Engineering, Technology & Applied Science Research Vol. 8, No. 3, 2018, 2923-2930 2923  
  

www.etasr.com Opanuga et al.: Hall Current and Joule Heating Effects on Flow of Couple Stress Fluid with … 
 

Hall Current and Joule Heating Effects on Flow of 
Couple Stress Fluid with Entropy Generation 

Abiodun A. Opanuga  
Department of Mathematics 

Covenant University 
Ota, Nigeria 

abiodun.opanuga@covenantuiversity.edu.ng  

Sheila A. Bishop  
Department of Mathematics 

Covenant University 
Ota, Nigeria 

sheila.bishop@covenantuiversity.edu.ng 

Hilary I. Okagbue  
Department of Mathematics 

Covenant University 
Ota, Nigeria 

hilary.okagbue@covenantuiversity.edu.ng  

Olasunmbo O. Agboola  
Department of Mathematics 

Covenant University 
Ota, Nigeria 

ola.agboola@covenantuiversity.edu.ng
 

 

Abstract—In this work, an analytical study of the effects of Hall 
current and Joule heating on the entropy generation rate of 
couple stress fluid is performed. It is assumed that the applied 
pressure gradient induces fluid motion. At constant velocity, hot 
fluid is injected at the lower wall and sucked off at the upper 
wall. The obtained equations governing the flow are transformed 
to dimensionless form and the resulting nonlinear coupled 
boundary value problems for velocity and temperature profiles 
are solved by Adomian decomposition method. Analytical 
expressions for fluid velocity and temperature are used to obtain 
the entropy generation and the irreversibility ratio. The effects of 
Hall current, Joule heating, suction/injection and magnetic field 
parameters are presented and discussed through graphs. It is 
found that Hall current enhances both primary and secondary 
velocities and entropy generation. It is also interesting that Joule 
heating raises fluid temperature and encourages entropy 
production. On the other hand Hartman number inhibits fluid 
motion while increase in suction/injection parameter leads to a 
shift in flow symmetry. 

Keywords-Hall current; Joule heating; entropy generation; 
couple stress fluid; Adomian decomposition method 

I. INTRODUCTION  
The study of hydromagnetic flow has been extensively 

investigated in the past years due to its applications in MHD 
generators, flow control, shock damping in car absorbers, 
nuclear reactors, plasma studies, purifications of metal from 
non-metal enclosures, geothermal energy extractions, polymer 
technology and metallurgy. Relevant investigation was 
pioneered in the first half of the twentieth century. Thereafter 
several studies have been conducted. Author in [1] investigated 
the spontaneous magnetic field in a conducting liquid in 
turbulent motion. In [2], magnetohydrodynamics at high 
Hartmann numbers were investigated. Author in [3] considered 
the effect of a uniform magnetic field on the Eckman layer over 
an infinite horizontal plate at rest relative to an electrically 

conducting liquid rotating with uniform angular velocity about 
a vertical axis while authors in [4] analyzed the combined 
effect of free and forced convection on MHD flow in a rotating 
porous channel, authors in [5] investigated the radiation effect 
of magnetohydrodynamics flow of gas between concentric 
spheres. In [6], authors considered the radiative effect on 
velocity, magnetic and temperature fields of a 
magnetohydrodynamic oscillatory flow past a limiting surface 
with variable suction In [7], author considered the 
hydromagnetic natural convection flow between vertical 
parallel plates with time-periodic boundary conditions, Authors 
in [8] studied convection heat and mass transfer in a 
hydromagnetic flow of second grade fluid in the presence of 
thermal radiation and thermal diffusion. In the studies above 
where the effect of magnetic field is reported, small and 
moderate values of the magnetic field are assumed. However, 
the current trend of research is geared toward a strong magnetic 
field and a low density gas due to its numerous applications 
such as in space flight, nuclear fusion research, 
magnetohydrodynamic generators, refrigeration coils, electric 
transformers, Hall accelerators and biomedical engineering 
(e.g. cardiac MRI and ECG). Hall current occurs when the 
applied magnetic field is very strong or gas is ionized with low 
density leading to a reduction in conductivity normal to the 
magnetic field, as a result of the free spiraling of electrons and 
ions around the magnetic lines of force before collisions. This 
then induces a current in the direction of both electric and 
magnetic fields. This is referred to as Hall effect, the induced 
current is called Hall current [9-15]. 

In this study, the Hall current and Joule heating effects on 
the flow of couple stress fluid with entropy generation is 
considered. The study is essential due to the fact that entropy 
generation occurs in moving fluid with high temperature which 
can lead to loss of resources and effort if the inherent 
irreversibility in the fluid flow is not well addressed. Authors in 



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www.etasr.com Opanuga et al.: Hall Current and Joule Heating Effects on Flow of Couple Stress Fluid with … 
 

[16] submitted that all processes that produce, convert and 
consume energy must be re-examined very carefully and all 
available-work destruction mechanisms must be removed. To 
the best of our knowledge similar study has not yet been 
reported in literature, although various factors responsible for 
the entropy production have been studied, for instance in [17] 
authors considered effects of velocity slip and temperature 
jump on the entropy generation in MHD flow over a porous 
rotating disk. Also, authors in [18] presented entropy 
generation on MHD nanofluid blood flow due to peristaltic 
waves. Recently, authors in [19] studied the thermodynamic 
analysis of hydromagnetic third grade fluid flow through a 
channel filled with porous medium. In [20], the effect of 
thermal radiation on the entropy generation of hydromagnetic 
flow through porous channel was investigated. More studies on 
the factors responsible for entropy production are reported in 
[21-26]. Several techniques such as Homotopy perturbation 
[27], differential transform method [28-29], variational 
iteration technique [30], finite difference technique [31] etc. are 
available in literature. However, the Adomian decomposition 
method is applied in this work due to its simplicity in 
application and rapid convergence, (see Tables I and II). 
Furthermore, the method has been used to analyze various 
linear and nonlinear problems such as the fractional-order 
differential equations [32], the time dependent Edem–Fowler 
type equation [33], the Navier–Stokes equations [34], the 
evolution model [35], the Flierl–Petviashivili equation [36], the 
fourth-order wave equation [37], the peristaltic transport model 
[38], the Fokker–Planck equation [39] and the Bratu’s problem 
[40]. 

II. PROBLEM FORMULATION 
Fully developed, steady, incompressible and electrically 

conducting couple stress fluid between two parallel plates of 
distance h apart has been investigated. The coordinate system is 
taken such that the x-axis is along the lower plate in the flow 
direction, the y-axis is normal to the xy-plane while z-axis is 
made perpendicular to the plates. A constant pressure gradient 
is induced along the x-direction. Hot fluid is injected into the 
channel wall at the lower plate and sucked off at the upper plate 
with the same velocity. Following [41], the generalized Ohm’s 
law with Hall current is 

0

( ) ( )e eJ J B E q B
B

 
        (1) 

It is further assumed that if (jx, jy, jz) are the components of 
the current density J, the equations of conservation of electric 
charge 0J   shows that jz is constant which is assumed to 
be zero because jz=0 at the plates which are electrically non-
conducting. It then implies that jz=0, everywhere in the flow. 
Furthermore, the electrical field E=0 [42]. Following the given 
assumptions, (1) becomes: 

0x yj mj wB       (2) 

0y xj mj uB       (3)  

Note that e em    represents the Hall parameter. Solving 
(2) and (3) for jx and jy gives: 

 0 21x
B

j w mu
m


 

     
(4) 

 021y
B

j mw u
m


 


    (5) 

The governing equations for the flow following [43-44] are: 

Momentum equation along axis x:

  

 

* 2 * 4 *

* *2 *4

2
* *

2

2 * 2 *
* *

*2 *2

;
1

(0) (0) 0 ( ) ( ),

du dp d u d u

dy dx dy dy

M
u mw

m

d u d u
u u h h

dy dy

      




   



  (6) 

Momentum equation along axis y: 
 

 
* 2 * 4 * 2

* *
* *2 *4 2

2 * 2 *
* *

2 2

;
1

(0) (0) 0 ( ) ( ),

dw d w d w M
w mu

dy dy dy m

d w d w
w w h h

dy dy

     


   

  (7) 

Energy equation: 

 2

2 2* 2 * * *

0 * *2 * *

2 22 * 2 *
2 *2 *
0*2 *2

* *

;

(0) ( ) 0.

p

dT d T du dw
c v k

dy dy dy dy

d u d w
B w u

dy dy

T T h

 

 

    
       

     
    

      
     

 

 (8) 

Introducing the following dimensionless variables, 
** * * 2

0

0 0 0 0

2
0 0

0

, , , ,

, , , ,

h

h

T Ty u w h dP
y u w G

h v v T T v dx

T T v hh
a v s

T v






 


     




    

  (9) 

Equations (6), (7) yield the following dimensionless form; 

 
2 4 2

2 2 4 2

2 2

2 2

1
,

1

(0) (0) 0 (1) (1),

du d u d u M
s G u mw

dy dy a dy m

d u d u
u u

dy dy

    


   
 (10) 

 
2 4 2

2 2 4 2

2 2

2 2

1
,

1

(0) (0) 0 (1) (1),

dw d w d w M
s w mu

dy dy a dy m

d w d w
w w

dy dy

   


   

 (11) 

 

2 22

2

2 22 2
2 2

2 2 2
;

(0) 0, (1) 1.

r

d d du dw
sp Br

dy dy dy dy

Br d u d w
Jh u w

a dy dy

 



 

    
       

     
    
           
   

 (12) 



Engineering, Technology & Applied Science Research Vol. 8, No. 3, 2018, 2923-2930 2925  
  

www.etasr.com Opanuga et al.: Hall Current and Joule Heating Effects on Flow of Couple Stress Fluid with … 
 

where 

2 2 2
0

2

2 2
2 2 0

Pr , , ,
( ) ( )

, .

p G

h h

c T h E
Br Ns

k k T T k T T

B h
Jh BrM M

 




  
 

 



 
  (13) 

III. ADOMIAN DECOMPOSITION  
The Adomian decomposition method is applied by writing 

(10), (11) and (12) in integral form as: 

 

32
1

2

2

2
0 0 0 0

2

( )
3!

,

1

y y y y

b
u y b y y

du d u
s G

dY dY
dYdYdYdY

M
u mw

m

  

 
    

 
 
  



  (14) 

 

32
1

2 2

2 2
0 0 0 0

( )
3!

,
1

y y y y

f
w y f y y

dw d w M
s w mu dYdYdYdY

dY dY m

  

 
   

 
  

(15) 

 

and  

 

3 4

2 2

2 22 2

2 2 2
0 0

2 2

( )

Pr

+ .
y y

y b b y

d du dw
s Br

dy dy dy

Br d u d w
dYdY

a dy dy

Jh u w







  

    
      

    
 

                
 

 
 
 


 (16) 

 

Using the partial sum in (17) and the Adomian polynomials 
for the non-linear terms in (18), (14)-(16) yield (19)-(21). The 
final stage for the implementation of ADM is the coding of 
equations (19)-(21) in symbolic Mathematica software, which 
yields a very large symbolic solution. 

0 0

0

( ) ( ),   ( ) ( ),  

( ) ( ),

n n
n n

n
n

u y u y w y w y

y y 

 

 





 



 


  (17)  

2 2 2
0 0 1 0 1 2 1 0 2

2 2 2
0 0 1 0 1 2 1 0 2

: , 2 , 2
.

: , 2 , 2

n

n

A u A u A u u A u u u

B w B w B w w B w w w

     


       (18) 

 

32
1

0

2

2

2
0 0 0 0

2

( )
3!

,

1

r

n
n

y y y y

b
u y b y y

du d u
s G

dY dY
dYdYdYdY

M
u mw

m



  

 
    

 
 
  





 (19) 

 

32
1

0

2 2

2 2
0 0 0 0

( )
3!

,
1

r

n
n

y y y y

f
w y f y y

dw d w M
s w mu dYdYdYdY

dY dY m



  

 
   

 





 (20) 

 

3 4
0

2 2

2 22 2

2 2 2
0 0

( )

Pr

+ .

r

n
n

y y

n n

y b b y

d du dw
s Br

dy dy dy

Br d u d w
dYdY

a dy dy

J A B









  

    
      

    
 

                
 

 
 
 





 (21) 

A. Results Verification 
The approximate solution obtained by ADM can be verified 

by comparing our result and the one in [44] as displayed in 
Table I. 

B. Entropy Generation Analysis 
The local entropy generation expression [35] for the flow is 

given as 

 2 2

2 2 2* * *

2 * * *
0 0

2 2 22 * 2 *
* *0

*2 *2
0 0

G

k dT du dw
E

T dy T dy dy

Bd u d w
w u

T dy dy T v








      
                

    
           

 (22) 

Using (9) in (22) yields  

 

2 2 2

2 22 2
2 2

2 2 2

d Br du dw
Ns

dy dy dy

Br d u d w Jh
w u

a dy dy





      
                

    
            

 (23) 

Entropy generation within the flow can be analysed by 
letting 

 

2 2 2

1 2

2 22 2
2 2

2 2 2

,
d Br du dw

N N
dy dy dy

Br d u d w Jh
w u

a dy dy





      
                

    
            

  (24) 

If Bejan number (Be) is less than one-half, irreversibility 
due to viscous dissipation dominates entropy generation and 
when (Be) is greater than one-half irreversibility due to heat 
transfer dominates the fluid flow. Bejan number (Be) equal to 
one-half indicates that both contribute equally to entropy 
generation. The Bejan number can be written as: 



  

Fig
Br
(J)
vel
y i
the
fro
inc
eff
ma
dem
tre
suc
shi

is 
du
usu
tha
the
par
tem
ho
flu

Engineerin

www.etasr

1 1

1s

N
Be

N
 



TABL

G=1, s=0.1,
S/N 

0 
0.1 0.0
0.2 0.0
0.3 0.0
0.4 0.0
0.5 0.0
0.6 0.0
0.7 0.0
0.8 0.0
0.9 0.0
1 2.20

TABLE II.  

S/N 
0 

0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
1 

I

This section
gures 1-13 by 
r=0.5. To expla
), suction/inje
locity profile, 
s presented in
e influence of
om these figur
creases. This i
fect of the Lo
agnetic field 
monstrate that

end is revers
ction/injection
ift in flow sym

In Figures 3
displayed. We

ue to the fact 
ually generate
at retards fluid
e variation in 
rameter is d
mperature is r
wever Hall cu

uid temperature

 

ng, Technology

r.com 

2

1

,   .
N

N
 

  

LE I.  RESUL

α=1, Μ=0 
UADM 

0 −
03714182 
07024039 
09612623 
11255559 
11818879 
11257781 
09616293 
07027825 
03716603 

02672×10-11 -

RESULTS (EXACT
G = 1, S = 0

UEXACT 
6.94E-15 

0.003714182 
0.007024039 
0.009612623 
0.011255559 
0.011818879 
0.011257781 
0.009616293 
0.007027825 
0.003716603 

9.98E-15 

IV. RESULT
n presents res

fixing parame
ain the effect o
ection (s) and
 the non-dime

n Figures 1 to 
f Hall current 
res that fluid m
is due to the f
orentz resistiv

on fluid m
t fluid motion 
sed at the 
n parameter (
mmetry. 

(a) and 3(b), m
e can see that
that applicati

es a resistive ty
d flow.  The r

Hall parame
epicted in F
educed for di

urrent does no
e.  

y & Applied Sci

Op

LTS FOR VELOCITY

[44
UEXACT 

−3.91321×10−14 
0.00371418 
0.00702404 
0.00961262 
0.0112556 
0.0118189 
0.0112578 
0.00961629 
0.00702782 
0.0037166 

3.91321×10-14 

T SOLUTION) FOR 
0.1, A = 1, M = 0 

UADM 
0 

0.00371418 
0.00702404 
0.00961262 
0.01125556 
0.01181888 
0.01125778 
0.00961629 
0.00702782 
0.0037166 
2.0267E-11 
 

TS AND DISCUS

sults of the a
eters Pr=0.71,
of Hall curren
d magnetic (
ensional veloc
3. Figures 1(a
on fluid velo

motion is enha
fact that Hall 
ve force impo
motion. Figur

accelerates at
upper wall 

(s>0) increase

magnetic effe
t fluid motion 
ion of transve
ype of force c

response of tem
eter, Joule hea
igures 4-6. I
fferent values

ot have much s

ience Research

panuga et al.: H

 

(25

Y PROFILES 

4] 
UADM 

0 
0.00371418 
0.00702404 
0.00961262 
0.0112556 
0.0118189 
0.0112578 
0.00961629 
0.00702783 
0.0037166 

-7.66586×10-18

VELOCITY PROFIL

 
6.94E-15 
1.20E-12 
2.22E-12 
2.91E-12 
3.05E-12 
2.44E-12 
8.78E-13 
1.87E-12 
6.13E-12 
1.22E-11 
2.03E-11 

SION 

analysis as plo
 m=0.5, Jh=2,

nt (m), Joule he
(M) paramete
city u and w a
a) and 1(b) illu
ocity. It is app
anced as hall c

current reduc
osed by the ap
res 2(a) and 
t the lower wa
with injectio

es, this indica

ect on fluid ve
is inhibited, t

erse magnetic
called Lorentz
mperature pro
ating and ma
In Figure 4, 
s of Hall param
significant effe

h V

Hall Current an
 

5)  

 

LES  

ots in 
, G=1, 
eating 

ers on 
against 
ustrate 
parent 
urrent 
es the 
pplied 

2(b) 
all, the 
on as 
ates a 

elocity 
this is 

c field 
z force 
file to 
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Vol. 8, No. 3, 20

nd Joule Heatin

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erent m 

 2.  (a): Prima
different s 

018, 2923-2930 

ng Effects on F

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Flow of Couple 

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2926  

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Vol. 8, No. 3, 20

nd Joule Heatin

  

Fig. 6

Figure 5 dem
rease in the J
ect of varying

mperature. It i
gnetic field in

ating which e
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ating and su
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018, 2923-2930 

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ent, Joule he
on Bejan numb
and 13 revea
f Hall curren
trend is rever

ease. It is conc
ansfer contribu

r different s 

ent m 

rent J 

s of hall curren
uple stress flu
ns were obt

mian decompo
to compute en

h V

Hall Current an
 

eating, 
ber are 
al that 
nt and 
sed as 
cluded 
ute to 

nt and 
uid has 
tained, 
osition 
ntropy 

gen
exp









Vol. 8, No. 3, 20

nd Joule Heatin

neration and 
plain the physi

 From the resu

Hall current i
while the tem

Increase in 
temperature, e

Hartman nu
temperature a

Both viscous
entropy gener

F

Fi

u' 

w' 

μ 

p 

T' 

T0 

k 

Br 

G 

θ 

u, w 

v0 

s 

018, 2923-2930 

ng Effects on F

irreversibility
ics of the flow

ults it is conclu

increases fluid
mperature and B

Joule heatin
entropy genera

umber retards
and Bejan num

s dissipation a
ration. 

ig. 12.  Bejan n

g. 13.  Bejan nu

NOMEN

velocity c

velocity c

dy

flu

initia

thermal c

Br

dimensio

dimen

the dim

constant veloc

suction

Flow of Couple 

y ratio. Plots 
. 

uded that: 

d velocity and 
Bejan number 

ng paramete
ation and Beja

s fluid flow
mber. 

and heat tran

 

number for differe

umber for differen

NCLATURE 

component along 

component along 

ynamic viscosity 

fluid pressure 

uid temperature 

al fluid temperatur

conductivity of the

rinkman number 

onless pressure gra

sionless temperat

mensionless veloci

city of fluid suctio

n/injection parame

2928  

Stress Fluid w

are presente

entropy gener
are reduced. 

r enhances 
an number. 

w, increases 

sfer contribut

 

ent s 

nt M 

x-axis 

y-axis 

re 

e fluid 

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ture 

ities 

on/injection 

eter 

with … 

ed to 

ration 

fluid 

fluid 

tes to 



Engineering, Technology & Applied Science Research Vol. 8, No. 3, 2018, 2923-2930 2929  
  

www.etasr.com Opanuga et al.: Hall Current and Joule Heating Effects on Flow of Couple Stress Fluid with … 
 

(s>0 is suction and s<0 is injection) 

v kinematic viscosity 

NS dimensionless entropy generation rate 

Be Bejan number 

EG  entropy generation parameter 

Ω temperature difference parameter 

Cp specific heat at constant pressure 

η fluid particle size effect due to couple stresses 

h channel width 

ρ fluid density 

Β0 uniform transverse magnetic field 

Μ magnetic field parameter 

Jh Joule heating parameter 

a couple stress parameter 

Pr Prandtl number 

m Hall current parameter 

REFERENCES 
[1] G. K. Batchelor, “On the spontaneous magnetic field in a 

conducting liquid in turbulent motion”, Proceedings of the Royal 
Society of London. Series A, Mathematical and Physical Sciences, 
Vol. 21, No. 1066, pp. 406-416, 1950 

[2] J. C. R. Hunt, J. A. Shercliff, “Magnetohydrodynamics at high 
Hartmann number”, Annual Review of Fluid Mechanics, Vol. 3, 
pp. 37-62, 1971 

[3] S. Gupta, “Magnetohydrodynamic Ekmann layer”, Acta 
Mechanica, Vol. 13, pp. 155-160, 1972 

[4] D. R. V. Prasada Rao, D. V. Krishna, L. Debnath, “Combined 
effect of free and forced convection on MHD flow in a rotating 
porous channel”, International Journal of Mathematics and 
Mathematical Sciences, Vol. 5, No. 1, pp. 165-182, 1982 

[5] J. A. Gbadeyan, A. S. Idowu, “Radiation effect of 
magnetohydrodynamic flow of gas between concentric spheres”, 
Journal of the Nigerian Association of Mathematical Physics, Vol. 
10, pp. 305-314, 2006 

[6] O. Areo, P. O. Olaleye, J. A. Gbadeyan, “The radiative effect on 
velocity, magnetic and temperature fields of a magneto 
hydrodynamic oscillatory flow past a limiting surface with variable 
suction”, European Journal of Applied Sciences, Vol. 3, No. 3, pp. 
102-112, 2011  

[7] S. O. Adesanya, E. O. Oluwadare, J. A. Falade, O. D. Makinde, 
“Hydromagnetic natural convection flow between vertical parallel 
plates with time-periodic boundary conditions”, Journal of 
Magnetism and Magnetic Materials, Vol. 396, pp. 295–303, 2015 

[8] I. Olajuwon, “Convection heat and mass transferr in a 
hydromagnetic flow of second grade fluid in the presence of 
thermal radiation and thermal diffusion”, International 
Communications in Heat and Mass Transfer, Vol. 38, No. 3, pp. 
377–382, 2011 

[9] Raptis, P. C. Ram, “Effects of hall current and 
rotation”,”Astrophysics and Space Science, Vol. 106, No. 2, pp. 
257-264, 1984 

[10] M. Abd El-Aziz1, T. Nabil, “Homotopy analysis solution of 
hydromagnetic mixed convection flow past an exponentially 
stretching sheet with hall current”, Mathematical Problems in 
Engineering, Vol. 2012, Article ID 454023, 2012 

[11] S. Das, S. L. Maji, R. N. Jana, “Hall Effects on unsteady 
hydromagnetic flow induced by a porous plate”, International 
Journal of Computer Applications (0975 – 8887), Vol. 57, No. 18, 
pp. 37-44, 2012 

[12] K. Jha, C. A. Apere, “Combined effect of hall and ion-slip currents 
on  unsteady MHD Couette flows in a rotating system”, Journal of 
the Physical Society of Japan, Vol. 79, No. 10, 2010 

[13] Pal, B. Talukdar, I. S. Shivakumara, K. Vajravelu, “Effects of Hall 
current and chemical reaction on oscillatory mixed convection-
radiation of a micropolar fluid in a rotating system”, Chemical 
Engineering Communications, Vol. 199, pp. 943–965, 2012 

[14] S. Asghar, M. R. Mohyuddin, T. Hayat, “Effects of Hall current 
and heat transfer on flow due to a pull of ecentric rotating disks”, 
International Journal of Heat and Mass Transfer, Vol. 48, No. 3-4, 
pp. 599-607, 2005 

[15] M. Aboeldahab, E. M. E. Elbarbary, “Hall current effect on 
magnetohydrodynamic free convection flow past a semi-infinite 
vertical plate with mass transfer”, International Journal of 
Engineering Science, Vol. 39, No. 14, pp, 1641-1652, 2001 

[16] Arikoglu, I. Ozkol, G. Komurgoz, “Effect of slip on entropy 
generation in a single rotating disk in MHD flow”, Applied 
Energy, Vol. 85, No. 12, pp. 1225-1236, 2008 

[17] M. M. Rashidi, N. Freidooni Mehr, “Effects of velocity slip and 
temperature jump on the entropy generation in MHD flow over a 
porous rotating disk”, Journal of Mechanical Engineering, Vol. 1, 
No. 3, pp. 3-14. 2012 

[18] M. M. Rashidi, M. M. Bhatti, M. A. Abbas, M. E. Ali, “Entropy 
generation on MHD blood flow of nanofluid due to peristaltic 
waves”, Entropy, Vol. 18, No. 4, 2016 

[19] S. O. Adesanya, J. A. Falade, “Thermodynamics analysis of 
hydromagnetic third grade fluid flow through a channel filled with 
porous medium”, Alexandria Engineering Journal, Vol. 54, No. 3, 
pp. 615–622, 2015 

[20] A. Opanuga, J. A. Gbadeyan, S. A. Iyase, H. I. Okagbue, “Effect of 
thermal radiation on entropy generation of hydromagnetic flow 
through porous channel”, Pacific Journal of Science and 
Technology, Vol. 17, No. 2, pp. 59-68, 2016 

[21] S. O. Adesanya, J. A. Falade, S. Jangili, O. A Beg, “Irreversibility 
analysis for reactive third-grade fluid flow and heat transfer with 
convective wall cooling”, Alexandria Engineering Journal, Vol. 56, 
No. 1, pp. 153–160, 2017 

[22] S. O. Adesanya, O. D. Makinde, “Irreversibility analysis in a 
couple stress film flow along an inclined heated plate with 
adiabatic free surface”, Physica A, Vol. 432, pp. 222–229, 2015 

[23] A. Opanuga, H. I. Okagbue, O. O. Agboola, “Irreversibility 
analysis of a radiative MHD Poiseuille flow through porous 
medium with slip condition”, Proceedings of the World Congress 
on Engineering, London, UK, Vol. 1, pp. 167-171, July 5-7, 2017 

[24] M. Pakdemirli, B. S. Yilbas, “Entropy generation in a pipe due to 
non-Newtonian fluid flow: Constant viscosity case”, Sadhana, Vol. 
31, No. 1, pp. 21–29, 2006 

[25] A. Opanuga, H. I. Okagbue, O. O. Agboola, O. F. Imaga, “Entropy 
generation analysis of buoyancy effect on hydromagnetic 
poiseuille flow with internal heat generation”, Defect and 
Diffusion, Vol. 378, pp. 102-112, 2017 

[26] L. B. Erbay, Z. Altac, B. Sulus, “An analysis of the entropy 
generation in a square enclosure”, Entropy, Vol. 5, No. 5, pp. 496-
505, 2003 

[27] A. Opanuga, E. A. Owoloko, O. O. Agboola, H. I. Okagbue, 
“Application of homotopy perturbation and modified Adomian 
decomposition methods for higher order boundary value 
problems”, Proceedings of the World Congress on Engineering, 
London, UK, Vol. 1, July 5-7, 2017 

[28] M. C. Agarana, M. E. Emetere, “Solving non-linear damped driven 
simple pendulum with small amplitude using a semi analytical 
method”, ARPN Journal of Engineering and Applied Sciences, 
Vol. 11, No. 7, pp. 4478-4484, 2016 

[29] M. C. Agarana, A. N. Ede, “Application of differential transform 
method to vibration analysis of damped railway bridge on 
Pasternak foundation under moving train”, Proceedings of The 
World Congress on Engineering and Computer Science 2016, 
London, UK, June 29–July 1, 2016, 2016 

[30] O. O. Agboola, J. A. Gbadeyan, A. A. Opanuga, M. C. Agarana, S. 
A. Bishop, J. G. Oghonyon, “Variational Iteration Method for 
Natural Frequencies of a Cantilever Beam with Special Attention 



Engineering, Technology & Applied Science Research Vol. 8, No. 3, 2018, 2923-2930 2930  
  

www.etasr.com Opanuga et al.: Hall Current and Joule Heating Effects on Flow of Couple Stress Fluid with … 
 

to the Higher Modes”, Proceedings of The World Congress on 
Engineering and Computer Science 2017, London, UK, July 5-7, 
2017 

[31] A. A. Opanuga, E. A. Owoloko, H. I. Okagbue, O. O. Agboola, 
“Finite difference method and Laplace transform for boundary 
value problems”, Proceedings of The World Congress on 
Engineering and Computer Science 2017, London, UK, July 5-7, 
2017 

[32] V. Daftardar-Gejj H. Jafari, “Solving a multi-order fractional 
differential equation using Adomian Decomposition”, Applied 
Mathematics and Computation, Vol. 189, No. 1, pp. 541-548, 2007 

[33] A. M. Wazwaz, “Analytical solution for the time-dependent 
Emden–Fowler type of equations by Adomian decomposition 
method”, Applied Mathematics and Computation, Vol. 166, No. 3, 
pp. 638-651, 2005 

[34] K.Haldar, “Application of Adomian's approximations to the 
Navier-Stokes equations in cylindrical coordinates”, Applied 
Mathematics Letters, Vol. 9, No. 4, pp. 109-113, 1996 

[35] B. Zhang, Q.-B. Wu, X.-G. Luo, “Experimentation with two-step 
Adomian decomposition method to solve evolution models”, 
Applied Mathematics and Computation, Vol. 175, No. 2, pp. 1495-
1502, 2006 

[36] A. M. Wazwaz, “Padé approximants and Adomian decomposition 
method for solving the Flierl–Petviashivili equation and its 
variants”, Applied Mathematics and Computation, Vol. 182, No. 2, 
pp. 1812-1818, 2006 

[37] H. Haddadpour, “An exact solution for variable coefficients fourth-
order wave equation using the Adomian method”, Mathematical 
and Computer Modelling, Vol. 44, No. 11–12, pp. 1144-1152, 
2006 

[38] S.Nadeem, N. S. Akbar, “Effects of heat transfer on the peristaltic 
transport of MHD Newtonian fluid with variable viscosity: 
Application of Adomian decomposition method”, Communications 
in Nonlinear Science and Numerical Simulation, Vol. 14, No. 11, 
pp. 3844-3855, 2009 

[39] M. Tataria, M. Dehghana, M. Razzaghi, “Application of the 
Adomian decomposition method for the Fokker–Planck equation”, 
Mathematical and Computer Modelling, Vol. 45, No. 5–6, pp. 639-
650, 2007 

[40] S. O. Adesanya, E. S. Babadipe, S.A. Arekete, “A new result on 
Adomian decomposition method for solving Bratu’s problem”, 
Mathematical Theory and modeling, Vol. 3, No. 2, pp. 116-120, 
2013 

[41] T. G. Cowling, Magnetohydrodynamics, Interscience, Inc, New 
York, 1957 

[42] R. C. Meyer, “On Reducing Aerodynamic Heat-Transfer Rates by 
MagnetohydrodynamicTechniques”, Journal of the Aerospace 
Sciences, Vol. 25, No. 9, pp. 561-566, 1958 

[43] S. Das, R. N. Jana, “Effects of hall current on entropy generation in 
a porous channel with suction/injection”, International Journal of 
Energy & Technology, Vol. 5, No. 25, pp. 1–11, 2013 

[44] S. O. Adesanya, O. D. Makinde, “Effects of couple stresses on 
entropy generation rate in a porous channel with convective 
heating”, Computational and Applied Mathematics, Vol. 34, No. 1, 
pp. 293–307, 2015 

[45] A. Bejan, Entropy Generation through Heat and Fluid Flow, Wiley, 
New York, 1982