DUAL SOLUTIONS OF HYBRID NANOFLUID FLOW OVER A CONE WITH THE INFLUENCE OF THERMAL RADIATION AND CHEMICAL REACTION AND ITS STABILITY ANALYSIS †

The main intention of this study is to differentiate the stable and realisable solutions between the dual solutions of the water-based hybrid nanofluid flow driven by a solid cone along with energy transfer in the form of heat and mass by employing a new approach called stability analysis. The deviation of thermal radiation, chemical responses and heat absorption/generation are reserved into account. The leading equations which support the mathematical representation of this study are renovated by utilizing a set of similarity variables and solved by the MATLAB built-in bvp4c solver scheme. The outcomes of this study are presented both graphically and numerically. From this study, two kind of flow solutions have been achieved where one of them is related to the time-independent solutions and stable in nature. Also, the speed of the hybrid nanofluid can be controlled by applying magnetic field, but we should keep in mind that excessive amount of magnetic parameter may damage the system by burning.


INTRODUCTION
In recent times, physics of nanofluid flow has achieved important significance due to their various significance in diverse areas such as medicine, electronics and heat transfer devices etc. Nanofluids have better performance in different thermo-physical properties compared to the base fluids like water and oil etc.The mixture of nanoparticles with base fluids are termed as "Nanofluid" which was first coined by Choi [1].
The nanoparticles with different oxidation stages such as metallic or non-metallic particles like , , , ,   and  etc that are typically used in a base fluid like water, ethylene glycol, Kerosene and different bio-fluids to form nanofluids. Due to the vast applications of nanofluids, researchers have motivated to study the importance of nanofluids and interpreted different kind of results associated with nanofluids and their properties.Recently, Mishra et al. [2] have investigated the water based nanofluid containing -nanoparticle under different slip effects.Chanie et al. [3] have explored the flow behaviour of water-based nanofluid containing  and  particles and found that the motion of the fluid containing  is more effective than the  −  .
Hybrid nanofluids are new kind of fluids that are made up of two or more different nanoparticles with traditional base fluids.This kind of fluid has advance features than the general nanofluids.An individual matter can never has all the required characteristic that is the material may be omitted or deficient some properties.The hybrid nanoparticles can be customized in such a way that it can process better significant than the other nanofluids.  + ,   + ,  + &  +  etc are some examples of hybrid nanomaterials.Turcu et al. [4] and Jana et al. [5] were the foremost authors who have studied hybrid nanofluid experimentally.They have examined that the rate of heat transfer of the hybrid nanofluids is noticeably superior than the general nanofluids.This kind of fluids have better significant thermo-physical properties than the nanofluids.
In this study, we have investigated the water-based hybrid nanofluid containing  and   nanoparticles by considering a solid cone with the influence of both thermal and mass transmissions.This type of hybrid nanoparticles is used for oxygen storage, production and many other industrial applications [6].There are many applications of fluid flow due to a solid cone in different industrial and engineering sciences such as the solder tip, the conical heater and the continuous variable transmission (CVT) in modern car [7].Recently, many authors [2,7,8,9,10,11] have investigated the nanofluid and hybrid nanofluid flow caused due to a solid cone and given different importance outcomes and characteristics of cone.
Different physical areas recognise the importance of the synchronised effects of heat and mass transmissions on the flow of magnetised fluid under various flow geometries.Many industrial and engineering processes such as annealing and thickening of copper wire, paper production, MHD pump and MHD generators etc need both heat and mass transfers phenomenon with magnetic field effects.Dey and Borah [12] and Dey et al. [13,14 15] have investigated the boundary-layer fluid flow under the influence of both heat and mass transfers by considering different geometries.Alzahrani et al. [16] have examined the flow behaviour of hybrid nanofluid over a flat plate with the effects of both heat and mass transfers.Devi and Anjali Devi [17] and Khashi'ie et al. [18] have looked towards improving thermal transmission of  −   hybrid nanofluid flow over a extending surface.Researchers [2,7,19] have investigated the hybrid nanofluid flow due to a solid and rotating cone under different flow factors such as heat and mass transfers etc. Abdullah et al. [20] have analysed the convective heat transfer characteristics of hybrid nanofluid driven by square enclosure.
Also, the thermal radiation and chemical reaction effects on the fluid flow problems have played an important role in different physical fields.From the several decades, many researchers have given attention on these two flow factors as because of their multifarious application in different industrial, engineering and medical applications.Chamkha et al. [21] have examined the effects of radiation on mixed convection fluid flow by suspending nanoparticles.Sulochana et al. [22] have explored the influence of both thermal radiation and chemical reaction on magnetohydrodynamics nanofluid flow due to moving surface.Recently, Sharma et al. [23] have explored the effects of radiation parameter on the hybrid nanofluid flow over an extending surface with Joule heating.Dey et al. [24] have analyzed the stability analysis of magnetized fluid flow with the effects of chemical reaction.Saleem et al. [25] have investigated the MHD nanofluid flow over a rotating cone under the influence thermal radiation.
Due to the lack of information about the smoothness of the surface of the considering geometry and mathematical tools with assumptions, some initial complexity in the flow is observed which may develop non-uniqueness flow solutions.This initial complexity in the flow classifies the flow solutions into two categories, one solution is stable and physically tractable.Markin [26] was the first author who has explored the idea of dual solutions and their stability.Recently, Ghosh and Mukhopadhyay [27], Dey and Borah [12], Dey et al. [14], Dey et al. [24], Waini et al. [28] and Dey et al. [29] have explored the nature of non-uniqueness solutions and their stability behaviour by considering different fluids model.
The present work is all inspired by the above literatures and its immense relevance in different physical fields.We were able to ascertain via the literature review that this study has novelty like effects of thermal radiation, considering nanoparticles to form hybrid nanofluid, dual solutions and its stability analysis and will have a significant influence on other experts in the field.To the authors' awareness, this type of fluid model obtained by inserting the nanoparticles  +   with water which is driven by a solid cone along with energy transfers has not yet been taken into account while analysing dual solutions and their stability.
The intention of this work is to analyze the dual solutions and their stability of the water-based hybrid nanofluid driven by a solid cone which is sited in a porous medium under the influence of thermal radiation and chemical reaction with different slip flow effects such as velocity, thermal and concentration slips.Here, we have initially suspended the  solid nanoparticle of volume fraction  = 0.06 into the water base fluid to form / nanofluid.Again,   solid nanoparticle of volume fraction  = 0.1 is added into / nanofluid and achieved the   − / hybrid nanofluid.Also, a harmonized magnetic field is applied in the normal direction of the conical surface which plays an important role to enhance the thermal properties of the fluid.The leading equations which support the mathematical model of this problem are renovated by utilizing a set of similarity variables and solved by the MATLAB built-in bvp4c solver technique.Stability analysis is executed between the flow solutions to characterise the stable and physically achievable solution.Also, for the verification of our numerical codes, we established a reasonable uniformity when we compared our findings to the previously published article Mishra et al. [2].

FORMULATION OF THE PROBLEM
We have constructed a mathematical model of this study by considering a permeable cone of radius () which is immersed in a steady, incompressible and two-dimensional hybrid nanofluid.The flow diagram and its coordinate system are shown in Fig. 1.Where, the axis is measured along the surface of the cone and axis is taken in the normal direction of the conical surface such that vertex of the cone is taken as the origin of the system.In this study, we have considered the influence of both heat and mass transfers such that  and  prescribe the constant wall temperature and concentration respectively and  & the temperature and concentration at free stream region.A uniform magnetic field of strength  is applied in the normal direction of the conical surface.Here, we have considered the hybrid nanofluid which is formed by adding   nanoparticles into the /  nanofluid and hence   − /  hybrid nanofluid is found.Table 1 discusses the thermo-physical characteristics of the base fluid and solid particles.
Here, we have initially suspended the  solid nanoparticle of volume fraction  = 0.06 into the water base fluid to form /  nanofluid.Again,   solid nanoparticle of volume fraction  = 0.1 is added into /  nanofluid and achieved the   − /  hybrid nanofluid.The properties of hybrid nanofluidis given below such that the base fluid and the nanoparticles are denoted with the subscript ,  & [7].
The related boundary conditions are: Where, 1 2 3 , & l l l are the slip factors and vanishing of 1 2 3 , & l l l implies the no slip flow in the system.To alter the nature of equations ( 1)-( 4), we have adopted the following set of variables [2]., () , s i n , () , () , From the Roseland approximation (following Prameela et al. [30]), the local radiative heat flux term for optically thick gray fluid is given by where (by Taylor series approximation).As a result, After utilizing the equations ( 6) and ( 8) into the equations ( 1)-( 4), we have achieved the following set of equations and the equation (1) which represents the conservation of mass identically satisfies the similarity variables (6).
Where, ( 1,2,3,4,5,6,7) i A i = 's are defined in the following way: ) The dimensionless parameters are: During this study, we have observed three types of physical quantities which play an important role in different physical fields by computing shear stress, rate of heat transfer and rate of mass development at the surface.These physical quantities are defined in the following way [2], [7]: Now, using the equation ( 6) into the equation ( 13), we have achieved the following normalize form of the above quantities as: ( ) ( ) ( ) 2 Re ''(0), Re '(0), Re '(0), where, the local Reynolds number,  = .
FLOW STABILITY Due to the considering geometry and lack of mathematical tools and assumptions, some initial disturbances in the flow have been occurred which are decay or growth with time.These disturbances classify the flow solutions into two categories, one of them converges to its time-independent flow solutions as the initial complexity in the flow decay with time.The flow stability is needed to characterize the stable and unstable solutions between the dual solutions.To implement the stability analysis of this problem, the unsteady governing equations are essential which are obtained by adding the terms , & into the equations ( 2), ( 3) & ( 4) respectively.Due to the presence of time variable, equation ( 6) takes a modified form as given below: The time-dependent governing equations after using equation ( 15) become: (1 + ) The surface restrictions become:
Where,  is an unknown eigenvalue parameter, , & the small associated to time-independent solutions.Inserting equation ( 20) into the equations ( 16)-( 18), fixing  → 0 and simplifying the equations, we have achieved the following linearized eigenvalue problems.

𝐴 𝐹
Relevant boundary conditions are: Following Mishra et al. [31] and Dey and Borah [12], the equations ( 21)-( 23) are solved together with adjusted boundary conditions (referring Wahid et al. [28] and Dey and Borah [32]).It is noticed that this eigenvalue problem gives an infinite number of eigenvalues  <  <  < ⋯, where  signifies the smallest eigenvalue.The flow stability can be examined with the help of this smallest eigenvalue.If  > 0, then the initial complexity in the flow decay with time and the flow solution becomes stable.Otherwise, the flow solution to be unstable due to the escalation of complexity in the flow with time.

RESULT AND DISCUSSION
The set of equations ( 9)-( 11) and ( 21)-( 23) along with their two-point boundary conditions have been solved numerically by utilizing the MATLAB built-in bvp4c solver scheme.This technique performs the three-stage Lobatto IIIa formula and executes the finite difference method.It controls and adjusts the error upto 10 − by its residuals.The numerical explanations of this study have been achieved for the velocity  ′ (), thermal fraction()and mass fraction () profiles of the hybrid nanofluid as an outcomes of different novel flow parameters and have been displayed in figures ( 2)- (9).In this problem, we have achieved two types of solutions, first solution is represented by the solid line and it is related to the time-independent solution.The dashed line signifies the second solution which is converged slowly to its free stream region due to the presence of flow disturbance.
Before conferring the numerical results, we afford confirmation of our numerical code by solving the model presented in [2] and comparing the present numerical results (first solution) with the results reported in [2].Mishra et al. [2] have analysed the water based nanofluid flow containing the  nanoparticle over a solid cone with the influence of heat and mass transfers.In the non-appearance of thermal radiation and hybrid nanofluid, our leading equations during steady case of this study are matched with Mishra et al. [2] works.Table 2 reflects the comparison of the numerical code in terms of local Nusselt number and gives a good conformity for our results.
Table 3 is developed to check the flow stability between the dual solutions with the help of evaluated smallest eigenvalues for different values of suction parameter ().From this table, it is noticed that the smallest eigenvalues are positive and negative for first solution and second solution respectively.Due to the positive values of least eigenvalues, the initial disturbances in the flow lie down as time evolves and the flow solution converges quickly to its timeindependent solution.Hence, the first solution becomes stable and physically achievable.But, negative values of least eigenvalues develop the initial disturbances in the flow and hence the flow solution (second solution) is being as unstable behaviour and slowly converges to its free stream region.
Table 4 is inserted to show the skin friction coefficient numerically during time-dependent and time-independent cases for developing values of the Prandtl, Schmidt and Eckert numbers.These quantities help to evaluate the effects of shear stress at the surface of the cone.It appears from this data that the skin friction coefficient has been experienced a reduction from the noble gas ( = 0.015) to sea-water ( = 13.5).Again, the effects of shear stress at the surface have been enhanced from the hydrogen ( = 0.22) to water vapour ( = 0.60).It is also seen from this table is that increasing values of the Eckert number () raises the influence of shear stress during both the solutions at the surface of the cone.It is also observed that the effects of shear stress of the fluid at the surface of the cone during time-dependent case (second solution) is fewer than the first solution.Table 5 is inserted to show the heat transfer rate of the fluid at the surface of the cone for developing values of Prandtl and Eckert numbers.From this table, it is achieved that the Prandtl number develops the rate of heat transfer of the fluid during both the cases.But the Eckert number reduces the rate of heat transfer at the surface of the cone.The effect of  on the hybrid nanofluid's velocity and thermal fraction is shown in Figs.
(2) and (3).From these figures, it is believed that the hybrid nanofluid's velocity has been decelerated with the increasing amount of during both the cases, whereas the thermal fraction of the fluid has been developed with .A resistive type force known as the "Lorentz force" is produced as a result of the application of a magnetic field and has the power to control fluid motion.As a result, the hybrid nanofluid's velocity is decreased with increased values of .Again, due to the effects of magnetic field, the velocity of the fluid has been decelerated and a frictional force is developed between the fluid and surface of the cone which generates additional energy in terms of heat at the surface of the cone and hence the thermal fraction of the hybrid nanofluid is enhanced with.From Figure ( 2), it appears that the hybrid nanofluid moves more slowly in the time-dependent (second solution) situation than in the time-independent (first solution) case.Due to this reason, fluid's temperature during second solution superiors than the first solution in the vicinity of the surface of the cone.Figures ( 4)-( 6) are depicted to show the effects of porosity of the porous medium on the velocity, thermal fraction and mass fraction of the hybrid nanofluid.
From Figure (4), it is noticed that the speed of the fluid has been enhanced with the increasing values of  during both the cases.As a result, the overall pressure of the hybrid nanofluid reduces and the thermal fraction and mass fraction of the fluid are directly proportional to the pressure and hence the thermal fraction and mass fraction are experienced reduction with  (see Fig. 5 and Fig. 6).Also, the second solution of the velocity distribution and mass fraction of the fluid are comparatively fewer than the first solution.
Figure (7) shows how  affects the thermal fraction of the hybrid nanofluid.It can be seen from this figure that when  increases, the thermal portion of the hybrid nanofluid decreases in both cases.The cause of this occurrence is that the thermal radiation parameter () is directly proportional to the third power of the temperature of fluid at free stream region.Therefore, as  increases, the hybrid nanofluid's thermal fraction over the cone decreases.Also, it is seen that the thermal fraction of the fluid during the second solution is larger than the first solution near the surface of the cone.Figure (8) reflects the impact of heat generation parameter () on the thermal fraction of the fluid.From this figure, it is perceived that the thermal fraction of the hybrid nanofluid has been increased with .Influence of  on the mass fraction of the hybrid nanofluid is presented in Fig. 9.The chemical reaction is occurred mainly due to the effects of suction/injection of the fluid flow and presence of flow slips effects.The chemical reaction parameter has the capacity to fall down the mass fraction of the considering fluid.It can also be observed that the concentration level of the hybrid nanofluid during the second solution is fewer and slowly conversed to its free stream region over the first solution.
We have found from this study that all flow profiles display dual solutions up to a specific area of the similarity variable () and asymptotically meet the far-field boundary conditions.

CONCLUSION
This paper analyzes  −   /water hybrid nanofluid flow driven by a solid cone which sited in a porous medium.The energy transfer in terms of heat and mass is encountered with the effects of magnetic field, thermal radiation and chemical reaction.The leading equations are solved through numerical method called "three-stage Lobatto IIIa formula" by using MATLAB built in bvp4c solver scheme.Major results have been obtained as follows: 1) The velocity of the hybrid nanofluid has been decelerated with the improving amount of , but it has enhanced the thermal fraction of the fluid during both the cases.2) Due to the increasing amount of  , the velocity of the fluid has accelerated.Whereas, thermal and mass fractions of the hybrid nanofluid have been controlled by employing  .3) Thermal fraction of the hybrid nanofluid flow is a decreasing function of the thermal radiation () and an increasing function of the heat generation () parameters.4) Mass fraction of the hybrid nanofluid during the second solution is fewer than the first solution with the increasing amount of .5) Effects of shear stress is dropped down for incremental values of the Prandtl number.But we should keep in mind that the excessive amount of the Prandtl number may damage the system by developing the rate of heat transfer at the surface of the cone.6) "According to the stability point of view, the first solution, which converges to the steady flow solution more quickly, is stable, whereas the second solution is unstable and impractical.

Figure 1 .
Figure 1.Flow diagram and its coordinate system.

Figure 2 . 3 .
Figure 2. Velocity profile for incremental amount of  Figure 3. Sketch of thermal fraction for incremental amount of