A computational convection analysis of SiO2/water and MoS2‐SiO2/water based fluidic system in inverted cone

A complete shape factor investigation of water‐based mixture type hybrid nanofluid in a permeable boundary with the impact of magnetic field, thick dissemination, and warm radiation is presented in this article. A computational convection analysis of an inverted semi vertical cone with a porous surface in the form of SiO2$$ {\mathrm{SiO}}_2 $$ /water nanofluid and MoS2‐SiO2$$ {\mathrm{MoS}}_2\hbox{-} {\mathrm{SiO}}_2 $$ /water hybrid nanofluid transport is developed. The system of differential equations is presented and solved numerically by the Lobatto IIIA method. The temperature distributions and fluid velocity are studied along with the coefficient of skin friction and the Nusselt number, taking into account the form of distinct nano‐particles. The flow problem's results are approximated by using several embedding variables. Tables and graphs are constructed for a variety of scenarios including maximum residual error, mesh points, and Nusselt numbers. We conclude that boundary film thickness reduces and the fluid flow is resisted by magnetic field presence. Fluid flow slows down as λ$$ \lambda $$ increases, and this reduction is more evident in nanofluids than in hybrid nanofluids. With an increment in S$$ S $$ , velocity drops. A detailed analysis of the proposed ordinary differential equations, boundary conditions, and numerical data of skin friction is given both in tabular and graphical forms. Additionally, it is observed that the fluid flow slows down more for the hybrid nanofluid than for the SiO2$$ {\mathrm{SiO}}_2 $$ /water nanofluid. Moreover, it is clear that the temperature increase for the SiO2$$ {\mathrm{SiO}}_2 $$ /water nanofluid is substantially greater. The authors deduce that the existence of a magnetic field resists fluid flow for hybrid nanofluid forms and decreases the thickness of the viscous boundary layer.

vulnerability, a new class of fluid called NFs has been introduced.Such NFs are frequently utilized for macroscopic cooling because of the crucial characteristics of heat transport.The single-phase modeling approach of NFs is especially important for lubricant refining, coolants, and applications in daily life including portable computer systems, air conditioners, coolers, and nanostructures. 1In the manufacturing industry, solar thermal collectors are recognized as a highly straightforward and environmentally friendly way to convert UV irradiation to heat energy. 2,3In such collectors, various fluids such as glycol and water are used to improvise their performance and power. 4,5Recently, numerically and experimentally, several researchers have researched the use of NFs in many kinds of solar collectors. 6,7Many researchers and engineers have recently shown a strong interest in the subject of two-dimensional axially symmetric fluid motion inside a quasi-inverted cone.These flows offer important applications in a variety of engineering and manufacturing processes, such as the design of sustainable power plant equipment, gas turbines, propulsion systems for aircraft, spacecraft, missiles, satellites, storage of grains, the disposal of nuclear waste, the spread of chemical contaminants through water-saturated soil, and other processes that involve the movement of moisture through air trapped in fiber insulations. 8,9 hybrid nanofluid (HNF) covers nano-particles of various sorts.Synergistic results provide the HNF to integrate advantages from component nano-particles but avoid their drawbacks.This feature results in the properties of HNF as compared to mono-particle-based NFs. 10 Because of the minute size of the nano-particles and their very large common surface area, NFs have useful properties such as high-temperature conductivity, durable stability, minimum obstruction in flow passes, and homogeneity. 11,12Choi and Eastman developed the idea of NFs for the suspension of ultrafine particulate liquids. 13Khan and Pop have found the issue of flow over the stretch sheets in their initial work on NF. 14 In this research, a complete shape factor investigation of water-based mixture NF in a permeable boundary alongside the impact of transverse attractive field, thick dissemination, and warm radiation is given.This article presents a computational convection analysis of an inverted cone with a porous surface in the form of SiO 2 /water NF and MoS 2 -SiO 2 /water hybrid NF transport.Temperature distributions and NF velocity are studied, along with the coefficient of Nusselt number and the skin friction, taking into account the form of distinct nano-particles.The system of differential equations was simplified and numerically solved using the Lobatto IIIA method.The main findings from the present research are that the existence of a magnetic field resists flow for NF and HNF forms and decreases the thickness of the viscous boundary layer.
Ramzan described the modeling across a linear and significantly stretching layer of such a two-dimensional natural flow assessment of Williamson fluid. 15NFs are arranged by dissolving the nano-particles in the base fluid and can considerably improve the heat conduction and get a heat transfer rate resembling that of pure fluids.NFs introduced by Choi have thermal conductivities in magnitudes, more than the conductivities of the base fluids, and with sizes suggestively smaller than 100 nm.The presence of nano-particles upgrades the warmth move execution of the base fluid fundamentally.
In 1979, Chiou calculated the laminar convection flow in the cone frustum.It was assumed that the temperature and the heat flux of the wall not changing and then reached a conclusion that the temperature was inversely proportional to the Prandtl number when moving along the wall of frustum. 16Yih studied heat transfer and mass transfer of heat (convection) over an inverted cone with the porous medium have to change wall temperature and flux. 17L-Aziz studied the numerical solution to investigate the effects of the time-dependent chemical reactions on the heat and mass transfer of NF and stagnation point flow on the stretching surface. 180][21][22] Recently, Ahmad et al. has worked on a HNF and NF models by using the Lobatto IIIA technique for obtaining the solution of fluidic models. 23 few years ago, Noghrehabadi discussed convection through NF over standing plates pasted in permeable medium surface heat flux in Reference 24.Water flow, glycol, and ethylene-based NF flow having natural convection through vertical cone on a PM are given in Reference 25.This research was extended by Ellahi, he found the effects of shape and size of nano-particles on entropy. 26Namburu calculated copper oxide's viscosity in nano-particles suspended in ethylene glycol and H 2 O mixtures.Later, he found a relationship between particle volume viscosity and temperature based on the data.They studied the rheological comportment of NFs. 27It was observed by Chen that the shear thinning behavior of NFs was dependent on the thickness of particles, viscosity, and fluid flow rate. 28Chan and Dinge studied the behavior of NFs having titanite as constituent nano-particles.They concluded that titanite has very strong thinning properties for NFs and also, they have a huge impact on the concentration and temperature of the medium. 29Then the results obtained were matched with other results of experiments on various kind of particles, and it showed a very good resemblance with their results. 30Afshaari and Akbaree examined the nano-particles effects on the viscosity of NFs where it was determined that viscosity increases with the rise of volume fraction of nano-particles. 31Kaang and Kime discussed natural convection which is a type of heat transfer in which fluids flow does not depend on an outer source.They also discussed some examples related to the research work. 32obatto IIIA method is applied for the numerical investigation of differential equations which are described by the use of approximations to the solution at endpoints.Due to strong stability properties, Lobatto IIIA methods have been considered for boundary value problems (BVP).We use the Lobatto IIIA method for numerical treatment since it is stiffly accurate.In this work, the authors have presented the analysis for the various values of mesh points and real tolerance to check the efficiency, stability, and reliability of proposed technique.In this research work, we analyzed the results of cross diffusion into a porous medium on the skin friction coefficient, mass and heat transfer from cone into a porous medium.A Lobatto IIIA method is used to find numerical solutions for energy, concentration, and momentum equations.A complete implementation strategy of the Lobatto IIIA method is demonstrated step by step in Figure 1 in a comprehensive manner.The Figure 1 expresses the graphical illustration of the proposed methodology by stating the problems, both NF and HNF, and its numerical computing procedure in an extensive manner.Using the Lobatto IIIA method, Nagoor 33 explained numerically the effect of different physical constraints on velocity and temperature fields for Darcy-Forchheimer HNF flow in revolving frames.The above-mentioned experiments in which many researchers presumed different fluids with various nano-particles and observed interesting findings for their thermophysical behavior are the motivation behind this proposed study.There is substantial research being conducted on the numerical approach to the problem of NF flow, [34][35][36][37] but relatively few researchers have attempted to solve the problem of hybrid flow of NF using special numerical methods.The present article offers a detailed form study of MoS 2 -SiO 2 /water water-dependent HNF inside a semi vertical and inverted cone having porous boundary as well as the impact of the presence of the magnetic field, viscous dissipation including thermal radiation.
The following are a few of the important aspects of this research: • The radiative flow of HNFs MoS 2 -SiO 2 /water over a semi-inverted cone with porous boundary effects has been modeled in a novel way, using the power of sufficient similarity transformations, PDEs describing the flow model are converted into a system of ODEs.
• A computational investigation by exploiting the Lobatto IIIA approach is implemented for the dynamical analysis of a hybrid nanofluidic system passed over a semi-inverted cone with a porous boundary.
• The primary approach of boundary layer approximations is applied to systemize the governing nonlinear PDEs of the nanofluidic and hybrid nanofluidic model and these PDEs are converted into ODEs by the competence of similarity variables.
• The implementation of a numerical approach namely Lobatto IIIA technique that relates to the categories of finite difference numerical techniques for examining the nanofluidic model, is a novel work.
• The graphical illustration of each fluidic parameter on velocity and temperature fields is portrayed along with the numerical computed data of skin friction coefficient and Nusselt-number for assessment analysis.
The rest of the sections of the manuscript are cataloged as follows: Section 2 describes flow equation modeling, Section 3 provides numerical results and discussion and Section 4 summarizes the fluidic system's conclusion and recommendations.

MODELING AND SIMULATION OF NANOFLUIDIC EQUATIONS
The development of physical problems involves an isothermic cone with an inverted porous medium in which the flow of an NF is a 2D steady-state and incompressible, is presented. is a semi-vertical angle along with the cone.The origin of the cartesian coordinates system is taken from vertex of cone.Moreover, we chose the x-axis and y-axis as parallel and perpendicular to surface of inverted cone, respectively.The physical abstract of proposed model is illustrated in Figure 2. Considering the existence of viscous properties and the effects of thermal energy, heat transmission is involved on surface of the cone (inverted).Figure 2 expresses the problem statement of the inverted cone model in x, y coordinate system.Table 1 presents the experimental values for different parameters of fluidic systems.Tables 2 and 3 represents the experimental values and properties of various parameters for the case of nanofluidic and hybrid nanofluidic systems, respectively.Also, we take temperature T as a constant value. 38The temperature of the wall of an inverted cone is defined as T w = ax  + T ∞ where a(a > 0) is constant and T ∞ is temperature distant from the cone surface and its property is T ∞ > T ∞ .Also,  is power index law.Under these assumptions, along with the approximations of boundary layers, the governing equations are determined as follow: y

Properties NF
Density Thermal conductivity s TA B L E 3 Experimental values for HNF system.

Properties
Hybrid NF Thermal cond.
The attached boundary conditions are: where u and v are components of velocity along x and y directions, with thin boundaries r = x sin .The density of the HNF is  hnf .The dynamic viscosity of HNF is  hnf .T is the temperature of HNF and q r is a flux which depends on the temperature of the body.g is a gravitational acceleration and v w is a suction velocity on the surface of inverted cone.According to the Rosseland approximation 39,40 radiative heat flux is defined as: here  * * and k * * are the Stefan-Boltzmann constant and coefficient of mean proportion.Also we introduced the stream function  as flows: Now we introduce the following similarity transformations as: x ,  = x y Gr x f (), where Gr x is Rayleigh number and mathematically given by: Since Equation ( 1) is already satisfied and now we substitute the transformations defined in Equation ( 4) into the Equations ( 1) to (3) and we get following coupled ODEs: The mathematical expressions of Φ 1 , Φ 2 , and Φ 3 are defined as: The mathematical expression of prominence fluidic parameter of interest for proposed nanofluidic system are defined as: where, Pr represents Prandtl number, Ec is Eckert number, M is Hartman number, and N is radiation parameter respectively.Here f () = S when  = 0 with S < 0 being suction case and S > 0 shows injection case.The Nusselt number Nu and skin friction coefficient C f are defined as: The expression of shear stress on the surface of the cone is indicated by  w and given as: The q w shows the heat flux from the surface of the cone and defined as: After solving from Equations ( 16) to (18), we obtain the dimensionless form of Nusselt number Nu x and skin-friction x which are given as:

RESULTS AND DISCUSSION
Here, the empirical results of fluidic parameters on fluid velocity and temperature distributions are analyzed graphically.Furthermore, tabular forms are created for relevant physical parameters namely, the Nusselt number and skin friction coefficient for various nano-materials shape factors throughout the conduct of HNF MoS 2 -SiO 2 /water and dotted lines will be used to show SiO 2 /water NF results, respectively.For this purpose, the coupled nonlinear ordinary differential system of the nanofluidic model is derived from governing PDEs by the exploitation of the non-dimensional similarity variables approach.The transformed nanofluidic differential system of ODEs presented in Equations ( 9) and ( 10) are tackled numerically along with attached boundary conditions Equation ( 12) by employing the Lobatto IIIA approach with MATLAB routine "bvp4c."Power law  is The influence of power law index on f ′ ().

F I G U R E 4 Influence of Hartmann number on f ′ (𝜂).
analyzed through Figure 3.The fluid is shown to decelerate with such addition in  and this reduction in the fluid flow for NF compared to HNF is more pronounced.
Figure 4 shows the effect of a magnetic field parameter M on f ′ (), so it suffices that the existence of the magnetic field resists fluid flow for NF and HNF. Figure 5 is sketched to analyze the influence of N thermal radiation parameter on the velocity profile of NF and HNF, respectively.Here, N relates to the acceleration of flowing fluid and the density of the NF and HNFs at the boundary layer of momentum.Figure 6 shows the influence of parameter S on SiO 2 /H 2 O and MoS 2 -SiO 2 /H 2 O NFs velocity.It is notable that S presents two cases, if S > 0, it presents the injection case and when S < 0, it presents the suction case.Moreover, the value of f ′ (), almost reaches 0.1 for S = 0.2 which is far greater than the value of f ′ () for SiO 2 /H 2 O at the same value of S. This figure clearly shows that as S is increased, the velocity reduces as a result of the surface suction acting in opposition to the flow of fluid, which results in a reduction in velocity.Additionally, NF exhibits a far greater velocity decrease than HNF.It is notable that this difference is more significant as compared to other values of S. Furthermore, decreased NF velocity is significantly greater than the HNF.The influence of the volumetric fraction of both NFs is examined by Figure 7.It is shown that the velocity of fluid grows with the change in all volumetric fractions of NPs.
The SiO 2 volumetric-fraction of HNF has been settled at 1%.The increasing trend in fluid velocity is related to the point that HNF and NF dynamic viscosity is in inverse relation to volumetric fractions of nano-particles.As a result, the increase in

F I G U R E 7
Impact of volumetric fractions  1 and  2 on f ′ ().

F I G U R E 8
The effect of Eckert number on f ′ ().

F I G U R E 9 Impact of Eckert number on 𝜃(𝜂).
seen from this figure that the increased Eckert number leads to the acceleration of both forms of NF flow.Moreover, in the case of HNF, this change is greater.From these graphical figures, it is also noted that the distributions of NF and HNF temperatures rise via an increase for both N and .The influence of volumetric fractions of the nano-particles is shown in Figure 12.It is noticed that throughout the presence of NF and HNF, the rise in volumetric fractions  1 and  2 contributes to increased temperature, respectively.
Table 4 reported the computed values of the Nusselt number in tabular form.The local Nusselt number upgraded by raising , Qh, We, Pr, and Bi while decay by increments in M * ,  w , , Sc, K 1 , Nr, Nt, A * , , and  * .The computed numerical data of Skin friction is rendered in Table 5.The skin friction grows for the magnetic parameter, Schmidt, and local Weisenberge number while declines for the temperature ratio parameter, unsteadiness, and shear rate parameter respectively.
The complexity of the proposed computational approach is interpreted in terms of tabulated data for each fluidic parameter.The mesh points analysis of the considered fluidic model acquired by the proposed numerical approach is  tabulated in Table 6 while Table 7 depicted the computed values of residual error by Lobatto IIIA approach for each sundry physical fluidic parameter, enlisting five different scenarios and three cases for NF and HNF.A comparative analysis is presented for the proposed method with other numerical methods.In Reference 41, the authors have presented the solution of the current problem by using the shooting method and Runge-Kutta method of order six and in Reference 40, the authors have presented the solutions by using the shooting algorithm.Here, the authors have compared the results for particular values of parameters of volumetric fractions and Nusselt numbers in Tables 8  and 9 for different values of .The current study has obtained better results as compared to Runge-Kutta and shooting methods and performed almost similar to both techniques while using a simplified and less complicated algorithm.

CONCLUSION, NOVELTY, AND RECOMMENDATIONS
This article portrays a numerical investigation of porous surfaces for convection.The nanofluidic model is based on the inverted cone along with porous boundary for both NF SiO 2 /water and HNF MoS 2 -SiO 2 /water respectively.We observed that fluid velocity reduces with a rise in  and as compared to HNF.The decrease in the flow of fluid is greater in the case of NF.It is also observed that the flow of fluid for SiO 2 /water NF decelerates more as compared to the HNF.The fluid flow decelerates as  increases, and NFs shows the deceleration more pronouncedly than HNFs.With a rise in S > 0, velocity decreases.Moreover, it has been observed that fluidic flow slows down more for SiO 2 /water NF than for HNF.Both forms of NF flow are accelerated by a rise in Eckert number.In the case of HNF, this rise is also much greater.The momentum and fluid flow boundary layer thickness of NF and HNFs are accelerated by a rise in the heat radiation parameter N.For both NF and HNFs, the skin friction coefficient and Nusselt number rise as the volumetric fractions of nanoparticles increase by  1 and  2 , respectively.
1 and  2 means a reduction in the base fluid viscosity and this accelerates the fluid flow.Furthermore, the fluid velocity is observed more in the case of the HNF.Eckert number Ec influences are indicated in Figure 8.It can be F I G U R E 5 Effect of radiation parameter on f ′ ().

F I G U R E 6
Effect of suction/injection parameter on f ′ ().

Figure 9
considering four different values of the nano-particles shaping factors for NF and also HNFS is arranged for various values of Eckert number.From such a figure it has been shown that the distribution of temperatures in both forms of NFs increases with the Eckert number.Figures 10 and 11 are plotted for increasing values of power law index  and N (thermal radiation parameter) respectively.

F I G U R E 10
Impact of power law index on ().F I G U R E 11 Impact of radiation parameter (N) and m on ().F I G U R E 12The impact of volumetric fractions  1 and  2 on ().
Experimental values for fluidic system.Experimental values for NF system.
Illustration of physical problem in cartesian coord.TA B L E 1

TA B L E 4
The numerical data analysis of local Nusselt number for SiO 2 /H 2 O and MoS 2 -SiO 2 /H 2 O.The numerical data analysis of skin friction for SiO 2 /H 2 O and MoS 2 -SiO 2 /H 2 O.The effect of mesh points on convergence of SiO 2 /H 2 O and MoS 2 -SiO 2 ∕H 2 O.
The illustration of residual error analysis of SiO 2 /H 2 O and MoS 2 -SiO 2 /H 2 O.The comparison of C f for volumetric fractions  1 = 0 =  .