The present analysis addressed the thermal slip and radiation effects on electrically conducting flow phenomena of the convective heat transfer along the vertical symmetric heated surface with porous medium and magnetohydrodynamics impacts. The nonlinear-coupled PDE equations for the above fluid flow mechanism are formulated and then converted into non-similar formulations by applying an appropriate and well-known similarity transformation for integration. The final non-similar forms are integrated numerically by employing the Keller box scheme method. The transformed algebraic equations are plotted graphically and numerically on the MATLAB software package. The behavior of physical quantities such as velocity graph, magnetic field graph, and temperature graph along with their slopes, that is, skin friction, magnetic intensity, and transfer of heat under the effect of different parameters included in the flow model, is discussed.
Figure 2a–c present the influence of thermalslip number with various values
=
,
,
and
along the thermally and magnetized surface. To check the behavior of physical properties, the fluid velocity, magnetic field, and temperature of the fluid by keeping some constant variables havebeen deduced. In
Figure 2a, it is obtained that velocity
is increased at lower
=
but the smaller quantity of
velocity is obtained at large
=
. It is also noted that suitable variations are obtained at each value of
with a certain height and then approach asymptotically to the given boundary condition. Due to slip flow, the frictional resistance between the viscous fluid and the surface is eliminated, and the fluid velocity booststhe heat transfer and skin friction along the surface. It can be predicted that an increase in skin friction corresponds to a thinning of the velocity boundary layer. In
Figure 2b, the magnetic effects in the fluid are maximum at larger
=
and a smaller quantity is observed at lower
=
. The magnetic profile of the fluid obtained in suitable variation at each value of the
. It occurs because magnetic diffusion is reduced by increasing the magneticPrandtl number, which is responsible for the above-said phenomena. From
Figure 2c, it is concluded that the slip temperature is maximum for a small value of
=
but smaller quantity is explored at large
=
in a prominent way. The prominent variations in temperature
with prominent slip effects in the presence of
=
.
Figure 3a–c demonstrates the velocity profile, magnetic profile, and temperature
for various quantities of
=
and
along magnetized heated geometry. In
Figure 3a, the velocity graph is maximum at a smaller value of
and minimum value at larger
=
with prominent amplitude and then approachesits given condition asymptotically. Increasing
means the medium is more porous, and the fluid permeability in the porous layer is increased and thus yields resistance in the fluid flow. In
Figure 3b, it is concluded that the magnetic profile is increased at a higher value of
=
and is minimum at a smaller value of
=
with suitable variations. It occurs because the magnetic force parameter is the ratio of magnetic energy to kinetic energy, so with the increase in the magnetic force parameter
, the magnetic energy is increased, while the kinetic energy is reduced. In
Figure 3c, it is examined that the temperature profile is increased at a higher value of
=
but decreased at a smaller value of
=
with better thermal slip. In
Figure 4a, the maximum amplitude in velocity is obtained at larger
R and minimum velocity is obtained at lower
R in the presence of magnetic force and strong magnetic Prandtl number. As the magnetic Prandtl number increases, the viscosity of fluid increases, and fluid becomes thicker and, consequently, the boundary layer thickness decreases. The prominent variations are obtained in the magnetic profile for each value of
R in
Figure 4b. The prominent thermalslip response is observed in temperature for each
R in
Figure 4c. An excellent and favorable slip phenomenonis observed in the temperature graph and presentssuitable behavior. The suitable variations are obtained in the velocity profile graph at each value of the
R and then approached asymptotically to the given boundary condition.
Figure 5a–c are presented the influence of magnetic force number M with various values
=
and
along the heated surface. In
Figure 5a, it is noticed that
velocity is increased at a lower quantity of
=
but a smaller quantity of
velocity is noticed at large
=
with suitable amplitude in the presence of thermal slip. This result was expected because an increase in
means an enhancement in the Lorentz forces, which opposes the flow and velocity of the fluid decreases. In
Figure 5b, the magnetic profile is increased at the lower magnetic force parameter
=
but the smaller quantity of magnetic profile is obtained at large
=
with prominent variations. In
Figure 5c, it is concluded that the magneticforce effects in the fluid temperature aremaximum at
=
and smaller quantity is found at large
=
with excellent thermalslip for the given phenomenon and approached to the given boundary conditions asymptotically. This behavior of the aforementioned variables supports the physical hypothesis that as the magnetic field grows stronger, more resistance is formed inside the fluid flow domain, which reduces the velocity distribution and improves the temperature profile.
Figure 6a–c are presented against the various values of the Prandtl parameter
=
and
along the magnetic plate. In
Figure 6a, it is observed that the velocity
is maximum with suitable amplitude response at a smaller value of Prandtl parameter
=
but the smaller quantity of
is obtained at higher
=
. It is also noted that
profile shows suitable behavior along the surface of geometrical shape in the presence of mixed convection. In
Figure 6b, the magnetic field profile showed suitable variations for each value of
. From
Figure 6c, it is concluded that the temperature profile along the surface of the given shape is maximum at a small value of
=
while the temperature profile is minimum at a large value
=
. This phenomenon is expected because an increase in
tends to increase density variation with temperature, which enhances the buoyancy force. The thermal boundary layer thickness is reduced due to an increase in
. The prominent thermal slip with suitable variations isobtained in the temperature graph at each value of
. Due to the fluid’s poor thermal conductivity and decreased heat transfer as
enhanced, the temperature of the fluid flow domain was reduced. In
Figure 7a, it is noted that the velocity profile shows suitable amplitude effects along the heated plate in the presence of thermal slip and radiations. In
Figure 7b, the magnetic profile is increased at lower mixedconvective parameter
=
but the smaller quantity of magnetic profile is obtained at large
=
with prominent variations. In
Figure 7c, it is noted that the temperature profile with excellent thermalslip is obtained for the given phenomenon and approachesthe given boundary conditions asymptotically. Physically, it was expected because larger values of
correspond to stronger buoyancy forces, which leads to an increase the acceleration of fluid flow. Due to conducting phenomena, the magnetic effects are strongly observed exactly at the surface but far from the surface are zero for each value.
Table 1 presents the comparison of skin friction for three values of magnetic Prandtl number
=
at the leading edge of the magnetized heated surface. In
Table 1, Mehmood et al. [
20] explored skin friction along a magnetized wedge, Ilyas et al. [
21] obtained skin friction along a magnetized cone, but prominent results of skin friction are deduced in the current analysis along the magnetized heated surface. The values of skin friction are approximately matched with the previous results by using lower magnetic force
in the presence of porous medium and found suitable agreement in the present numerical results of skin friction. It can be seen that the maximum skin friction is obtained at lower
but lower skin friction is computed at maximum
fromphysical point of view. This result was expected because an increase in
means an enhancement in the Lorentz forces, which opposes the flow and velocity of the fluid decreases. It was also expected because as the magnetic Prandtl number increases, the viscosity of fluid increases, and fluid becomes thicker, and consequently, the boundary layer thickness decreases. From
Table 2, it is concluded that the skin friction
is increased at smaller
=
but the smaller quantity of skin friction is obtained at higher
=
with free/force convective parameter
=
and magnetic force
. With the increase in value of
, the magnetic field becomes stronger along the surface, which is a clear indication that the Lorentz force is more effective in this case. It is noted that the magnetic intensity
is maximum at small
=
but the smaller quantity of magnetic intensity is examined at large
=
under the impact of porous parameter
=
. It is also depicted that the heat transfer is increased at lower
=
but the smaller quantity of heat transfer is depicted at large
under the impact of maximum
=
. Physically, it is possible because thermal conductivity decreases as
increases, which has a lower magnitude of frictional forces between the viscous layers.
Table 3 is indicated the influence of porous
numbers with diverse
=
and
along a vertical plate to check the behavior of physical properties the
and for
of the fluid while some parameters are fixed
=
,
M =
and
=
. From
Table 3, it is presumed that the
is maximum at a smaller value of
=
but the smaller quantity of skin friction is depicted at a higher quantity of
=
in the presence of temperatureslip number
=
. Increasing
means the medium is more porous, and the fluid permeability in the porous layer is increased and thus yields resistance in the fluid flow. However, due to the strong buoyancy number
which acts like a pressure gradient and dominatesover the resistance, skin friction is increased, and slight changes in heatand magnetic intensityare noted. It is depicted that the magnetic intensity is increased at smaller
=
but the smaller quantity of magnetic intensity is explored at higher
=
in the presence of
=
. It is also mentioned that the heat transfer
is maximum at a lower value of
=
but the small quantity of heat transfer is noticed at larger
=
in the presence of thermal slip parameter
=
. In
Table 4, the skin friction is increased at larger
R but reduced at lower
R with temperatureslip and porous effects. The magnetic intensity is increased at higher
R but decreased at lower
R numerically, but the heat transfer is increased at
R in the presence of a strong magnetic Prandtl number. The reason behind this is that increasing the value of
is equivalent to decreasing magnetic diffusivity, and consequently, the strength of the magnetic field becomes loose. Physically, it is accurate to say that adding thermal radiation to a flow model raises the temperature of the fluid flow domain.
Table 5 presents the comparison of heat transfer with Hirschhorn et al. [
32], which verified the
Table 2 results due to a strong magnetic field. The magnetized surface insulates the heat and reduces the excessive heating along the surface in the presence of a magnetic Prandtl number. In addition, the skin friction is maximum at lower thermal slip due to minimum friction resistance between the surface and fluid’s layers. So, the given results are valid and in suitable agreement froma physical point of view.