1. Introduction
Over the past few decades, a series of discussions regarding convection has been tackled by researchers. Initially, the process of convection ensues once the heat is transmitted, hence exchanging the heat from the hot to the cool position. Convection can occur in three different ways: forced, free (natural), and mixed. The most favorable problem is in mixed convection, which combines forced and natural convection, manifesting most prominently in the movement of air boundary layers, solar collectors, heat exchangers, electronic devices, as well as nuclear reactors. When buoyancy forces in forced convection or forced flow in free convection are significant, these processes occur. [
1] pioneered researchers in exploring the problem towards a vertical surface. The author investigates the two-dimensional (2D) flow of a uniform stream past an impermeable vertical surface embedded in a saturated porous medium. Later, [
2] extended the knowledge into 2D stagnation flows, taking into account both cases of arbitrary wall temperature as well as arbitrary surface heat flux variations. Furthermore, [
3] took into account the effect of viscous dissipation, which affects both shear stress and heat transfer at the surface, in their research. Meanwhile, [
4] portrayed the issue in sphere coordinates, which took into account viscoelastic nanofluid as its nanoparticles. The findings show that the velocity and shear stress is reduced when the elasticity parameter is prominent in the flow. [
5] revealed their discovery of the problem of a semi-infinite upward flat plate for both opposing as well as assisting buoyancy-induced flow cases. The authors discovered that dual solutions occurred in the opposing flow case. More discussions can be found [
6,
7,
8,
9,
10] regarding mixed convection problems that have been intensively enfolded by numerous researchers in various surfaces and effects.
Due to its numerous applications in engineering and science, and technology, stagnation point flow represents among the most significant problems in fluid dynamics. The early idea of formulation analytically 2D stagnation point flow originated by [
11] and soon [
12] considered the problem of steady flow near the surface of a cylinder in a stream. Additionally, researchers added the problem over cylinder geometry. The issue of axisymmetric stagnation flow over an infinite stationary circular cylinder’s precise solution was originally revealed by [
13]. Compilation of study conducted by [
14,
15,
16] highlighted the stagnation point flow on steady and unsteady flow over moving and circular cylinders. Later, [
17] proceeded with his work and found out that lengthening the curvature’s surface helps in delaying the blow-up time. Furthermore, [
18] examines the lower stagnation point flow over a horizontal circular cylinder of brinkman-viscoelastic fluid. [
19] explore the use of homogeneous and heterogenous reaction over a permeable cylinder. The work on stagnation flow with the effect of slip continued to be investigated by [
20]. Extended problems that were covered by other researchers by taking into account different effects and surfaces can be found in this literature [
21,
22,
23,
24,
25] for more details.
Since the initial discovery of carbon nanotubes (CNTs) was made by the Russian researchers [
26], followed by [
27,
28] in 1991 and 1993, respectively. Due to their remarkable structure and exceptional physical properties, they have garnered significant amount of interest from researchers over the past few decades. The theoretical study has been published by numerous researchers with regards to different surfaces and effects. Through the use of RSM, [
29] investigate the marginal impact of thermal radiation on the enhancement of heat transfer associated with Darcy-Forchheimer (DF) flow in carbon nanotubes along a stretched rotating surface. [
30] examined magnetic effect over a vertically moving rotating disc. Demonstration on long chain hydrocarbons flows through carbon nanotubes (CNTs) was covered by [
31]. Meanwhile, [
32] discourse the attractive aspects of suction/injection and heat generation/absorption effects in the flow. It is noted that the suction effect and heat-absorption parameters increase the range of the solution. In addition, the rate of heat transfer for hybrid carbon nanotube was higher when compared with carbon nanotube and ordinary fluid.
However, the use of carbon nanotubes in biomedical, biotechnology and bio-engineering field are not a new dawn to scientists and researchers. Instead, the carbon nanotubes have the potential to be used as therapeutically beneficial nanostructures due to their extremely large surface area, rich electronic polyaromatic structure, low weight but excellent mechanical strength, also exceptional chemical and thermal stability [
33,
34,
35]. In some circumstances, the CNTs show a remarkable ability when we combined with external magnetic properties. They will be high functionalized in biomedical application especially as a targeted drug delivery agent [
36,
37]. Furthermore, another discernible advantage of using magnetic CNTs is to look similar to a needle shape, which allows easy penetration and movement through cell walls and bloods, respectively, since they have less resistance [
38]. The outstanding reviews made by [
39] covers on latest advance in the synthesis and application of magnetic carbon nanotubes.
Thermal radiation is a process by which energy is emitted directly from the radiated surface in the form of an electromagnetic wave in all direction. From the engineering and physical point of view, thermal radiation effect has a pivotal role in the flow of different liquid and heat transfer. Thermal radiation is found to be useful in engineering processes which require high operating temperature. These include the design of the nuclear plant, gas turbine, aircraft, space vehicle, reliable equipment, satellite etc. [
40] pioneered the study on aircraft structure. Series of literature with regards on aircraft structure can be seen in [
41,
42,
43]. However, the theoretical work on unsteady stagnation point nanofluid flow with convective boundary conditions was carried out by [
44]. Later, [
45] analyzed the mixed convection problem by taking into account the effect of Soret and Dufour. [
46] studied simultaneous impact of thermal radiation and thermophoresis near an inclined porous plate.
Motivated by the above aforementioned works, this paper decomposes the novelty in (i) utilizing hybrid carbon nanotubes as an essential component in improving the heat transfer effectiveness [
32], (ii) adding the hydromagnetic effect for this problem due to its high potential capability especially in drug and gene delivery, cell separation and manipulation in bio-medical field [
47,
48], (iii) analyzing the thermal radiation effect in this current study as it enhances the rate of heat transfer [
49], (iv) performing the stability analysis in order to identify the stability of the solution since non-unique solution is expected to exist. This contribution may benefit the scientists and academicians and give add-on value to their expertise.
2. Mathematical Framework
A vertical stretching/shrinking cylinder is passed by a 2D stagnation point flow that is submerged in an SWCNT-MWCNT/water hybrid nanofluid. The following presumptions apply to the illustration shown in
Figure 1:
A uniform magnetic field is employed in the radial direction. Assuming that we have the low Reynold number, Rm, then the magnetic field formed by induction is trivial compared to the applied magnetic field.
The fluid velocities are expressed by u and that are in axial coordinate, x and radial coordinate, r, accordingly.
The surface temperature, is higher than the ambient temperature, .
The cylinder with radius R is assumed to move in linear velocity, while the free stream velocity is in which L resembles the cylinder’s characteristic length.
The flow is characterized by the Buoyancy parameter, in which Gr denotes Grashoff number, and Re denotes Reynolds number. Here, the mixed convection regime is generally expressed as the range of , in which and are the upper and lower bounds of the mixed convection flow regime.
In this present problem, we are using various volumes of solid fraction SWCNT dispersed in a constantly 0.01 volume fraction of MWCNT/water to produce SWCNT-MWCNT/ water.
Table 1 lists the thermophysical characteristics of the hybrid nanofluid.
The related governing equation to model the fluid flow for our problem in presence of magnetic field and thermal radiation is as follows: [
51,
52]
with assumptions of boundary conditions (BCs) [
53,
54],
where
u and
denote the component of velocity for
x and
r axes, accordingly. Meanwhile,
and
T referring to radiative heat flux and temperature, respectively. Meanwhile,
and
are kinematic viscosity and magnetic field. Furthermore, the term
in Equation (3) may be derived by means of the idea from Rosseland approximation for an optically thick layer [
55] which leads to,
The
denotes Stefan-Boltzmann constant while
refers to the mean absorption coefficient. In an intense absorption flow, it is claimed that Equation (5) is effective at a position far from the boundary layer surface. Moreover, the term
may be understood as a linear function of temperature
T with the assumption of the temperature distinction within the flow is very minute. Employing Taylor’s expansion of
around
which denotes the ambient temperature while ignoring the higher order terms, we obtain,
Hence, the right-hand side of Equation (3) can be abridged as,
where,
is thermal diffusivity and
is radiation parameter [
56].
Table 2 condenses all the thermophysical properties that correspond to these parameters.
From
Table 2, the description for hybrid nanofluids, nanofluids as well as fluids are denoted by
hnf,
nf and
f, respectively. To present the two distinct nanoparticles’ solid volume fractions,
is for SWCNT meanwhile,
for MWCNT. Apart from that, s1 is for SWCNT, while s2 is for MWCNT nanoparticles in which S = s1 + s2.
The following similarity variables are added to the model in order to convert the governing equations above into ordinary differential equations [
59,
60],
where
and
which identically satisfied Equation (1). Hence, the new momentum and energy equations which similar to the become,
along the BCs,
where
is the velocity ratio parameter. Moreover, for case
> 0 (cylinder),
corresponds to the shrinking flow case, and
correlates to the stretching flow case, meanwhile
for static flow. Moreover, the related parameter
refers to curvature,
denotes magnetic,
resembles mixed convection in which
< 0 for opposing flow,
for assisting flow as well as
for forced convection flow, whereas
Pr denotes Prandtl number. These parameters may be written explicitly as,
where,
and
.
The physical quantities interest in this research are skin friction coefficient,
and local Nusselt number,
which are expressed as,
The reduction to ODE for Equation (13) are obtained by using the similarities solution in (8). Hence, the simplified expressions are;
3. Solutions on Stability Analysis
In dealing with the boundary layer flow problem, we might deal with possibility of having zero, unique, multiple, or even ghost solution(s) [
61,
62,
63,
64] under some boundary conditions. Commonly, the solution that initially satisfies the far field boundary condition is designated as the first solution for case of multiple solutions (non-unique). Numerous researchers [
65,
66,
67,
68] showed that the first solution is stable and reliable solution. However, [
69] testified that in certain existence problems, second solution is stable. Due to this, a stability analysis is important to validate the reliability of the solutions. With respect to [
70], a temporal stability analysis is discovered. Three vital steps are performed to identify the solutions’ stability. First, by inaugurating a new dimensionless parameter,
with a new set of similarity variables in terms of
and
. Second, we implement the linear eigenvalues equations. Third, to attain the range of possible eigenvalues, the BCs need to be relaxed.
Introduce a new dimensionless time variable,
and the similarity transformations with respect to
are as follows [
71]
In this context, the term t in the expression is the time since the disturbance may decay or grow with time.
By differentiating Equation (15), Equations (2) and (3) are reduced to,
subject to BCs,
According to [
72], eigenvalues may be employed as a method of solving the ODEs. Given a set of eigenvalues equations;
in which
as well as
are set as the small relative with respect to
and
, accordingly. Meanwhile,
refers to an unknown parameter of eigenvalue. The eigenvalues problem with respect to Equations (16) and (17) yields an infinite inequality given by
This indicates any early decay
or growth of disruptions
To investigate the solution’s stability, a small perturbation also needs to be taken into consideration in which
and
. Moreover, upon differentiating Equation (19) and equating it to Equations (16) and (17) with
yields the linearized equations given below:
together with;
The possible eigenvalues range may be obtained by relaxing one of the BCs, which is either
or
[
73]. In this problem, the condition that we relax is
and the linear eigenvalues problems (20)–(22) are unfold as
. Here, numerical findings for stability were calculated by placing an algorithm in bvp4c MATLAB.
5. Analysis of Results
The intention of this section is to discuss on the comprehension of the highlighted parameters’ effect in the model, for instance, the SWCNT/MWCNT concentration volume (
, thermal radiation (
Nr), magnetic (
M), curvature parameter,
and mixed convection parameter,
towards
and our quantities interest of study;
the local Nusselt number
including the temperature,
profiles and velocity,
are ingrained in the form of graph from
Figure 2,
Figure 3,
Figure 4,
Figure 5,
Figure 6,
Figure 7,
Figure 8,
Figure 9,
Figure 10,
Figure 11,
Figure 12,
Figure 13,
Figure 14,
Figure 15,
Figure 16,
Figure 17,
Figure 18,
Figure 19,
Figure 20,
Figure 21,
Figure 22,
Figure 23,
Figure 24,
Figure 25 and
Figure 26. Through our graphical computation, the interrelation between the effects with respect to the physical quantities is apparent. In addition, the parameters’ identification that affects the parting of the boundary layer flow may be observed via
Figure 2,
Figure 3,
Figure 4,
Figure 5,
Figure 6,
Figure 7,
Figure 8 and
Figure 9 as well as
Figure 16. A comparison of the values of
when
with the previous studies are presented in
Table 3. The data are clearly discernible to be in good agreement, despite the fact that the solutions provided in earlier works are solved in various approaches. As a result, we are confident that the method we picked is appropriate and that the model we developed, the numerical calculations, and the results below are accurate. It is important to note that this confirms the novelty and originality of the model and the findings presented in this research.
In this study, two parameters are set to be fixed,
and
. While the other governing parameters such as
,
,
and
vary for further computation. Graph for local skin friction,
and local Nusselt number,
are highlighted in
Figure 10,
Figure 11,
Figure 12,
Figure 13,
Figure 14,
Figure 15 and
Figure 17. Profiles for velocity,
and temperature,
are pictorial in
Figure 18,
Figure 19,
Figure 20,
Figure 21,
Figure 22,
Figure 23,
Figure 24,
Figure 25 and
Figure 26. Since the thermophysical properties equations relate with the different volume of nanoparticles,
hence the range for the thermophysical properties values are:
and
.
Different concentration of
and
are selected to illustrate three behavior of the fluid, for instance,
(viscous fluid),
(SWCNT/water nanofluid) as well as
(SWCNT-MWCNT/water hybrid nanofluid) are plotted in
Figure 2 and
Figure 3. The graphs clearly show that the range of solution widens as the volume concentration of nanoparticles is added. Moreover, the hybrid nanofluid assist in delaying the detachment of the first and second solutions for
and
, when
by means that the opposite movement between the hybrid carbon nanotubes and the plate, respectively where
,
and
.
Figure 4 and
Figure 5 exhibit the
and
variation with respect to mixed convection parameter,
for multiple values of
M. It is noticed that the range of solution deteriorating as
M is prominent in the flow for both graphs. The
denotes the detachment between the first and second solution, which may be seemed in the opposing flow
. Furthermore, the increment of
M fastens the separation of boundary layer flow, i.e.,
and
This figure was supported by
Figure 10 and
Figure 11, where the value of
and
were reduced as the strength of
M increased. As a result of the increase of
M, there will be more resistance and a drag force known as the Lorentz force. In turn, this force might lessen heat transfer and skin friction by opposing and slowing the hybrid nanofluid’s velocity. On top of that, it is important to note that the increment in concentration of hybrid nanofluid increased the value of
and
. Basically, the strong interaction between the hydromagnetic and the concentrated nanoparticles in the fluid led the fluid to be gradually dragged in the direction of the nanoparticle’s motion, consequently increasing both
and
.
Figure 6 and
Figure 7 illustrate the influence of different values of
with the velocity parameter
ε. It can be seen that a dual solution occurs when the governing parameter is inserted. The dual solution appears when the region is shrunk
. The hybrid nanofluid seems to accelerate the boundary layer’s separation at the flat plat
quicker in comparison to when
and
. Moreover, the presence of
causes the widening of the range solution and increasing of
and
. Additionally, the increment of curvature parameter by means decreasing the radius of the cylinder. Therefore, more particles can fit into the cylinder surface area offering greater resistance to the particles of the fluid. Thus, boost up the heat transfer. These results are supported by
Figure 12 and
Figure 13. Consequently, come to a consensus with the aforementioned [
22] that the performance of heat transfer is enhanced provided that the value of
increases.
Variation
and
with respect to
for distinct values of
is given in
Figure 8 and
Figure 9. The duality of the solution exists at the buoyancy opposing flow
in which
as well as
and unique solution appears at buoyancy assisting flow,
. The additional on nanoparticles help in dispensing the energy in the form of heat [
58]. The collision between the hybrid fluid nanoparticles caused by the increase of volume fraction with respect to SWCNT also results in the increment of the thermophoretic force and heat conduction caused by shear; leading to increasing of
and
.
The impact of
over
and
can be seen in
Figure 12 and
Figure 13. This control parameter indicates reliable support, as reported in
Figure 6 and
Figure 7, in which the findings increased when
is prominent. Obviously, the larger
causes the reduction of the radius of the cylinder and less resistive force between the fluid and surface. Furthermore, increasing the friction between the surface shear stress, causing the thickness of the boundary layer. Here, the drag friction produces energy in the form of heat, leading to an increase in
.
Figure 14 and
Figure 15 explain the improvement in
for
and
. The
and
improved with the increment of
in the flow. Note that the edge of the dotted line of
, when
represents the SWCNT-water. It shows that the value of
which means lower than the value of MWCNT-water as
increase. Surprisingly,
and
increased drastically when
increase with the increase of
. When the nanoparticles of SWCNT be more concentrated, the hybrid nanofluid’s velocity will be improved, raising the shear stress and therefore enhancing the skin friction coefficient. This phenomenon happens due to the attractive force called Van der Waals forces that occurred between the molecules, resulting in the changes in the thermophoretic force. As we can see the increase of nanoparticle volume fraction for
is not giving much significant on the reduced Nusselt number. To the best author’s view, this is due to the influence of the
Nr parameter in the local Nusselt number which explained the insignificant result that occurs when the volume of nanoparticles,
increased.
The effect of several values of thermal radiation,
Nr on
are portrayed in
Figure 16. Duality of the solution exists in the shrinking region,
between the range
a unique solution occurs at
and no solution was discovered at
. It may also be observed from the graph that the critical value,
, such as
,
and
. The large
Nr impede the separation of the boundary layer. Thermal radiation is the process where heat is transferred by electromagnetic waves without the presence of any constituents. It is noticed from
Figure 17 that the
reduces with the increment in radiation parameter
. The mark of
in the flow district is to reduce the temperature substantially. The increment in
implies the release of heat energy from the flow region and which improve the random movement of the nanoparticles. However, in this problem, the lowest value of
is said enough to have a high ability in heat transfer.
Profiles on velocity,
, temperature,
and concentration,
are given in
Figure 18,
Figure 19,
Figure 20,
Figure 21,
Figure 22,
Figure 23,
Figure 24,
Figure 25 and
Figure 26 with respect to the essential controlled parameter aimed in this research which is
Nr, as well as
For the first and second solutions, the profiles show features that are opposing. Note that, we only present the result for the local temperature profile due to the fact that the parameter
Nr only exist in Equation (10). Every profile published satisfies the BCs asymptotically (11). It is intriguing that the first solution reaches zero faster than the second solution. Practically, the boundary layer thickness of the second solution is thicker compared to the first solution.
The temporal stability analysis approach introduced by [
73] is via solving the linearized eigenvalue problem in (19)–(22) with introductory of new BC,
. To define the two solutions’ stability, the
value is vital. Numerical values achieved are shown in
Table 4 and
Table 5. This table demonstrates that the second solution exhibits the positive values of
, meanwhile the first solution illustrates the positive values with respect to
. The smallest eigenvalues
against
were plotted in
Figure 27. The results from the table above, where we saw stable solutions for both systems, were ultimately supported by this figure. As opposed to earlier literature, it is worth noting that some of the positive values of
in the second solution for our problem are considered to be stable solution. When there are non-unique solutions, it is crucial to identify stable solutions in order to accurately anticipate the flow behaviour.