Abstract--Our motive is to examine the impact of thermal radiation and suction or injection with viscous dissipation on an MHD boundary layer flow past a vertical porous stretched sheet immersed in a porous medium. The set of the flow equations is converted into a set of non-linear ordinary differential equations by using similarity transformation. We used MATLAB bv4c technique to solve the set of equations. The impact of non-dimensional physical parameters on flow profiles is analyzed and depicted in graphs. We observe the influence of non-dimensional physical quantities on the Nusselt number, the Sherwood number, and skin friction presented in tables. A comparison of the obtained numerical results with existing results in a limiting sense is also presented. We enhance radiation to observe the deceleration of fluid velocity and temperature profile for both suction and injection. While enhancing porosity parameter accelerates velocity whereas decelerates temperature profile. As the heat source parameter increases, the temperature of the fluid decreases for both suction and injection, it has been found. The skin friction and the heat transfer rate decrease with the increasing values of the radiation parameter. Increasing magnetic parameter decelerates the skin friction, Nusselt number, and Sherwood number.
Keywords--Suction/Injection, Heat and Mass Transfer, Thermal Radiation, MHD, Heat Source, Porous Medium, MATLAB bv4c.
I. INTRODUCTION
Heat and mass transfer through a viscous fluid flow in a porous medium is applicable in various industrial fields such as drying porous solid, enhanced oil recovery, geothermal reservoirs, and several such research areas. Free convective flow and heat transfer through stretching porous medium has an impact on polymer and metallurgy technology. A porous sheet's stretching capacity is responsible for the rate of heat and mass transfer through a viscous and incompressible fluid. Therefore, the flow over a stretching porous sheet has evolved as a vital area of research. Crane (1970) developed the study of flow over stretching sheet focusing on the result that in polymer industries standard of the finishing product depends on the heat and mass transfer rate over a stretching sheet. Subhas and Veena (1998) studied the flow of heat and visco-elastic fluid over an impermeable stretching sheet inserted in a saturated porous medium. Dogonchi et al. (2018) investigated the Joule heating, and thermal radiation effect in MHD flow of nanofluid over a stretching surface. Ghadikolaei et al. (2018a) approached numerically to observe the flow of non-linear nanofluid passes through a porous stretching sheet. Khan et al. (2019) investigated the Williamson nanofluid flow analytically over a stretchable surface in the presence of Joule heating. Nayaka et al. (2019) observed the mass transfer of an axisymmetric flow through a non-linear radially stretched sheet. Ibrahim and Gadisa (2020) considered a stretching sheet to evaluate the non-linear convection flow of Oldroyd-B fluid. Recently (Manjunatha et al., 2020; Rajasekhar et al., 2020; Vaida et al., 2020; Prasad et al., 2020a; Divya et al., 2021; Vaida et al., 2021), various studies have been conducted on MHD peristaltic flow considering heat and mass transfer in both uniform and non-uniform medium with small variable viscosity.
Suction or injection influences the heat transfer rate in heat and mass transfer of a fluid flow. Mostly in aerodynamics, suction and injection theory is applied to control the pollution and cost-effectiveness of a commercial aircraft by reducing fuel consumption. Suction and injection at the boundary wall are responsible for the change in the rate of mass and heat transfer. Agarwal and Roy (1979) observed the distinction of stagnation and pressure point for both suction and injection. Jha and Aina (2018) observed the influence of suction or injection on a mixed convection flow in a microchannel. Rehman et al. (2019) studied the combined influence of suction/injection, magnetic field, slippage, and surface tension at the boundary to improve the boundary layer thickness. Recently, Pal and Mandal (2020) scrutinized the combined convective heat and mass flow in the presence of suction and injection. Prasad et al. (2020b) observed the two-dimensional MHD boundary layer flow through a stretching rotating disk in the presence of suction/injection.
The influence of the radiation parameter on fluid flow significantly impacts conductive gray fluids flow, such as liquid metal fluids, high-temperature plasmas, power generator systems, and nuclear reactors' cooling. Sharma and Konwar (2016) examined the impact of thermal radiation on an MHD flow over a rotating cone. Sarma et al. (2014) observed the thermal radiation and chemical reaction effect on an unsteady MHD flow over an infinite accelerated vertical plate considering a porous medium. Recently Rajput et al. (2020) examined the magnetohydrodynamics boundary layer flow through an exponentially stretched surface in a porous medium with radiation effect. Kumar et al.(2020) reported the impact of thermal radiation on heat and mass flow over both divergent and convergent channels to conclude that the heat transfer rate is higher in divergent channels than in the convergent channel. Haq et al. (2020) studied the radiation effect on fluid flow over a shrinking surface in the presence of suction and injection. Sarma and Pandit (2015) studied Hall current, rotation, and Soret effects on MHD natural convective combined heat and mass flow over an accelerated vertical plate through a porous medium. Pal and Mandal (2017) analysed the impact of radiation on mixed convective heat and mass flow through a stretching and shrinking sheet at a stagnation point. Ghadikolaei et al. (2018b) studied the effect of thermal radiation and Joule heating on magnetohydrodynamics Casson nano-fluid flow through an inclined porous stretched surface. Khader (2018) developed a numerical formula to express the thin liquid film's flow and heat transfer. Megahed (2019) observed the Carreau flow over a non-linear stretched surface in the presence of thermal radiation. Waini et al. (2020) investigated the flow and heat transfer of hybrid nano-fluid over a stretched or shrunk surface in the presence of a magnetic field and thermal radiation. Pal et al. (2020) examined the heat and mass flow through a stretched surface with thermal radiation.
The works mentioned above depict the importance of studying the influence of thermal radiation on the MHD free and mixed convective heat and mass transfer flow over a porous stretched surface considering suction and injection. In this study, we observed the influence of both suction and injection on a two-dimensional, radiative flow of viscous electrically conducting fluid over a stretching porous surface. Our work is an extension of the work of Jaber (2016), where he observed the influence of Joule heating and viscous dissipation on the exact configuration without the presence of suction and injection. Here we have observed the impact of radiation parameters along with other involved parameters in fluid profiles.
II. MATHEMATICAL FORMULATION
We consider the steady two-dimensional flow of an
electrically conducting with viscous and incompressible fluid passes through a
porous stretching sheet issuing from a slit at the origin of the rectangular
co-ordinates as shown in Fig. 1. The sheet is assumed to be horizontal and coincides
with the plate 
. The plate is continuously stretching
by two equal powers with opposite directions. The speed is assumed to be
proportional to its distance from the slit point. The flow is considered to be
in the 
-direction
which is taken along the plate, see Fig. 1. We have examined the effect of heat
generation, suction/in-
Figure 1: Mathematical formulation of the problem.
jection, thermal radiation, and viscous dissipation
irrespective of time on the flow field. The plate is subjected to a power law
heat flux in presence of heat source. The sheet moves along the -axis with the
velocity 
, where 
is the
stretching rate considered a positive integrand of the velocity 
in the 
-axis
direction. A uniform strong magnetic field of strength 
is subjected
normal to the plate. This magnetic field generates an electric field for the
electrically conducting fluid. We assumed a negligible Reynolds number to
overcome the effect of the induced magnetic field. The governing equations Jaber (2016) under boundary layer and
Boussinesq’s approximations for flow over the stretching surface embedded in a
porous medium are, given below,
Continuity equation:
(1)
Equation of Motion:
(2)
Equation of Energy:
(3)
Equation of Mass-deposition:
(4)
The boundary conditions are (Kumar, 2009):
(5)
For an optically
thick fluid, in addition to emission, there is also self-absorption, and
usually, the absorption coefficient is wavelength dependent and significant, so
we can adapt the Rosseland approximation for the radiative heat flux vector 
. Thus is given by
(6)
We assume that
the temperature differences within the flow are sufficiently small so that 
can be expressed as a linear function. By using Taylor's
series, we expand about the free stream temperature. 
and neglecting higher-order terms. This results in the
following approximation:
(7)
We introduced the similarity transformations as follows,
,
(8)
After using the similarity transformations, we have:
(9)
(10)
(11)
where,
,
M=
.
The boundary conditions:
;
(12)
III. NUMERICAL METHOD OF SOLUTION
The MATLAB built-in bvp4c solver scheme is implemented to solve the resulting coupled Eqs. (9)-(11) along with the boundary condition (12). It is a boundary value problem solver technique and gives the results by employing finite difference code. This technique is already built-in MATLAB and solves the problem by indirectly resolving the error, providing the solutions at each mesh point of the given interval. The user has to offer the following three essential functions to get the results:
(i) Initial (guess) solutions:
This scheme shows promising results with good guess values. Although, by getting low guess values, this scheme also shows the output.
(ii) System of first-order equations:
The resulting Eqs. (9)-(11) can be written in the system of first-order equations by introducing the new variables as follows:
Then, we have achieved the following first-order system of equations
(iii) Relevant boundary conditions:
The boundary conditions 11 (a-b) can be written in the following way:
;
.
IV. RESULT AND DISCUSSION
We solved non-linear
differential Eqs. (9)-(11) under boundary conditions (12) using MATLAB
built-in bvp4c solver scheme. The influence of the various flow physical
parameters on flow profiles is depicted in graphs, and on the skin friction
coefficient, local Nusselt and Sherwood numbers are presented in tabular form,
respectively. We assumed
, so that the fluid becomes
equivalent to water.
First, we
present the result for variation of the mag
Figure 2: Velocity profiles for various values of 
.
Figure 3: Temperature profiles for various values of .
Figure 4: Velocity profiles for various values of 
.
netic parameter
. Figure 2 depicts that an increase in the
values of the magnetic parameter reduces the velocity boundary layer for both
suction/injection. Since an increase in the magnetic field develops the
opposite force to the flow direction, named Lorentz force, it reduces the
boundary layer thickness.
Figure 3 represents the temperature profile for different
values of the magnetic parameter. The non-dimensional temperature increases for increasing
values of 
for both suction and injection. The temperature at a point
on the sheet rises significantly with the increase in , i.e., the heat transfer rate decreases with the
increasing magnetic parameter .
Figures 4-5
depict the influence of thermal and concentration buoyancy forces on the fluid
velocity. From Fig. 4, we can observe the accelerated fluid velocity 
for increasing 
for both suction and injection. Fig.
5 states that the fluid velocity increases on accelerating
. 
is the ratio of thermal buoyancy
force to viscous force, and 
is the ratio of concentration
buoyancy force to viscous force. This implies a decrease of on accelerating thermal buoyancy
force and an increase of on enhancing the concentration
buoyancy force. Here the free convective flow is caused due to thermal and
concentration buoyancy forces; that is why thermal buoyancy force
Figure 5: Velocity profiles for various values of 
.
Figure 6: Velocity profiles for various values of 
.
Figure 7: Temperature profiles for various values of .
reduces the fluid velocity, whereas concentration buoyancy force increases the fluid velocity throughout the boundary layer region.
Figures 6 and 7 depict the impact of the permeability parameter on fluid velocity and temperature profile, respectively. We observed from Figs. 6 and 7 that the fluid velocity increases on increasing permeability parameter, but the temperature profile decreases for both suction and injection, respectively. From the flow configuration, it is evident that an increase in porosity of the medium assists the flow along the flow direction, causing the fluid velocity to increase due to its orientation through the porous medium.
Figures 8 and 9 demonstrate the
influence of thermal radiation parameter (
) on fluid
velocity and fluid temperature, respectively. It is evident from Figs. 8 and 9
that the thermal radiation leads to a decrease in the fluid velocity and fluid
temperature for both suction/injection. Physically, thermal radiation causes a
fall in the temperature of the fluid medium and thereby causes a fall in the kinetic
energy of the fluid particles. This results in a corresponding decrease in the
fluid velocity. Thus, Figs. 8 and 9 are in excellent agreement with the laws of
Physics.
Figure 8: Velocity profiles for various values of 
.
Figure 9: Temperature profiles for various values of .
Figure 10: Temperature profiles for various values of β.
Next, we show the influence of thermal radiation on velocity and temperature profiles.
Figure 10 depicts the variations in temperature profile for multiple values of heat generation/absorption parameter. Temperature profile decreases with increasing heat generation/absorption parameter. Physically, the presence of heat generation/absorption coefficient tends to reduce the fluid temperature. This causes the thermal buoyancy effects to decrease, resulting in a net reduction in the fluid velocity.
The influence of Schmidt
number (
) on concentration profiles are depicted in Figs.
11. It is noticed from Figs. 11 that concentration profiles decrease on
increasing the values of . The Schmidt number embodies the ratio of the
momentum to the mass diffusivity. The Schmidt number, therefore, quantifies the
relative effectiveness of momentum to mass transport by diffusion in the
concentration (species) boundary layers. As the Schmidt number increases, the
concentration decreases. This causes the concentration buoyancy effects to
reduce. This behavior is clear from Figs 11.
Figure 11: Concentration profiles for various values of 
.
Figure 12: Concentration profiles
for various values of .
Table 1: Variation of 
for various values of Magnetic parameter when 
, 
, 
, 
=0, 
.
|
|
Jaber works ( |
Present results |
|
1 |
2.00007 |
2.0115 |
|
3 |
2.56155 |
2.5612 |
|
5 |
3.0 |
3.0 |
Figure 12
illustrates the effect of permeability of porous medium (
) on concentration profile. It is perceived from
Fig. 12 that concentration profile decreases on increasing permeability
parameter.
The magnitude of 
, 
and 
are given in Tables 1-2. We have compared our
results with other works reported in the literature (Jaber, 2016) with an excellent
agreement, as shown in Table 1. Table 2 is prepared to show the value of , and different values of 
and . The magnitude of skin friction coefficient
increases when increased. In contrast, it has the reverse
effect on the local Nusselt number and the local Sherwood number. The magnitude
of skin friction coefficient and the local Nusselt number increases when is increased. The magnitude of the skin friction
coefficient decreases with 's increased value, increasing with 's increased value. The local Nusselt number and
the local Sherwood number increase with the increased value of , whereas it decreases with the increased value
of . It has also been found that the magnitude of
skin friction coefficient decreases with the increased value of porosity
parameter . In contrast, the local Nusselt number and the
local Sherwood number increase with the increased value of .
Table 1 compares the skin friction of Jaberworks with our present results and obtain a fair agreement.
Table2: depicts the variations in 
and 
for the
different values of physical parameters.
V. CONCLUSION
An investigation of heat and mass transfer and thermal radiation on an MHD boundary layer flow past a vertical porous stretching surface embedded in a porous medium with viscous dissipation and suction/injection is carried out. Closed-form expressions for velocity, temperature, concentration, the skin friction coefficient, the rate of heat transfer expressed as a Nusselt number, and the rate of mass transfer expressed as a Sherwood number are generated from our mathematical model. We observed
