Abstract-- The current article aims to investigate Soret and Dufour's effects on MHD flow of heat and mass transfer nanofluid over a vertical stretching sheet with activation energy and radiation. Using Similarity transformations, the non-linear momentum, energy, solute, and nanoparticle concentration boundary layer equations are simplified. The resulting boundary layer equations are solved numerically with bvp4c and along with the shooting technique. The impact of various parameters on velocity, temperature, concentration, and nanoparticle volume fraction, Skin friction, local Sherwood and local Nusselt numbers are presented through graphs. It is found that the temperature, fluid velocity, and nanoparticle volume fraction enhanced with both Dufour and Soret effects.
Keywords-- Chemical reaction, Activation energy, MHD, Soret/Dufour, Nanofluid.
I. INTRODUCTION
Activation energy along with Chemical reaction, play an important role in oil and water emulsions, oil reservoirs, geothermal engineering, and chemical engineering. At a given temperature, the lower the activation energy, the greater the chemical reaction rate, according to the Ar-rhenius concept. Usman et al. (2020) investigated that on the boundary layer chemical species concentration in-creased by Soret number and Activation energy. Yesodha et al. (2021) reported that the concentration values are reduced and temperature gradients are enhanced by the effect of a rise in activation energy. Dhlamini et al. (2019) studied the impact of activation energy and chemical reaction on unsteady convective flow over an infinite length boundary layer. Punith Gowda et al. (2021) explicated the combined effect of activation energy, magnetic field, and chemical reaction on Marangoni driven non-Newtonian nanofluid flow. Sivasankaran et al. (2017) found that, When the value of the chemical reaction parameter is enhanced, the mass transfer rate enhances, and the heat transfer rate is reduced.
Awais et al. (2021) analyzed the effect of activation energy along with the uniform magnetic field, entropy optimization, and mixed convection rheology on hyper-bolic tangent nanofluid. Kotresh et al. (2021) investigated the structural modifications of nanofluid flow past a stretchable rotating disc with Arrhenius activation energy and chemical response. Bai et al. (2021) studied the un-steady fractional MHD Maxwell fluid flow near un-steady stagnation-point over a stretching plate. They found that the temperature boundary layer is thicker than the velocity profile.
The energy flux induced by a mass concentration gradient as a combined consequence of irreversible processes is the Dufour effect. It is the inverse phenomenon of the Soret effect. These effects are important in geo-sciences and chemical engineering. Few works (Ullah et al., 2017; Jawad et al., 2021; Khan et al., 2020) are highlighted with Soret and Dufour's effects. Niranjan et al. (2017) studied, the Soret & Dufour effects on MHD flow of a viscous fluid in the presence of slip, chemical reaction, and radiation past a vertical plate in a porous medium. Kasmani et al. (2017) investigated the impact of Soret and Dufour's effects on the flow of double-diffusive boundary layer nanofluid in the presence of suction over a moving wedge. Reddy and Krishna (2018) discussed the influence of Soret and Dufour's effects on MHD fluid flow past a linearly stretching sheet in a porous medium. Sreedevi et al. (2017) explored that the transfer rate of heat reduces with an increase in the Soret parameter. Idowu and Falodun (2019) examined the velocity of the fluid and temperature profiles increase with the Dufour parameter. Jagan et al. (2018) examined the solutal boundary layer thickness enhances with Soret number. Waini et al. (2021) discussed the effects of Soret and Dufour on Al2O3-water nanofluid over a moving thin needle by using Tiwari and Das model.
Thermal radiation has various numerous applications in the chemical industry, physics, and engineering ap-plications such as space technology, nuclear reactors, glass production, polymer processing, etc. Reddy et al. (2018) formulated the impact of radiation, Soret, and Dufour effects on nanofluid flow for dual solutions. Okuvade et al. (2018) formulated the effect of Dufour, Soret, and thermal radiation on unsteady MHD convective incompressible fluid past a vertical plate. MHD mixed convective three-dimensional boundary-layer flow on a bidirectional stretching sheet with radiation, Soret, and Dufour effects was analyzed by Prasannakumara et al. (2018). Rasool et al. (2020) investigated the impact of thermal radiation, Soret–Dufour effects, and chemical reaction on the steady incompressible flow of nanofluids. Many authors discussed the impact of radiation on heat transfer of non-Newtonian fluids past a stretching sheet (Huang, 2018; and Hayat et al., 2017). The effect of radiation, suction and stretching/shrinking parameter on axisymmetric flow and heat transfer of nanofluid over a sheet explored numerically by Rosca et al. (2021). They found the higher heat of nanofluid and mass transfer rate for stretching flow. The combined impact of thermal radiation and activation energy on MHD non-Newtonian tangent hyperbolic nanofluid over a moving surface with electrical and slip features was studied by Khan et al. (2019).
The main aim of the present article is to explore the influences of radiation and activation energy on MHD mixed convection flow of nanofluid over a vertical stretching sheet with Soret and Dufour effects. The governing equations are converted into ordinary differential equations by using similarity transformations. Then the resultant equations are solved numerically by using MATLAB bvp4c. The effect of various flow parameters on solute concentration, nanoparticle volume fraction, temperature, velocity, skin friction, heat and mass transfer coefficients are discussed and presented graphically
II. MATHEMATICAL ANALYSIS
The steady, laminar, incompressible, MHD flow of
chemically reacting nanofluid flow past a vertical sur-face with radiation is
considered. In this problem, the combined influence of Soret/Dufour effects and
activation energy are taken into account. The coordinate sys-tem and physical
model of the problem as shown in Fig.1. The coordinate system 
, 
is chosen
along the stretching sheet and 
is chosen
normal to the stretching sheet. The velocity components 
and 
are taken as
along and directions,
respectively. The strength of the magnetic field 
is applied
normal to the flow field. Let us assume that the linear velocity 
stretches the
sheet surface in the vertical direction, where 
denotes the
stretching rate constant. At the wall the temperature, solute concentration,
nanoparticle volume fraction are denoted as 
, 
, 
, and far from
the wall, the temperature, solute concentration, and nanoparticle volume
fraction are denoted as as 
, 
, 
,
respectively. The governing equations for mass, momentum, thermal energy,
solute and nanoparticles can be written as
, (1)
(2)
(3)
(4)
(5)
Figure 1: Physical model of the flow problem and
coordinate system.
where 
and 
.
Subject to the boundary conditions are,
, 
, 
, 
at 
. (6)
, 
, 
, 
as 
. (7)
The fluid kinematic viscosity is 
. The modified
Arrhenius equation is 
. Where the
reaction rate is 
, the
temperature is 
, the activation energy is 
, the
Boltzmann constant is 
eV/K, the
fitted rate constant is 
and its range
generally 
.
The radiative heat flux 
is given by
. (8)
The heat flux () radiative
term in Eq. (3) simplified by using the Rosseland approximation. Using Taylor’s
series, we can expand 
about and by
ignoring higher-order terms, showing as a linear function of temperature . i.e.,
, then 
. Finally, we
get
. (9)
Now we define the following non-dimensional functions 
), 
), 
), 
) and
similarity variable 
as
, 
, 
(10)
The Stream function 
is defined as
and 
(11)
The governing Eqs. (1) to (5) are transformed into the ordinary differential equations by using the Eqs. (9) to (11), as follows
(12)
(13)
(14)
(15)
The boundary conditions (6) and (7) become
, 
, 
,
, 
at 
(16)
, 
, 
,
as 
(17)
where the nondimensional variables are defined as
, 
, 
, 
,
, 
, 
,
, 
, 
, 
,
, 
, 
,
, 
III. PHYSICAL QUANTITIES OF ENGINEERING INTEREST
The physical quantities describing the skin friction (
, the Local
Nusselt number 
), the local
Sherwood number (
, and the local
nanoparticle Sherwood number (
are shown
below.
, 
,
, 
(18)
where the shear stress (
), surface heat flux (
), surface
mass flux (
), and surface
nanoparticle mass flux (
) are defined
as
, 
,
, 
.
Using Eq. (10), we obtained the dimensionless form of physical quantities are as follows
, 
,
, 
(19)
where 
is the local
Reynold’s number.
IV. NUMERICAL METHOD
The nonlinear ODEs (12-15) subjected to the boundary conditions (16-17) are solved numerically by the shooting method. For this purpose, let
, 
, 
, 
, 
, 
, 
, 
. (20)
Equations (12-15) are converted into following first order ordinary differential equations
(21)
(22)
(23)
(24)
Subject to the boundary conditions
, 
, 
,
, 
at (25)
, 
, 
,
as (26)
These simplified equations are used in MATLAB bvp4c software to analyze variation of variables on subsequent profiles.
III. RESULTS
In the present article, we investigate the influence of thermal radiation, Soret and Dufour's effects on MHD heat transfer nanofluid flow past a vertical surface with activation energy. The governing equations (12-15) are solved through the MATLAB bvp4c by applying boundary conditions (16-17). The impact of various dimensionless parameters on velocity, temperature, concentration, and nanoparticle volume fraction, Skin friction, local Nusselt number and local Sherwood numbers, analyzed through graphs. The numerical results are compared with the results of Khan et al. (2019) and Ibrahim (2017) and they are shown in the Table 1. These results provided a very good agreement.
Figures 2 to Fig.4 depicts the impact of the Dufour effect (
) on velocity
profile (
), temperature
profile (
) and solute
concentration (
), respectively. Figure 2 represents that the velocity of fluid
enhances with . Since the
Dufour parameter as in energy equation, which affects the temperature. The
temperature profile and the width of the boundary layer enhanced with the
Dufour number values represented in Fig.3. Figure 4
Figure
2: Behaviour of velocity profile with 
.
Figure
3: Behaviour of temperature profile with .
Figure 4: Behaviour of solute
concentration profile with .
Figure 5: Behaviour of
velocity profile with 
.
depicts that, the solute concentration decreases upon the increase of the Dufour number.
Figures 5 to Fig.7 illustrates the influence of the Soret effect
(
) on velocity
profile, temperature profile, and solute concentration respectively. Figure 5
shows that the fluid velocity () increases as
the Soret number () increases. Fluid temperature () enhanced
with Soret number ( depicted in Fig.6. The larger the Soret number, the greater the
temperature variance and the faster the gradient. As a result of the increased
Soret effect, the
Figure
6: Behaviour of temperature profile with .
Figure
7: Behaviour of solute concentration profile with .
Figure
8: Behaviour of velocity profile with 
.
Figure 9: Behaviour of temperature
profile with 
fluid velocity increases. Solute concentration () of the fluid
also enhanced with Soret number illustrated in Fig.7. As a result, in the study
of problems in mixed convection, we can conclude that Dufour and Soret's
effects are more enthusiastic. The effects of Soret and Dufour appear to be
important in combined convection in the occurrence of slip parameter, chemical
reaction, and radiation in a porous medium.
Figures 8 to Fig.11 presented to visualize the
behaviour of the radiation (
) on velocity
profile, temperature profile, nanoparticle volume fraction, and solute concen-
Figure
10: Behaviour of nanoparticle volume fraction profile with .
Figure
11: Behaviour of solute concentration profile with .
Figure
12: Behaviour of temperature profile with 
.
Figure 13: Behaviour of nanoparticle
volume fraction profile with .
tration respectively. Figure 9 reveals the enhancement of fluid
velocity () with . The
temperature profile () and width of
the boundary layer increase with represented
in Fig. 9. That is the radiation effect increases heat transfer. To facilitate
cooling, the radiation effect should be kept to a minimum. From Fig. 10, we
noticed that the radiation parameter enhances the nanoparticle volume fraction
profile (
). Figure 11
shows the decreasing phenomena in solute concentration () with
Figure
14: Behaviour of solute concentration profile with .
Figure
15: Behaviour of solute concentration profile with 
.
Figure
16: Behaviour of Temperature profile with 
.
Figure 17: Behaviour of nanoparticle
volume fraction profile with .
radiation (
. The number
of particles and solutes dissolved in a solution is referred to as solute
concentration.
The decrease in solute concentration causes an increase in fluid velocity. Because of this, the transport rate increased.
Figure 12 to Fig. 14 portrayed the behaviour of the
activation energy (
) on temperature profile (),
nanoparticle volume fraction (), and solute
concentration profile (),
respectively. From Fig. 12 and Fig. 13 the temperature profile, and
nanoparticle volume frac-
Figure 18: Behaviour of local Nusselt
number with 
and 
.
Figure
19: Behaviour of local Sherwood number with and .
Figure 20: Behaviour of skin
friction with and .
tion reduced with activation energy (), but solute
concentration enhanced as shown in Fig. 14. The increase in solute
concentration causes a decrease in fluid velocity. From this, we conclude that
increased values of activation energy reduce the fluid velocity.
The impact of Schmidt number (
) on solute
concentration profile () is portrayed
in Fig. 15. It represents solute concentration decreases with Schmidt number;
hence, the fluid velocity increases. The variation of temperature profile () and
nanoparticle volume fraction () by Biot
number 
in elucidated
by Fig. 16 and Fig. 17. Here both temperature and nanoparticle concentration in
the boundary layer are enhanced with Biot number.
Figure 18 represents the effect of Soret number () and Schmidt
number () on Nusselt
number (
). The heat
transfer rate () increases with the Soret number but decreases with the Schmidt
number. Here the width of the boundary layer increases gradually with Schmidt
number. The impact of Soret number and Schmidt number on Sherwood number (
) illustrated
in Fig. 19. The mass transfer coefficient () was reduced
with the Soret number () and enhanced
with the Schmidt number (). The width
of the boundary layer increases gradually. Figure 20 enables the influence of
radiation () and Schmidt
number () on Skin
friction (
). The Skin
friction enhances with both radiation and Schmidt number. Hence, the fluid
velocity decreases. Because friction and velocity are inversely proportional to
each other. Here the width of the boundary layer is constant.
IV. CONCLUSIONS
This study explores the numerical description of the collective impact of thermal radiation, Soret and Dufour's effects, and activation energy on MHD chemically reacting nanofluid flow past a vertical sheet. As a result of our investigation, we have discovered the following:
· Fluid velocity (, and
temperature () enhanced
with Dufour effect (), but solute
concentration () reduced.
· Solute concentration increases on increasing the Soret number () increases.
· Temperature enhanced with higher values of ().
· Solute concentration increases on increasing the activation energy parameter.
· The heat transfer rate () of the fluid
increases with the Soret effect (), but a
reverse trend happens in the mass transfer () of the
fluid.
· The Skin friction () enhances
with higher values of thermal radiation ().
NOMENCLATURE
Strength
of magnetic field
Biot
number
Nanoparticle
concentration
Specific
heat at constant pressure
Concentration
susceptibility
Brownian
diffusion coefficient
Dufour
number
Mass
diffusivity
Thermophoretic
diffusion coefficient
Non-dimensional
activation energy
Dimensionless
velocity
Local
Grashof number
Gravitational
acceleration
Thermal
diffusion ratio
Mean
absorption coefficient
Thermal
conductivity
Magnetic
field parameter
Brownian
diffusion parameter
Regular
buoyancy parameter
Buoyancy
ratio parameter
Thermophoresis
parameter
Prandtl
number of base fluid
Thermal
radiation parameter
Local
Reynolds number
Solute
concentration
Schmidt
number
Soret
number
Temperature
Greek symbols
Mixed
convection parameter
Dimensionless
reaction rate
Temperature
difference parameter
Kinematic
viscosity
Dynamic
Viscosity
Density
Electrical
conductivity
Stefan-Boltzmann
constant
Density
of the fluid
Coefficient
of thermal expansion
Nanoparticle
density
Thermal
diffusivity of the base fluid
Ratio
of the effective heat capacity of the nanoparticle material and the heat capacity
of the fluid
Heat
capacity of the base fluid
Effective
heat capacity of the nano particle material
Dimensionless
temperature
Dimensionless
solute concentration
Dimensionless
nanoparticle concentration
Subscripts
Condition
at a wall
Condition
at free stream
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Received: November 26, 2021
Sent to Subject Editor: December 15, 2021
Accepted: January 13, 2022
Recommended by Subject Editor Fabio Giannetti