Jaya Presentation for Colloquium.pptx

Welcome
Date : 21.10.2016
Time : 2.30 pm
18 December 2017 Jayachandra Babu M (Colloquium) 1

Research Supervisor
Dr. N. Sandeep,
Assistant Professor,
Department of Mathematics,
VIT, Vellore.
By
Jayachandra Babu M,
Research Scholar,
14PHD0216 (EPT).
NUMERICAL STUDY OF SOME MHD FLOWS OVER A
STRETCHING SURFACE

Chapter 1 : Introduction and background
Chapter 2 : Effect of nonlinear thermal radiation on non-aligned bio-convective
stagnation point flow of a nanofluid over a stretching sheet
Chapter 3 : MHD non-Newtonian fluid flow over a slendering stretching sheet in the
presence of cross-diffusion
Chapter 4 : Magnetohydrodynamic dissipative flow across a slendering stretching sheet
with temperature dependent viscosity
Chapter 5 : Three-dimensional MHD slip flow of nanofluids over a slendering stretching
sheet with thermophoresis and Brownian motion
Chapter 6 : 3D MHD flow of ferrofluid over a slendering stretching sheet with
thermophoresis and Brownian motion
Content

Chapter - 1
Introduction and Background

Motivation to study the MHD Flow:
 Magnetohydrodynamics (MHD) is the study of the magnetic properties of
electrically conducting fluids.
 The electrically conducting non-Newtonian fluids can be opted as a cooling liquid
as their flow can be regulated by external magnetic field, which regulates the heat
transfer to some extent.
 The interaction between the conducting fluid and the magnetic field radically
modifies the flow, with consequent effects on such important flow properties as
pressure drop and heat transfer
Examples: plasmas, liquid metals, and salt water or electrolytes.
Applications: Useful in Cooling of Nuclear Reactors (Nuclear fission reactors are
often cooled by liquid sodium. Liquid sodium is often pumped around using
electromagnetic forces), wireless mouse, magnetic reasoning scanning, aeronautical
plasma flows, chemical engineering and electronics, magnetic material processing,
geophysics and control of the cooling rate etc.
18 December 2017
Jayachandra Babu M (Colloquium)
5

Classification Of Fluid Flows
 Viscous flows and inviscid flows: significance of viscosity
 Compressible and incompressible flows : change in density
with the change in pressure
 Laminar and Turbulent flows : The highly
ordered/disordered fluid motion
 Steady and unsteady flows: change in flow conditions
(velocity, cross section ..) with time
 Natural and forced flows : fluid motion is due to natural
means or forced

One, Two, and Three-Dimensional Flows
• A flow field is best characterized by its velocity distribution.
• A flow is said to be one-, two-, or three-dimensional if the flow
velocity varies in one, two, or three dimensions, respectively.
• However, the variation of velocity in certain directions can be
small relative to the variation in other directions and can be
ignored.
The development of the velocity profile in a circular pipe. V = V(r, z) and thus the
flow is two-dimensional in the entrance region, and becomes one-dimensional
downstream when the velocity profile fully develops and remains unchanged in
the flow direction, V = V(r).
18 December 2017
7

Newtonian Fluid
 Newtonian fluid is a fluid in which the velocity gradient is directly
proportional to the shear stress.
 Though many fluids are Newtonian fluids over a wide range of
temperatures and pressures, there are some fluids which are having the
departure from the simple Newtonian relationships. They are called non-
Newtonian fluids.
𝜏𝑦𝑧 = −𝜇
𝑑𝑢
𝑑𝑦

Non-Newtonian Fluids

Dilatant fluid:
The fluid in which the viscosity increases as the velocity gradient increases.
Ex: pastes, suspensions like corn starch in water
Pseudoplastic fluid:
The fluid in which the viscosity decreases as the velocity gradient increases.
Ex: Nail polish, blood, paint
Bingham fluid:
A viscous fluid that possesses a yield strength which must be exceeded before the fluid will
flow.
Ex: Lava

• Nanomaterials can be metals, ceramics, polymeric materials or composite
materials whose size is in the range of 1-100 nanometers (nm).
• Nanofluids are suspensions of metallic or nonmetallic nano powders in
base liquid and can be employed to increase heat transfer rate in various
applications
Nanomaterials and fluids

Heat Transfer
 Heat is a form of energy transfer from a high temperature location to a low temperature location.
The three main methods of heat transfer - conduction, convection and radiation.
 The rate of heat transfer is of great importance because of
the frequent need to either increase or decrease the rate at which
heat flows between two locations.
 Ex: electricity generation. Household electricity is most frequently
manufactured by using fossil fuels or nuclear fuels. The method involves
generating heat in a reactor. The heat is transferred to water and the water carries the heat to a
steam turbine (or other type of electrical generator) where the electricity is produced. The
challenge is to efficiently transfer the heat to the water and to the steam turbine with as little loss
as possible. Attention must be given to increasing heat transfer rates in the reactor and in the
turbine and decreasing heat transfer rates in the pipes between the reactor and the turbine.

Mass Transfer
 Mass transfer is the net movement of mass from one location, usually meaning stream, phase,
fraction or component, to another.
 Mass transfer occurs in many processes, such as absorption, evaporation, drying, filtration and
distillation.
 Some common examples of mass transfer processes are the evaporation of water from a pond to
the atmosphere, the purification of blood in the kidneys and liver, and the distillation of alcohol.
 In industrial processes, mass transfer operations include separation of chemical components in
distillation columns, absorbers such as scrubbers or stripping.
 Mass transfer is often coupled to additional transport processes, for instance in industrial cooling
towers. These towers couple heat transfer to mass transfer by allowing hot water to flow in contact
with hotter air and evaporate as it absorbs heat from the air.

Soret and Dufour effects: The energy flux caused by concentration gradient is known as Dufour or
diffusion-thermo effect, whereas the mass diffusion due to temperature gradient is called Soret or
thermo-diffusion effect. They have their own moment in the fields such as hydrology and
geosciences. The Soret effect is employed in the detachment of isotopes and in mix among the
gases with light molecular weight (Hydrogen or Helium) and medium molecular weight (Hydrogen
or Air).
Thermophoresis: It is a mass transfer mechanism of movement of small particles due to thermal
gradient. It cause small particles to lodge on the bleak surfaces.
Brownian motion: Random motion of particles suspended in a fluid resulting from their collision
with the fast-moving atoms or molecules in the gas or liquid.
Ex: These have numerous applications in various fields like nuclear reactor safety, aerosol
technology, environmental and atmosphere pollution.
Other physical effects used

Numerical Methods
 When analytical solution of the mathematically defined problem is possible but it is time-consuming
and the error of approximation we obtain with numerical solution is acceptable, then we can go for
numerical methods.
 Analytical methods have limitations in case of nonlinear problem. In such cases numerical methods
works very well.

Shooting Technique
 Convert the BVP Into initial value problem
 Guess a value for the auxiliary conditions at one point of time.
 Solve the initial value problem using Euler, Runge-Kutta, Newton’s
method…
 Check if the boundary conditions are satisfied, otherwise modify the guess
and resolve the problem.
 Use interpolation in updating the guess.
 It is an iterative procedure and can be efficient in solving the BVP.
 This method is effective when the interval is [t0, tf] (initial and boundary
conditions)

Runge-Kutta-Fehlberg Method
 The trickiest part of using Runge-Kutta methods to approximate the solution of a
differential equation is choosing the right step size.
 Too large a step size and the error is too large and the approximation is inaccurate.
 Too small a step size and the process will take too long and possibly have too much
round off error to be accurate. Furthermore, the appropriate step size may change during
the course of a single problem.
 What we need is an algorithm which includes a method for choosing the appropriate step
size at each step. The Runge-Kutta Fehlberg methods do just this, which is why they have
largely replaced the Runge-Kutta methods in practice.

Uniqueness of the thesis:
 Present work
• In Ch.2, we considered non-aligned stagnation flow. This
has applications in medical device filtration technologies
and microbial fuel cell technology areas
• In Ch. 3 & 4, we considered the 2D flow over a slendering
sheet. This has applications such as polymer extrusion,
metallurgical processes and machine design.
• In Ch. 5 & 6, we considered the 3D flow over a slendering
sheet. This has applications in several fields including
aerosol technology.
 Existed works
• Existed works focused on aligned stagnation flow
• Existed works focused on 2D flow over a uniform
thickness sheet
• Existed works focused on 3D flow over a uniform
thickness sheet

Chapter - 2
Effect of nonlinear thermal radiation on non-aligned bio-convective
stagnation point flow of a nanofluid over a stretching sheet
This paper was published in Alexandria Engineering Journal (ELSEVIER)

Aim of the study
The current study covers the relative study of MHD bioconvective stagnation point flow
of a nanofluid comprising gyrotactic microorganisms across a stretching sheet in the
presence of nonlinear radiation parameter.
Bioconvection is the process of spontaneous pattern construction in suspension of
microorganisms which are a little denser compare to water and swim upward.
Bioconvection is useful in bio-reactors, fuel cell technology and bio-diesel fuels.
Now a days, to generate or raise convection in nanofluids, inquiries are produced on the
deployment of microorganisms. As nanoparticles are not having motility, their movement
will be drive by thermophoresis and Brownian motion. Thus the movement of
microorganisms is free of the movement of nanoparticles. This combination attracted
researchers because of its use in medical filtration device technologies and microbial fuel
cell technology.

Physical model of the problem
Fig. 2.1

Governing Equations:
2
2 *
3
2
16
3 *
T
f B
f p
D
T T T T T C T
u v T D
x y y C k y y y y T y

 
 
 
   
       
 
    
   
 
       
   
 
2 2
2 2
T
B
D
C C C T
u v D
x y T
y y

 
   
    
   
 
2
2
c
n
w
dW
n n n C
u v D n
x y y C C y y

 
    
    
     
 
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
𝜕𝑢
𝜕𝑥
+
𝜕𝑣
𝜕𝑦
= 0,
𝑢
𝜕𝑢
𝜕𝑥
+ 𝑣
𝜕𝑢
𝜕𝑦
= 𝑈∞
𝜕𝑈∞
𝜕𝑥
+
1
𝜌𝑓
𝜕
𝜕𝑦
𝜇𝑓
𝜕𝑢
𝜕𝑦
−
𝜎𝐵0
2
𝑢 − 𝑈∞
𝜌𝑓
,
,

By using the following similarity transformations (2.7) , we have converted the governing
equations (2.2) – (2.5) as a set of nonlinear ordinary differential equations:
 
   
       
'( ) , ,
1 1 , ,
f f f
w
w w
u c x c f G v c f
C C n n
T T
C C n n
     
      
 

 

  


 
    
  
    
(2.7)
The corresponding boundary conditions are
 
( ) , 0, , , n at 0
, , , n as
w f w w
T
u U x cx v k h T T C C n y
y
u U ax by T T C C n y

   
 
         
 

        
(2.6)

The following equations are the resultant transformed equations:
 
2 2
1 1
''' '' ' '' ' ' 0
f f e ff f M f

   
 
      
 
 
2 2
''' '' ' '' ' ' ' 0
G G e fG f G M G S

    
      
 
 
'' Pr ' '' 0
Nt
Le f
Nb
  
  
 
'' ' ' '' '' Pr ' 0
Pe Lb f
      
    
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
     
3 2 2
1 1 1 '' 3 1 1 1 Pr ' Pr ' ' Pr ' 0
w w w
Ra Ra Nt Nb f
         
   
          
   
   
   
And the corresponding boundary conditions (2.7) changed as
       
   
1 2
0, 0, ' 1, ' 0, ' 1 , 0, 0 at 0
' , '' , 0, 0, 0 as
f G f g Bi
f G
       
       
         


       

(2.13)

The Magnetic field parameter, Viscosity variation parameter, Radiation parameter, Prandtl
number, Thermophoresis parameter, Brownian motion, Lewis number, Pecklet number,
Bioconvection constant and Bioconvection Lewis number are given as follows:
 
 
 
2 3
0
1
16 *
, , , ,Pr , ,
3 *
, , , ,
f T w
f f
f f
B w c
B n w n
D
B T
a
M m T T S Ra Nt
c kk
c
D C C dW n
Nb Le Pe Lb
D D n n D
  
 

  

 





 

 
       


 
     
 
(2.14)
For engineering design interest, the abbreviated Nusselt number, Sherwood number and the
density number of the motile microorganisms (after non-dimensionalization) are given by
           
1 2 1 2 1 2
' 0 , ' 0 , ' 0
x x x x x x
Nu Pe Sh Pe Nn Pe
  
      (2.15)

Effect of Magnetic field parameter on velocity, temperature and concentration profile
Fig. 2.2 Fig. 2.3 Fig. 2.4
Results and Discussion

Fig. 2.5 Effect of thermophoresis on
temperature profile
concentration profile

Fig. 2.8 Effect of bioconvection Lewis
number on gyrotactic microorganism
profile
Fig. 2.7 Effect of Pecklet number
on gyrotactic microorganism
profile

Fig. 2.9 Effect of radiation parameter on
temperature profile
Fig. 2.10 Effect of Brownian motion parameter on

Table 2.1 Values of reduced Nusselt number, reduced Sherwood number and the mass
transfer rate of the motile microorganisms for diverse parameters in oblique flow
𝑀 R𝑎 𝑁𝑡 𝑃𝑒 𝐿𝑏 𝐵𝑖 −𝜃′(0 −𝜙′(0 −𝜒′ 0
1 0.090411 1.654470 1.888114
3 0.089468 1.548341 1.763810
6 0.088414 1.441098 1.640789
1 0.090410 1.654493 1.888125
1.5 0.088766 1.655390 1.887038
2 0.087159 1.655816 1.885704
1 0.089672 1.616068 1.868808
2 0.087665 1.556681 1.839729
3 0.084420 1.528600 1.826674
0.3 0.090421 1.654328 1.576237
0.5 0.090421 1.654328 1.888383
0.7 0.090421 1.654328 2.207772
0.5 0.090421 1.654328 1.888383
0.6 0.090421 1.654328 2.005665
0.7 0.090421 1.654328 2.115546
0.1 0.090411 1.654470 1.888114
0.15 0.127607 1.632295 1.872169
0.2 0.158914 1.613592 1.857638

Table 2.2 Values of reduced Nusselt number, reduced Sherwood number and the mass
transfer rate of the motile microorganisms for diverse parameters in free stream flow
𝑀 R𝑎 𝑁𝑡 𝑃𝑒 𝐿𝑏 𝐵𝑖 −𝜃′(0 −𝜙′(0 −𝜒′ 0
1 0.090029 1.619865 1.844518
3 0.088532 1.478636 1.677737
6 0.086624 1.328780 1.505927
1 0.090025 1.619880 1.844427
1.5 0.088175 1.619641 1.842178
2 0.086314 1.618945 1.839627
1 0.089231 1.578762 1.823705
2 0.087010 1.512457 1.790153
3 0.083315 1.474527 1.770132
0.3 0.090059 1.619764 1.536353
0.5 0.090059 1.619764 1.845458
0.7 0.090059 1.619764 2.161277
0.5 0.090059 1.619764 1.845458
0.6 0.090059 1.619764 1.963004
0.7 0.090059 1.619764 2.073377
0.1 0.090029 1.619865 1.844518
0.15 0.126614 1.594756 1.825741
0.2 0.156903 1.572792 1.807985
0.090059 1.619764 1.845458
0.090059 1.619764 1.961908
0.090059 1.619764 2.078358

Table 2.3 Comparison of the present results for −𝜃′(0 and −𝜙′(0 when Pr = 𝐿𝑒 = 10,
, 𝐵𝑖 = 0.1, 𝛽 = 𝑅𝑎 = 𝜆1 = 𝑃𝑒 = 𝑀 = 𝐿𝑏 = 0
Nt −𝜃′(0
Nb=0.1
(O.D.
Makinde
et al.
2011)
−𝜃′(0
Nb=0.1
(Khan et
al. 2016)
Present
results
−𝜙′(0
Nb=0.1
(O.D.
Makinde
et al.
2011)
−𝜙′(0
Nb=0.1
(Khan et
al. 2016)
Present
results
0.1 0.0929 0.09291 0.092912 2.27741 2.27741 2.277412
0.3 0.0925 0.09252 0.092521 2.2228 2.22281 2.222811
0.5 0.0921 0.09212 0.092120 2.1783 2.17834 2.178341

Conclusions
Rising values of the Magnetic field, thermophoresis and Biot number increases the
temperature field.
The density of the motile microorganisms is a decreasing function of Peclet number
and Bioconvection Lewis number.
Brownian motion and Lewis number both suppress the concentration profile.
Increasing the external magnetic field lessen the heat and mass transfer rate.

Chapter - 3
MHD non-Newtonian fluid flow over a slendering
stretching sheet in the presence of cross-diffusion
This paper was published in Alexandria Engineering Journal (ELSEVIER)

Aim of the study
The boundary layer flow across a slendering stretching sheet has gotten awesome
consideration due to its inexhaustible pragmatic applications in nuclear reactor
technology, acoustical components, chemical and manufacturing procedures, for example,
polymer extrusion, and machine design.
In this paper, we focused on the Soret and Dufour effects on the Williamson fluid flow
over a stretching sheet with variable thickness in the presence of velocity slip parameter.
The governing nonlinear partial differential equations of this problem transformed as the
nonlinear ordinary differential equations by using suitable transformations and then solved
numerically using Runge-Kutta fourth order method based shooting technique.
18 December 2017
37

Fig. 3.1
 
1
2
m
y A x b

 

0
u v
x y
 
 
 
2 2
2 2
m T
p s p
D k
T T k T C
u v
x y C C C
y y

   
  
   
2 2
2 2
m T
m
m
D k
C C C T
u v D
x y T
y y
   
  
   
(3.1)
(3.2)
(3.3)
(3.4)
And the corresponding boundary conditions are
 
       
*
1
* *
2 3
( , ) ( ) , , 0,
, , ,
0, ,
w
w w
u
u x y U x h v x y
y
T C
T x y T x h C x y C x h
y y
and
u T T C C at y
 

 

   
 

  

   
 
    
   
 
   


     

(3.5)
2 2 2
2 2
( )
2
u u u B x u u u
u v
x y y y y

 

    
    
    

(3.6)
(3.7)
(3.8)
0 ( ) '( )
m
u U x b f 
  ,
1
0
1 1
( ) '( ) ( )
2 1
m
m m
v U x b f f
m
   

   
 
   
 
 

 
 
,
1
0
1 ( )
,
2
m
m x b
y U



 

( )
w
T T
T x T
 



 ,
( )
w
C C
C x C
 




Then the resultant transformed equations are
2
2
(1 '') ''' ' '' ' 0
1
m
f f f ff M f
m
     

1
'' Pr( ' ' '') 0
1
m
f f Du
m
   

   

1
'' ( ' ' '') 0
1
m
Sc f f Sr
m
   

   

(3.9)
   
   
1 1
2 3
1
(0) 1 ''(0) , '(0) 1 ''(0) ,
1
(0) 1 '(0) , (0) 1 '(0) ,
'( ) 0, ( ) 0, ( ) 0
m
f h f f h f
m
h h
f

   
 
 
 
   
  

  

    

      


(3.10)

3 1
3
0
2
0
0
( )
( 1) ,
2 ( )
,Pr , ,
( 1) ( )
( )
,
( )
m
p m T w
s p w
m T w
m m w
x b
m U
C
B D k C C
M Du
U m k C C T T
D k T T
Sc Sr
D T C C



 









    


 
   
  



 
 

(3.11)
Williamson fluid parameter, Magnetic field parameter, Prandtl number, Dufour number,
Schmidt number and Soret number are given as follows:
For engineering design interest, the skin friction coefficient, the local Nusselt number and the
local Sherwood number (after non-dimensionalization) are given by:
   
 
0.5 0.5
0.5
1 1
2 Re ''(0). Re '(0)
2 2
1
Sh Re '(0)
2
f x x x
x x
m m
C f Nu
m


 
 
   


 
  

(3.12)

Fig. 3.2 Soret effect on concentration
profile
Fig. 3.3 Soret effect on temperature
profile

Fig. 3.4 Dufour effect on temperature
profile
Fig. 3.5 Dufour effect on concentration
profile

Fig. 3.6 Velocity slip parameter
effect on velocity profile
Fig. 3.7 Temperature jump parameter
effect on temperature profile

Table 3.1: Values of the skin friction coefficient, local Nusselt number and local Sherwood number for different parameters in both
cases
𝑀 𝑆𝑟 𝐷𝑢 ℎ1
ℎ2
ℎ3
𝐶𝑓𝑥 −𝑁𝑢𝑥 −𝑆ℎ𝑥
𝛬 = 0 𝛬 = 0.2 𝛬 = 0 𝛬 = 0.2 𝛬 = 0 𝛬 = 0.2
1 -2.180932 -2.232527 0.491866 0.487824 0.621523 0.612501
1.5 -2.459006 -2.522645 0.455531 0.450817 0.542038 0.529592
2 -2.747013 -2.822965 0.421898 0.416492 0.452003 0.435928
0.1 -2.747014 -2.822971 0.392954 0.388011 0.828698 0.812607
0.2 -2.747014 -2.822971 0.395267 0.390203 0.732428 0.716213
0.3 -2.747014 -2.822971 0.397443 0.392227 0.634924 0.618492
0.1 -2.747014 -2.822971 0.614913 0.502342 0.576201 0.562753
0.2 -2.747014 -2.822970 0.593745 0.484776 0.507034 0.491797
0.3 -2.747014 -2.822971 0.572666 0.467279 0.436474 0.419239
0.1 -3.933728 -4.167604 0.450826 0.443378 0.579201 0.560291
0.3 -2.747014 -2.822971 0.401337 0.395672 0.436474 0.419239
0.5 -2.127096 -2.162213 0.369257 0.364538 0.335768 0.321009
0.1 -2.747014 -2.822971 0.423148 0.416551 0.412101 0.395545
0.3 -2.747014 -2.822971 0.401337 0.395672 0.436474 0.419239
0.5 -2.747014 -2.822971 0.381665 0.376787 0.458457 0.440670
0.1 -2.747014 -2.822971 0.415999 0.409827 0.510434 0.487900
0.6 -2.747014 -2.822971 0.385889 0.380608 0.358546 0.346166
1.2 -2.747014 -2.822971 0.367188 0.362161 0.264204 0.256686
M
Sr
Du
1
h2
h3
h x
Cf x
Nu
 x
Sh
 0
  0.2
  0
  0.2
  0
  0.2
 

Table 3.2 Comparison of the values of when M=Du=Sc=Sr=0, m=0.5
𝑓′′(0
𝜆 ℎ1 Khader and
Megahed [2003]
Present study
0.2 0 -0.924828 -0.924829
0.25 0.2 -0.733395 -0.733396
0.5 0.2 -0.759570 -0.759570

Conclusions
Soret and Dufour numbers increases the concentration and shows the opposite behavior
on temperature profile.
The effect of velocity slip parameter is quite opposite on velocity and temperature. It
depreciated the velocity field and enhanced the temperature field.
Velocity slip parameter enhances the skin friction coefficient, Sherwood number and
depreciates the Nusselt number.
Dufour number reduce both the heat and mass transfer rates.

Chapter - 4
Magnetohydrodynamic dissipative flow across the slendering
stretching sheet with temperature dependent viscosity
This paper was published in Results in Physics (ELSEVIER)

Aim of the study
• In this study, we analyzed the two-dimensional MHD flow across a slendering stretching
sheet in the presence of variable viscosity and viscous dissipation. The sheet is thought to
be convectively warmed. Convective boundary conditions through heat and mass are
employed.
• Similarity transformations used to convert the governing nonlinear partial differential
equations as a group of nonlinear ordinary differential equations. Runge-Kutta based
shooting technique is utilized to solve the converted equations.

Fig. 4.1

0
u v
x y
 
 
 
(4.1)
(4.2)
(4.3)
(4.4)
2
( )
u u u
u v B x u
x y y y
  
   
   
  
   
   
   
2
2
2
p p
T T k T u
u v T
x y C y c y

  
 
 
   
 
    
 
   
 
 
2
0
2
( )
m
C C C
u v D k C C
x y y

  
   
  
 
   
1 2
( , ) ( ), , 0,
, at 0
0, ,
w
w B w
u x y U x v x y
T C
k h T T D h C C y
y y
and
u T T C C at y
 
  

  
       
  


     

(4.5)
)
(     
1 1
* 1 w
T a b T T
   
   
 
 

(4.6)
       
1
1
1 ( )
, ( ) ,
2
1 1
( ) ( ) ,
2 1
( ) , ( )
n
n
n
w w
n x c df
y b u b x c
d
n df n
v b x c f
d n
T T T x T C C C x C

 
  

   


   

 
   


 
  
 
    
 
 

 
  

     



The resultant transformed equations are
 
 
3 2 2 2
1 3 2 2 2
2 2
1 0
1 1
d f n d f d f df d d f
a A f M A
d n d d n d d d


     
      
 
2
2 2
2 2
1
Pr 0
1
d d n df d f
f Ec
d d n d d
 

   
 
 

 
   
 
 
  
 
2
2
1 2
0
1 1
d d n df
Sc f Kr
d d n d n
 
 
  
 

   
 
 
 
(4.7)
(4.8)
(4.9)

   
0
1 2
0 0
1
(0) , 1,
1
1 (0) , 1 (0) ,
0, ( ) 0, ( ) 0
n df
f
n d
d d
d d
df
d

 



 
   
 
 


 



 
  
 

  


      



    


(4.10)
Variable viscosity parameter, Magnetic field parameter, Prandtl number, Eckert
number, Schmidt number and Chemical reaction parameter are given by:
 
 
 
2
0
1
5 1
2
2
0
1
, ,Pr ,
, ,
p
w
n
n
p m
C
B
A b T T M
b k
b x c k
Ec Sc Kr
C D b x c








    


 
   
 
(4.11)

The essential physical measures of concern, the skin friction coefficient, the local Nusselt number and the
Sherwood numbers are indicated as below:
   
 
0.5 0.5
2
1/2 1/2
2
0
0
0.5
1/2
0
1 1
2 Re , Re
2 2
1
Sh Re
2
f x x x
x x
n d f n d
C Nu
d d
n d
d




 







 
   
   
   
    

 
 
   
  
(4.12)
Numerical Procedure
The set of transformed equations (4.7) - (4.9) with the boundary conditions (4.10) are solved by using Runge-
Kutta based shooting technique. To analyze the three basic profiles (velocity, temperature, and concentration)
through graphs for the impacts of different parameters, for example, viscosity variation parameter, we utilize
the following:
1 1 2
2, 1, 0.65, 0.5, 0.3, 1, 0.5, 0.3, 0.2, 0.2
M a n A Ec Sc Kr   
         

Fig. 4.2 Effect of Eckert number on
temperature profile
Fig. 4.3 Effect of chemical reaction
parameter on concentration profile

Effect of variable viscosity parameter on temperature and concentration profiles
Fig.4.4 Fig.4.5

Effect of wall thickness parameter on velocity, temperature and concentration profiles
Fig.4.6 Fig.4.7 Fig.4.8

Fig. 4.9 Effect of heat transfer Biot
number on temperature profile
Fig. 4.10 Effect of mass transfer
Biot number on concentration
profile

Table 4.1: The effect of different parameters on the skin friction coefficient, local Nusselt
and the Sherwood numbers in terms of values
𝑀 𝐸𝑐 𝜆 𝑆𝑐 𝛽1 𝛽2 𝐾𝑟 𝐴 𝑓′′ 0 −𝜃′(0 −𝜙′(0
1 -1.308178 0.115245 0.168546
2 -1.662102 0.100166 0.167888
3 -1.965126 0.086884 0.167393
0.2 -1.648289 0.107210 0.167870
0.3 -1.679902 0.089271 0.167814
0.4 -1.715608 0.070135 0.167753
1 -1.711883 0.107082 0.169993
2 -1.786025 0.115823 0.172771
3 -1.863899 0.123301 0.175137
2 -1.649401 0.108530 0.177203
3 -1.649401 0.108530 0.181474
4 -1.649401 0.108530 0.184057
0.2 -1.681797 0.088135 0.167811
0.3 -1.707280 0.112911 0.167772
0.4 -1.727150 0.131196 0.167741
0.2 -1.670091 0.095216 0.167840
0.3 -1.670091 0.095216 0.233025
0.4 -1.670091 0.095216 0.289180
0.2 -1.670091 0.095216 0.160787
0.3 -1.670091 0.095216 0.163714
0.4 -1.670091 0.095216 0.165992
0.5 -1.889172 -0.007535 0.167478
0.6 -1.900625 -0.008296 0.167467
0.7 -1.913637 -0.009229 0.167454

Table 4.2: Validation of the numerical technique by comparing with the others for −𝜃′(0
M RKS Bvp4c Bvp5c
1 0.115245 0.1152451342 0.1152451342
2 0.100166 0.1001668710 0.1001668711
3 0.086884 0.0868845644 0.0868845643
4 0.006452 0.0064522314 0.0064522314

Conclusions
 Magnetic field parameter, Eckert number, heat transfer Biot number and variable viscosity
parameter are useful to enhance the temperature field of the flow over a slender sheet.
 Wall thickness parameter improves both heat and mass transfer rates.
 Mass transfer Biot number and variable viscosity parameter increase the concentration field.
 Rising values of the wall thickness parameter and Schmidt number lessen the concentration.
 Increasing values of the Eckert number lessen the heat transfer rate.

Chapter - 5
Three-dimensional MHD slip flow of nanofluids over a slendering
stretching sheet with thermophoresis and Brownian motion
This paper was published in Advanced Powder Technology (ELSEVIER)

Aim of the study
We considered water based Cu and CuO nanofluids. With the assistance of
similarity transformations, we changed the derived governed equations as
ordinary differential equations.
We exhibit and explain the graphs for various parameters of interest. We
discussed the skin friction coefficient, reduced Nusselt and reduced Sherwood
numbers for the influence of the pertinent parameters with the assistance of
tables separately for two nanofluids. (Cu-water and CuO-water).

Fig. 5.1

0
u v w
x y z
  
  
  
 
2
2
2
B
p nf B
nf
D
T T T T C T T
c u v w k D
x y z z z T z
z
 

 
 
      
 
    
 
   
 
     
  
   
2 2
2 2
T
B
D
C C C C T
u v w D
x y z T
z z

    
   
    
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
 
       
* *
1 1
* *
2 3
( , ) ( ) , , ( ) ,
, , ,
0, v=0, ,
w w
w w
u v
u x y U x h v x y V x h
z z
T C
T x y T x h C x y C x h
z z
and
u T T C C at y
 

 
   
   
    
 
    

 
   
    
   
 
   


     
2
2
2
( )
nf nf
u u u u
u v w B x u
x y z z
  
 
   
   
 
   
 
2
2
2
( )
nf nf
v v v v
u v w B x v
x y z z
  
 
   
   
 
   
 

Nanofluid parameters are given by:
( )
1
, 1 , (1 )
(1 2.5 ) ( )
3( 1) 3( 1)
1 , 1
2 ( 2) ( 1)
( )
where , , , ,
( )
nf nf p nf
f f p f
nf nf
f f
p s
s s s
f p f f f
c
r e
c
k k
k k
c k
r e k
c k
  
   
   

  
   

 

  

       
 

  
    
    


   


(5.7)
 
0.5
1 *0.5
( ) , ( )
2 1 1
( ) ( )
1 2 2
n n
n
f g
u a x y c v a x y c
a n n f g
w x y c f g
n
 


 

  
      
  

 
 
   
    
      
 
     
  
    
 
 
0.5
1 *0.5
( 1)
( )
2
n
n a
z x y c




 
  
 
  ,  
( )
w
T T T x T 
 
   ,  
( )
w
C C C x C 
 
  
(5.8)

 
   
 
   
2
3 2
3 2
3 1
1 1 1
1 1 0
1 2.5 2 2 2 1
n f f f g n f f
r n n f g M
 
 
         
   

 
       
         
   
 
   
 
         
   
 
 
   
 
   
2
3 2
3 2
3 1
1 1 1
1 1 0
1 2.5 2 2 2 1
n g g f g n g g
r n n f g M
 
 
         
   

 
       
         
   
 
   
 
         
   
 
 
2
2
2
2 1 1
Pr 1 ( ) 0
1 2 2
nf
f
k n f g n
Nb Nt d f g
k n
    
  
      
 
   
        
        
  
   
 
       
   
 
 
2 2
2 2
1
0
1
Nt f g n
Le f g
Nb n
  

    
 
 
     
     
 
 
     
 
 
(5.9)
(5.10)
(5.11)
(5.12)
And the changed boundary conditions are
(5.13)
   
   
   
1 1
1 1
2 3
1
(0) 1 ''(0) , '(0) 1 ''(0) ,
1
1
(0) 1 ''(0) , '(0) 1 ''(0) ,
1
(0) 1 '(0) , (0) 1 '(0) ,
'( ) 0, '( ) 0, ( ) 0, ( ) 0
n
f h f f h f
n
n
g h g g h g
n
h h
f g


   
 


 
   
  

  


  
   
  

  

   

        


2
0 0 0
,Pr , , ,
f p B T
f B
B C D C D T
M Nb Nt Le
a k k T k D
    
 
    
(5.14)
Magnetic field parameter, Prandtl number, Brownian motion, Thermophoresis and Lewis
number are given by :
For engineering concern, the skin-friction coefficient, the local Nusselt number and the
Sherwood numbers (after non-dimensionalization) are given by:
   
 
0.5 0.5
1/2 1/2
0.5
1/2
1 1
2 Re ''(0). Re '(0)
2 2
1
Sh Re '( )
2
0
f x x x
x x
n n
C f Nu
n


 
 
   
   
   
    


  
   
  
(5.15)
Numerical Procedure
Equations (5.9) - (5.12) subject to the conditions (5.13) are figured out with the assistance of shooting process
numerically. For numerical solutions, we assigned the values for non-dimensional parameters as
1 2 3
6.2, 0.65, 0
0.1, 0.1, 2,
.3, 0.1, 0.3, 0.3, 0.3, 1.
Pr n h h h Le
Nb Nt M
 
  
       

temperature profile

Fig. 5.4 Effect of Brownian motion
on temperature profile
Fig. 5.5 Effect of Brownian motion
on concentration profile

Effect of volume fraction parameter on velocity, temperature and concentration profiles
Fig. 5.6 Fig. 5.7 Fig. 5.8

Effect of velocity slip parameter on velocity profiles
Fig. 5.9 Fig. 5.10

Fig. 5.12 Effect of temperature jump
parameter on temperature profile
Fig. 5.13 Effect of concentration jump
parameter on concentration profile

Table 5.1: Skin friction coefficient, reduced Nusselt number and reduced Sherwood number
values of various parameters for the mixture of water and Cu
𝑁𝑏
Nt 𝜑 Le ℎ1 ℎ2 ℎ3 𝑓′′ 0 −𝜃′(0 −𝜙′(0
0.1 -1.210278 1.053276 1.416400
0.2 -1.210278 0.952262 1.568673
0.3 -1.210278 0.856031 1.618353
0.1 -1.210278 1.053276 1.416400
0.2 -1.210278 0.965144 1.232196
0.3 -1.210278 0.884537 1.106288
0.01 -1.801297 1.409106 1.618712
0.04 -1.853671 1.371369 1.626106
0.1 -1.912716 1.298678 1.643243
5 -1.208579 1.065172 0.911661
7 -1.208579 1.057251 1.160033
10 -1.208579 1.053273 1.415687
0.3 -1.208491 1.053303 1.415697
0.4 -1.062894 1.003678 1.367307
0.5 -0.950393 0.960988 1.324892
0.3 -1.210278 1.053276 1.416400
0.5 -1.210278 0.881066 1.455965
0.7 -1.210278 0.754801 1.485609
0.3 -1.210278 1.053276 1.416400
0.6 -1.210278 1.082283 0.933785
0.9 -1.210278 1.096642 0.696436

Table 5.2: Skin friction coefficient, reduced Nusselt number and reduced Sherwood number
values of various parameters for the mixture of water and CuO
𝑁𝑏
Nt 𝜑 Le ℎ1 ℎ2 ℎ3 𝑓′′ 0 −𝜃′(0 −𝜙′(0
0.1 -1.183941 1.065687 1.427469
0.2 -1.183941 0.964142 1.579332
0.3 -1.183941 0.867279 1.628888
0.1 -1.183941 1.065687 1.427469
0.2 -1.183941 0.976766 1.244759
0.3 -1.183941 0.895344 1.120616
0.01 -1.786772 1.410073 1.619656
0.04 -1.800547 1.375104 1.629519
0.1 -1.800750 1.307401 1.650247
5 -1.181989 1.077793 0.925419
7 -1.181989 1.069970 1.172807
10 -1.181989 1.066072 1.427291
0.3 -1.181884 1.066125 1.427333
0.4 -1.043149 1.016615 1.379317
0.5 -0.935197 0.973837 1.337055
0.3 -1.183941 1.065687 1.427469
0.5 -1.183941 0.889670 1.467211
0.7 -1.183941 0.761051 1.496913
0.3 -1.183941 1.065687 1.427469
0.6 -1.183941 1.094992 0.939183
0.9 -1.183941 1.109451 0.699769

Table 5.1
Thermo-physical attributes of water and nanoparticles
𝜌 𝑘 𝑔 𝑚
2
𝐶𝑝 𝐽 𝑘𝑔 𝐾 𝑘 𝑊 𝑚 𝐾
Pure water 997.1 4179 0.613
Copper 8933 385 401
Copper oxide 6320 531.8 76.5

Table 5.3 : Comparison of the present results for reduced Nusselt number when
1 2 3
0, 0, 6.2, 1, 0, 0.1, 0
0,
N h h h
b Nt Pr n Le  
         
Reduced Nusselt number −𝜃′(0
Rashidi et al. [2014] Present study
M Cu-water CuO-waterr Cu-water CuO-water
1 0.92890862 0.94800696 0.9289085 0.948006
2 0.88544087 0.90160918 0.8854404 0.901609
3 0.84801483 0.86211196 0.8480148 0.862111

Conclusions
 Heat and mass transfer performance of Cu-water nanofluid is high when compared with
CuO-water nanofluid.
 Rising values of Nb and Le lessen the concentration field.
 Increasing values of ϕ, h1 depreciate velocity field.
 Nt and Nb both reduce the local Nusselt number but shows opposite behavior on
Sherwood number.
 Rising values of h1 enhances the skin friction coefficient but lessen the heat and mass
transfer rate.

Chapter - 6
3D MHD flow of ferrofluid over a slendering stretching
sheet with thermophoresis and Brownian motion
This paper was published in Journal of Molecular Liquids (ELSEVIER)

Aim of the study
The purpose of this paper is the theoretical analysis of the steady, three-
dimensional, MHD and slip flow of a nanofluid across a slendering stretching
sheet with thermophoresis and Brownian motion.
Our considerations are water as base fluid, graphene and magnetite as
nanoparticles and we did the simultaneous study of them on various profiles. R-
K fourth order based shooting method is enforced to resolve the altered
governing non-linear equations.

Fig. 6.1

0
u v w
x y z
  
  
  
2
2
2
( )
nf nf nf
u u u u
u v w B x u
x y z z
  
 
   
   
 
   
 
2
2
2
( )
nf nf nf
v v v v
u v w B x v
x y z z
  
 
   
   
 
   
 
 
2
2
2
B
p nf B
nf
D
T T T T C T T
c u v w k D
x y z z z T z
z
 

 
 
      
 
    
 
   
 
     
  
   
2 2
2 2
T
B
D
C C C C T
u v w D
x y z T
z z

    
   
    
(6.1)
(6.2)
(6.3)
(6.4)
(6.5)
 
       
* *
1 1
* *
2 3
( , ) ( ) , , ( ) ,
, , ,
0, v=0, ,
w w
w w
u v
u x y u x j v x y v x j
z z
T C
T x y T x j C x y C x j
z z
and
u T T C C at y
 

 
   
   
    
 
    

 
   
    
   
 
   


     
(6.6)

Nanofluid parameters are given by:
( )
1
, 1 , (1 )
(1 2.5 ) ( )
3( 1) 3( 1)
1 , 1
2 ( 2) ( 1)
( )
where , , , ,
( )
nf nf p nf
f f p f
nf nf
f f
p s
s s s
f p f f f
c
r e
c
k k
k k
c k
r e k
c k
  
   
   

  
   

 

  

       
 

  
    
    


   


(6.7)
 
0.5
1 *0.5
( ) , ( )
2 1 1
( ) ( )
1 2 2
n n
n
f g
u a x y c v a x y c
a n n f g
w x y c f g
n
 


 

  
      
  

 
 
   
    
      
 
     
  
    
 
 
0.5
1 *0.5
( 1)
( )
2
n
n a
z x y c




 
  
 
  ,  
( )
w
T T T x T 
 
   ,  
( )
w
C C C x C 
 
  
(6.8)

 
   
 
   
2
3 2
3 2
3 1
1 1 1
1 1 0
1 2.5 2 2 2 1
n f f f g n f f
r n n f g M
 
 
         
   

 
       
         
   
 
   
 
         
   
 
 
   
 
   
2
3 2
3 2
3 1
1 1 1
1 1 0
1 2.5 2 2 2 1
n g g f g n g g
r n n f g M
 
 
         
   

 
       
         
   
 
   
 
         
   
 
 
2
2
2
2 1 1
Pr 1 ( ) 0
1 2 2
nf
f
k n f g n
Nb Nt d f g
k n
    
  
      
 
   
        
        
  
   
 
       
   
 
 
2 2
2 2
1
0
1
Nt f g n
Le f g
Nb n
  

    
 
 
     
     
 
 
     
 
 
(6.9)
(6.10)
(6.11)
(6.12)
And the changed boundary conditions are
2 2 2
1 1 1
2 2 2
0 0 0
2
1 2 3
2
0 0
0
1 1
(0) 1 , '(0) 1 , (0) 1 ,
1 1
'(0) 1 , (0) 1 , (0) 1 ,
0, 0,
n f f n g
f j f j g j
n n
g
g j j j
f g
  
 

  
 
 
  
 
  
 

     
    
   
       
     
   
    
   
     
     
     
  
     
     
  
     
   
 
 
 
 
0, 0 as
  










  



(6.13)

2
0 0 0
,Pr , , ,
f p B T
f B
B C D C D T
M Nb Nt Le
a k k T k D
    
 
    
(6.14)
Magnetic field parameter, Prandtl number, Brownian motion, Thermophoresis and Lewis number are
given by :
For engineering concern, the skin-friction coefficient, the local Nusselt number and the Sherwood
numbers (after non-dimensionalization) are given by:
   
 
0.5 0.5
1/2 1/2
0.5
1/2
1 1
2 Re ''(0). Re '(0)
2 2
1
Sh Re '( )
2
0
f x x x
x x
n n
C f Nu
n


 
 
   
   
   
    


  
   
  
(6.15)
Numerical Procedure
Equations (6.9) to (6.12) subject to the conditions (6.13) are solved with the assistance of Runge-Kutta based
shooting technique numerically. For numerical solutions, we assigned the values for non-dimensional
parameters as
1 2 3
6.2, 0.65,
0.1, 0.1, 2, 0.3, 0.3, 0.3
0.3, 0 1
,
.1,
Nb Nt M j j j
Pr n Le
 
     
    

Fig. 6.2 Fig. 6.3
Fig. 6.4
Fig. 6.5
Effect of nanoparticle volume fraction on velocity,
temperature and concentration profiles

Fig. 6.6 Fig. 6.7
Effect of thermophoresis parameter on temperature and concentration profiles

Fig. 6.8 Fig. 6.9
Effect of Brownian motion parameter on temperature and concentration profiles

Table 6.1: Effect of various parameters on friction factor, Nusselt number and
Sherwood number for the mixture of water and magnetite
𝜑 Nt Nb 𝑗1 𝑗2 𝑗3 Le 𝑓′′ 0 −𝜃′(0 −𝜙′(0
0.01 -0.875967 1.281648 0.322377
0.05 -0.938080 1.225730 0.320365
0.1 -0.991571 1.162464 0.322377
0.1 -0.991571 1.162464 0.322377
0.2 -0.991571 1.090804 -0.156461
0.3 -0.991571 1.022743 -0.545024
0.1 -0.991571 1.162464 0.322377
0.2 -0.991571 1.071778 0.645183
0.3 -0.991571 0.982329 0.752139
0.3 -0.991571 1.162464 0.322377
0.6 -0.732947 1.067327 0.270555
0.9 -0.586166 0.998284 0.235931
0.3 -0.991571 1.162464 0.322377
0.6 -0.991571 0.872439 0.463977
0.9 -0.991571 0.695397 0.551459
0.3 -0.991571 1.162464 0.322377
0.6 -0.991571 1.169280 0.249784
0.9 -0.991571 1.173591 0.203874
1 -0.991419 1.203844 -0.141120
1.5 -0.991419 1.179337 0.124039
2 -0.991419 1.162620 0.321750

Table 6.2: Effect of various parameters on friction factor, Nusselt number and
Sherwood number for the mixture of water and graphene
𝜑 Nt Nb 𝑗1 𝑗2 𝑗3 Le 𝑓′′ 0 −𝜃′(0 −𝜙′(0
0.01 -0.868158 1.283956 0.331300
0.05 -0.906959 1.235448 0.336819
0.1 -0.942829 1.178522 0.348779
0.1 -0.942829 1.178522 0.348779
0.2 -0.942829 1.108635 -0.126864
0.3 -0.942829 1.041974 -0.513716
0.1 -0.942829 1.178522 0.348779
0.2 -0.942829 1.090898 0.667649
0.3 -0.942829 1.004273 0.773376
0.3 -0.942829 1.178522 0.348779
0.6 -0.707089 1.085372 0.295228
0.9 -0.570068 1.016777 0.295228
0.3 -0.942829 1.178522 0.348779
0.6 -0.942829 0.880978 0.490583
0.9 -0.942829 0.700614 0.577627
0.3 -0.942829 1.178522 0.348779
0.6 -0.942829 1.185611 0.269088
0.9 -0.942829 1.190062 0.219039
1 -0.942607 1.217297 -0.112125
1.5 -0.942607 1.194245 0.152575
2 -0.942607 1.178656 0.348644

Table 6.3
Thermo-physical attributes of water and nano particles
Pure water Graphene Magnetite
𝜌 𝑘 𝑔 𝑚2 997 2250 5180
𝐶𝑝 𝐽 𝑘𝑔 𝐾 4076 2100 670
𝑘 𝑊 𝑚 𝐾 0.605 2500 9.7
𝜎 𝑆 𝑚 0.005 1x107 1x105

Conclusions
 The temperature increases with the increase in the thermophoresis and Brownian motion
parameters.
 Nanofluid volume fraction parameter improves both temperature and concentration but showed
opposite behavior on velocity profiles.
 All profiles (velocity, temperature, concentration) decreased with the enhancement in the
corresponding slip parameter values.
 Nanoparticle volume fraction, thermophoresis parameter, Brownian motion parameter and
velocity slip parameters reduce the heat transfer rate.

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buoyancy force, International Journal of Numerical Methods for Heat & Fluid Flow 25 (7) (2015) 1638-1657.
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Fluid Mechanics 9 (2) (2016) 683-692.
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Brownian motion effects, Advanced Powder Technology 27(5) (2016) 2039-2050.
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Mechanics 9(2) (2016) 683-692.
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stretching sheet with space and time dependent internal heat source/sink, Applications and Applied Mathematics, 10 (1), 2015, 312-327.

Future Plan
 Linear and nonlinear Stability analysis
 Statistical mechanics
 Compressible flow modeling

My publications
1. Heat and Mass transfer in MHD Eyring-Powell nanofluid flow due to cone in porous medium, International
Journal of Engineering Research in Africa, 19 (2016) 57-74.
2. Effect of variable heat source/sink on chemically reacting 3D slip flow caused by a slendering stretching
sheet, International Journal of Engineering Research in Africa, , 25(2016) 58-69.
3. Three-dimensional MHD slip flow of nanofluids over a slendering stretching sheet with thermophoresis and
Brownian motion effects , Advanced Powder Technology, 27.5 (2016) 2039-2050.
4. UCM flow across a melting surface in the presence of double stratification and cross-diffusion effects,
Journal of Molecular Liquids, 232 (2017) 27-35.
5. 3D MHD slip flow of a nanofluid over a slendering stretching sheet with thermophoresis and Brownian
motion effects, Journal of Molecular Liquids, 222 (2016) 1003-1009.
6. Magnetohydrodynamic dissipative flow across the slendering stretching sheet with temperature dependent
variable viscosity, Results in Physics, 2017 May 25.
7. Free convective MHD Cattaneo-Christov flow over three different geometries with thermophoresis and
Brownian motion, Alexandria Engineering Journal, 2017 Feb 3.
8. Effect of nonlinear thermal radiation on non-aligned bio-convective stagnation point flow of a magnetic-
nanofluid over a stretching sheet, Alexandria Engineering Journal, 55.3 (2016) 1931-1939.
9. MHD non-Newtonian fluid flow over a slendering stretching sheet in the presence of cross-diffusion Effects,
Alexandria Engineering Journal, 55.3 (2016) 2193-2201.
10. Nonlinear Thermal Radiation and Induced Magneticfield Effects on Stagnation-Point Flow of Ferrofluids,
Journal of Advanced Physics, 5 (2015) 1-7.

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