70 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

 

  

 

 

Analytical Study of Soret and Dufour effect in the Electro-osmotic 

peristaltic flow of Rabinowitsch fluid model 
 

 

 

 

 

 

 

 

 

 

 Abstract  

     The present paper concerned with study the of combined electro-osmotic peristaltic 

transport with heat and mass transfer which is represented by the Soret and Dufour 

phenomenon with the presence of the Joule electrothermal heating through a microchannel 

occupy by Rabinowitsch fluid. The unsteady two-dimensional governing equations for flow 

with energy and concentration conservation have been formed in a Cartesian coordinate 

system and the lubrication theory is applied to modify the relevant equations to the problem. 

The Debye-Hukel linearization approximation is utilizing to modify the 

electrohydrodynamics problem. The expressions for the axial velocity, the temperature 

profile, the concentration profile, and the volumetric flow rate are obtained analytically to 

gain exact solutions, while the numerical integration is used to analyze the pressure rise. The 

attitude of the Helmholtz-Smoluchowski velocity, the electro-osmotic parameter, the Joule 

electrothermal heating, the Dufour number, the Soret number, is studied and graphically 

analyzed. It is showed that the presence of the electric field increased the velocity of the fluid 

and pressure rise, and it is revealed that the temperature and concentration profile congregate 

in the center of the channel with increase in the Soret and Dufour number. 

Keyword: Peristaltic flow, Electro-osmosis, Axial electric field, Rabinowitsch fluid, Soret 

and Dufour effect, Joule heating. 

1.Introduction  

     For many times, attention has been paid to studying a mechanism called peristalsis, which 

is a special type of fluid transport, it is created by the propagation of expansion or contraction 

progressive waves along the flexible wall of the conduit. Peristaltic mechanics find its 

Ibn Al Haitham Journal for Pure and Applied Science 

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

 

Doi: 10.30526/34.2.2629 

 

 

Article history: Received, 15,June,2020, Accepted29,November,2020, Published in April 2021 

 

Saba S. Hasen 

sabaoday79@gmail.com 

Department of Mathematics, College 

of Sciences, University of Baghdad, 

Baghdad, Iraq 

 

Ahmed M. Abdulhadi 

ahm6161@yahoo.com             

Department of Applied Science, 

University of Technology, Baghdad, 

Iraq 

 

 

mailto:sabaoday79@gmail.com
mailto:ahm6161@yahoo.com


  

71 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

importance in biological science in a general way and hydrodynamics in (an especial way). 

The literature detects that various studies have illustrated the phenomenon under diverse 

influences. Numerous researches are cited in [1-5]. 

Electro-osmosis is defined as the movement of an ionized liquid relative to a stationary 

charged superficies by means of an applied electric potential. The Electro-osmosis field 

allows the fluid infusion and control of flow using external electric fields, rescinding the 

requirement for mechanical pumps or valves with moving components. The velocities of flow 

correlating for electro-osmosis are not controlled by the size of the channel, with condition 

that the electrical double layer (EDL) is much smaller than the distinctive length scale of the 

channel. Electro-osmosis is considered a significant phenomenon in many biological and 

medical processes, where it has been invested in techniques of industrial separation in 

biotechnology and especially in medical micro-pumps, which have become common in 

microfluidics, the designs of electro-osmotic can produce considerable pressures and flow 

with no need to moving parts of mechanical and also in devices of capillary electrophoresis, 

pumping of the electro-osmotic can fulfill high effectiveness in capillaries lower than 100 lm 

and this is advantageous to deployment in systems of miniaturized chemical analysis. The 

pumps Electro-osmotic have analogous features to the pumps of electrohydrodynamic (EHD) 

and the pumps of a traveling wave, where the electrical force imposed generate the influence 

of pumping with no need for any mechanical parts and Thus, maintenance and replacement of 

other parts can be alleviated. The continuous amelioration in the design of the electro-osmotic 

pump has stimulated great attention in both experimental prototype testing and also 

computational and mathematical modeling. These two approaches have confirmed to be 

extremely important and complimentary in accelerating the appearance of next-generation 

electro-osmotic micro-pumps. 

In the last few decades, literature has been made on electro-osmotic peristaltic transports. 

Pumps of the peristaltic when actuated associate with the electrostatic process achieves better 

results. Peristalsis with Electroosmotic pumping grants us an enormous range of applications 

in every area. Several mathematical models with different physical applications and diverse 

flow systems have been investigated. Chakraborty [6] is one of the first investigations in this 

field, as he has been proven that the axial electric field can raise the rates of microfluidic 

transport in the peristaltic pumps of in the microtubules. Tripathi et al. [7] discussed the effect 

of the external electric field on the Newtonian model with peristaltic flow through the 

microchannel and they concluded that the applied electric field can control the velocity and 

pressure gradient of the fluid during the flux, however, the impact of Joule electrothermal 

heating and buoyancy on nanofluids in a two-dimensional microchannel under electroosmotic 

peristaltic pumping is investigated in Refs.[8,9]. Further studies in this field are cited in [10-

13]. While Prakash et al. [14] studied the electroosmotic peristaltic transport impact on 

biorheological fluid in the microchannel with consideration of the existence of Joule heating, 

convective boundary condition, and thermal radiation. Investigations of the influence of 

electro-magnetohydrodynamics (EHD) characteristics together with heat transfer have been 

reported in Refs. [15-18]. Some other praiseworthy investigation regarding the topic can be 

seen via Refs. [19-25]. 

Heat transfer Occupies a substantial role in the cooling processes of medical and industrial 

applications and is currently deemed as a significant field of study since is a very important 

heat transfer in the human body. Also, heat transfer includes many intricate processes like 



  

72 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

evaluating skin burns to an organism, the destruction of unwanted tissues as a result of cancer, 

the technique of dilution in check blood flow, making of paper, processing of food, 

vasodilation, and radiation between the surface and its environment. Mass transfer is another 

substantial phenomenon in industry and physiology. It has numerous applications such as the 

process of membrane separation, diffusion of nutrients’ out from the blood to neighboring 

tissues, osmosis of reverse, the process of distillation, the process of combustion, and 

chemical impurities diffusion. Several attempts can be viewed in the available reference list 

[26-34]. 

This paper displays a new hydrodynamic model to simulate the effect of the external electric 

field and the electric double layer thickness (EDL) on unsteady peristaltic transport of 

Rabinowitsch fluid through a symmetric microchannel with consideration the influence of 

Soret and Dufour phenomenon. Also, the effect of Joule electrothermal heating. Accordingly, 

the paper is organized as follows: in Section. 2, the mathematical model is formulated for this 

problem and The exact solution for the axial velocity, the temperature profile, concentration, 

and the volumetric flow rate is offered, whereas resort to a numerical integration to find the 

pressure rise. In Section.3, the impacts of the physical parameters are examined in detail via 

graphical representations.  Finally, in Section 4, Key observations of the results are 

concluded. 

 

2.Modeling   

     Let's start with the movement of the Non-Newtonian Rabinowitsch fluid model subject to 

the  electroosmotic peristaltic transport through two-dimensional a symmetric horizontal 

microchannel of width 2𝑎 as is illustrated in Figure (1). The fluid is under the influence of 

heat and mass transfer, with the interest in studying the effect of the Sort and Dufour 

phenomenon. The flow results from a sinusoidal peristaltic wave along the channel with wave 

speed 𝑐 and wavelength 𝜆. The Cartesian coordinates were used to investigate the problem 

where the wave propagates parallel to the 𝑋-direction while 𝑌-axis represented the 

perpendicular to the wave . 

The mathematical formula of the channel can be expressed as follows 

�̅� = ±�̅�(�̅�, 𝑡̅) = ±(𝑎 − �̅� cos2
𝜋

𝜆
 (�̅� − 𝑐 𝑡̅))                                                                                                                            

(1) 

Where 𝑎 represents the half-width of the channel, �̅� is the wave amplitude, 𝑐 wave speed, 𝑡̅ is 

the time and �̅�(�̅�, 𝑡̅) the transverse displacement of the upper and lower wall respectively. 

 
Figure 1. The geometry of the channel 

 



  

73 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

The governing equations for unsteady flow in two dimensional of incompressible non-

Newtonian Rabinowitsch fluid under an axially applied electro kinetic body force in the 

laboratory frame (�̅�, �̅�, 𝑡)̅ can be reviewed as follows. 
𝜕�̅�

𝜕�̅�
+

𝜕�̅�

𝜕�̅�
= 0                                                                                                                                           (2) 

𝜌 (
𝜕�̅�

𝜕𝑡̅
+ �̅�

𝜕�̅�

𝜕�̅�
+ �̅�

𝜕�̅�

𝜕�̅�
) = −

𝜕�̅�

𝜕�̅�
+

𝜕�̅��̅��̅�
𝜕�̅�

+
𝜕�̅��̅��̅�

𝜕�̅�
+ �̅�𝑒 �̅��̅� + 𝜌𝑔𝐵𝑇 (�̅� − �̅�0) + 𝜌𝑔𝐵𝑐 (𝐶̅ − 𝐶0̅)                  (3) 

𝜌 (
𝜕�̅�

𝜕𝑡̅
+ �̅�

𝜕�̅�

𝜕�̅�
+ �̅�

𝜕�̅�

𝜕�̅�
) = −

𝜕�̅�

𝜕�̅�
+

𝜕�̅��̅��̅�
𝜕�̅�

+
𝜕�̅��̅��̅�

𝜕�̅�
                                                                                         (4) 

𝜌𝑐𝑝 (
𝜕�̅�

𝜕𝑡̅
+ �̅�

𝜕�̅�

𝜕�̅�
+ �̅�

𝜕�̅�

𝜕�̅�
) = κ (

𝜕2�̅�

𝜕�̅�2
+

𝜕2�̅�

𝜕�̅�2
) +

𝐷𝐾𝑇

𝐶𝑠
(

𝜕2𝐶̅

𝜕�̅�2
+

𝜕2𝐶̅

𝜕�̅� 2
) + 𝜎�̅��̅�

2
                                                  (5) 

(
𝜕𝐶̅

𝜕𝑡̅
+ �̅�

𝜕𝐶̅

𝜕�̅�
+ �̅�

𝜕𝐶̅

𝜕�̅�
) = 𝐷 (

𝜕2𝐶̅

𝜕�̅�2
+

𝜕2𝐶̅

𝜕�̅�2
) +

𝐷𝐾𝑇

𝑇𝑚
(

𝜕2�̅�

𝜕�̅�2
+

𝜕2�̅�

𝜕�̅�2
)                                                                     (6) 

And the constituent equation for the Rabinowitsch fluid model is formulated  as follows: 

𝑆�̅̅��̅� +  ̅𝑆̅
3

�̅��̅� = 𝜇
𝜕�̅�

𝜕�̅�
                                                                                                                            (7) 

where 𝜌 the density of the fluid, �̅�, �̅� are the axial velocity and transverse velocity 

respectively, �̅� is the pressure, 𝑔 is the gravitational acceleration, 𝐵𝑇  is the coefficient of 

thermal expansion, �̅� is temperature,  𝐵𝑐  is the coefficient of concentration expansion, 𝐶̅ is the 

concentration, 𝑐𝑝 is the specific heat at the constant pressure, κ is thermal conductivity,  𝐾𝑇  is 

the thermal-diffusion ratio, 𝜎 is electrical conductivity,  𝐶𝑠is the concentration 

susceptibility, 𝑇𝑚 is the mean temperature. And �̅�𝑒 and �̅��̅�  are denote to the electric field and 

the net charge density in a unit volume of the fluid respectively. the Poisson's-Boltzmann 

equation for the electrical potential is used due to the electric double layer in the microchannel 

and is expressed as  

∇2Φ̅ = −
�̅�𝑒

𝜀0
                                                                                                                                           (8) 

where the charge density is given by 

 �̅�𝑒 = 𝑒 𝑧(�̅�
+ − �̅�−)                                                                                                                              (9) 

Where 𝑒 is the electronic charge, 𝑧 is charge balance, �̅�+and �̅�− are the number of densities of 

positive (cations) and negative (anions) respectively. To obtain the potential distribution it 

necessary to determine the charge number density, therefore Nernst-Planck equation [35] 

useful for this purpose, which can represent as follows  

(
𝜕�̅�±

𝜕𝑡̅
+ �̅�

𝜕�̅�±

𝜕�̅�
+ �̅�

𝜕�̅�±

𝜕�̅�
) = 𝐷(

𝜕2�̅�±

𝜕�̅�2
+

𝜕2�̅�±

𝜕�̅�2
) ±

𝐷𝑒𝑧

𝑇𝑒𝐾𝐵
[

𝜕

𝜕�̅�
(�̅�±

𝜕Φ̅

𝜕�̅�
) +

𝜕

𝜕�̅�
(�̅�±

𝜕Φ̅

𝜕�̅�
)]                                 (10) 

Where 𝐷 the diffusion coefficient, 𝐾𝐵 the Boltzmann constant, 𝑇𝑒  is the average temperature 

of the electrolytic solution.  

And the boundary conditions accompanying the wall are  
𝜕�̅�

𝜕�̅�
= 0, �̅� = 𝑇0, 𝐶̅ = 𝐶0,        𝑎𝑡        �̅� = 0                                                                                        (11) 

�̅� = 0 ,  �̅� = 𝑇1, 𝐶̅ = 𝐶1, 𝑎𝑡    �̅� = ±�̅�(�̅�, 𝑡̅)                                                                                      (12) 

Defining the following quantities to non-dimensionalize the equations governing the flow 

problem 

𝑥 =
�̅�

𝜆
, 𝑦 =

�̅�

𝑎
,    𝑡 =

𝑐𝑡̅

𝜆
,     𝑝 =

𝑎2�̅�

𝑐𝜇𝜆
 , 𝛿 =

𝑎

𝜆
   ,   𝑢 =

�̅�

𝑐
,   𝑣 =

�̅�

𝛿𝑐
, 𝑅𝑒 =

𝜌𝑐𝑎

𝜇
,      

𝑆𝑖𝑗 =
𝑎𝑆�̅�𝑗

𝑐𝜇
, 𝜃 =

�̅� − 𝑇0
𝑇1 − 𝑇0

,   𝑛 =
�̅�

𝑛0
, =

𝑐2𝜇2 ̅

𝑎2
, 𝜑 =

�̅�

𝑎
 , 𝐶 =

𝐶̅ − 𝐶0
𝐶1 − 𝐶0

 , 𝑈𝐻𝑆 = −
�̅��̅� 𝜉 0

𝜇 𝑐
,   

Φ =
Φ̅

𝜉
,  𝑆𝑐 =

𝜇

𝜌 𝐷
,  𝑃𝑟 =

𝜇𝑐𝑝  

𝜅
, ℎ =

�̅�

𝑎
 𝑆 =

𝜎�̅��̅�
2𝑎2

𝜅(𝑇1 − 𝑇0)
, 𝐺𝑟𝑡 =

𝜌𝑔𝛽𝑇 𝑎
2(𝑇1 − 𝑇0)

𝜇 𝑐
 



  

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

𝐺𝑟𝑐 =
𝜌𝑔𝛽𝐶𝑎

2(𝐶1−𝐶0)

𝜇 𝑐
 , 𝐷𝑢 =

𝐷𝐾𝑇 (𝐶1−𝐶0)

𝑐𝜌𝐶𝑠𝜇(𝑇1−𝑇0)
, 𝑆𝑟 =

𝜌𝐷𝐾𝑇 (𝑇1−𝑇0)

𝜇𝑇𝑚(𝐶1−𝐶0)
                                                                (13) 

 

Where 𝑥 the non-dimensional axial coordinate, 𝑦 the non-dimensional transverse coordinate, 𝑡 

the non-dimensional time, 𝑝 the non-dimensional pressure, 𝛿 the wavenumber, 𝑢, 𝑣 the non-

dimensional axial and transverse velocity components, 𝑅𝑒 the Reynolds number, 𝑆𝑖𝑗 the non-

dimensional shear stress, 𝜃 the non-dimensional temperature, 𝑛 non-dimensional bulk 

concentration (number density) of the ions in the electrolyte,  𝑛0 the concentration of ions at 

the bulk,  the coefficient of pseudo-plasticity, 𝐶 the non-dimensional concentration,  𝑈𝐻𝑆 the 

Helmholtz-Smoluchowski velocity,  Φ the non-dimensional electric potential, 𝜉 the zeta 

potential,  𝑆𝑐  the Schmidt number,  𝑃𝑟 the Prandtl number,  ℎ the non-dimensional transverse 

wall displacement, 𝑆 the normalized generation term that represents the ratio of Joule heating 

to surface heat flux(for constant wall temperature), 𝐺𝑟𝑡 the local temperature Grashof number, 

𝐺𝑟𝑐 the local concentration Grashof number, 𝐷𝑢the Dufour number,  𝑆𝑟the Soret number. 

After employing Eq.(13) in Eqs. (2)-(7) and (10), with rearrangement, the following forms 

will be produced. 

 

𝛿(
𝜕𝑢

𝜕𝑥
+

𝜕𝑣

𝜕𝑦
) = 0                                                                                                                                   (14) 

𝑅𝑒𝛿 (
𝜕𝑢

𝜕 𝑡
+ 𝑢

𝜕𝑢

𝜕𝑥
+ 𝑣

𝜕𝑢

𝜕𝑦
) = −

𝜕𝑝

𝜕𝑥
+ 𝛿

𝜕𝑆𝑥𝑥

𝜕𝑥
+

𝜕𝑆𝑥𝑦

𝜕𝑦
−

𝑎2

𝜇 𝑐

𝜇 𝑐

𝜉𝜀0
𝑒 𝑧𝑛0(𝑛

+ − 𝑛−)𝑈𝐻𝑆 + 𝐺𝑟𝑡 𝜃 + 𝐺𝑟𝑐  𝐶     (15) 

𝑅𝑒𝛿 3 (
𝜕𝑣

𝜕 𝑡
+ 𝑢

𝜕𝑣

𝜕𝑥
+ 𝑣

𝜕𝑣

𝜕𝑦
) = −

𝜕𝑝

𝜕𝑦
+ 𝛿 2

𝜕𝑆𝑦𝑥

𝜕𝑥
+ 𝛿

𝜕𝑆𝑦𝑦

𝜕𝑦
                                                                         (16) 

𝑅𝑒𝑃𝑟 𝛿(
𝜕𝜃

𝜕𝑡
+ 𝑢

𝜕𝜃

𝜕𝑥
+ 𝑣

𝜕𝜃

𝜕𝑦
) = δ2

𝜕2𝜃

𝜕𝑥2
+

𝜕2𝜃

𝜕𝑦2
+

𝐷𝑚𝐾𝑇 (𝐶1−𝐶0)

𝐶𝑠𝜅(𝑇1−𝑇0)
𝛿 2

𝜕2𝐶

𝜕𝑥2
+

𝐷𝑚𝐾𝑇 (𝐶1−𝐶0)

𝐶𝑠𝜅(𝑇1−𝑇0)

𝜕2𝐶

𝜕𝑦2
+

𝑎2

𝜅(𝑇1−𝑇0)
𝜎�̅��̅�

2
  (17) 

𝑅𝑒 𝑆𝑐 𝛿 (
𝜕𝐶

𝜕  𝑡
+ 𝑢

𝜕𝐶

𝜕𝑥
+ 𝑣

𝜕𝐶

𝜕𝑦
) = 𝛿 2

𝜕2𝐶

𝜕𝑥2
+

𝜕2𝐶

𝜕𝑦2
+

𝐾𝑇(𝑇1−𝑇0) 

𝑇𝑚(𝐶1−𝐶0)
𝛿 2

𝜕2𝜃

𝜕𝑥2
+

𝐾𝑇 (𝑇1−𝑇0) 

𝑇𝑚(𝐶1−𝐶0)

𝜕2𝜃

𝜕𝑦2
                                (18) 

𝑅𝑒 𝑆𝑐 𝛿(
𝜕𝑛±

𝜕 𝑡
+ 𝑢

𝜕𝑛±

𝜕𝑥
+ 𝑣

𝜕𝑛±

𝜕𝑦
) = 𝛿 2

𝜕2𝑛±

𝜕𝑥2
+

𝜕2𝑛±

𝜕𝑦2
± 𝛿 2  

𝑒𝑧

𝑇𝑒𝐾𝐵

𝜕

𝜕𝑥
(𝑛±

𝜕ξΦ

𝜕𝜆𝑥
) +

𝑒𝑧ξ 

𝑇𝑒𝐾𝐵

𝜕

𝜕𝑦
(𝑛±

𝜕Φ

𝜕𝑦
)              (19) 

Applying the lubrication theory, as is customary for peristaltic hydrodynamics, the following 

emerging linearized conservation equations are obtained 

−
𝜕𝑝

𝜕𝑥
+

𝜕𝑆𝑥𝑦

𝜕𝑦
−

𝑎2

𝜇 𝑐

𝜇 𝑐

𝜉𝜀0
𝑒 𝑧𝑛0(𝑛

+ − 𝑛−)𝑈𝐻𝑆 + 𝐺𝑟𝑡 𝜃 + 𝐺𝑟𝑐  𝐶 = 0                                                        (20) 

 
𝜕𝑝

𝜕𝑦
= 0                                                                                                                                                 (21) 

𝜕2𝜃

𝜕𝑦2
+ 𝑃𝑟 𝐷𝑢

𝜕2𝐶

𝜕𝑦2
+ 𝑆 = 0                                                                                                                      (22) 

𝜕2𝐶

𝜕𝑦2
+ 𝑆𝑐  𝑆𝑟

𝜕2𝜃

𝜕𝑦2
= 0                                                                                                                             (23) 

And the non-dimensional constituent equation for the Rabinowitsch fluid model is 

𝑆𝑥𝑦 +  𝑆𝑥𝑦
3  =

𝜕𝑢

𝜕𝑦
                                                                                                                                (24) 

The non-dimensional associated boundary conditions are 
𝜕𝑢

𝜕𝑦
= 0, 𝜃 = 0, 𝐶 = 0, 𝑆𝑥𝑦 = 0         𝑎𝑡        𝑦 = 0                                                                             (25) 

𝑢 = 0, 𝜃 = 1, 𝐶 = 1,           𝑎𝑡       𝑦 = ±ℎ(𝑥, 𝑡)                                                                               (26) 

And the non-dimensional Nernst-Planck equation becomes  
𝜕2𝑛±

𝜕𝑦2
+

𝑒𝑧ξ 

𝑇𝑒𝐾𝐵

𝜕

𝜕𝑦
(𝑛±

𝜕Φ

𝜕𝑦
) = 0                                                                                                                (27) 



  

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It is necessary to calculate the charge number density to obtain the value of the potential 

distribution and this is done by finding the solution of equation. (27) with help the bulk 

conditions 𝑛± = 1 at Φ = 0 and 
𝜕𝑛

𝜕𝑦
= 0 at 

𝜕Φ

𝜕𝑦
= 0, and the result is  

𝑛± = 𝑒
+̅

𝑒𝑧ξ 

𝑇𝑒𝐾𝐵
Φ

                                                                                                                                    (28) 

By substituting (28) in equation.(20), we obtain the electrical distribution as follows, we 

obtain the electrical distribution as follows 

𝜕2Φ

𝜕𝑦2
= −

𝑎2𝑒 𝑧 𝑛0

𝜉𝜀0
(𝑒

+
𝑒𝑧ξ 

𝑇𝑒𝐾𝐵
Φ

− 𝑒
−

𝑒𝑧ξ 

𝑇𝑒𝐾𝐵
Φ

)                                                                                             (29) 

We can write in another form as 
𝜕2Φ

𝜕𝑦2
= −

𝑎2𝑒 𝑧 𝑛0

𝜉𝜀0
sinh(

𝑒𝑧ξ 

𝑇𝑒𝐾𝐵
Φ)                                                                                                             (30) 

Now by employing the 𝐷𝑒𝑏𝑦𝑒 − 𝐻�̈�𝑐𝑘𝑒𝑙 linearization approximation the form of the 

Poisson-Boltzmann equation is transformed as follows  

 
𝜕2Φ

𝜕𝑦2
= −𝑚2Φ                                                                                                                                     (31) 

Where 𝑚 = 𝑎 𝑒 𝑧  √
2𝑛0

𝜀0𝑇𝑒𝐾𝐵
=

𝑎

𝜆𝑑
  and is called the electro-osmotic parameter, 𝜆𝑑 is Debye 

length or characteristic thickness of the electrical double layer (EDL). 

By solving the equation.(31) which subject to boundary conditions 

 
𝜕Φ

𝜕𝑦
|

𝑦=0
= 0   and Φ|𝑦=ℎ = 1,  the potential distribution function is found as follows  

Φ =
cosh(𝑚𝑦)

cosh(𝑚ℎ)
                                                                                                                                       (32) 

Now Eq.(20) can be written in the following form 

−
𝜕𝑝

𝜕𝑥
+

𝜕𝑆𝑥𝑦

𝜕𝑦
+ 𝑚2Φ𝑈𝐻𝑆 + 𝐺𝑟𝑡 𝜃 + 𝐺𝑟𝑐  𝐶 = 0                                                                                  (33) 

 

3.Analytical solutions 

     The temperature and concentration equations are obtained when solving equation. (22) and 

equation.(23) simultaneous using the boundary conditions equation (25)and (26)as follows 

𝜃 =
−2𝑦−ℎ2𝑦 𝑆+ℎ 𝑦2𝑆+2 𝑦 𝐷𝑢 𝑃𝑟 𝑆𝑐 𝑆𝑟

2ℎ(−1+𝐷𝑢 𝑃𝑟 𝑆𝑐 𝑆𝑟)
                                                                                                      (34) 

𝐶 =
−2𝑦+ℎ2𝑦 𝑆 𝑆𝑐 𝑆𝑟−ℎ𝑦

2𝑆 𝑆𝑐 𝑆𝑟+2 𝑦 𝐷𝑢 𝑃𝑟 𝑆𝑐𝑆𝑟

2ℎ(−1+𝐷𝑢𝑃𝑟𝑆𝑐𝑆𝑟)
                                                                                          (35) 

Substitute equation.(34) and equation (35) in equation (33) and integrating it with using the 

boundary conditions equation.(25), the following result yield  

𝑆𝑥𝑦 =
1

12ℎ(−1+𝐷𝑢𝑃𝑟𝑆𝑐𝑆𝑟)
(−12ℎ

𝜕𝑝

𝜕𝑥
𝑦 + 6𝑦2𝐺𝑟𝑐 + 6𝑦

2𝐺𝑟𝑡 + 3ℎ
2𝑦2𝑆 𝐺𝑟𝑡 − 2ℎ𝑦

3𝑆𝐺𝑟𝑡 −

3ℎ2𝑦2𝑆 𝐺𝑟𝑐  𝑆𝑐 𝑆𝑟 + 2ℎ𝑦
3𝑆 𝐺𝑟𝑐  𝑆𝑐 𝑆𝑟 +. ..                                                                                                                             

(36) 

The axial velocity is found by compensation equation (36) in (24) and using the remaining 

conditions, the 𝑢 is as follows 

𝑢 = (362880ℎ4𝑚6
𝜕𝑝

𝜕𝑥
Cosh[ℎ𝑚] + 181440ℎ6𝑚6

𝜕𝑝

𝜕𝑥

3
Cosh[ℎ𝑚] + 120960ℎ4𝑚6

𝜕𝑝

𝜕𝑥
Cosh[3ℎ𝑚]+ 

60480ℎ6𝑚6
𝜕𝑝

𝜕𝑥

3
Cosh[3ℎ𝑚] − 120960ℎ4𝑚6Cosh[ℎ𝑚]𝐺𝑟𝑐 −

217728ℎ6𝑚6
𝜕𝑝

𝜕𝑥

2
Cosh[ℎ𝑚]𝐺𝑟𝑐 – …                                                                                                             (37) 

The volumetric flow rate can be defined as follows 

𝑄 = ∫ 𝑢 𝑑𝑦
ℎ

0
                                                                                                                                     (38) 



  

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The relationship between the laboratory frame and the wave frame can be represented by the 

following transformation 

�̅� = 𝑥 − 𝑡,    �̅� = 𝑦,    �̅� = 𝑢 − 𝑐,   �̅� = 𝑣,                                                                                           (39) 

It can express the volumetric flow rate in the wave frame as follows 

�̅� = ∫ �̅�(�̅�, �̅�)𝑑�̅� = ∫ (𝑢 − 1)𝑑�̅�
ℎ(𝑥)

0

ℎ(�̅�)

0
                                                                                           (40) 

From equations (38) -(40) yield  

�̅� = 𝑄 − ℎ                                                                                                                              (41) 

 Averaging volumetric flow rate over one time period, we obtain   

𝑄∗ = ∫ 𝑄 𝑑𝑡 = ∫ (�̅� + ℎ)
1

0
𝑑𝑡 = �̅� + 1 −

𝜑

2

1

0
                                                                                       (42) 

Rewrite equation (55) as  

𝑄∗ = 𝑄 − ℎ + 1 −
𝜑

2
                                                                                                                           (43) 

The dimensionless form of the volumetric flow rate in the laboratory frame  

by virtue of equation (38), is 

𝑄 = (362880ℎ4𝑚6
𝜕𝑝

𝜕𝑥
Cosh[ℎ𝑚] + 181440ℎ6𝑚6

𝜕𝑝

𝜕𝑥

3
Cosh[ℎ𝑚] + 120960ℎ4𝑚6

𝜕𝑝

𝜕𝑥
Cosh[3ℎ𝑚] +

60480ℎ6𝑚6
𝜕𝑝

𝜕𝑥

3
Cosh[3ℎ𝑚] − 120960ℎ4𝑚6Cosh[ℎ𝑚]𝐺𝑟𝑐 − 217728ℎ

6𝑚6
𝜕𝑝

𝜕𝑥

2
Cosh[ℎ𝑚]𝐺𝑟𝑐 −

40320ℎ4𝑚6Cosh[3ℎ𝑚]𝐺𝑐 − 72576ℎ
6𝑚6

𝜕𝑝

𝜕𝑥

2
Cosh[3ℎ𝑚]𝐺𝑟𝑐 + ⋯                                               (44) 

From the above equation, we can derive the pressure gradient by integrating Eq. (44)    

 
𝜕𝑝

𝜕𝑥
= (5(−42ℎ5𝑚3 Cosh[ℎ𝑚]𝐺𝑟𝑐 − 14ℎ

5𝑚3 Cosh[3ℎ𝑚]𝐺𝑟𝑐 − 42ℎ
5𝑚3+ Cosh[ℎ𝑚]𝐺𝑟𝑡 

−9ℎ7𝑚3 𝑆Cosh[ℎ𝑚]𝐺𝑟𝑡 − 14ℎ
5𝑚3 Cosh[3ℎ𝑚]𝐺𝑟𝑡 − 3ℎ

7𝑚3 𝑆Cosh[3ℎ𝑚]𝐺𝑟𝑡 

+9ℎ7𝑚3 𝑆Cosh[ℎ𝑚]𝐺𝑟𝑐 𝑆𝑐 𝑆𝑟 + 3ℎ
7𝑚3 𝑆Cosh[3ℎ𝑚]𝐺𝑟𝑐 𝑆𝑐 𝑆𝑟 + 42ℎ

5𝑚3 Cosh[ℎ𝑚]𝐷𝑢𝐺𝑟𝑐𝑃𝑟 𝑆𝑐 𝑆𝑟 +

14ℎ5𝑚3 Cosh[3ℎ𝑚]𝐷𝑢𝐺𝑟𝑐 𝑃𝑟 𝑆𝑐 𝑆𝑟 + 42ℎ
5𝑚3 Cosh[ℎ𝑚]𝐷𝑢𝐺𝑟𝑡 𝑃𝑟 𝑆𝑐 𝑆𝑟 + ⋯.                                 (45) 

The pressure difference across one wavelength is calculated from the previous equation 

∆𝑝 = ∫
𝜕𝑝

𝜕𝑥
 𝑑𝑥

1

0
                                                                                                                                    (46) 

The stream function is related to the velocity components in the wave frame by the following 

formula after resorting to  equation (37) 

�̅� =
𝜕𝜓

𝜕�̅�
, �̅� = −

𝜕𝜓

𝜕�̅�
                                                                                                                             (47) 

The heat transfer coefficient at the wall is defined by  

Ζ(𝑥) = ℎ𝑥 𝜃𝑦 (ℎ)                                                                                                                                (48) 

 

4.Numerical results and discussion 

 The main purpose of this section is to monitor the physical behavior of the 

Rabinowitsch fluid model flow as it passes through a very small channel and this behavior is 

embodied by graphs and how the parameters are associated with the problem affect the flow. 

Mathematica program is used to visualize flow behavior. 

 

4.1. Velocity Profile  

    The velocity profile response to variation in pertinent parameters as external electric field 

𝑈HS,  Joule heating 𝑆,  the Dufour number 𝐷𝑢, the Soret number 𝑆𝑟 and the coefficient of 

Rabinowitsch fluid  is depicted in Figures (2)(a-k). The snapshots of the axial velocity 

exhibit asymmetric and skewed shape about the midline of the channel also are observed 

deviations of the axial velocity from the foreseeable parabolic shape. Figure (a) shows the 

impact of the external electric field which can be simulated by maximum electro osmotic 



  

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velocity or Helmholtz-Smoluchowski 𝑈𝐻𝑆 = −
𝐸𝑥𝜉𝜀0

𝜇 𝑐
 on the axial velocity. It is clear from the 

diagram when the value of electric field high leads to boost the Helmholtz-Smoluchowski, in 

turn to superfat the electro-kinetic body force which is assistive to acceleration flux fluid in 

the center of the channel,  while it is inhibitive whenever to get closer to the channel wall. 

Obviously, the summit of the parabola velocity distribution increases and become more 

consistent with an increase 𝑈𝐻𝑆 which means an increase in the external electric field. Among 

the benefits of the electric field is the acceleration of the flow in the electric kinetics, as it is 

found that it is very sensitive to the photoelectric flow, which leads to relatively low changes 

in the intensity of the electric field. This nature is considered a benefit in practical, analog, 

micro-electric pumps, as very good acceleration in the basic flow that can be accomplished 

with relatively slight adjustments in the electric field. This, in turn, reduces costs at the same 

time and addresses the problems of electromagnetic compatibility. Figure (b) illustrates the 

role of Joule heating on the axial velocity. It is clear that the velocity accelerate  toward the 

charge carrying walls with a significant drop off towards the center of the conduit when 

increasing the positive value 𝑆 with 𝑈HS = 1 and vice versa, a slowdown in the axial velocity 

occur when increasing negative values 𝑆. A concentration gradient of positive ions is created 

near the wall surface due to the existence of negative ion charges on the wall surface and as a 

result. There is an electrical potential distribution in the electrolyte which is called the electric 

double layer. The height of the (axial) electric field leads to a quadratically increase in the 

Joule parameter(for constant temperature difference) also the electro-kinetic body force 

significantly boosts, in Outcome, there will be an acceleration in the axial flux when Joule 

heating is positive, While deceleration occurs in flux when  Joule heating is negative. It is 

evident that the Joule parameter incites a significant modification in flux distribution through 

the channel and confirms the stellar ability of electro-osmotic phenomena in controlling 

velocity profiles at the microchannel walls. Figure (c)-(d) depict the variation of the axial 

velocity against the Soret number 𝑆𝑟 for two different value to 𝑈HS, where one can see from 

Figure (c) that increasing the Soret number with 𝑈HS = 1, causes inhibitive axial velocity on 

the left side of the channel, while the effect is almost non-existent on the other side. When 

𝑈HS is negative the maximum velocity emerges at the conduit walls while the velocity is 

minimum at the core region that is clear in Figure (d). Figure (e) reveals the impact of the 

Dufour number 𝐷𝑢 on the velocity profile. The increase in value 𝐷𝑢 leads to enhance the axial 

velocity in the region near the wall with 𝑈HS = 1. The velocity accelerates at the center core 

zone of the conduit with increasing 𝐷𝑢 with case the 𝑈HS = −1 see Figure (f). The last 

diagram Figure (g) shows the effect of the coefficient of Rabinowitsch fluid  on the axial 

velocity. The effect of the parameter is limited to the sides of the channel. It reduces the axial 

velocity in this region, while it has almost no effect in the core region. 

 

 

 

 



  

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(a)                                                              (b) 

Figure2.the axial velocity profile for variation of (a): the Helmholtz-Smoluchowski velocity𝑈𝐻𝑆, (b): the Joule 

heating 𝑆 with (𝑈HS = 1) ,and for fixed values of parameters  (𝑄
∗ = 0.5, 𝜑 = 0.2, = 0.2, 𝐷𝑢 = 0.6, 𝑃𝑟 =

0.2, 𝑆𝑐 = 0.5, 𝑆𝑟 = 0.3, 𝑚 = 2, 𝐺𝑟𝑐 = 0.5, 𝐺𝑟𝑡 = 0.3, 𝑥 = 0.1, 𝑡 = 0.1). 

 
(c )                                                          (d) 

Figure2.the axial velocity profile for variation of , (c): the Soret number 𝑆𝑟 ,( 𝑈HS = 1) (d): the Soret number 

𝑆𝑟 ,( 𝑈HS = −1), and for fixed values of parameters ( 𝑄
∗ = 0.5, 𝜑 = 0.2, = 0.5, 𝑚 = 2, 𝐷𝑢 = 0.6, 𝑃𝑟 =

0.2, 𝑆𝑐 = 0.5, 𝑆𝑟 = 1, 𝑆 = 2, 𝐺𝑟𝑐 = 0.5, 𝐺𝑟𝑡 = 0.7, 𝑥 = 0.5, 𝑡 = 0.1)                 

 
      (e)                                                              (f) 

Figure2. the axial velocity profile for variation of , (e): the Dufour number 𝐷𝑢 (𝑈HS = 1), (f): the Dufour 

number 𝐷𝑢  (𝑈HS = −1), and for fixed values of parameters ( 𝑄
∗ = 0.5, 𝜑 = 0.2, = 0.5, 𝑚 = 2, 𝐷𝑢 = 0.6,

𝑃𝑟 = 0.2, 𝑆𝑐 = 0.5, 𝑆𝑟 = 1, 𝑆 = 2, 𝑈HS = 1, 𝐺𝑟𝑐 = 0.5, 𝐺𝑟𝑡 = 0.7, 𝑥 = 0.5, 𝑡 = 0.1)                                

 
    (g) 

Figure2. the axial velocity profile for variation of, (g): the coefficient of Rabinowitsch fluid, and for fixed 

values of parameters(𝑄∗ = 0.5, 𝜑 = 0.5, 𝑚 = 2,  𝐷𝑢 = 0.6, 𝑃𝑟 = 1.2, 𝑆𝑐 = 0.5, 𝑆𝑟 = 0.5, 𝑆 = 2,  𝑈HS =

1,   𝐺𝑟𝑐 = 0.5,  𝐺𝑟𝑡  = 0.7, 𝑥 = 0.2, 𝑡 = 0.1). 

 

4.2. Temperature Distribution 

     The parameters effect on the temperature profile is analyzed via Figures (3)(a-c) and from 

the first look at the graphs, it is noted that with the positive values of Joule heating, the 

temperature decreases and is concentrated in the middle area of the channel and its values are 

in the negative part of the axis of 𝑦. Where Figure (a) reveals the impact of the positive and 

negative values of Joule heating 𝑆 on the distribution of temperature. It is observed that the 

highest temperature between 𝑦 ∈ [0,1] when 𝑆 is positive but the heat is progressively 

decreases until its values are negative which is means induces refrigeration in the electrolyte. 

The influence will be reversed with the negative value of  𝑆 where the temperature increases 

and its value becomes positive. It is also observed that in the absence of the influence of Joule 



  

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heating 𝑆 = 0, the temperature profile would be linear. It can be observed from Figure (b) that 

the  Dufour  number 𝐷𝑢   has a negative effect on the temperature profile since a decrease in 

temperature is observed in the areas near the wall. Figure (c) shows the receding of 

temperature toward  the core area, with increasing of the Soret number 𝑆𝑟 which makes the 

sides cooler than the middle.   

 

 

( a)                                                    (b) 

 

( c) 

Figure3. the temperature distribution  for variation of, (a): the Joule heating 𝑆 ,(b): the Dufour number 𝐷𝑢 , (c): 

the  Soret number 𝑆𝑟 , and for fixed values of parameters(𝜑 = 0.5, 𝑃𝑟 = 1.5, 𝑆𝑐 = 0.3,    𝑥 = 0.2, 𝑡 = 0.1).    

 

  4.3. Concentration profile 

 The concentration is affected by the diversity of the governing parameters, and this is 

shown in illustrations Figures (4) (a-c). It is observed that the joule heating  𝑆 clearly affect 

the concentration when its values differed between positive and negative. We note from 

Figure (a) with the positive values of joule heating that the concentration increases gradually 

and has positively parabolic. While negative values of 𝑆 show an opposite effect, where the 

concentration distribution takes negatively parabolic shape. In the absence of joule heating, 

the concentration is linear. The impact of the Dufour number 𝐷𝑢 is illustrated in Figure (b) 

which tells us when the value of number getting more, the concentration increases and the 

speed of response to the change in the left side of the channel is more than on the right side. 

Figure (c) presents the influence of the Soret number 𝑆𝑟 on the concentration, the graph 

indicates  that  𝑆𝑟 enhance concentration.  

 

(a)                                                      (b) 



  

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    ( c ) 

Figure4. Concentration profile for variation of, (a): the Joule heating 𝑆, (b): the Dufour  number 𝐷𝑢 , (c): the  

Soret number 𝑆𝑟 , and for fixed values of parameters (𝜑 = 0.5, 𝑃𝑟 = 1.5,  𝑆𝑐 = 0.5,    𝑥 = 0.2, 𝑡 = 0.1).        

      

4.4. Heat Transfer Coefficient 

 In this part of the analysis, the reaction of Heat transfer Ζ(𝑥) with the diversity of 

different interesting parameters is recorded. The first vision to the graphs shows the wavy 

behavior to Heat transfer rate, and this is returned to the relaxing movement and contraction 

resulted from the peristaltic flow. Figures (5)(a-c) reveals the Heat transfer profile vs. axial 

coordinate. The behavior of heat transfer suffers fluctuations in the reaction, as the area 

(0.1 ≤ 𝑥 ≤ 0.6) witnessed a decrease in the rate and an increase at  (0.6 ≤ 𝑥 ≤ 1.1). The 

presence Dufour number has a slight effect on the Heat transfer rate and this is clear in Figure 

(a). The influence of Soret number on the Heat transfer rate is indicated in Figure (b). It is 

shown that the wave shape is more consistent with an increase 𝑆𝑟. while the behavior reverses 

with Joule heating the wave be more harmonious with decreases the value of  S. This is 

illustrated in Figure (c).  

 

( a)                                                             (b) 

 

    (c ) 

Figure5. the Heat Transfer Coefficient for variation of (a): the Dufour  number 𝐷𝑢 , (b): the  Soret number 𝑆𝑟 , 

(c): the Joule heating 𝑆, and for fixed values of parameters (𝜑 = 0.5, 𝑃𝑟 = 1.5,  𝑆𝑐 = 0.3, 𝑡 = 0.1).  

 

4.5. Pressure rise  

 The complicated integral that appears in equation (46) has been processed numerically 

with the help of  Mathematica to obtain pressure rise ∆𝑝. Figures (6)(a-d) illustrate the 

behavior of the pressure rise across one wavelength against the averaged volumetric flow rate 



  

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for the Helmholtz-Smoluchowski velocity 𝑈𝐻𝑆, the Joule heating 𝑆, the electro-osmotic 

parameter 𝑚, and  the coefficient of pseudo-plasticity. There are three regimes of pumping 

that can be categorized. The positive pumping or the pumping region ( ∆𝑝 > 0 ) which 

considers the peristaltic pumping, the second regime namely the co-pumping region or 

augmented pumping region (∆𝑝 < 0) and the last the free pumping region. In general, we 

notice from the drawings that the highest value of pressure rise is in the pumping region with 

a negative the averaged volumetric flow rate. It is evident that the relation between the 

pressure rise and the averaged volumetric flow rate is an inverse relationship, the flow rate 

increases with the decrease in the pressure rise. Figure (a) shows the presence of the applied 

electric field on the pressure rate. The positive value of 𝑈𝐻𝑆 obviously, enhances the pumping 

rate (∆𝑝)  in the peristaltic pumping and augmented pumping region, while the negative 

values of 𝑈𝐻𝑆  depresses the pressure. In the free region we notice that the flow rate takes the 

highest value when 𝑈𝐻𝑆 > 0, but in the absence of the external electric field, the flow rate 

becomes zero. The influence of Joule heating has been illustrated in Figure (b). Poor 

improvement in the pumping rate in the peristaltic pumping is induced by raising the 

magnitude of the Joule heating 𝑆. Decrease the thickness of the electrical double layer EDL 

i.e. increasing  𝑚 (which has an inverse relationship with 𝜆𝑑), lead to increase in the pressure 

rate ∆𝑝, this is clear in Figure (c). It can be noticed from Figure (d) that the increment in the 

coefficient of Rabinowitsch fluid, the pumping rate decreases in the peristaltic pumping 

(∆𝑝 > 0, −2 ≤ 𝑄∗ ≤ −0.5), whereas the attitude of pumping rate changes in (∆𝑝 > 0, 𝑄∗ >

0), the pressure rise increases. 

 

( a)                                                          (b) 

Figure6. Pressure rise upon the mean flow rate 𝑄∗ for variation of (a): the Helmholtz-Smoluchowski 

velocity𝑈𝐻𝑆, (b): Joule heating 𝑆,and for fixed values of parameters ( 𝜑 = 0.5, = 0.5, 𝑚 = 2,  𝐷𝑢 =

0.6,  𝑃𝑟 = 1.2,  𝑆𝑐 = 0.3,  𝑆𝑟 = 0.5, 𝐺𝑟𝑐 = 0.5, 𝐺𝑟𝑡 = 0.7, 𝑡 = 0.5). 

 

(c)                                                           (d) 

Figure6. Pressure rise upon the mean flow rate 𝑄∗ for variation of (c): electro-osmotic parameter 𝑚, (d): the 

parameter of Pseudoplastic fluid , and for fixed values of parameters( = 0.5, 𝐷𝑢 = 0.6, 𝑃𝑟 = 1.2, 𝑆𝑐 =

0.3, 𝑆𝑟 = 0.5, 𝑆 = 2, 𝑈HS = 1, 𝐺𝑟𝑐 = 0.5, 𝐺𝑟𝑡 = 0.7, 𝑡 = 0.5).                                          

4.6. Trapping formulation 

 Streamlines visualizations in the conduit for various values of pertinent parameters for a 

fixed time (𝑡 = 0.1)  is illustrated in Figures (7) (a-d), and this visualization lets a better 

examination of the so-called trapping phenomenon, wherein an internally circulating bolus of 



  

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the fluid is formed by closed streamlines. In all the graphs, there is a considerable shortage of 

symmetry about the midline (𝑦 = 0). In a particular, evident distortion of streamlines emerges 

in Figure (a) when the magnitudes of the external electric field or (the Helmholtz-

Smoluchowski velocity𝑈𝐻𝑆) differ between positive and negative values. At 𝑈𝐻𝑆 = −1, the 

numbers of trapping bolus increase at high transverse coordinate value, whereas, in the lower 

coordinates,  the streamlines testify a shift from the bolus to the curves. Figure (b) displays 

the effect of the Dufour number 𝐷𝑢 on streamline. It is revealed that the number of the 

trapping bolus decreases for a larger value of  𝐷𝑢. The impact of the Soret number 𝑆𝑟 on the 

bolus is sketched in Figure (c). It is evident from the diagram that ascending values of 𝑆𝑟  lead 

to an increase in the number and size of the bolus. It is revealed in Figure (d) that the number 

of trapping bolus increases in the upper zone of the channel, whereas the bolus begins 

gradually shrinking in the number and expanding in the size at a lower region with 𝑆 < 0. 

  

( a) 

Figure7. Streamlines for variation of the Helmholtz-Smoluchowski velocity 𝑈𝐻𝑆, (a): 𝑈𝐻𝑆 = 1, 𝑈𝐻𝑆 = −1 and 

fixed (𝑄∗ = 0.5, 𝜑 = 0.5, = 0.5, 𝑚 = 2, 𝐷𝑢 = 0.6, 𝑃𝑟 = 1.2, 𝑆𝑐 = 0.5, 𝑆𝑟 = 0.3, 𝑆 = 2, 𝐺𝑟𝑐 = 1.5, 𝐺𝑟𝑡 = 2.5)  

  

(b) 

Figure7. Streamlines for variation of the Dufour number 𝐷𝑢 , (b):  𝐷𝑢  = 0.4, 𝐷𝑢 = 1 and fixed (𝑄
∗ = 0.5, 𝜑 =

0.5, = 0.5, 𝑚 = 2, 𝑃𝑟 = 1.2, 𝑆𝑐 = 0.5, 𝑆𝑟 = 0.3, 𝑆 = 2, 𝑈HS = 1, 𝐺𝑟𝑐 = 1.5, 𝐺𝑟𝑡 = 2.5)  

  

( c) 

Figure7. Streamlines for variation of the Soret number 𝑆𝑟 , (c):  𝑆𝑟 = 0.5 , 𝑆𝑟 = 1 and fixed (𝑄
∗ = 0.5, 𝜑 =

0.5, = 0.5, 𝑚 = 2, 𝐷𝑢 = 0.6, 𝑃𝑟 = 1.2, 𝑆𝑐 = 0.5, 𝑆 = 2, 𝑈HS = 1, 𝐺𝑟𝑐 = 1.5, 𝐺𝑟𝑡 = 2.5). 

  

( d) 

Figure7. Streamlines for variation of the Joule heating 𝑆, (d): 𝑆 = 2 , 𝑆 = −2 and fixed(𝑄∗ = 0.5, 𝜑 = 0.5, =

0.5, 𝑚 = 2, 𝐷𝑢 = 0.6, 𝑃𝑟 = 1.3, 𝑆𝑐 = 0.5,  𝑆𝑟 = 0.3,    𝑈HS = 1,  𝐺𝑟𝑐 = 1.5,  𝐺𝑟𝑡 = 2.5)      



  

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5.Conclusion 

     The mathematical and theoretical studies on the unsteady electro-osmotic peristaltic 

pumping in an isothermal finite length micro-channel have been conducted to investigate the 

effect of Soret and Dufour phenomenon and Joule electrothermal heating on Rabinowitsch 

fluid model. Mathematica program has been employed to evaluate numerical solutions 

derived for the axial velocity, the temperature profile, concentration, the heat transfer 

coefficient, and the pressure rise after simplified the electrokinetic transport equations via the 

lubrication theory. Key observations are 

1- The axial velocity is increasing functions of the external electric field 𝑈HS and this result 

agrees with offered in Tripathi et.al. [8,10] while the opposite behavior is presented in  

Jayavel [14].  

2- The axial velocity shows flat parabolic behavior in nature. This behavior increases at a left 

wall via the positive Joule heating 𝑆,  the Dufour number 𝐷𝑢.  While decreasing behavior 

is observed due to the Soret number 𝑆𝑟 .the negative Joule heating 𝑆 accelerates the axial 

velocity in the center of the channel whereas inhibited at the wall. 

3- With increasing 𝐷𝑢 and 𝑆𝑟, the temperature distribution drops and the positive joule 

heating raises the heat in mid of the channel while it slows down the heat near the wall. 

With the negative joule heating, the temperature takes exactly reversal behavior. 

4- The concentration field is an increasing function of the applied electric field (which 

represented by Joule heating 𝑆 =
𝜎�̅�

�̅�
2 𝑎2

𝜅(𝑇1−𝑇0)
), 𝐷𝑢 and 𝑆𝑟. 

5- The pressure rise increases with 𝑆, electro-osmosis parameter 𝑚, whereas the pressure rise 

testifies a changeful behavior with  .  

6- The trapping bolus distort with negative the external electric field 𝑈HS. 

7- The trapping bolus expands with increasing 𝑆𝑟 , and 𝑆 < 0, whilst the number of boluses 

decreases with 𝐷𝑢  

 

 

 

 



  

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