Radiative Flow and Heat Transfer of Chemically Reacted Casson
Nanofluid Over a Wedge Under the Influence of MHD and Viscous
Dissipation
KAVITHA G1, VITTAL CH2, VIJAYALAXMI TANKASALA 3, *, DHANALAXMI V 1
1. Department of Mathematics, Osmania University, Hyderabad-500007, Telangana, INDIA.
2. Department of Mathematics, University College of Science, Saifabad, Osmania University,
Hyderabad-500007, Telangana, INDIA.
3. Department of Mathematics, NTR Govt. Degree & P.G. College for Women, Mahabubnagar-509001,
Telangana, INDIA.
* Corresponding author email: vijaya9966998024@rediffmail.com
Abstract: The effect of thermal radiation, viscous dissipation with magnetohydrodynamic and chemically
reacted Casson Nano fluid flow across a moving wedge and convective boundary condition in existence
of internal heat generation/absorption is studied, the obtained results are presented. The simple governing
partial differential equations are altered into non-linear ordinary differential equations through employing
suitable similarity transformations. The obtained non-linear ordinary differential equations are effectively
elucidated numerically through Keller-Box method by use of MATLAB software. The variation of the
pertinent constraints on velocity, temperature and nanoparticle volume fraction are explored through
graphs. The graphs of skin friction coefficient, Nussult number and Sherwood number are plotted as a
function of physical parameters for different values. The present results are well matched with the previous
reported work. The present work demonstrates the nanoparticle volume fraction profile decrease by
enhance of chemical reaction, Brownian motion, Schmidt parameter values and increases with increment
in thermophoresis parameter values.
Key-words: Nanoparticle, Thermophoresis, Brownian motion, Schmidt number, Chemical reaction,
Wedge.
Received: March 14, 2024. Revised: August 7, 2024. Accepted: September 9, 2024. Published: October 17, 2024.
1. Introduction
Non-Newtonian fluids attracted many
researchers, scientists and engineers due to its
wide range of applications in industry,
manufacturing processing, biological fluids etc.
Due to many potential and technological
applications, the non-Newtonian fluids are
considered the most important than viscous fluids.
Non-Newtonian fluids exhibits variable viscosity
as a result of applied force. To analyze the
behavior of non-Newtonian liquids, numerous
significant difficulties are encountered, comprises
highly nonlinear governing boundary layer
equations and a higher degree of complexity than
Newtonian fluids. Hence, few reports are
available [1-7] and need further investigation for
better understanding of the flow characteristics of
non-Newtonian fluids. Casson fluid model is one
of the simplest, non-Newtonian fluid model of
differential type. The reported studies revealed the
Casson fluid model is fit to the rheological data
and it is better than general viscoplastic models
for many materials. Casson fluid exhibits a yield
stress. The Casson model is accurate in very low
and very high shear rate. This model is used in
industries to get more accurate representation of
high shear rate viscosities when only low and
intermediate shear rate data are available.
Besides, Casson fluids witnessed significant
applications in polymer processing industries and
biomechanics. Recently, many researchers are
International Journal of Chemical Engineering and Materials
DOI: 10.37394/232031.2024.3.6
Kavitha G., Vittal Ch.,
Vijayalaxmi Tankasala, Dhanalaxmi V.
E-ISSN: 2945-0519
39
Volume 3, 2024
considered the boundary layer flow of Casson
fluid over dissimilar geometries. Studies on
magnetohydrodynamics (MHD) flow of a Casson
fluid over an exponentially shrinking sheet has
made by Nadeem et al. [8]. The reported literature
available on flow analysis of Casson fluids is
made by Hayat et al. [9], Bhattacharyya et al. [10],
and Mukhopadhyay et al. [11], Nadeem et al. [12],
Haq et al. [13].
The convective flow past a wedge has been
widely studied by researchers due to its
applications in aerodynamics, heat exchangers,
geothermal systems, etc. Furthermore, the wedge
flow is important due to the fact that each value of
wedge angle yields a diverse pressure profile,
thereby offer insight into boundary layer behavior
in number of situations. The Falkner-Skan
(F-S) wedge flow for a non-Newtonian fluid with
a variable free stream condition is reported by
Postelnicu and Pop [14]. Kafoussias group [15]
explored the MHD laminar boundary-layer flow
of a non-Newtonian fluid over a permeable
wedge. Swati Mukhopadhyay and group
members [16] examined the boundary layer
forced convection flow of a Casson fluid past a
symmetric wedge. The observed results
demonstrates rise of the Casson fluid parameter,
the fluid velocity increases and the temperature
decreased. The fluid velocity is suppressed with
the increase of suction. Kishan et al. studied the
magnetohydrodynamic heat transfer of non-
Newtonian power-law fluids flowing over a
wedge [17, 18]. Wedge flow analysis at different
thermophysical conditions is reported by many
researchers [19-23]. Su et al. demonstrate an
MHD mixed convection flow of viscous fluid via
a porous stretching wedge [24]. Hossain et al.
reported the flow of an unsteady mixed-
convection boundary layer through a symmetric
wedge with a changing surface temperature [25].
Deka and Sharma examined the
magnetohydrodynamic mixed convection flow
through a wedge with changing temperature and
chemical reaction using Falkner-Skan
transformations [26]. Srinivasacharya et al.
investigated heat, mass transport characteristics
are stable in a laminar MHD flow across a wedge
in the presence of changing magnetic fields [27].
Hall effects on MHD squeezing flow of a
water based nanofluid through a saturated porous
medium in between two parallel disks is studied
as consider the Hall current into the account by
Veera Krishna et al [28]. Study on effects of heat,
mass transfer on free convective flow of
micropolar fluid over an infinite vertical porous
plate in presence of inclined magnetic field with
an angle of inclination, constant suction velocity
by Veera Krishna and team members [29]. Recent
studies on the effects of various non-dimensional
parameters on velocity, temperature and
concentration within the boundary layer are
examined through Soret and Joule effects with
MHD mixed convective flow of an
incompressible, electrically conducting viscous
fluid past an infinite vertical porous plate, the
Perturbation technique is used to solve the non-
dimensional equations [30]. Hall and ion slip
effects on magnetohydrodynamic (MHD) free
convective rotating flow of nanofluids in a porous
medium past a moving vertical semi-infinite flat
plate are investigated by Chamkha et al in the year
2020, and found the impact of thermal convection
of nanoparticles has increased the temperature
distribution, which helps in destroying the cancer
cells during the drug delivery process [31].
Chamkha and team studied the effect of MHD
flow of an electrically conducting second-grade
fluid through porous medium over a semi-infinite
vertical stretching sheet by consider the
thermophoresis, thermal radiation, and convective
boundary conditions. The fluid velocity,
temperature in the boundary layer area become
considerably higher with enhance the values of
the thermal radiation parameter. The Nusselt
number is improved by increase of surface
convection parameter [32]. Heat transfer on the
peristaltic magnetohydrodynamic flow of a
Jeffrey fluid through a porous medium in a
vertical echelon under the influence of a uniform
transverse magnetic field normal to the channel
International Journal of Chemical Engineering and Materials
DOI: 10.37394/232031.2024.3.6
Kavitha G., Vittal Ch.,
Vijayalaxmi Tankasala, Dhanalaxmi V.
E-ISSN: 2945-0519
40
Volume 3, 2024
and its effect by consider the Hall current into
account is made by Chamkha team in the year
2018. The size of the trapping bolus decreases by
enhance in Hartmann number or permeability
parameter and raises with increase of Hall
parameter or Jeffrey number [33].
Recently, cooling of electronic devices,
gadgets, etc. is the major industrial requirement.
Due to the lower thermal conductivity rate of
ordinary base fluids like water, ethylene glycol,
oil, etc. the nanoscale solid particles are mixed
with the fluids which results in vary the
thermophysical characteristics of these fluids and
improved heat transfer rate dramatically. In the
year 1996 the first report on the nature of the
colloidal suspension, studied by Choi and team
members [34]. Recently, advancement of
nanofluids, its mathematical modelling, etc. plays
a vital role in industrial and nanotechnology.
Nanofluids are used in various applications like
cooling of electronics devices / industry, nuclear
reactor safety, heat ex-changer, biomedicine,
hyperthermia, vehicle thermal management,
engine cooling, many others. Two mechanisms
have been processed to elucidate the increase
effective thermal conductivity in nanofluids, one
is due higher thermal conductivity of the
nanoparticles and second one is contribution from
the Brownian motion of the nanoparticles. The
implication of the two contributions is closely
associated to the bulk temperature of the
nanoparticle’s suspension, the size and the
volume fraction of the nanoparticles in the
nanofluid, as well as the thermophysical
properties of the nanoparticles and the base fluid
materials. Many researchers in fluid dynamics are
extended an interest in nanofluids over the last
few years owing to its potential applications. It
directed the mixing of fluids with metal
nanoparticles towards improve the heat transfer
capacity of the fluids. Brownian motion of
nanoparticles at molecular and nanoscale levels is
an important nanoscale mechanism governing its
thermal behavior. In nanofluid, due to its size of
the nanoparticles, the Brownian motion takes
place, which could influence the heat transfer
properties. Brownian motion aids to warm the
boundary layer and simultaneously aggravates
particle deposition away from the fluid regime,
thereby accounting for the reduced concentration
magnitudes. Choi and team members [35] studied
addition of small amount of nanoparticles to the
base fluid which results in increased thermal
conductivity. Investigation on boundary layer
flow of Casson nanofluid over a vertical
exponentially stretching cylinder is made by
Malik et al. [36]. Haq et al., reported on heat
transfer and MHD effects on Casson nanofluid
flow over a shrinking sheet, results show the trend
of velocity is identical for MHD, Casson fluid and
shrinking parameters [37]. Studies on MHD
mixed convection stagnation-point flow of a non-
Newtonian power-law nanofluid for a stretching
surface is done by Madhu and Kishan, results
states the effect of magnetic field parameter
reduces the velocity profiles [38]. Mustafa
examined the MHD flow of Casson nanofluid past
a non-linearly stretching, the observations states
the velocity decreases and skin friction coefficient
increases as strength of magnetic field is
increased. The velocity, boundary layer thickness
decreased with the Casson fluid parameter, Skin
friction coefficient for Casson fluid is greater than
that for Newtonian fluid [39]. The effect of slip,
convective boundary condition on magneto-
hydrodynamic stagnation point flow and heat
transfer due to Casson nanofluid past a stretching
sheet is reported by Ibrahim and Makinde [40].
Chamkha et al. observed the influence of radiation
on mixed convection in the presence of a wedge
implanted in a porous media filled with a
nanofluid, the observed results are very good
agreement with the reported results. It is found
that the local Nusselt number increases when any
of the buoyancy ratio, the Brownian motion, the
thermophoresis, the radiation–conduction and the
surface temperature parameters, and the Lewis
number increases. Whereas the local Sherwood
number is increased as the buoyancy ratio,
Brownian motion parameter, Lewis number,
wedge angle parameter, radiation–conduction
parameter or the surface temperature parameter
International Journal of Chemical Engineering and Materials
DOI: 10.37394/232031.2024.3.6
Kavitha G., Vittal Ch.,
Vijayalaxmi Tankasala, Dhanalaxmi V.
E-ISSN: 2945-0519
41
Volume 3, 2024
increases [41]. Boundary layer flow through a
moving wedge in a nanofluid is explored by Khan
and Pop, found that the dimensionless velocity at
the surface increases / decreases with stretching /
shrinking parameters, and dimensionless
temperature increases with both Brownian motion
and thermophoresis parameters [42]. Mahdy team
studied on unsteady MHD boundary layer flow of
a moving stretched porous wedge containing
tangent hyperbolic two phase nanofluid [43].
Studies on heat and mass transfer nature of
Casson nanofluid flow over a moving wedge filed
is made by Amar and Kishan [44], and concluded
that the flow velocity increases with enhance in
wedge angle and the thickness of the boundary
layer is decline with increase of magnetic field
parameter in wedge positions. The flow velocity
is decrease with increase of Casson parameter and
permeability parameter, whereas the skin friction
coefficient enhances with increase of magnetic
field parameter. Jamal Shah and team studied the
MHD flow of Casson nanofluid for the
applications of gold nanoparticles flow [45], the
results conclude that increase of the volume
percentage of gold nanoparticles from 0 to 0.04
percent towards an increase of up to 3.825 % in
the heat transfer rate. Recently, investigation of
the three-dimensional Casson hybrid nanofluid
(ZnO + Ag) flow under the influence of an applied
changing magnetic flux is reported, results show
the hybrid nanofluid performed excellent transfer
of heat, could use for cooling purposes of the
system [46]. Chamkha group investigated the
dissipative MHD free convective nanofluid flow
past a vertical cone under radiative chemical
reaction with mass flux studies [47]. The flow,
heat and mass transfer nature of Casson nanofluid
past an exponentially stretching surface with
activation energy, Hall current, thermal radiation,
heat source/sink, Brownian motion and
thermophoresis studies are made by Suresh
Kumar group [48].
To the best of author’s knowledge, no
reports are available in the literature on the
Casson nanofluid flow over a wedge under the
influence of chemical reaction with magnetic field
effect. In the present study, investigation on
chemical reaction influence, flow and heat
transfer characteristics of a Casson nanofluid on a
wedge under the influence of uniform transverse
magnetic field is made, the observed results are
presented. The effects of the various flow
controlling parameters on velocity, temperature
and nanoparticle volume fraction have been
investigated numerically and analysed with the
help of their graphical representations.
2. Model description
The present work is formulated as mentioned
below.
Consider the two - dimensional steady
incompressible flow and MHD heat transfer
of a chemically reacted Casson nanofluid
over a moving wedge in the occurrence
viscous and radiation effect.
The induced magnetic field is ignored due
to applied magnetic field is very high.
The moving wedge surface maintains
constant temperature due to convective heat
transfer and the lower surface of the wedge
heated by convection from a hot fluid at
temperature  with heat transfer quantity
.
A wedge surface is extending with a
constant velocity 󰇛󰇜 is subject to the
laminar boundary layer that is indicated by
󰇛󰇜.
When 󰇛󰇜 positive that is the direction of
the extending wedge is in the identical way
to the fluid flow, while it negative indicates
the shrinking wedge is in the reverse
direction.
The 󰇛󰇜 here  are fixed
as shown in the figure.
International Journal of Chemical Engineering and Materials
DOI: 10.37394/232031.2024.3.6
Kavitha G., Vittal Ch.,
Vijayalaxmi Tankasala, Dhanalaxmi V.
E-ISSN: 2945-0519
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Volume 3, 2024
Figure 1: Flow configuration and co-ordinate
system.
Based on N. Amar et al [49] & M. Madhu
et al [50] the governing equations for above
assumptions can be stated as
Continuity Equation is


 (1)
Momentum Equation




󰇡
󰇢

󰇡
󰇛󰇜
󰇢󰇛󰇛󰇜󰇜(2)
Equation of Thermal Energy




󰇡
󰇢
󰇛󰇜
󰇡
󰇢 (3)
Equation of Species Concentration




󰇛󰇜
(4)
The associated boundary conditions are

󰇛󰇜
󰇛󰇜





󰇛󰇜
󰇛󰇜
(6)
Where
u
and
v
denotes the velocities in the
direction of
x
-and
-respectively, is the
Casson fluid parameter,
is the kinematic
viscosity,
f
is the density of the base fluid,
is
the electrical conductivity,
is the thermal
diffusivity,
p
c
is the effective heat capacity of
the Nano particles,
f
c
is the heat capacity of
the base fluid,
f
p
c
c
)(
)(
is the ratio of the Nano
particle heat capacity and base fluid heat capacity,
B
D
is the Brownian motion diffusion coefficient
and
T
D
is the thermophoresis diffusion
coefficient, 󰇛󰇜
󰇛󰇜 is the chemical
reaction parameter, with rate constant , where
for destructive reaction, for
generative reaction and for no reaction.
We consider that the magnetic field
)(
0xBxB
,
where the constant magnetic field is
0
B
.
Using the Rosseland [51] approximation the
radiative heat flux is simplified as:


 󰇛󰇜
We assume that the temperature
differences within the flow region, namely, the
term can be expressed as a linear function of
temperature. The best linear approximation of
is obtained by expanding it in a Taylor series
about is

󰇛󰇜
󰇛󰇜
󰇛󰇜
Neglecting higher order terms of 󰇛󰇜 from
equation (8) and attained as


󰇛󰇜
Using equation (9) into equation (7) we get


󰇛󰇜
The modified equation of (3) by the
assistance of equation (10) is
International Journal of Chemical Engineering and Materials
DOI: 10.37394/232031.2024.3.6
Kavitha G., Vittal Ch.,
Vijayalaxmi Tankasala, Dhanalaxmi V.
E-ISSN: 2945-0519
43
Volume 3, 2024





󰇛󰇜



󰇛󰇜
The stream function 󰇛 󰇜 is defined as
below and satisfied the equation (1)

 and 
 󰇛󰇜
The desired conversions are described as
󰇧󰇛󰇜󰇛󰇜
 󰇨󰇡
󰇢
󰇧󰇛󰇜
󰇛󰇜󰇨󰇡
󰇢󰇛󰇜󰇛󰇜
󰇛󰇜

󰇧󰇛󰇜󰇛󰇜
 󰇨󰇡
󰇢󰇩󰇛󰇜
󰇛󰇜
󰇛󰇜
󰇪

󰇛󰇜
By employing the similarity
transfigurations and equation (12) the governing
equations (2), (3), (4) reduced to the following
ordinary differential equations along with the
boundary conditions
2
2
1
1 '"( ) ( ) "( ) 1 '( ) '( ) 1 0 (14)
4
1 "( ) Pr ( ) '( ) Pr "( ) 2 Pr ( ) 0 (15)
3
"( ) ( ) '( ) "( ) ( ) 0 (16)
f f f f M f
Rf Ec f Q
Nt
Scf Sc
Nb






(0) , '(0) , '(0) 1 (0) , ' ' 0 0
'( ) 1, ( ) 0, ( ) 0 (17)
f s f bi Nb Nt at
f at
The involved physical parameters are
Casson fluid parameter , wedge parameter ,
Velocity Ratio Parameter relates to flow in
the reverse direction in outside flow, relates
to flow in the same direction on a permeable
extending wedge wall, and relates to a
surface of a stationary wedge. Magnetic field ,
is the suction and is the injection,
Prandtl Number, is the heat source or sink
parameter,  is the Eckert number, R is the
thermal radiation parameter,  is the thermal slip
parameter,  is the Schmidt number, Brownian
number  , Thermophoresis parameter , is
the Chemical reaction parameter.
The shear stress at the surface of wedge is
intended as follows:

 󰇛󰇜
 󰇡
󰇢󰇧
󰇨
The physical quantity of major interest is the local
skin-friction coefficient 󰇛󰇜, local Nusselt
number and local Sherwood number in
non-dimensional form is given by

 
󰆒󰇛󰇜
Where 󰇡
󰇢

󰇛󰇜

International Journal of Chemical Engineering and Materials
DOI: 10.37394/232031.2024.3.6
Kavitha G., Vittal Ch.,
Vijayalaxmi Tankasala, Dhanalaxmi V.
E-ISSN: 2945-0519
44
Volume 3, 2024

󰆒󰇛󰇜
󰇛󰇜


󰆒󰇛󰇜
Where 
be the Reynolds number
3. Numerical scheme
By adopting Keller-Box method and
utilize of MATLAB software, the
following set of steps are implemented to
get the solutions.
In order to solve and transform the system
of higher ordinary differential equations
(14) – (16) with their corresponding initial
and boundary conditions (17),
introduction of new independent variables
of 󰇛󰇜 󰇛󰇜 󰇛󰇜󰇛󰇜 󰇛󰇜
󰇛󰇜󰇛󰇜󰇛󰇜are made and
transformed into first order differential
equations and also the boundary
conditions are changed, as mentioned in
below.
󰆒󰆒󰆒󰆒
Hence, the equations (14) to (16) and (17)
could be written as

󰆒󰇛󰇜󰇛󰇜

󰆒
󰇛󰇜
󰆒
󰆒
The boundary conditions in equation (17)
has been modified as
󰇛󰇜 󰇛󰇜
󰇛󰇜󰇛󰇜
󰇛󰇜󰇛󰇜󰆒
󰇛󰇜 󰇛󰇜 󰇛󰇜
The resultant differential equations are first
expressed in the form of finite difference
and then linearized through Newton’s
technique.
To find the solution, block tri-diagonal
elimination procedure is used to set of linear
equations which are arranged into the
matrix form.
To get the accuracy, the following
appropriate initial guesses have been
chosen.
󰇛󰇜󰇛󰇜
󰇛󰇜 

󰇛󰇜

 

The choices of the above initial guesses
depend on the convergence criteria and
transformed boundary conditions equation
of (17). The step size 0.001 is used to obtain
the numerical solution with suitable decimal
place of accuracy as the criterion of
convergence.
4. Results and discussion
In order to authorize the present numerical
method and the obtained numerical results are
confirmed with the results obtained by N. Amar et
al [49]., Ahmad et al [52] and Ullah et al [53] in
Table 1 and Table 2 denotes the compared results
with Ahmad et al [52], Kuo [54] and N. Amar et
al [49]. Comparisons are found in good with the
outcomes.
Table 1: Deviations of  for different values
of and the other values are
Pr 0Nb Nt Sc Ec Q bi R
Ahmad
et al.52
Ullah
et al.
53
Amar et
al. 49
Present
study
-1
0.75655
0.7566
0.756572
0.756434
-0.5
0.96922
0.9692
0.969230
0.969021
0
1.23258
1.2326
1.232588
1.232865
0.5
1.54175
1.5418
1.541751
1.541754
1
1.88931
1.8893
1.889314
1.889314
International Journal of Chemical Engineering and Materials
DOI: 10.37394/232031.2024.3.6
Kavitha G., Vittal Ch.,
Vijayalaxmi Tankasala, Dhanalaxmi V.
E-ISSN: 2945-0519
45
Volume 3, 2024
Table 2: Deviations of  for different
values of and the other values are
Pr 0Nb Nt Sc Ec Q bi R
Ahmad
et al. 52
Kuo et
al. 54
Amar et
al. 49
Present
study
0
0.46959
0.469600
0.469600
0.469594
0.1
0.58703
0.587889
0.587035
0.587123
0.3
0.77475
0.775524
0.774755
0.774755
0.5
0.92768
0.927905
0.927680
0.927685
1
1.23258
1.231289
1.232588
1.232589
1.6
1.52151
1.518488
1.521514
1.521518
2
1.68721
1.683095
1.687218
1.687215
In ordered to understand the mathematical
model the computational results are presented
graphically for velocity, temperature and Nano
particle volume fraction profiles for different
values of flow controlling parameters in Figures 2
to 15. The skin friction co-efficient 󰇡
󰇢󰇛󰇜Nusselt number 󰆒󰇛󰇜 and Sherwood
number 󰇛󰇜 origin graphs are presented
through Figures 16-18.
The Casson parameter 󰇛󰇜 has an
influence on the velocity and temperature fields
depicted in Figures 2(a) and 2(b) and it is noticed
that the increase of the Casson parameter 󰇛󰇜
enhances the velocity and drops the temperature.
Figure 2(a): 󰆒󰇛󰇜 for picked values of
Figure 2(b): 󰇛󰇜 for picked values of
The impact of changing the wedge
parameter () on the velocity and temperature are
demonstrated in Figures 3(a) and 3(b). These
figures exhibit that enhancing the wedge
parameter values enriches the velocity and
temperature profiles due to the pressure exerted
on the flow, which accelerates the velocity area
and the temperature profile.
Figure 3(a): 󰆒󰇛󰇜 for picked values of
0 1 2 3 4 5 6 7
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
f'()
= 0.3, 0.6, 1, 10
Fig. 2(a)
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
()
22.2
0.09
0.1
0.11
()
= 0.3, 0.6, 1, 10
Fig. 2(b)
0 1 2 3 4 5
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
f'()
= 1, 2, 3, 4
Fig. 3(a)
International Journal of Chemical Engineering and Materials
DOI: 10.37394/232031.2024.3.6
Kavitha G., Vittal Ch.,
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E-ISSN: 2945-0519
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Volume 3, 2024
Figure 3(b): 󰇛󰇜 for picked values of
Figure 4 illuminate the effect of different
magnetic field parameter 󰇛󰇜 values on non-
dimensional velocity include the case of
pure hydrodynamic flow means nonexistence of
magnetic field. It is shown that the velocity profile
decreases with the increasing of is implies that
the momentum boundary layer thickness becomes
thinner. Physically, increasing the values lead
to strong Lorentz force along the vertical direction
which offers more resistance to the flow.
Figure 4: 󰆒󰇛󰇜 for picked values of .
The effect of thermal radiation parameter
on the width of temperature profile is explained
in the Figure 5 and its stats that enhancing the
radiation factor rises the thickness of the thermal
boundary layer due to rate of heat transferred to
fluid by increased radiation.
Figure 5: 󰇛󰇜 for picked values of
The ratio of momentum to thermal
diffusivity in the boundary layer is known as
Prandtl number and it identifies a desperate
thermophysical property of a fluid. From the
Figure 6 if the values of  rises, the momentum
diffusivity is larger than thermal diffusivity which
occurs in low-conductivity fluids and as a result,
both the thickness of the thermal boundary layer
and temperature distribution decline also rise in
heat transfer rate.
Figure 6: 󰇛󰇜 for picked values of 
Figure 7 shows the effect of erratic viscous
dissipation parameter 󰇛󰇜 on temperature
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
()
= 1, 2, 3, 4
Fig. 3(b)
0 1 2 3 4 5 6 7
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
f'()
M = 0, 0.25, 0.5, 1
0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
()
R = 0, 2, 4, 6
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
()
Pr = 0.71, 3, 7, 10
International Journal of Chemical Engineering and Materials
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Kavitha G., Vittal Ch.,
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profile. The Eckert number articulate the rapport
between the kinetic energy in the flow and the
enthalpy, also causes the transformation of kinetic
energy into internal energy by the work that is
done against the viscous fluid stresses. While
increase in Eckert number resulting to increase the
kinetic energy which leads to increases the
collision of molecules and such increased
collision molecules increases the dissipation of
heat in the boundary layer region and therefore
increases the temperature profile.
Figure 7: 󰇛󰇜 for picked values of 
The heat generation/absorption parameter
shows the significant effect on the
dimensionless temperature profile of the fluid.
The presence of external heat source causes to
increase in both the temperature distribution and
thermal state of the fluid and hence increases the
thermal boundary layer thickness. So, the
temperature profile increases with an increase in
the values of which shown in Figure 8.
Figure 8: 󰇛󰇜 for picked values of
The effect of Schmidt number 󰇛󰇜 on
concentration profile is shown in Figure 9 and
which describes decreases nanoparticle volume
fraction with an increase in . Due to the
Schmidt number being the ration of momentum
diffusivity to mass diffusivity, an increase in 
corresponds to the decrease in Brownian diffusion
coefficient and higher momentum diffusivity can
result in more nanoparticle penetration.
Figure 9: 󰇛󰇜 for picked values of 
The impact of thermophoresis on the
nanoparticle concentration is depicted in Figure
10. From this we conclude that, when
thermophoresis increases, the nanoparticle
concentration decreases. Effect of Brownian
motion on the nanoparticle concentration is
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
()
1.2 1.4 1.6 1.8
0.18
0.2
0.22
0.24
()
Ec = 1, 2, 3, 4
0 1 2 3 4 5 6 7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
()
Q = -3, -2, -1, 0
0 1 2 3 4 5 6
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
()
Sc = 0.5, 1, 1.5, 2
International Journal of Chemical Engineering and Materials
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Kavitha G., Vittal Ch.,
Vijayalaxmi Tankasala, Dhanalaxmi V.
E-ISSN: 2945-0519
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Volume 3, 2024
shown in Figure 11. Here, we can see that, as the
Brownian parameter increases, nanoparticle
concentration increases throughout the flow
domain.
Figure 10: 󰇛󰇜 for picked values of 
Figure 11: 󰇛󰇜 for picked values of 
Effect of chemical reaction parameter 󰇛󰇜
on nanoparticle volume fraction profile is shown
in Figure 12 for different positive values of
which reveals that the nanoparticle volume
fraction decreases for constructive chemical
reaction parameter.
Figure 12: 󰇛󰇜 for picked values of
The impact of suction parameter on
velocity and temperature profile of the nanofluid
is depicted via Figure 13(a) and 13(b) respectively
and we observed that with enrichment of , the
nanofluid velocity increased because of
improvement in momentum boundary layer
thickness due to enrichment impact of suction
parameter but the decreasing behavior was
perceived over the temperature profile because of
an upsurge in the rate of heat transfer.
Figure 13(a): 󰆒󰇛󰇜 for picked values of
0 1 2 3 4 5 6
-0.4
-0.3
-0.2
-0.1
0
0.1
()
Nt = 0.2, 0.4, 0.6, 0.8
0 1 2 3 4 5 6 7
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
()
Nb = 0.3, 0.5, 0.7, 0.9
0 1 2 3 4 5 6 7
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
()
= 1, 2, 3, 4
0 1 2 3 4 5 6 7 8 9
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
f'()
s = -2, -1, 0, 1, 2
Fig.13(a)
International Journal of Chemical Engineering and Materials
DOI: 10.37394/232031.2024.3.6
Kavitha G., Vittal Ch.,
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E-ISSN: 2945-0519
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Volume 3, 2024
Figure 13(b): 󰇛󰇜 for picked values of
The influences of velocity ratio parameter
on flow velocity, temperature and concentration
profiles are revealed through Figures 14(a), 14(b)
and 14(c) respectively. As the values of velocity
ratio upsurge, the boundary layer thickness rises
and the flow has boundary layer structure. The
graph of velocity is possible when the free steam
velocity is less than or equal to the velocity of
stretching sheet. That is when velocity ratio is less
than or equal to one. But as the value of velocity
ratio parameter increases, thermal boundary layer
thickness decreases as well as the same result will
reflect on concentration profile of the nanofluid.
Figure 14(a): 󰆒󰇛󰇜 for picked values of
Figure 14(b): 󰇛󰇜 for picked values of
Figure 14(c): 󰇛󰇜 for picked values of
Figure 15(a) gives the influence of thermal
Biot number 󰇛󰇜 on the temperature profile. Due
to large Biot number simulates a strong heat
convection surface which leads to more heat to the
surface and hence the fluid temperature increases
with the increment in . From Figure 15(b) it is
observed that increases the fluid concentration
effectively with increasing values of .
0 1 2 3 4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
()
s = -2, -1, 0, 1, 2
Fig. 13(b)
0 1 2 3 4 5 6 7
0
0.2
0.4
0.6
0.8
1
1.2
f'()
= 0, 0.2, 0.4, 1, 1.2
Fig. 14(a)
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
()
= 0, 0.2, 0.4, 1, 1.2
Fig. 14(b)
0 1 2 3 4 5 6
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
()
= 0, 0.2, 0.4, 1, 1.2
Fig. 14(c)
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DOI: 10.37394/232031.2024.3.6
Kavitha G., Vittal Ch.,
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E-ISSN: 2945-0519
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Volume 3, 2024
Figure 15(a): 󰇛󰇜 for picked values of 
Figure 15(b): 󰇛󰇜 for picked values of 
The Variation of skin-friction coefficient
󰇡
󰇢󰇛󰇜 against Wedge parameter for
different values of magnetic parameter and
Casson fluid parameter reveals in the Figure 16
and it is very clear that the skin-friction
coefficient is increased with both the increase of
and it also observed that as increases skin-
friction coefficient increases. Figure 17 gives the
plots of Nussult number 󰇛󰇜 versus the Eckert
number  for different values of heat source or
sink parameter and thermal radiation parameter
. It is observed that the Nussult number increases
with both the increased values of and . In
Figure 18 the effect of thermophoresis parameter
 on Sherwood number 󰇛󰇜 is depicted for
different values of Brownian motion parameter
 and Schmidge number  and it is clearly
explaining that Sherwood number diminishes
with both the enhanced values of  and . It is
also observed that Sherwood number increases
rapidly for larger values of .
1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Cf
M = 0, 
M = 0.25, 
M = 0.5, 
Figure 16: Effect of and on skin friction
coefficient
1.0 1.5 2.0 2.5 3.0 3.5 4.0
-0.5
-0.4
-0.3
Nux
Ec
R = 2, Q = -3
R = 4, Q = -2
R = 6, Q = -1
Figure 17: Effect of and  on heat
transfer rate
0 1 2 3 4 5 6
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
()
bi = 0.1, 0.3, 0.5, 0.7
Fig. 15(a)
0 1 2 3 4 5 6
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
()
bi = 0.1, 0.3, 0.5, 0.7
Fig. 15(b)
International Journal of Chemical Engineering and Materials
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E-ISSN: 2945-0519
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Volume 3, 2024
0.1 0.2 0.3 0.4 0.5
0.0
0.5
1.0
1.5
2.0
Shx
Nt
Nb = 0.1, Sc = 0.5
Nb = 0.3, Sc = 1.5
Nb = 0.5, Sc = 2.0
Figure 18: Effect of  and  on mass
transfer rate
5. Conclusion
The present study talks about the analysis
of viscous dissipation and radiation effects on
MHD heat transfer flow of Casson nanofluid
through a moving wedge by incorporating the
effects of Brownian and thermophoresis under the
convective boundary condition and internal heat
absorption/generation. Governing non-linear
partial differential equations are turned into non-
linear ordinary differential equations by using
appropriate similarity transformations. The
following specific conclusions have been drawn
from above established results.
The results are confirmed with N. Amar et al,
Ahmad et al, Ullah et al, Kuo in good
agreement.
The effect of Casson fluid parameter ,
wedge parameter enhances the velocity
profile whereas magnetic field parameter
reduces the velocity profile.
Both the velocity ratio parameter and the
suction parameter augments the velocity
profile but these two parameters show the
differing effect on temperature profile.
Temperature distribution increases with
increase of thermal radiation , wedge
parameter and internal heat absorption
and decreases with increase of Prandtle
number  and Schmidge number .
The nanoparticle profile decreases with
increase of Brownian motion , Schmidge
number  and chemical reaction parameter
where as it increase with increase of
thermophoresis .
The skin-friction coefficient values increase
with all the increase of and .
The rate of heat transfer increases with both
the increased values of and .
Sherwood number diminishes with both the
enhanced values of  and  and
increases rapidly for larger values of .
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International Journal of Chemical Engineering and Materials
DOI: 10.37394/232031.2024.3.6
Kavitha G., Vittal Ch.,
Vijayalaxmi Tankasala, Dhanalaxmi V.
E-ISSN: 2945-0519
53
Volume 3, 2024
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_US
International Journal of Chemical Engineering and Materials
DOI: 10.37394/232031.2024.3.6
Kavitha G., Vittal Ch.,
Vijayalaxmi Tankasala, Dhanalaxmi V.
E-ISSN: 2945-0519
54
Volume 3, 2024