Aspects of Non-unique Solutions for Hiemenz Flow Filled with Ternary Hybrid Nano ﬂ uid over a Stretching/Shrinking Sheet

This study is carried out to scrutinize the Hiemenz ﬂ ow for ternary hybrid nano ﬂ uid ﬂ ow across a stretching/shrinking sheet. This study aims to inspect the impacts of variations in the stretching/shrinking parameter and the volume fraction of nanoparticles on key aspects of the ternary hybrid nano ﬂ uid ﬂ ow, speci ﬁ cally the skin friction, Nusselt number (which relates to heat transfer), velocity pro ﬁ les, and the temperature pro ﬁ les. The ﬂ ow equations transform into a system of ordinary differential equations (ODEs) using a similarity transformation. Subsequently, the system is numerically solved using the MATLAB software ’ s 4th-order accuracy boundary value problem solver, known as “ bvp4c ” . Numeric ﬁ ndings reveal that skin friction values exhibit variations based on the magnitude of the stretching/shrinking parameter. Moreover, in the speci ﬁ c context of the ﬂ ow problem being studied, the heat conduction ef ﬁ ciency of the hybrid (ternary) nano ﬂ uid surpasses that of the hybrid nano ﬂ uid. The system yields two distinct solutions within a speci ﬁ c shrinking/stretching parameter interval. Through an examination of the temporal stability of the solutions, it was determined that only one remained stable over an extended period. Remember that these current ﬁ ndings hold solely for the combination of copper, alumina, and titania.


Introduction
Recently, researchers in fluid dynamics have emphasized exploring the properties of stagnation point flow.The fluid flow around the region positioned at the front end of a bluntnosed object is commonly referred to as stagnation point flow.The phenomena mentioned above can be observed on any solid entity that is submerged in a fluid medium.In the stagnation region, the highest pressure results from force buildup due to halted flow, accompanied by rapid heat transfer and significant accumulation of particles or substances from the fluid, making it a critical location.The researchers have been attracted by the varied array of applications of stagnation point flow, including the process of extrusion, the polymer industry, and concerns linked to aerodynamics [1,2].The investigation of stagnation point flow is also crucial to understand physiological fluid dynamics, such as the way blood clots form artificial organs [3,4].The phenomenon of stagnation point flow over a stretching/shrinking sheet is commonly observed in the real-world application of the plastic extrusion.Extrusion is a manufacturing method that uses pressure to propel a material through a die, forming a consistent and uninterrupted profile.Throughout this procedure, the fluid dynamics experts utilize the velocity and temperature profile, skin friction, and heat transfer rate to guarantee the final products' quality.Hiemenz [5] was among the pioneering researchers investigating stagnation point flow, and later, the problem was extended to the axisymmetric flow case [6].Then, Mahapatra and Gupta [7] discovered the impact of heat transmission on the stagnation point flow past a plate that is stretched radially.Afterward, Wang [8] extended the idea to the axisymmetric stagnation point flow over a shrinking sheet, and the author's research highlights under the case of shrinking sheet, dual solutions occur in a particular shrinking parameter range.Following that, numerous captivating studies were conducted exploring the Hiemenz flow under the same boundary conditions, comprising a variety of physical parameters [9][10][11][12].
Water, ethylene glycol, ethanol, and glycerin play a crucial role in heating and cooling, power generation, and chemical processes.Nevertheless, as mentioned earlier, the heat transfer process is hindered due to the limited heat transfer of conventional fluids.Therefore, a new fluid type referred to as nanofluid was created by Choi and Eastman [13] to overcome the limitations of conventional fluids.Nanofluids consist of a kind of nanoparticle ranging in size from 1 to 100 nm that are dispersed in the base fluid.They have demonstrated that using nanofluids can improve thermal properties and enhance heat transport rates.Additionally, several relevant sources discussing the application of nanofluids to enhance heat transfer can be found in the investigations conducted by Mehobbiand Rashidi [14][15][16].Later, researchers shifted their attention toward hybrid nanofluids, which are mixtures of nanoparticles from two distinct categories with the base fluid.These extended nanofluids demonstrated commendable heat conductivity characteristics and can boost the heat transfer rate in comparison to single nanofluids [12,[17][18][19][20].
The need for enhanced thermal conductivity and heat transmission has driven the development of ternary hybrid nanofluids, known as stable mixtures of three distinct varieties of nanoparticles in a base fluid, offering enhanced properties compared to single and hybrid nanofluids.This new extended nanofluid can help optimize heat transmission in various engineering and industrial settings [21].Ramadhan et al. [22] dispersed Al 2 O 3 (alumina), TiO 2 (titania), and SiO 2 (silica) in ethylene glycol to create a ternary hybrid nanofluid.In their experimental work, the authors found that ternary hybrid nanofluid possessed the best effective thermal conductivity when the concentration volume was 0.3% compared to 0.05%−0.2%.At the same time, Sundar et al. [23] suspended GO (graphene oxide), Fe3O4 (iron oxide), and TiO 2 nanoparticles in ethylene glycol to make ternary hybrid nanofluid.Their findings indicated that the system thermal efficacy of ternary hybrid nanofluids has increased to 14.32%.Besides, Animasaun et al. [24] evaluated the stagnation point flow concerning a stretching sheet involving ternary hybrid nanofluid Ag-Al 2 O 3 -Al/water (argentum-alumina-aluminium/ water).The results revealed that increasing the Biot number between 0.1 and 3.0 increases the heat transfer rate of ternary hybrid nanofluids in situations involving heat generation/ absorption.Gupta and Rana [25] examined the 3-D stagnation point flow of Ag-Al 2 O 3-Al/water over a rotating and stretching disk.The research concluded that the ternary hybrid nanofluid possesses an enhanced heat conductivity compared to water, leading to an increased heat transfer rate.
The excellent thermophysical characteristics and rheological behavior of the Al 2 O 3 -Cu-TiO 2 /water (alumina-copper-titania/water) ternary nanofluid were observed in an experiment done by Xuan et al. [26].According to the authors in [25], when large Cu nanoparticles are present in a fluid, there is a notable physical separation or distance between these particles and the surrounding water molecules.This gap contributes to a high level of thermal contact resistance, which can hinder efficient heat transfer.When smaller sized Al 2 O 3 and TiO 2 nanoparticles are introduced into the fluid, they fill in the gaps between the larger Cu nanoparticles.This action reduces the interparticle space and, as a result, improves the fluid's thermal properties.
As far as the authors know, no reported studies discuss stagnation point flow utilizing Al 2 O 3 -CuTiO 2 /water as a ternary hybrid nanofluid.Hence, drawing inspiration from [25], we analyzed the heat transfer characteristics of the ternary hybrid nanofluid (Al 2 O 3 -Cu-TiO 2 /water) in the context of stagnation point flow over a stretching/shrinking sheet.Additionally, this study may offer insights to engineers and researchers on strategies to increase or decrease heat transfer rates by controlling the parameters' values.Besides, the authors are concerned that this flow problem will produce more than one solution; hence, a stability analysis is conducted to determine the long-run stability of the solutions.Skin friction and Nusselt number comparison analyses are also conducted between ternary hybrid nanofluid and hybrid nanofluid.

Problem Formulation
This research focuses on examining the Hiemenz flow of a ternary hybrid Al 2 O 3 -Cu-TiO 2 /water nanofluid over a stretching/shrinking sheet.Here, the flow overview is demonstrated in Figure 1.
The free stream and surface velocities are taken as u e ðxÞ ¼ ax, and u w ðxÞ ¼ bx, respectively, with a and b are considered as constants.The ambient temperature T 1 and the wall temperature T w also considered as constant.
Mathematically, the equations govern the flow problem are as follows [27]: Here, ðu; vÞ resembles the velocity component along the ðx; yÞ-axis while T refers to the temperature.Meanwhile, the thermophysical properties with respect to hybrid nanofluid and ternary hybrid nanofluid are displayed in Tables 1 and 2 indicates the thermophysical properties of nanoparticles as well as water.The thermal conductivity, dynamic viscosity, density, as well as the heat capacity is symbolized as k; μ; ρ, and ðρC p Þ, respectively.It is essential to mention that subscripts nf ; f ; hnf as well as thnf resembles the nanofluid, base fluid, hybrid nanofluid and ternary hybrid nanofluid, accordingly.Moreover, Al 2 O 3 (subscript n1), Cu (subscript n2), and TiO 2 (subscripts n3) are the nanoparticles used in this research work and their volume fractions are depicted by φ 1 ; φ 2 as well as φ 3 , accordingly.
Then, a set of dimensionless variables are introduced as follows [30]: Heat capacity Thermal conductivity where By utilizing similarity variables in Equations ( 5) and ( 6), Equations ( 1)-( 3) reduce and are expressed in the following form: Subsequently, the initial and boundary conditions in Equation ( 4) experience a transformation in yielding the expressions given below: in which λ ¼ b a refers to the parameter of shrinking/stretching.The negative values of λ represent the case of shrinking, while the stretching case is represented by λ possesses positive values, and λ ¼ 0 specifies the nonmoving surface.
Meanwhile, Pr in Equation ( 8) is the Prandtl number that is defined as follows: The coefficient of skin friction C f and the local Nusselt number Nu x resemble the crucial physical quantities in consideration which are defined as follows [27]: The shear stress τ w as well as the heat flux q w at the surface are expressed as follows: By implementing Equations ( 5) and ( 12) into Equation (11), gives: where the local Reynolds number is given by Re x ¼ u e x=ν f .

Stability Analysis
Based on the results obtained, Equations ( 7)-( 9) allow the appearance of the dual solutions.Therefore, by applying the discovery from Merkin [31] and Weidman et al. [32] about the stability of the solutions, long-term stability and dependability of the solution can be determined.To do this, the case of unsteady flow of the boundary value problem Equations ( 2) and ( 3) is considered and a new dimensionless time variable τ, is introduced.
Then, a new set of dimensionless variables is introduced as follows: Substituting (16) into Equations ( 14) and ( 15), the converted differential equations are attained.
Next, the exponential perturbation functions are introduced to as follows [32]: with steady flow solution of f ¼ f 0 ðηÞ and θ ¼ θ 0 ðηÞ while FðηÞ as well as GðηÞ are relatively small in comparison with f 0 ðηÞ and θ 0 ðηÞ while ε is the unknown eigenvalue.Substituting Equation (20) into Equations ( 17)-( 19) gives: subject to the boundary conditions: According to Khashi'ie et al. [33], there is no eigenvalues ε can be generated from homogenous boundary conditions.As we can see, Equation ( 23) is a set of homogenous boundary conditions.Hence, to overcome this problem, nonhomogenous boundary must be considered as a way of attaining the eigenvalues ε.As suggested by Harris et al. [34], the boundary condition F 0 ðηÞ→0 is relaxed and replaced by F 00 ð0Þ ¼ 1, without loss of generalization.The new set of boundary conditions to be considered in this problem is as follows: In the meantime, when the eigenvalue problems ( 21) and ( 22) subjected to boundary conditions in Equation ( 24) are solved, an infinite series of eigenvalues ε 1 <ε 2 <ε 3 … can be obtained.The stability of a solution's flow is determined by the positivity of the smallest eigenvalue ε, indicating an initial decay of perturbation.Conversely, if ε is negative, signifying an initial growth of perturbation, the flow is considered unstable.

Results and Discussion
The solutions to the mathematical model, as outlined in Equations ( 7) and ( 8), subject to the specified boundary conditions in Equation ( 9), were accomplished by employing a bvp4c solver in the MATLAB software, and the details are explained in Shampine et al. [35].This solver occupies a finite difference method that employs the three-stage Lobatto IIIa formula with 4-th order accuracy.Beside generating the values of physical quantities for fixed values of parameters, this solver can do the reverse, where it can be used to find the values of the parameters by fixing the values of physical quantities of interest.
We explored the impact of the stretching/shrinking parameter and volume fraction on skin friction, heat transfer, velocity, and temperature profiles.The values were selected through a combination of referencing existing literature and trial-and-error approaches due to the presence of dual solutions.As indicated in Table 2, a Prandtl number value of 6.2 was employed, representing water as the base fluid at a temperature of 25°C following Oztop and Abu-Nada [36].Therefore, in this study, the parameter Pr is held constant at 6.2, while the parameters φ are systematically adjusted within the range of 0.01-0.05,and the parameters λ are modified within the range of −1.2465 to 2.
The ternary hybrid nanofluid flow, which contains alumina, copper, and titania in water, is considered.The volume fractions of alumina, copper, and titania used in this study are 0.01, 0.03, and 0.05, respectively.The volume fraction of each nanoparticle is evenly distributed at 1/3 to form the ternary hybrid nanofluid, as per the experiment carried out by Ramadhan et al. [22].Investigating non-unique solutions to the equations governing the current problem is another focus of this work.Additionally, comparison results between the hybrid nanofluid and the ternary hybrid nanofluid are conducted.To ascertain the veracity of the present study results for various values of λ when φ 1 ¼ φ 2 ¼ φ 3 ¼ 0 are compared with the previous studies, as depicted in Table 3.
The values of Re 1=2 x C f toward λ when φ 1 ¼ φ 2 ¼ φ 3 ¼ 0:01; 0:03 and 0:05 are depicted in Figure 2.Here is an apparent trend of increasing skin friction values as the volume fractions of nanoparticles increase within a specified range of − 0:65<λ<1.The viscosity of a fluid tends to increase as the volume fractions of solid nanoparticles inside it increase.An elevated viscosity leads to a rise in flow resistance and an increase in skin friction at the interface between the fluid and the surface.Besides, it also can be seen from Figure 2, the skin friction decreases when λ>1 and no skin friction when λ ¼ 1: Figure 2 also demonstrates the presence of non-unique solutions for the shrinking sheet case in ternary hybrid nanofluid is ranging between − 1:2465<λ< − 0:9, where λ c ¼ − 1:2465 serves as the critical value distinguishing the first and second solutions.Simultaneously, this problem attains a unique solution when λ> − 1:2465: Figure 3 illustrates the observed trend of the Nusselt number exhibiting an increase when the volume fractions are progressively elevated.The Nusselt number is employed to characterize the convective heat transport of a fluid.The Advances in Mathematical Physics escalating volume fractions observed in the stretching/ shrinking sheet scenario led to the increased presence of nanoparticles within the same fluid volume, which can enhance the fluid's heat conduction properties due to more particles being available to carry heat energy.Therefore, this phenomenon can potentially increase the rate of heat transfer or the Nusselt number of the fluid.
Figure 4 depicts the velocity profile that elucidates the thinning momentum boundary layer as φ 1 ; φ 2 , and φ 3 increase.Thus, the ternary hybrid nanofluid's velocity gradients increase.This result correlates with the growing pattern of the skin friction coefficient (Figure 2), as it is known that an upsurge in fluid velocity corresponds to the increased surface shear stress.The temperature profile of ternary hybrid nanofluid when φ 1 ; φ 2 , and φ 3 vary from 0.01 to 0.05 is depicted in Figure 5. Physically, it is evident that as φ 1 ; φ 2 and φ 3 increase, the temperature gradient of the ternary hybrid nanofluid increases, leading to an enhancement in the fluid's convective properties.This outcome is consistent with the conclusion depicted in Figure 3, where an augmentation in convective characteristics leads to an upward trend in the rate of heat transfer in the fluid.
The skin friction and the Nusselt number with respect to a hybrid nanofluid ðφ 1 ¼ φ 2 ¼ 0:05Þ and ternary hybrid nanofluid ðφ 1 ¼ φ 2 ¼ φ 3 ¼ 0:05Þ past a shrinking/stretching sheet with the critical point, λ c ¼ − 1:2465 are shown in Figures 6 and 7, respectively.We can see that both nanofluids possess same critical point, suggesting that certain parameters or conditions are influencing the behavior of both types of nanofluids in a similar manner.The incorporation of φ 3 indicates the development of a ternary hybrid nanofluid, Al 2 O 3 -Cu-TiO 2 /water, derived from the hybrid Al 2 O 3 -Cu/water nanofluid.This addition boosts the Re 1=2 x C f   x Nu x showing an increasing pattern.This outcome validates the hypothesis that, when φ 3 is added, the nanoparticle density can be optimized to enhance the nanofluid heat transfer rate.Hence, ternary hybrid nanofluids evidently demonstrate a superior thermal conductivity rate in comparison to the hybrid nanofluids.Figure 9 depicts the temperature profile, which illustrates the variations in temperature as the hybrid Al 2 O 3 -Cu/water nanofluid transforms into the ternary hybrid Al 2 O 3 -Cu-TiO 2 /water nanofluid.As the temperature increases, ternary hybrid nanofluid thermal conductivity rises, which may be due to the increase energy conveyed by the rise in nanoparticle fraction of volume over the state of shrinking/stretching sheets.Additionally, it is evident from Figures 4, 5, 8 and 9 that the far-field boundary criteria in Equation ( 9) were met asymptotically.As emphasized by Ishak [38] it is imperative that velocity and temperature profiles adhere to the specified boundary conditions.This ensures the validation of numerical results as accurate and reliable.
Lastly, the variations of eigenvalue against shrinking/ stretching parameter for the ternary hybrid nanofluid ðφ 1 ¼ φ 2 ¼ φ 3 ¼ 0:05Þ are presented in Figure 10.From the figure, the sign of smallest eigenvalue ε for the first solution is positive, signifies the stability characteristics of the solutions.Meanwhile, the values of smallest eigenvalue ε for the second     Advances in Mathematical Physics solution ia negative, indicates the second solution long-term instability.

Conclusion
The work on the Hiemenz flow with respect to ternary hybrid nanofluid flow over a stretching/shrinking sheet was concluded.The following insights have been attained:

FIGURE 6 :
FIGURE 6: Variants of skin friction coefficient with respect to ternary hybrid nanofluid and hybrid nanofluid against λ.

FIGURE 7 :
FIGURE 7: Variants of local nusselt number with respect to ternary hybrid nanofluid and hybrid nanofluid against λ.
(i) Within a specific range of shrinking intensity which is − 1:2465<λ< − 0:90, the dual solutions exist while a unique solution exists for λ> − 0:90.The separation between first and second solutions happened at λ c ¼ − 1:2465 for all volume fraction values considered.(ii) The stability analysis verified the long-term stability of the first solution, while the second solution was identified as unstable.(iii) As the nanoparticle volume fraction of ternary hybrid nanofluid Al 2 O 3 -Cu-TiO 2 /water increases, the heat transmission rate rises substantially.(iv) Based on the findings in comparative analysis, it can be inferred that ternary hybrid nanofluids possess an enhanced heat transfer rate compared to hybrid nanofluids.

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