UKH Journal of Science and Engineering | Volume 5 • Number 1 • 2021 38
Stagnation Point Heat Flow and Mass Transfer in a Casson
Nanofluid with Viscous Dissipation and Inclined Magnetic
Field
Wasiu Toyin Akaje
a*
, Olajuwon B. I
b
Department of Mathematics, Federal University of Agriculture, Abeokuta, Nigeria
E-mail:
a
akajewasiu@gmail.com,
b
ishola1@gmail.com
1. Introduction
The growing attention of many researchers recently on flow and heat transfer of a viscous and incompressible fluid
flowing through an elongating surface is because of its numerous engineering applications, especially in industry and
manufacturing processes. For example, glass fibre production, paper production, extrusion of polymers, refrigeration,
cooling of electronic equipment, copper wires drawing, and crude oil purification. To obtain the production quality for
such processes, the heat transfer, and the flow field are highly needed. Various researchers recently are engaged in using
the non-Newtonian heat transfer characteristics flowing through a stretching sheet. In 1970, Crane was the first author
to work on the flow of fluid over a stretching sheet with linear order. In the work, the steady-state similarity solution
was obtained. Mahapatra & Gupta in 2001 studied an incompressible viscous electrically conducting fluid past a
stretching sheet. The influence of variable thermal conductivity and heat source on MHD boundary layer flow of
electrically conducting fluid is investigated by (Alsedais, 2017). It was found that at A = 2.0 the skin-friction at the
surface increases and decreases at A = 2.0 with an increase in the Casson parameter. Zaimi & Ishak in 2016 reported
the slip effects on stagnation-point flow through a stretching vertical sheet.
Pavlov in 1974 considered the influence of the external magnetic field in the hydromagnetic flow over a stretching
sheet. Bhattacharyya & Layek in 2010 investigated the effect of suction/blowing on hydromagnetic boundary layer flow
over a permeable stretching sheet.
The influence of chemical reaction on slip flow of magnetohydrodynamic heat and mass Casson fluid over a stretching
sheet is analyzed by Kumar & Gangadhar in 2015 and it was observed that the reduction of some parameters such as
magnetic and momentum slip reduce the fluid flow. Rao & Sreenadh in 2017 studied the exponentially inclined
Access this article online
Received on: November 6, 2020
Accepted on: Janaury 7, 2021
Published on: June 30, 2021
DOI: 10.25079/ukhjse.v5n1y2021.pp38-49
E-ISSN: 2520-7792
Copyright © 2021 Akaje&Olajuwon. This is an open access article with Creative Commons Attribution Non-Commercial No Derivatives License
4.0 (CC BY-NC-ND 4.0)
Abstract
Influence of slip and inclined magnetic field on stagnation-point flow with chemical reaction are studied.
Implementation of the similarity transformations, transformed the fluid non-linear ordinary differential equations
and numerical computation is performed to solve those equations using Spectral Collocation Method. Various
pertinent parameters on fluid flow, temperature and concentration distributions of the Casson nanofluid flow as
well as the local skin friction coefficient, local Nusselt number, and Sherwood number are graphically displayed. The
results indicate that thermophoresis parameter N_t enhanced the temperature and nanoparticle concentration
profiles, because a rise in thermophoresis parameter enhances the thermophoresis force within the flow regime.
Values of both local Nusselt and Sherwood numbers are enhanced with an increase in Hartman number (magnetic
field parameter). The present results are compared with previously reported ones and are found to be in excellent
agreement.
Keywords: Magnetic Field, Casson Nanofluid, Stagnation Point, Viscous dissipation.
Research Article
UKH Journal of Science and Engineering | Volume 5 • Number 1 • 2021 39
permeable flow of a Casson fluid. The result shows that temperature distribution reduces as Prandtl number, thermal
slip factor, suction parameter increase and enhances with radiation parameter, Eckert number, and Magnetic parameter.
Soid and Ishak et al. in 2009 studied MHD stagnation point flow over a stretching/shrinking sheet (Fang et al.,
2009 )obtained the exact solution on the magnetohydrodynamic flow and mass transfer with slip condition through a
stretching sheet.
Nanofluids are gaining more attention from many researchers today because of its significant applications of enhancing
fluid transfer performance properties, particularly concerning heat transfer Choi & Eastman in 1995 were the first author
to introduce the concept of Nanofluid where he proposed the suspension of nanoparticles. The suspension of particles
such as metals, metals oxides, carbides, nitrides, carbon, and nanotubes are dispersed in a continuous medium (base
fluids) such as water, ethylene, glycol, and engine oil of size less than 100nm is known as Nanofluids. After Choi, many
authors worked on Nanofluids and Casson Nanofluids. Meraj & Junaid in 2015 studied hydromagnetic Casson nanofluid
through a non-linearly stretching sheet, it was reported that Brownian motion does not have an impact on fluid
temperature and heat transfer rate from the sheet while both temperature and nanoparticle volume fraction are
decreasing functions of thermophoresis parameters. The hydromagnetic chemically reactive nanofluid flow past a
permeable flat plate in a porous medium was investigated by Reddy et al. in 2016. Analytic and numeric solutions were
examined by Awais et al. in 2015. It was observed that a rise in the magnetic field parameter results in a decrease in fluid
particles inter-molecular movement which leads to an enhancement of fluid temperature. Also, the presence of a heat
source in a system can enhance the temperature whereas a heat sink causes a decrease in temperature. The
hydromagnetic and heat transfer flow of Williamson nanofluid through a stretching sheet with variable thickness and
the variable thermal conductivity were presented in Reddy et al. in 2017. Nageeb et al. studied the unsteady free
convective hydromagnetic chemically reactive boundary-layer flow of nanofluid over-stretching surfaces using the
spectral relaxation method (Haroun et al., 2015). Mohamed and Afify investigated chemically reactive Casson Nanofluid
flow through stretching sheet with slip boundary condition, viscous dissipation in (Afify, 2017).
In this paper, the work of Afify (Haroun et al., 2015) is extended by including stagnation point, heat generation, and
the effect of inclined magnetic field on the problem of the steady Casson nanofluid flowing over a stretching sheet.
2. Mathematical Analysis
We analyzed the steady two-dimensional incompressible flow of electrically conducting and chemically reactive Casson
nanofluid bounded by a stretching sheet at with the flow being confined in
is the stretched
linear velocity where b is the positive constant. The strength of the inclined magnetic field is
, and here
are the ambient temperature and nanoparticle concentration fields with
. Thermophoresis and Brownian
motion of nanoparticles are taken into consideration. The rheological equation of state of an isotropic and
incompressible flow of Casson fluid is given by (Haroun et al., 2015):
(1)
where
is the plastic dynamic viscosity of the non-Newtonian fluid,
is the yield stress of fluid, is the product of
the component of deformation rate with itself, namely, =
is the
component of the deformation rate
and
is the critical value of based on the non-Newtonian model. The governing equations of momentum, energy,
and mass are:
UKH Journal of Science and Engineering | Volume 5 • Number 1 • 2021 40
With the given boundary conditions:
In this work, the induced magnetic field is ignored. Consider the following dimensionless transformations
Using the stream function
such that
Therefore the equation (2) is satisfied. From the given transformations mentioned above, 3 to 6 become
Here prime represents differentiation with respect to. The flow parameters are defined as below:
The physical quantities of engineering interest are the Skin friction coefficient (rate of shear stress), the Nusselt number
(rate of heat transfer), and the Sherwood number (rate of mass transfer).
The local Skin-friction
local Nusselt Number
and local Sherwood Number
which are defined as
UKH Journal of Science and Engineering | Volume 5 • Number 1 • 2021 41
In terms of dimensionless quantities (14) we have
where
is the local Reynolds number.
3. Numerical Solution
The systems of nonlinear differential equations (9-11) parallel to the boundary conditions (12) are solved numerically
using the Chebyshev spectral collocation method. In this method, the unknown functions,
is
approximated by the sum of the basic functions
() (Shen et al., 2011; Sun et al., 2012).
The basic functions are taken as the Chebyshev polynomials, in (17), (18), and (19) which defined in the interval
as
are unknown constant to be obtained. is the considered flow problem domain, which transformed
into the of the definition of basis functions, by using the below transformation
where
denotes the edge of the boundary layer, by substituting (17), (18), and (19) into (9-11), non-zero residuals
were obtained. The coefficient
,
were chosen in such a way that the obtained residues were minimized
throughout the domain.
Table 1 shows the Comparison of results for
with N
t
, N
b
and E
c
for β = 0.5, λ = y = δ = 0.2,
Pr = 4, Le = 5 and A = Q = H = 0.
Table 1. shows the Comparison of results for 〖-θ〗^' (0) and 〖- 〗^' (0) with Nt, Nb λ and Ec for β = 0.5, λ
= y = δ = 0.2, Pr = 4, Le = 5 and A = Q = H = 0.
N
t
N
b
λ
Ec
Ahmed and Afify
Present results
Ahmed and Afify
(Haroun et
al., 2015)
Present results
0.1
0.1
0.2
0.2
0.655854
0.654296
1.19213
1.19232
0.3
0.1
0.2
0.2
0.510010
0.508726
1.22598
1.22667
0.5
0.1
0.2
0.2
0.394448
0.393369
1.45703
1.45778
0.1
0.2
0.2
0.2
0.550359
0.551452
1.27665
1.27465
0.1
0.4
0.2
0.2
0.371446
0.371348
1.31245
1.30315
0.1
0.6
0.2
0.2
0.235575
0.235684
1.31891
1.31663
0.1
0.1
- 0.2
0.2
0.664530
0.66554
0.637467
0.636539
0.1
0.1
0.0
0.2
0.659002
0.658105
0.961861
0.960761
0.1
0.1
0.4
0.2
0.654009
0.653702
1.367830
1.366820
0.1
0.1
0.2
0.0
0.795783
0.794862
1.113521
1.112604
0.1
0.1
0.2
1.0
0.082093
0.083140
1.515737
1.516703
0.1
0.1
0.2
1.3
- 0.139103
- 0.138211
1.641030
1.640031
UKH Journal of Science and Engineering | Volume 5 • Number 1 • 2021 42
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
0 1 3
0.5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
0 0.8 4
0.5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
0 0.8 4
0.5
0
2
4
6
8
0.6
0.5
0.4
0.3
0.2
0.1
0.0
f
0 1 3
0.5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
f
0 1 3
0.5
Figure 1. Effect of
Figure 2. Effect of
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
0 1 3
0.5
Figure 3. Effect of
Figure 4. Effect of
Figure 5. Effect of
Figure 6. Effect of
UKH Journal of Science and Engineering | Volume 5 • Number 1 • 2021 43
’
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
0 0.5 3
0.5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
Nt 0.1 Nt 0.5 Nt 0.9
0.5
0
2
4
6
8
0.0
0.5
1.0
1.5
Nt 0.1 Nt 0.5 Nt 0.9
0.5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
Nb 0.1 Nb 0.3 Nb 0.6
0.5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
Nb 0.1 Nb 0.3 Nb 0.6
0.5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
X 0.3 X 0 X 0.5
Nb
0.1
Figure 7. Effect of
Figure 8. Effect of
Figure 9. Effect of
Figure 10. Effect of
F igure 11. Effect of
Figure 12. Effect of
UKH Journal of Science and Engineering | Volume 5 • Number 1 • 2021 44
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
X 0.3 X 0 X 0.5
Nb
0.5
0
2
4
6
8
0.0
0.5
1.0
1.5
2.0
X 0.3 X 0 X 0.5
Nt
0.5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
X 0.3 X 0 X 0.5
Nt 0.1
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
Nt 0.1 Nt 0.5 Nt 0.9
Le
1
0
2
4
6
8
0.0
0.5
1.0
1.5
2.0
2.5
Nt 0.1 Nt 0.5 Nt 0.9
Le
1
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
Nt 0.1 Nt 0.5 Nt 0.9
Le
5
Figure 13. Effect of
Figure 14. Effect of
Figure 15. Effect of
Figure 16. Effect of
Figure 17. Effect of
Figure 18. Effect of
UKH Journal of Science and Engineering | Volume 5 • Number 1 • 2021 45
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
Nt 0.1 Nt 0.5 Nt 0.9
Le
5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
Nb 0.1 Nb 0.3 Nb 0.6
Le
1
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
Nb 0.1 Nb 0.3 Nb 0.6
Le
5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
Nb 0.1 Nb 0.3 Nb 0.6
Le
5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
Ec 0.1 Ec 0.5 Ec 1
0.5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
Nb 0.1 Nb 0.3 Nb 0.6
Le
1
Figure 19. Effect of
Figure 20. Effect of
.
Figure 21. Effect of
Figure 22. Effect of
Figure 23. Effect of
Figure 24. Effect of
UKH Journal of Science and Engineering | Volume 5 • Number 1 • 2021 46
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
Q 0.1 Q 0.3 Q 0.5
0.5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
f
0.5 1 2
Le
5, Pr 4
0
2
4
6
8
0.4
0.3
0.2
0.1
f
A 0.2 A 0.4 A 0.6
0.5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
A 0.2 A 0.4 A 0.6
0.5
0
2
4
6
8
0.0
0.2
0.4
0.6
0.8
1.0
Ha 0.1 Ha 0.3 Ha 0.5
0.5
Figure 25. Effect of
Figure 26. Effect of
Figure 27. Effect of
Figure 28. Effect of
0.0, 0.1, 0.3
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Nb
0
Figure 29. Effect of
Figure 30. Effect of N_b and λ-θ(η).
UKH Journal of Science and Engineering | Volume 5 • Number 1 • 2021 47
0.0, 0.1, 0.3
0.00
0.05
0.10
0.15
0.20
0.25
0.30
1.25
1.30
1.35
1.40
1.45
1.50
1.55
Nb
0
Nb
0.1 , 0.3, 0.5
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Ha
0
Nb 0.1 , 0.3 , 0.5
0.1
0.2
0.3
0.4
0.5
1.35
1.40
1.45
Ha
0
Ha 0.1, 0.3, 0.5
0.1
0.2
0.3
0.4
0.5
0.55
0.60
0.65
0.70
0.75
A
0
Ha
0.1, 0.3, 0.5
0.1
0.2
0.3
0.4
0.5
1.26
1.28
1.30
1.32
A
0
A
0.2, 0.4, 0.6
0.1
0.2
0.3
0.4
0.5
1.3
1.2
1.1
1.0
0.9
Ha
1
1 f 0
Figure 31. Effect of
.
Figure 32. Effect of
Figure 33. Effect of
.
Figure 34. Effect of
.
Figure 35: Effect of
Figure 36: Effect of
UKH Journal of Science and Engineering | Volume 5 • Number 1 • 2021 48
4. Results and Discussion
The numerical solutions are obtained for velocity, temperature, and nanoparticle concentration profiles for various
values of physical parameters, such as the slip parameter (, the thermal slip parameter (), the concentration slip
parameter (), the thermophoresis parameter (
), the Brownian motion (
), the chemical reaction parameter (), the
velocity ratio parameter (A), the heat generation parameter (Q), the Eckert number (Ec), the Casson parameter (, the
Lewis number (Le) and the Hartman number (
. The results obtained are presented pictorially in figures 1- 34 for
dimensionless velocity, dimensionless temperature, and dimensionless nanoparticle concentration profiles respectively.
We compared our result with that of Ahmed and Afify (Haroun et al., 2015) by neglecting the effects of
, Q and A.
The comparison shows good agreement as presented in Tables 1.
Figures 1-4 displayed the effects of on the fluid flow, the magnitude of the skin friction coefficient,
, the
temperature, and the nanoparticle concentration, profiles. It is observed that an increase in the slip parameter
decreases the velocity of the fluid profile and the skin friction coefficient because the high value of the slip parameter
corresponds to a reduction in the surface of skin friction while the increase in slip parameter increases both the
temperature and nanoparticle concentration profiles. The effects of the thermal slip parameter on the temperature and
nanoparticle concentration profiles are depicted in Figures 5 and 6 it is noteworthy that an increase in the thermal slip
parameter reduces the thermal boundary layer and nanoparticle concentration distribution. Because heat transfer within
the boundary layer fluid reduces with the enhancement of the thermal slip parameter. Figure 7 represents the effects of
the concentration profile and it observed that the nanoparticle concentration decrease with an increased concentration
slip parameter. A rise in temperature and nanoparticle concentration profiles is noticed in Figures 8 and 9. The impacts
of the Brownian motion parameter on heat flow and the nanoparticle concentration distributions are presented
graphically in figures 10 and 11. It is observed that the heat flow increase with an increase in the Brownian motion
parameter, while the nanoparticle concentration profile decreases.
The dimensionless nanoparticle concentration profiles for various values of
are exhibited in figures 12-
15 respectively, and the nanoparticle concentration profiles reduce with a rise in chemical reaction for different values
of
. Figures 16-23, displayed graphical representative of temperature and nanoparticle concentration
distributions for various values of
Both the temperature and nanoparticle concentration profiles
increase with an increase in
when Lewis number is 1 or 5 in figures 16 -20, while the reverse is the case in
figures 21-23 with the same value of Lewis number. The influence of Eckert number and heat generation parameters
on the temperature profiles within the boundary layer region is shown in figures 24 and 25 and it is understood that the
temperature profile increased with an increase in the Eckert number. This happened due to viscous heating, the increase
in the fluid temperature is enhanced and appreciable for higher values of Eckert number. Physically, Eckert's number
relates the kinetic energy to the enthalpy of a fluid, while an increase in heat generation leads to an increase in the
temperature throughout the entire boundary layer in figure 25. The heat generation does not only increase the
temperature of the fluid but also increases the thermal bounder layer thickness. Figure 26 present the influence of the
Casson parameter on the velocity profile and it is worthwhile to note that the velocity profile decreases with an increase
in the values of
From figure 27 and 28, the effects of velocity ratio parameter on the
magnitude of the skin friction coefficient and the temperature profiles are presented, it is noticed from figure 27 that
the magnitude of the skin friction profile increases with an increase in velocity ratio parameter near the wall, whereas
the reverse trend is observed far away from the wall, while the increase in velocity ratio parameter decreased the
temperature profile in figure 28. Figure 29 is plotted to see the behavior of Hartman number
on the temperature
profile and it is seen that temperature profile increases by increasing the value of
. Figures 30-31 represent the effect
of slip parameter and Brownian motion
on local Nusselt number
, while an increase in
reduces the local Nusselt number and the reverse case is noticed in Sherwood number. Figures 32-33, are depicted to
see the behavior of Brownian motion and Hartman number on both
, and it can be seen from the
graphs that there is a decrease in local Nusselt number and an increase in Sherwood number with an increase in the
value of the Brownian motion parameter. Effect of Hartman and velocity ratio parameters are depicted in figures 34-
35, and both local Nusselt and Sherwood numbers increase with an increase in Hartman number, while figure 36
represents the effect of velocity ratio on the magnitude of the skin friction coefficient and rise in velocity ratio leads to
an increase the skin friction coefficient.
5. Conclusions
Two-dimensional flow of stagnation point heat flow and mass transfer in a Casson Nanofluid with thermodiffusion and
an inclined magnetic field is considered and the numerical computation is performed via the Spectral Collocation
Method. From the study, we analyzed that rise in the thermophoresis parameter enhances both the temperature and
UKH Journal of Science and Engineering | Volume 5 • Number 1 • 2021 49
nanoparticle concentration profiles, while the velocity profile decreases with an increase in the Casson parameter. The
increase in the magnetic parameter enhances the temperature and nanoparticle concentration profiles.
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