷𝑎". In Figures (10 and 11) (a ,b and c), as " 𝜙°" and  "𝜇" changes, so does the 

variation in streamlines.  

According to our research, we've found that bolus sizes decrease"𝜇"  and " 𝜙°"  increase. The 

Figure 12(a ,b and c) shows that the wall draws fluid in the widest section of the duct, but this fluid 

is pushed away from the wall in the narrower section and the bolus disappears in the central part 

as "𝛽"   increases. Graphing the non-uniformity parameter "𝑘"of the asymmetric channel as shown 

in Figure 13(a, b, and c), when it is raised, the trapped bolus decreases in size and migrates 

downstream. These figures 14and15 (a, b, and c) demonstrate how In increasing the Hartman 

number " ℋ " ,and couple stress "𝛾" , the incidence of trapped bolus diminishes in size and vanishes 

in the direction of downstream, Another effect that may help protect red blood cells and other 

elements is the tendency to reduce bolus volume. 

  
(a)                                             (b)                                             (c)   

Figure 5.  Distribution of streamlines " 𝜓°"  for  (a)  "Φ"  = π/4   (b)  "Φ" = π/3   (c) "Φ"= 2π/5 

 
                       (a)                                                     (b)                                                          (c) 

Figure 6. Distribution of streamlines  " 𝜓°"  for  (a)  " �̀�"  =1     (b) " �̀�"  =2         (c) " �̀�"  =4    

 
(a)                                     (b)                                                       (c) 

Figure 7.Distribution of streamlines  " 𝜓°"    for (a)  "𝜌"  = 0.5 (b)  "𝜌"  = 2     (c) "𝜌"  = 6 

 



IHJPAS. 36(2)2023 

327 
 

  
(a)                                          (b)                                              (c) 

Figure 8.  Distribution of streamlines " 𝜓°"  for (a)  "𝑑1"  =0.2   (b)  "𝑑1"  =0.8    (c) "𝑑1"  =1    

 

 
(a)                                     (b)                              (c) 

Figure 9 . Distribution of streamlines" 𝜓°"     for  (a)    " 𝐷𝑎"  =0.5    (b)   " 𝐷𝑎"  =2   (c)  " 𝐷𝑎"  =6 

  
(a)                                               (b)                                                    (c) 

Figure 10 . Distribution of streamlines " 𝜓°" for  (a)  "𝜇"  =0.1   (b)"𝜇"  =0.4     (c) "𝜇"  =1 

 
(a)                                                (b)                                                     (c) 

Figure 11.  Distribution of streamlines " 𝜓°"  for  (a)  "𝜙°"  =
𝜋

6
   (b) "𝜙°"  =

𝜋

3
    (c) "𝜙°"  =

𝜋

2
 



IHJPAS. 36(2)2023 

328 
 

               (a)

                                                (b)                                                   (c) 

Figure 12.  Distribution of streamlines " 𝜓°" for  (a)  "𝛽"  =0.1   (b)  "𝛽"  =0.5  (c) "𝛽"  =1   

 
(a)                                      (b)                                          (c) 

Figure 13.  Distribution of streamlines " 𝜓°"   for  (a)  "k"  =0.2   (b)  "𝑘" = 0.4  (c) "𝑘"=0.6 

 
(a)                                              (b)                                                             (c) 

Figure 14.  Distribution of streamlines " 𝜓°"  for  (a)  " ℋ "  =3   (b)  " ℋ "  = 4  (c) " ℋ "=8

  
(a)                                                 (b)                                                   (c) 

Figure 15.  Distribution of streamlines " 𝜓°"  for  (a)  "𝛾"  =2   (b)  "𝛾"  =3     (c) "𝛾"  =4    

 

5.  Conclusions  

           In this research, we studied the effects of  rotation and incline magnetohydrodynamics to 

investigate the effects of heat transfer and couple stress fluid as they move via an inclined 

asymmetric channel and porous medium under low Reynolds approximations and long-

wavelength assumptions in the transport of bodily fluids by the use of non-Newtonian fluid 

models. Analytically, using Mathematica software. This investigation focused on studying the 

distribution of velocity, the pumping characteristics, the distribution of temperature, and the 

trapping phenomena. 



IHJPAS. 36(2)2023 

329 
 

1. It notes that the temperature and profile of velocity are parabolic. 

2. There is a decrease in axial velocity "W"   in the central region when increasing the 

viscosity  "𝜇"  ,  "𝛾" , " ℋ ","𝛽", and "𝑘"", but there is a rise in velocity at the boundary of 

the channel wall.                       

3. There is an increase in axial velocity "W"  in the central region when increasing the 

 rotation parameter " �̀�", "𝜙°" , density "𝜌" , "Φ ", "d1" and " 𝐷𝑎", but there is a decrease in 

velocity at the boundary of the channel wall.                      

4. When the viscosity "𝜇"  , and Hartmann number  " ℋ "are increased, the pressure gradient 

" δP∗  " increases, while " �̀�", "Φ" , "β", " 𝛼 ̃", "G𝑟", " 𝐷𝑎",  "𝛾", "𝜙°", "𝜌", "𝑘", "d1" , "𝑅𝑒" 

,and "θ° " decrease. 

5. The connection between pressure rise "  Δ𝑝∗ "and volumetric flow rate for each wavelength 

is seen to be linear.                                                                                                                                                                

6. In retrograde pumping, increases "   Δ𝑝∗ " pressure rise with the increasing  values the 

viscosity "𝜇"  , "G𝑟","𝑅𝑒" , " H ",  "θ° " ," 𝛼 ̃" and  "𝜙°", whereas  it decreases with the rising 

values the  rotation parameter " �̀�", "Φ", " 𝐷𝑎" , "𝛾" ,  "𝛽" ,density "𝜌" , "d1" ,and" k".        

7. The temperature "θ° " rises when the  rotation parameter " �̀�", density "𝜌" , "d1" , "𝑘" , 

 " 𝐷𝑎" , " ℋ" and Brinkman number"𝐵𝑟" all go up. It goes decrease when the inclination 

magnetic field angle "Φ", the slip parameter "𝛽" , couple-stress parameter"𝛾"  , phase 

difference "𝜙°"  , and the viscosity "𝜇"  all go up.  

8. When the values of" ℋ"and   "γ"  are increased, the trapped boluses are eliminated. The 

inclination magnetic field angle  "Φ", " �̀�", "𝜌" , "d1" and  " 𝐷𝑎"  have a increasing impact 

on the bolus size.                                                                                            

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