Abstract-- Analysis of steady, axisymmetric, hydromagnetic, viscoelastic fluid flow due to a circular sheet stretching radially has been made in presence of nonlinear thermal radiation, heat source, Arrhenius activation energy, Soret and Dufour effects. To study the viscoelastic fluid, Maxwell fluid model has been adopted. Governing simultaneous partial differential equations are reduced to ordinary differential equations by using similarity transformations. The ordinary differential equations are solved by using bvp4c method. The obtained results are shown graphically to illustrate the impact of pertinent physical parameters on velocity, temperature, concentration, skin friction, heat transfer and mass transfer.
Keywords-- Viscoelastic fluid, Maxwell fluid, hydromagnetic, Soret effects, Dufour effects, Arrhenius activation energy, Deborah number
I. INTRODUCTION
The problem of viscoelastic fluid flows have wider range of applications in industry and manufacturing processes as many fluids of industrial applications exhibit physical phenomena such as stress relaxation, shear dependent viscosity and normal shear stress. Viscoelastic fluid is a non-Newtonian fluid which posses both viscosity and elasticity property. Maxwell fluid model is a subclass of viscoelastic fluid which contains the characteristics of fluid relaxation time. Several researchers have given significant attention in exploring the features of Maxwell fluid. Wang and Hayat (2008) presented a two dimensional fluctuating flow of Maxwell fluid over an infinite porous plate. Hayat and Qasim (2010) presented the impact of thermal radiation and Joule heating on magnetohydrodynamic flow of Maxwell fluid. Zheng et al. (2011) examines the exact solutions for an unsteady rotating flow of a Maxwell fluid on coaxial cylinders. Hayat et al. (2012) discussed homotopy simulations for a Maxwell fluid in a stretching sheet considering melting heat transfer. Abel et al. (2012) numerically studied the MHD flow of a Maxwell fluid in a isothermal stretching surface. Nandy (2015) discussed the Maxwell fluid on a permeable shrinking surface in presence of nanoparcticles and Navier slip. Heat and mass transfer on MHD flow for Maxwell fluid due to exothermal and isothermal stretching disks has been analyzed by Khan et al. (2016). Mustafa et al. (2016) numerically discussed the Maxwell fluid flow due to a circular stretching elastic sheet with nonlinear thermal radiation. Several authors: Elbashbeshy et al. (2018), Imran et al. (2018), Farooq et al. (2019), Ibrahim and Negera (2020), Akolade (2021), Fetecau et al. (2021) have contributed in this line. Recently, Prasad et al. (2020), Vaidya et al. (2020a, 2020b) Dibya et al. (2021) and Vadya et al. (2021a, 2021b) studied the MHD flows in various situations.
The study of heat and mass transfer of MHD flow with Soret and Dufour effects is a significant subject owing to the several applications such as isotope separation, mixture of gases, solidification of binary alloys, chemical reactors, oil reservoirs and ground water pollutant migration. The diffusion of heat caused by temperature gradients (Soret effect) and by concentration gradients (Dufour effect) can play a significant role when the temperature and concentration gradients are large. Recently, Reddy et al. (2020), Imtiaz et al. (2020), Das (2021) explored the effects of Soret and Dufour in the flow phenomena.
In view of above discussion, the MHD flow of chemically reacting viscoelastic fluid characterized by Maxwell fluid in a circular stretching sheet with Soret- Dufour effects and Arrhenius activation energy is addressed in this study. The bvp4c method is used to solve the system of differential equations. Introduction of Soret and Dufour effects in this problem is the novelty of the present work. This study is an extension of Mustafa et al. (2016) to porous medium incorporating Soret and Dufour effects.
II. MATHEMATICAL FORMULATION
A steady hydromagnetic, axisymmetric flow of incompressible Maxwell
fluid caused by a circular sheet stretching through a binary mixture in presence
of Soret and Dufour effects have been considered. For the formulation of the
problem we use cylindrical coordinates 
in such a way
that the circular sheet lies in the 
-plane 
while the
fluid occupies semi infinite space 
of the
vertical axis as shown in Fig. 1. The surface is stretching in the radial
direction 
with velocity
, where 
is a
constant. Here, it is assumed that the fluid flow is symmetric with respect to
the coordinate 
A magnetic
field of uniform strength 
is applied
normal to the stretching surface. Non-linear heat radiation for temperature and
modified Arrhenius function for chemical reaction are considered.
The governing boundary layer equations of flow with heat and mass transfer (following Mustafa et al., 2016; Khan et al., 2016) are given by
, (1)
Fig. 1. Physical diagram of the problem.
, (2)
(3)
(4)
The boundary conditions are
, 
, 
at ;
, 
, 
at 
(5)
where 
and 
denote the
radial and axial velocity components and 
is fluid
relaxation time. The third term in the Eq. (4) is the modified Arrhenius
function in which 
is activation
energy and 
is a rate
constant lies in 
. The other variables are mentioned in the Nomenclature.
Following similarity variables are introduced:
, 
, 
,
, 
. (6)
Using (6) in Eqs. (2)-(4), we get
, (7)
, (8)
, (9)
The boundary conditions are
, 
, 
, 
at 
,
, 
, 
at 
. (10)
where non-dimensional parameters are 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
,
Table 1: Computational values of 
for
Newtonian fluid (
) with 
, 
, 
|
|
Ariel et al. (2006) |
Mustafa et al. (2016) |
Present (2016) |
|
0 |
1.178511 |
1.173721 |
1.736916 |
|
1 |
1.539601 |
1.535710 |
1.536425 |
|
2 |
2.313407 |
2.311718 |
2.312416 |
|
5 |
5.132002 |
5.131808 |
5.131972 |
. The
parameters are mentioned in the Nomenclature.
Local
Nusselt number (
) following Fourier law is defined as
, 
Local Sherwood number (
is defined as
, 
III. VALIDITY AND ACCURACY
The present study is well validated through the following investigations:
(i)
When 
, 
, 
, 
, present
study matched with that of Ariel et al. (2006).
(ii)
When , , , present
study matched with that of Mustafa et al. (2016).
The comparison of the present study for 
with Ariel et
al. (2006) and Mustafa et al. (2016) are presented in the Table 1.
It is found that the present result is in good agreement.
IV. RESULTS AND DISCUSSION
In calculations, the following default values of the parameters are
taken 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
.
Figures 2-4 has been plotted to highlight the impact
of Deborah number (
, Hartmann number (
) and porous
parameter (
) on velocity
profile 
,
respectively. It is seen that during the increasing of Deborah number leads to
decelerate the velocity profile. Deborah number characterizes the ratio of
fluid relaxation time to fluid characteristic material time. Small Deborah
number for fluid represent that the viscosity is dominant over elasticity. It
may be interpreted that higher values of De (higher elastic effect response),
less energy is stored and as a consequence speed might decrease and thus
reduces the thickness of the momentum boundary layer as seen in Fig. 2. Figures
3and 4 show that the presence of magnetic field and permeability parameter
increases the resistance to flow and thus decreases the thickness of momentum
boundary layer. Figure 5 depicts the effects of activation energy parameter (
) and chemical
reaction parameter (
) on temperature
profile 
. It reveals
that temperature increases with the increase in energy activation parameter () from 
to 
through . This might
be interpreted as increasing activation energy increases the collision between
molecules or atoms and thus enhances the temperature. Also, from the same Fig.
5 it is seen that the temperature reduces due to the advancement of chemical
reaction parameter. Figure 6
Fig. 2. Influence of 
on 
.
Fig. 3. Influence of 
on .
Fig. 4. Influence of 
on .
shows the influence of Eckert number (
) and Soret
number (
) on
temperature profile . It is
noticed that the growth in Eckert number and Soret number raises the fluid
temperature. Eckert number represents the ratio of kinetic energy to enthalpy
difference of heat. So, increasing Eckert number increases the kinetic energy
and thus causes enhancement in temperature. Figure 7 show the effect of heat
source parameter and radiation parameter on temperature . It is
noticed that the heat source parameter and radiation parameter enhances the
thickness of the thermal boundary layer. Thus the phenomenon of heat
advancement may be controlled by introducing heat source/ sink parameter
efficiently. Figure 8 portray the impact of Dufour number (
) on
temperature 
and
concentration 
profiles. Dufour
number is the ratio of concentration gradients to the thermal energy flux.
Hence, bigger Dufour number stands for larger concentration gradients, so increase
in reduces both
thermal and concentration boundary layer thickness. Figure
Fig. 6. Influence of 
, 
on 
.
Fig. 7. Influence of 
, 
on .
Fig. 8. Influence of 
on , 
.
Fig. 9. Influence of 
, 
on .
9 reveals the impact of activation energy parameter () and exponent
() on
concentration profile . It reveals
that the activation energy parameter enhances the species concentration within
the boundary layer. The influence of increasing exponent reduces the species
concentration. Figure 10 shows that the impact of Eckert
Fig. 10. Influence of , on .
Fig. 11. Influence of 
, 
on .
number () and Soret number () on
concentration profile . Eckert
number reduces the concentration profile.
This fact can be seen as Eckert number raises the
fluid temperature by increasing kinetic energy and thus diminishes the
concentration of the fluid. quantifies
the ratio of temperature gradient to concentration. Thus, bigger means larger
temperature gradient and so causes the species concentration to increase and
enhances the thickness of concentration boundary layer.
Figure 11 shows the impact of Schmidt number and
chemical reaction parameter on concentration profile
. Increasing
Schmidt number and chemical reaction parameter reduces the thickness of
concentration boundary layer. Physically, it can be said that an enhancement in
means a
decrease in molecular diffusivity and thus decrease in concentration boundary
layer.
The influence of 
on skin
friction coefficient 
against is shown in
Fig. 12. It reveals that the viscoelastic effects () enhances the
skin friction coefficient. Also, augmented magnetic drag force enhances the
skin friction and minimum is attained at the surface. The rate of heat transfer
for various
Schmidt number against is shown in
Fig. 13. Increasing Schmidt number reduces the rate of heat transfer whereas
rate of heat transfer augmented due to the influence of .
The impact on the rate
of mass transfer 
against is shown in
Fig. 14. It is observed that the influence of magnetic field raises the rate of
mass transfer, but mass transfer decreases exponentially due to the enhancement
of E.
Fig. 12. Influence of against on 
.
Fig. 13. Influence of against on 
.
Fig. 14. Influence of against on 
.
V. CONCLUSIONS
The main outcomes of this work are as below:
(i) Deborah number, Hartmann number and permeability parameter reduces the fluid velocity.
(ii) Fluid temperature increased via greater values of activation energy, Eckert number, Soret number, heat source parameter and radiation parameter, trend is opposite for chemical reaction parameter or Dufour effect.
(iii) Concentration profile decreased though greater values of Dufour number, Eckert number, Schmidt number, chemical reaction parameter, exponent of chemical reaction parameter, while reverse trend is seen for activation energy or Soret number.
(iv) Skin friction increased for the influence of Deborah number and Hartmann number.
(v) Rate of heat transfer can be decreased by increasing Schmidt number or decreasing Soret number.
(vi) Rate of mass transfer decreases exponentially with increasing activation energy.
The present study finds applications in extrusion processes, microchips, paint application, food processing and biological fluid handling. This problem can also be studied by considering variable viscosity and other viscoelastic fluid model such as Walters fluid model and second-order fluid model.
NOMENCLATURE
Stretching
rate, a positive constant
Strength
of magnetic field
Concentration
Specific
heat
Concentration
susceptibility
Deorah
number
Dufour
number
Mass
diffusivity
Eckert
number
Activation
energy
Hartmann
number
Permeability
parameter
Dimensional
permeability parameter
Mean absorption
coefficient
Chemical
reaction rate
Thermal
diffusion ratio
Prandtl
number
Dimensional
heat source/sink parameter
Radiation
parameter
Cylindrical
coordinates
Non-dimensional
heat source parameter
Schmidt
number
Soret number
A
rate constant
Fluid
temperature
Mean
fluid temperature
Dimensional
velocity component along -axis
Dimensional
velocity component along 
-axis
Greek symbols
Thermal
diffusivity
Non-dimensional
temperature
Temperature
ratio
Cylindrical
coordinate
Non-dimensional
concentration
Boltzman
constant
Chemical
reaction rate constant
Stefan
–Boltzman constant
Subscripts
Surface
condition
Free
stream condition
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Received: September 12, 2021
Sent to Subject Editor: November 2, 2021
Accepted: December 25, 2021
Recommended by Subject Editor Gianfranco Caruso