1. Introduction
Mixed convection flows or a combination of forced and free convections exists in numerous transport operations, both naturally occurring and in engineering applications. Such applications for example, include heat exchangers, solar collectors, nuclear reactors, atmospheric boundary layer flow, nanotechnology, electronic apparatus, etc. These operations occur during the effects of buoyancy forces in forced convections or the effects of forced flow in free convections become substantial. Over the past several decades, most research in mixed convection flow analysis has emphasised the occurrence of dual solutions for a particular range of the buoyancy (mixed convection) parameter in the opposing flow region, such as in the research by Ramachandran et al. [
1], Merkin and Mahmood [
2], Devi et al. [
3] and Lok et al. [
4]. In contrast to [
1,
2,
3,
4], Ridha and Curie [
5] continued the study by Merkin and Mahmood [
2] by establishing the existence of dual solutions in both the opposing and assisting flow regions. Furthermore, by implementing a stability analysis of the dual solutions for mixed convection flow in a saturated porous medium, Merkin [
6] demonstrated that the upper branch of the solutions is stable whereas the lower branch shows instability. Accordingly, various other researches have also stated the occurrence of dual solutions in the mixed convection flow in different configurations, namely by Roşca et al. [
7], Rahman et al. [
8] and recently by Abbasbandy et al. [
9]. An inclusive account of the theoretical research prior to 1987 for both laminar and turbulent mixed convection boundary layer flows may be found in the books by Gebhart et al. [
10], Schlichting and Gersten [
11], Pop and Ingham [
12] and Bejan [
13], for example.
The innovative idea of nanofluids was first brought up by Choi et al. [
14] in 1995, when the authors suggested a path for exceeding the performance of heat transfer fluids which were currently available. An extraordinary enhancement in the thermal properties of base fluids may be achieved just by utilizing a minimal amount of nanoparticles scattered uniformly and suspended stably in a base fluid. Nanofluids, as colloidal mixtures of nanoparticles (1–100 nm) along with a base liquid (nanoparticle fluid suspensions) are known, provide access to a new class of nanotechnology-based heat transfer media (Das et al. [
15]). Numerous techniques and methodologies, such as rising either the heat transfer surface or the heat transfer coefficient between the fluid and the surface that allows high heat transfer rates in a small volume, may be utilized to promote heat transfer. Notwithstanding, cooling turns out to be one of the most critical technical challenges faced by numerous and diverse industries, including microelectronics, transportation, solid-state lighting, and manufacturing. The addition of micrometre- or millimetre-sized solid metal or metal oxide particles to base fluids produces an increase in the thermal conductivity of the resultant fluids. On the other hand, apart from being applied in the field of heat transfer, nanofluids (nanometre-sized particles in a fluid) may also be synthesised for unique magnetic, electrical, chemical, and biological applications (see Manca et al. [
16]). Nanoparticles are produced from various materials such as copper (Cu), alumina (Al
2O
3), titania (TiO
2), copper oxide (CuO) as well as silver (Ag) (see Oztop and Abu-Nada [
17]). References on nanofluids are mentioned in the books written by Das et al. [
15], Nield and Bejan [
18], Minkowycz et al. [
19] and Shenoy et al. [
20], and also in the review papers written by Buongiorno et al. [
21], Kakaç and Pramuanjaroenkij [
22], Fan and Wang [
23], Mahian et al. [
24], Sheikholeslami and Ganji [
25], Groşan et al. [
26], Myers et al. [
27], etc. These review papers elaborate specifically on the production of nanofluids, the theoretical and experimental exploration of the thermal conductivity and viscosity of nanofluids, as well as the work conducted on the convective transport of nanofluids.
Interestingly, many studies investigating the boundary layer problem of mixed convection flow in a nanofluid are reported in the literature. Tamim et al. [
28] examined the effects of the magnetic field, suction/injection and solid volume fraction of nanoparticles on mixed convection about the stagnation-point flow of a nanofluid. On the other hand, Subhashini et al. [
29] investigated the mixed convection flow about the stagnation-point region over an exponentially stretching/shrinking sheet in a nanofluid for both suction and injection cases. Later, Mustafa et al. [
30] extended the study conducted by Tamim et al. [
28] in consideration of the combined effects of viscous dissipation and the magnetic field by gaining a unique solution for assisting and opposing flow cases. Recently, Ibrahim et al. [
31], Mabood et al. [
32] and Othman et al. [
33], similarly investigated the problem of mixed convection boundary layer flow in nanofluids under different physical conditions.
The impact of thermal radiation on heat transfer becomes increasingly important in the design of advanced energy conversion systems operating at high temperature. Moreover, thermal radiation has applications in numerous technological problems such as combustion, nuclear reactor safety, solar collectors, furnace design and many others (see Ozisik [
34]). Furthermore, the study of thermal radiation on flow and heat transfer characteristics in a nanofluid have attracted immense interest because nanofluids have different properties than those found in either the particles or the base fluid. Given this fact, many researchers have explored the impact of thermal radiation on flow and heat transfer in a nanofluid along with other various aspects. An important analysis by Hady et al. [
35] studied the boundary layer viscous flow and heat transfer characteristics of a nanofluid over a nonlinearly stretching sheet in the presence of thermal radiation in a single-phase model. In a separate study, Ibrahim and Shankar [
36] investigated the influences of thermal radiation, magnetic fields and slip boundary conditions on boundary layer flow and heat transfer past a permeable stretching sheet in a nanofluid. Notwithstanding, Haq et al. [
37] discussed the combined effects of thermal radiation, magnetohydrodynamic (MHD), velocity and thermal slip on the boundary layer stagnation-point flow of nanofluid and the effects over a stretching sheet. More recently, Daniel et al. [
38] investigated the effects of thermal radiation, magnetic fields, electrical fields, Ohmic dissipation, thermal and concentration stratifications on the flow and heat transfer of electrically conducting nanofluid past a permeable stretching sheet. In another recent study, Sreedevi et al. [
39] analysed the effect of thermal radiation, magnetic field and the chemical reaction on flow, heat and mass transfer of nanofluid over a linear and nonlinear stretching sheet saturated by the porous medium. Accordingly, several other studies have been undertaken on mixed convection boundary layer flow in nanofluids in the presence of thermal radiation, including works by Yazdi et al. [
40], Pal and Mandal [
41] and Ayub et al. [
42].
The heat source/sink effect in addition to the thermal radiation effect plays a vital role in governing the heat transfer in industrial operations in which the attributes of the output are dependent on the factors of heat control. Accordingly, many researchers have studied the impacts of a heat source/sink on the boundary layer flow and heat transfer of nanofluids along with different aspects. Rana and Bhargava [
43] numerically investigated the impact of the various types of nanoparticles on mixed convection flow of nanofluid along the vertical plate with a heat source/sink. Furthermore, Pal et al. [
44] analysed the combined impacts of internal heat generation/absorption, thermal radiation and suction/injection on mixed convection stagnation point flow of nanofluids over a stretching/shrinking sheet in a porous medium. In addition, Pal and Mandal [
45] discussed the impacts of microrotation and nanoparticles on boundary layer flow in nanofluids in the occurrence of non-uniform heat source/sink, suction, thermal radiation and magnetic fields. In another paper, Mondal et al. [
46] considered the influence of heat generation/absorption and thermal radiation on hydromagnetic three-dimensional mixed convection flow of nanofluid over a vertical stretching surface. Sharma and Gupta [
47] further investigated the effect of heat generation/absorption, MHD, thermal radiation, viscous dissipation on flow and heat transfer of Jeffrey nanofluids.
Interestingly, previous studies did not include the combined effects of thermal radiation, heat source/sink and suction on mixed convection flow of a nanofluid. Therefore, the primary aim of this article is to examine the impact of thermal radiation, heat source/sink and suction on mixed convection stagnation point flow over a stretching/shrinking sheet in a nanofluid, by applying a mathematical nanofluid model suggested by Tiwari and Das [
48]. In our opinion, the problem is relatively new, novel with no such articles reported at this stage in the literature. Suitable similarity transformations are employed to transform nonlinear partial differential equations into nonlinear ordinary differential equations. The equations are then solved numerically with the assistance of the bvp4c programme in MATLAB, and the results are graphically plotted and displayed in tables. The results from the study indicates that dual solutions exist for a particular range of parameters, namely, the mixed convection parameter, solid volume fraction of nanoparticles, thermal radiation parameter, heat source/sink parameter, suction parameter and stretching/shrinking parameter. Further, it is useful to mention, that the stability analysis of the dual solutions is conducted to investigate which solution is stable.
2. Mathematical Formulation
This study considers the two-dimensional steady mixed convection flow of a viscous and incompressible nanofluid near the stagnation-point past a permeable vertical stretching/shrinking surface with the velocity
and free stream velocity
, as illustrated in
Figure 1, where
and
denote the Cartesian coordinates evaluated along the surface of the stretching/shrinking sheet and normal to it, respectively. The fluid consists of a water-based nanofluid comprising three distinct types of nanoparticles which are copper (Cu), alumina (Al
2O
3) and titania (TiO
2). The thermophysical properties of water (the base fluid) and nanoparticles are shown in
Table 1. These thermophysical properties will be used in the numerical computations of this study. Also, it is assumed that the flow was subjected to the combined impact of thermal radiation and a heat source/sink. Another assumption made are such that the temperature of the stretching/shrinking sheet,
, and the temperature of the ambient nanofluid adopt a constant value
. Furthermore, it is also assumed that the water-based fluid and the nanoparticles are in thermal equilibrium and that no slip exists among them. The mathematical nanofluid model suggested by Tiwari and Das [
48] is applied in this case. It should be mentioned that this nanofluid model is a single-phase approach where the nanoparticles are assumed to have a uniform shape and size, and the interactions between nanoparticles and surrounding fluid are also neglected (Pang et al. [
49], Ebrahimi et al. [
50] and Sheremet et al. [
51]). This assumption is practical when the base fluid is easily fluidized, so it can be considered to behave as a single fluid, hence it applies to the justification of using single phase model in this study.
By taking into considerations of these assumptions together with the Boussinesq and the boundary layer approximations, the governing boundary layer equations of continuity, momentum and thermal energy in the existence of thermal radiation and the heat source or sink, are given as shown below:
and the associated boundary conditions to present the flow are:
where
and
represent the velocity elements along the
x and
y directions, respectively;
stands for the acceleration caused by gravity,
denotes the temperature of the nanofluid,
denotes the heat source/sink coefficient, with
corresponding to the heat source and
corresponding to the heat sink. Further,
represents the wall mass flux, with
corresponding to the suction. Moreover,
represents the dynamic viscosity of the nanofluid,
refers to the density of the nanofluid,
denotes the thermal expansion coefficient of the nanofluid as described in the Brinkman’s model,
denotes the heat capacitance of the nanofluid,
denotes the thermal diffusivity of the nanofluid,
represents the radiation heat flux, and lastly,
reflects the kinematic viscosity of the nanofluid. The relations of
,
,
,
,
and
are described in the following equations (see Oztop and Abu-Nada [
17]):
where
refers to the dynamic viscosity of the base fluid,
denotes the solid volume fraction of the nanoparticles,
and
represent the density of the base fluid and the density of the solid nanoparticle, respectively,
represents the thermal conductivity of the nanofluid, as approximated by the Maxwell-Garnett’s model, the subscript ‘
f’ represents the base fluid, and lastly, ‘
s’ reflects the solid nanoparticle.
Meanwhile, upon employing the Rosseland’s approximation, the radiation heat flux,
is given by Zheng [
52], which adopts the following form:
where
represents the Stefan-Boltzmann constant and
is the Rosseland mean spectral absorption coefficient. Furthermore, it is assumed that the temperature difference between the flow is such that
can be expanded using Taylor’s series as a linear combination of the temperature. Next, after the expansion of
into the Taylor’s series for
, the approximation was obtained by omitting the higher order terms, obtaining
. Therefore, upon substituting Equations (5) and (6) into Equation (3), the following equation is obtained:
To determine the similar forms of the Equations (1), (2) and (7), with boundary conditions (4), the terms are defined;
,
,
and
in the following form:
Here, a and b are constants, s denotes the suction parameter and represents the constant characteristic temperature, with indicating the cooled surface (opposing flow) while signifies the heated surface (assisting flow).
Furthermore, the governing Equations (1), (2) and (7) together with the boundary conditions (4) have been transformed into ordinary differential equations by the dimensionless functions
u,
v and
, in relation to the suitable similarity variable
as follows:
Note that denotes the dimensionless stream function, be the dimensionless velocity profile, represents the dimensionless temperature profile and the prime indicates the differentiation with respect to .
Equation (1) is therefore satisfied identically with the given similarity transformation (9). After substituting similarity transformation (9) into Equations (2) and (7), we obtain the following coupled nonlinear ordinary differential equations:
while the boundary conditions (4) adopt the new form:
where Pr denotes the Prandtl number,
represents the mixed convection parameter with the case of
corresponds to the opposing flow, whereas
corresponds to the assisting flow. Moreover,
Nr denotes the thermal radiation parameter,
K represents the heat source/sink parameter with the case
refers to the heat source and
refers to the heat sink. Further
c denotes the stretching/shrinking parameter, with
for a stretching sheet and
for a shrinking sheet, and
is the constant mass flux parameter, with
for suction and
for injection or withdrawal of the fluid, The parameters
,
λ,
Nr,
K,
s and
c can be expressed in the following equations as:
Here, the local Grashof number
and the local Reynolds number
are given by:
The interested physical quantities are the skin friction coefficient
and the local Nusselt number
which are expressed by:
where the shear stress at wall
and the constant surface heat flux
are expressed as:
Substituting (9) into (16) and using (15), the following is obtained:
3. Stability Analysis
The numerical results of the nonlinear ordinary differential equations given in Equations (10) and (11) together with the boundary conditions in Equation (12) indicates that for a particular range of the mixed convection parameter
λ, there exist dual solutions (upper and lower branch solutions) for the various values of the selected governing parameters. Therefore, to validate which solution is in the stable flow, the stability of the dual solutions is tested by accommodating the stability analysis shown in Merkin [
53]. To perform this, an unsteady form of the problem was considered. Equation (1) was retained, while Equations (2) and (7) were substituted by the following:
where
t represents the time. Analogous to the similarity transformation (9), the following new dimensionless functions
u,
v and
θ have been introduced in conjunction to the similarity variable
η which is the same as defined in (9), and the new similarity variable
τ as follows:
Of note, with variables
u and
v given in the above, the equation of continuity (1) is identically satisfied. Next, after substituting the new similarity transformation (20) into Equations (18) and (19), we obtained the following equations:
and were subjected to the boundary conditions:
Next, to study the stability of the dual solutions, small disturbances of the growth (or decay) rate
γ or better known as the unknown eigenvalue parameter, are taken in the form (see Weidman et al. [
54]):
where
and
satisfied the problem (10)–(12). Besides,
,
and all of the respective derivatives were assumed to be smaller when compared to
,
and its derivatives. By means of using (24), hence Equations (21) and (22) can be given as:
together with the following boundary conditions:
As proposed by Weidman et al. [
54], the initial growth or decay of the solutions (24) is identified, by setting
, thus, giving
and
. In this respect, the following linear eigenvalue problem was solved:
with the boundary conditions given by:
Indeed, it should be stated at this point, that the solutions
and
were determined from the problem depicted in Equations (10)–(12). Upon obtaining the results,
and
were again applied to Equations (28) and (29), and the linear eigenvalue problem (28)–(30) were solved. Harris et al. [
55] proposed to relax a suitable boundary condition on
or
to determine a better range of
. In the current study, the condition
is relaxed and for a fixed value of
γ, the linear eigenvalue problem (28)–(30) are solved, together with the new boundary condition;
. Notably, it is worth mentioning that the solutions of the linear eigenvalue problem (28)–(30) provides an infinite set of eigenvalues
where
refers to the smallest eigenvalue. Furthermore, a positive
reflects to an initial decay of disturbances and a stable flow. In contrast, a negative
indicates an initial growth of disturbances and unstable flow.
4. Results and Discussion
The derived nonlinear ordinary differential equations given in Equations (10) and (11) along with the boundary conditions given in (12) were solved numerically and were obtained using the bvp4c programme in MATLAB (Matlab R2015a, MathWorks, Natick, MA, USA) for the selected values of the mixed convection parameter
λ, solid volume fraction of nanoparticles
ϕ, thermal radiation parameter
Nr, heat source/sink parameter
K, suction parameter
s and stretching/shrinking parameter
c. The range of
ϕ values was taken as
, where
indicates a regular base fluid, while the value of the Prandtl number was considered as Pr = 6.2 (water), except for comparisons with the prior case. The correlative output of the results obtained for
, with the ones obtained in Bachok et al. [
56] for some values of
c and
ϕ with
λ =
Nr =
K =
s = 0 for Cu-water nanofluid, are presented in
Table 2. Also, it was achieved that the present results were in very good alliance, which confirms that the numerical approach applied in this study is perfect, and therefore, the obtained results were believed to be accurate and correct.
The variations of the reduced skin friction coefficient
and the reduced local Nusselt number
against
λ are shown in
Figure 2,
Figure 3,
Figure 4,
Figure 5,
Figure 6,
Figure 7,
Figure 8,
Figure 9,
Figure 10,
Figure 11,
Figure 12 and
Figure 13 for several values of
ϕ,
Nr,
K,
s and
c. It was observed that dual solutions (upper and lower branch solutions) occurred for Equations (10) and (11) subject to the boundary conditions (12) in the range of
, where
denotes the critical value of
λ. Note that no solutions exist for
while a unique solution exists when
. Also, it is obvious from these figures that the values of
increase as the parameters
ϕ,
s and
c increase, therefore suggesting that these parameters widen the range of occurrence of dual solutions. Accordingly, it is further confirmed that the presence of nanoparticles, heat sink, suction and stretching sheet could decelerate the separation of the boundary layer, while the presence of thermal radiation, heat source and shrinking sheet could accelerate the separation of the boundary layer. Also, the values of
are always positive for the upper branch solution which is due to the heat being transferred from the hot surface of the stretching/shrinking sheet to the cold fluid. The reverse trend is observed in the case of the lower branch solution, i.e.,
becomes unbounded as
and
.
The influences of the solid volume fraction of nanoparticles
ϕ on
and
are illustrated in
Figure 2 and
Figure 3.
Figure 2 illustrates that
increases smoothly with an increasing value of
. Brinkman [
57] explained that an increment in the nanoparticle volume fraction increases the fluid’s viscosity. Hence, this situation contributed towards increasing the skin friction along the surface.
Figure 3 depicts that as the volume fraction of nanoparticles increases which consequently decreases the rate of heat transfer at the surface of
. This occurs because the nanoparticles increase the viscosity, density and conductivity. However, they may decrease the heat capacitance of the nanofluid
(see MacDevette et al. [
58]). Therefore, there is a trade-off between the enhanced properties, increased viscosity, decreased
and possible decrease in the heat transfer coefficient.
Figure 4 and
Figure 5 display the impacts of the thermal radiation parameter
Nr on
and
, respectively.
Figure 4 also illustrates that the reduced skin friction coefficient increases as
Nr is reduced in the case of the opposing flow. The reverse trend is noted in the event of the assisting flow and an increment in thermal radiation enhances the transmission of energy between the nanoparticles on a heated surface (assisting flow). Eventually, the thickness of the momentum boundary layer becomes thinner and increases the wall shear stress which raise the values of
. The decrement trend in the rate of heat transfer at the surface leads to an increment in the thermal radiation. Notably, this is in accordance with the result shown in
Figure 5 where the reduced local Nusselt number
decreases with increasing
Nr. Notwithstanding, this fact can be explained as follows. As the influence of the thermal radiation becomes stronger, the thermal boundary layer thickness increases and further decreases the values of the rate of heat transfer. Therefore, the usage of nanofluids having thermal radiation cannot improve the cooling of the heated sheet.
Figure 6 displays the variations of
for various values of the heat source/sink parameter
K. It is noticeable that the decrement of
as the heat source/sink parameter increases from the negative (heat sink) to the positive values (heat source) in the event of the opposing flow. A contrary trend is noticeable in the event of the assisting flow. The influence of the heat source/sink parameter
K on the reduced local Nusselt number
is illustrated in
Figure 7. The results presented in this figure indicate that the rate of heat transfer at the surface decreases as the heat source/sink parameter increases from negative (heat sink) to positive values (heat source). Indeed, this is because the higher heat source effect can increase the thermal boundary layer thickness that reduces the rate of heat transfer.
Figure 8 illustrates the effect of the suction on the reduced skin friction coefficient. Further, it is observed that the values of
increase with the suction. This is because the influence of suction at the boundary that slows down the nanofluid motion and increases the velocity gradient at the surface. By observing
Figure 9, it is evident that the values of
that represents the heat transfer rate at the surface, increases with the suction. Precisely, the increase in the magnitude of the suction parameter consequently increased the rate of heat transfer. This is because the increasing suction decreases the thermal boundary layer thickness and in return increases the temperature gradient at the surface.
Figure 10,
Figure 11,
Figure 12 and
Figure 13 are shown to present the impacts of stretching/shrinking parameter
c on
and
. Notably, it is evident from these figures that
and
were higher in the event of the stretching sheet, in comparison with the case of the shrinking sheet. Hence, the stretching parameter provided the most significant effects upon the skin friction along the surface and the heat flux at the surface. Indeed, this suggests that the stretching sheet enhances the rate of heat transfer, whereas the shrinking sheet inhibits the effect of the heat transfer rate. Also, the increment in the value of the stretching parameter further increases the effect of free convection.
The variations of
and
for three distinct types of nanofluids with nanoparticles containing Cu, Al
2O
3 and TiO
2 are shown in
Figure 14 and
Figure 15 From the figures, it is evident that there is little difference in the value of the reduced skin friction coefficient and the reduced local Nusselt number for TiO
2- and Al
2O
3-water nanofluids. Further, it is also discovered that the values of
and
are highest for Cu-water nanofluid, followed by TiO
2- and Al
2O
3-water nanofluids which is due to Cu having the highest value of thermal conductivity in comparison to the other nanoparticles.
The velocity and temperature profiles for Cu-water nanofluid with different values of the volume fraction of the nanoparticles
ϕ are displayed in
Figure 16 and
Figure 17. It was discovered that an increment in the value of
ϕ, increased the velocity of the fluid. Since an increment in
ϕ is believed to increase the fluid’s viscosity as suggested by Brinkman [
57], hence it increases the skin friction along the permeable vertical shrinking flat plate. This statement can be proved by the velocity profiles as shown in
Figure 16, as an increment in
ϕ reduces the momentum boundary layer thickness. Furthermore, from
Figure 17, we can see that the temperature of the fluid increases with an increase in
ϕ, therefore, suggesting that the temperature of the nanofluids can be managed by increasing or decreasing the volume fraction of the nanoparticles in the base fluid.
The velocity and temperature profiles for Cu-, Al
2O
3- and TiO
2-water nanofluids are shown in
Figure 18 and
Figure 19, which indicate that by utilizing distinct types of nanofluids, the values of velocity and temperature change. Furthermore, we discovered that Cu-water nanofluid has higher velocity distribution and lower temperature distribution in comparison to the other two nanofluids for the upper branch solution. Also, it is discovered that the velocity and temperature profiles for Al
2O
3-water and TiO
2-water nanofluids nearly coincide with each other. Besides, the Cu-water nanofluid (compared to Al
2O
3-water and TiO
2-water nanofluids) has thinner momentum and thermal boundary layer thickness which is due to the fact Cu nanoparticles have the highest thermal conductivity value in comparison to the other two kinds of nanoparticles. Accordingly, the reduced value of thermal diffusivity causes higher temperature gradients and therefore, the higher enhancement in heat transfers. Notwithstanding, the Cu nanoparticle possess high values of thermal diffusivity, and therefore, lowers the temperature gradients that affects the performance of Cu nanoparticle.
From observing
Figure 16,
Figure 17,
Figure 18 and
Figure 19, is apparent that the velocity and temperature profiles for both the upper and lower branch solutions satisfied the far field boundary conditions (12) asymptotically. Thus, aids in validating and supporting the numerical results retrieved for the boundary value problem (10)–(12) and proved the occurrence of the dual nature of the solutions as shown in
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 and
Figure 15. Also, the lower branch solution for the corresponding profiles displayed a larger boundary layer thickness when compared to the upper branch solution.
A stability analysis was undertaken to test the stability of the dual solutions. Hence, the smallest eigenvalue
was determined by solving the linear eigenvalue problem (28)–(30) using the bvp4c programme in MATLAB.
Table 3 depicts the smallest eigenvalues for Cu-water nanofluid for some values of parameters
K,
Nr and
λ when
s = 1 and
c = −1. The results in
Table 3 further indicate that
is negative for the lower branch solution, while
is positive for the upper branch solution due to their correspondence to the initial decay of disturbances, thus signifying a stable flow. Furthermore, for both the upper and lower branch solutions, the values of
as
λ approaches
, is consistent with Merkin [
53], Weidman et al. [
54] and Harris et al. [
55].