Significance of magnetic field and chemical reaction on the natural convective flow of hybrid nanofluid by a sphere with viscous dissipation: A statistical approach


 Hybrid nanofluid, which is a combination of Propylene Glycol (PG) – Water (H2O) admixture and paraffin wax and sand, may be utilized as a standby for PG and (H2O) blend in solar thermal framework. Objective of this article is the exploration of the dissipative flow propylene-glycol and water mixture based hybrid nanofluid by a sphere with chemical reaction and heat source parameters. MATLAB in-built solver bvp4c is utilized to exhibit the impacts of various parameters on regular profiles including temperature. Correlation coefficient is utilized to elucidate the impact of pertinent parameters on engineering parameters of concern, such as, surface friction factor. Main findings of this work are magnetic field is having a negative association with friction factor and chemical reaction is consuming a significant positive relationship with Sherwood number. It is witnessed that heat source and Eckert number are useful to meliorate the fluid temperature. Furthermore, validation is performed among our results and earlier published outcomes. Good agreement is detected.


Introduction
Nanofluids have been introduced as a new interesting kind of heat transfer fluids to replace the use of regular fluids in industrial processes. They may diminish erosion and corrosion significantly owing to their small size. They have a broad range of applications including refrigeration, heat exchangers and cooling of electronic devices. Raju et al. [1] considered stretching cylinder and examined the Maxwell fluid flow with Cattaneo-Christov heat flux. Later, various researchers [2][3][4][5] assumed stretching sheet as a geometry and theoretically analysed distinct nanofluid flows with different parameters including viscous dissipation. Later, Updhya et al. [6][7][8] scrutinized various nanofluid (including Water + Iron Oxide) flows over varous geometries including moving surface. They detected that the heat transfer rate is high in the presence of ferrous oxide nanoparticles compared to the presence of aluminium oxide nanoparticles in the base fluid. Saba et al. [9] discussed the nanofluid (CNT based) flow by a curled elongating surface with heat absorption/generation and detected that the curvature parameter minimizes the friction factor. Durgaprasad et al. [10][11][12] inspected distinct three dimensional nanofluid flows, for instance, water and graphene mixture, across various geometries with different parameters including non-uniform heat source/sink. Sreedevi et al. [13] used FEM to solve the mathematical model related to nanofluid (water + Al 2 O 3 / water + TiO 2 ) flow by a wedge with thermal radiation. They observed that there is a reduction in the heat transfer rate for bigger values of radiation parameter. Upadhya et al. [14] used the combination of Runge-Kutta and Newton-Raphson procedures to numerically investigate the nanofluid flow among the stationary and porous disks with suction/injection. Eid [15] considered the Riga surface and scrutinized a Darcy-Forchheimer flow of Cu/CMC -Al 2 O 3 with heat sink/source. Newly, several researchers [16][17][18] applied different methods to analyze the various nanofluid flows through distinct geometries. They noticed that the chemical reaction parameter raises the rate of mass transfer and heat transfer rate is high in the absence of magnetic field when compared to its presence. Hybrid nanofluid is a modified version of mono nanofluid, which contains more than one nanoparticle. So, hybrid fluids are having better heat transfer features compared to mono fluids. These are having applications in many areas including solar collectors and military equipment. Ghadikolaei et al. [19] studied the shape factor of the nanoparticles in HNF (TiO 2 -Cu/water) flow by an elongating sheet with magnetic field and emphasized that the nanoparticles with platelet shape are more efficient. Tassaddiq et al. [20] elucidated the features of mass and heat transfer on the HNF (CNT + Fe 3 O 4 /H 2 O) by a revolving disk with magnetic field and noticed the amelioration in the fluid temperature with the raise in disk rotation. Abbas et al. [21] explained a HNF (Ag + Ni/water) flow of two models (Yamada-Ota and Xue) by an elongating cylinder with the aid of bvp4c technique. They observed that the Xue model has less heat transfer rate when compare to other model for temperature gradient. Ahmad and Nadeem [22] considered the upper part of horizontal surface and performed irreversibility analysis in the HNF (MWCNT + SWCNT/water) flow by using the same technique. They noticed that there is an escalation in the entropy with the raise in nanoparticle volume fraction. Later, with the aid of suction and injection, Nadeem et al. [23] provided a comparative analysis for the HNF (Cu -Al 2 O 3 /water) by an exponential curled sheet. Freshly, several researchers [24][25][26][27][28] made contribution to the analysis of hybrid nanofluid flows by considering various geometries.
Fluid flow by a sphere has practical applications in numerous areas of technology, for instance, mineralogy, food engineering and oilfield drilling. Yih [29] inspected the features of heat transfer on the non-Darcy dissipative flow of MHD fluid flow by a sphere immersed in a porous medium. He detected that the heat transfer rate shrinks with bigger Eckert number. Later, Molla et al. [30] and Alam et al. [31] considered sphere and elucidated the dissipative flow of MHD fluid with heat source. They noticed that the magnetic field parameter minimizes the friction coefficient and Nusselt number minifies with larger heat source parameter. Chamkha et al. [32] applied finite difference method to present the analysis of convective flow of a nanofluid through sphere with Brownian motion. Tham et al. [33] observed that bioconvection parameters have no significance on temperature in their investigation on the bioconvective flow of nanofluid (water based) through a sphere with thermophoresis. Amanulla et al. [34] considered the same geometry and theoretically explained the Darcy-Forchheimer flow of Prandtl-Eyring fluid with slip parameters. Later, Mahdy et al. [35] and Alwawi et al. [36] scrutinized the nanofluid flow (combined with the Casson fluid) by a sphere with magnetic field. They discovered that the Casson parameter minimizes the fluid velocity and magnetic field parameter enhances the entropy generation. Newly, several researchers [37][38][39][40][41][42] applied different strategies to examine the features of mass and heat transfer in various nanofluid flows through the sphere.
After cautious perception of the previously mentioned writing, we aim to discuss the significance of the viscous dissipation on the chemically reactive HNF (Propylene glycol -Water mixture + Paraffin Wax + Sand) flow with Ohmic heating. Results are explicated through plots and a statistical tool named correlation coefficient. Further, validation is performed among the current results and former outcomes and observed a decent accord.

Formulation
Natural convective dissipative flow of hybrid nanofluid (Water based Propylene glycol+ Sand + Paraffin Wax) flow by a sphere with Joule heating and chemical reaction is considered. Assumed that the flow is nearby the lower stagnation point of the sphere. Values of the thermophysical attributes of base liquefied and nanomaterials are exhibited in Table 1. Presumptions for this formulation are (i) Magnetic field of intensiveness B 0 is opposite to flow (y-direction) (see Figure 1).
is the radial length from symmetric axis to the surface.  Flow controlling equations for this problem are characterised as (Alam et al. [18]): and the conditions are specified as

Thermo physical characteristics of HNF
Following similarity transmutations are used to metamorphose the flow driven equations as a set of ODEs (Alam et al. [18]): Here Ψis the stream function, u, v are the constituents of velocity specified by in order to satisfy the continuity Eq. (1). With the aid of (6) and (7), Eqs (2-4) are transmuted to: − Kr ϕ = 0.
Since flow is near the stagnation point of the sphere, ξ ≈ 0, Eqs (8-10) can be rewritten as: On the other hand, conditions in (5) can be rewritten as where

Numerical Procedure
MATLAB built-in function bvp4c is used to resolve the altered Eqs (11)(12)(13) with the conditions (14). Since bvp4c solver is built-in function, it is simple to use this function.
As a pre-process to write the code, first we need to adopt the following assumptions (Waini et al. [31]): Then, using the Eqs (11)(12)(13) with conditions (14), we can develop a subsequent system of ODEs of first order: with the conditions After converting the above system as a MATLAB code, we can execute it to get the required outcomes in the form of graphs.

Interpretation of results
In this study, outcomes are offered for two cases i.e., PG -Water + Paraffin Wax + Sand and PG -Water + Paraffin Wax.

Velocity profiles
When we raise the volume fraction of nanoparticle, there is an escalation in the viscosity of the fluid, which obstructs the fluid flow. So, velocity minifies with the larger ϕ 1 (Figure 2). Fluid particles try to change their direction within the sight of magnetic field. Consequently, fluid velocity diminishes ( Figure 3). Figure 4 exhibits the natural behaviour (increment) of buoyancy ratio parameter on velocity profile.

Temperature profiles
When Ec increases, there is a change of increment in internal friction of the fluid. Due to that reason, fluid temperature upsurges ( Figure 5). Figure 6 elucidated the impact of ϕ 1 on temperature field. It is perceived that temperature ameliorates with the raise in ϕ 1 . This may be due to the reason that larger ϕ 1 adds more viscosity to the fluid, which in turn, enhances the friction between the particles. Raise in magnetic field parameter ameliorates the rate at which linear momentum is transferred from the electromagnetic field to the fluid particle. That means, there is an increase in the energy transfer from electromagnetic field to the fluid particle. So, enrichment in temperature observes with larger magnetic field parameter (Figure 7). Figure 8 displays the fact that the larger H raises the temperature profile. Usually, larger heat source parameter causes proliferation of additional heat within the fluid and in turn, assists to enhance the wideness of the thermal boundary.

Concentration profiles
Mass diffusivity minifies with larger Schmidt number. So, concentration minifies with larger Schmidt number (Figure 9). Typically, with the step-up in chemical reaction parameter, concentration decreases ( Figure 10). More entropy generation may be the reason for this behaviour. That means, they minify the heat transfer rate (Figures 13-14). From Figures 15-16, it is detected that Sc, Γ ameliorate the mass transfer rate. This may be due to reason that Sc, Γ minimize the concentration boundary layer thickness.

Correlation coeflcient and its significance
We have done a statistical analysis by using correlation coefficient to confirm the results related to physical parameters, such as, surface drag force. The Coefficient is a numerical quantity of relationship in the midst of two elements. Worth of the measurement lies among +1 and −1 where former represents positive affiliation and latter represents negative affiliation.
is the expression to assess the correlation coefficient for two variables y and z. P.E (Probable Error) of correlation coefficient helps with choosing the exactness and reliability of the coefficient value. Correlation between y and z is significant if |ryz| P E > 6 (or |ryz| > 6P E ) and unimportant otherwise.
√ p is the expression to assess the Probable Error.
From Table 2, it is clear that magnetic field parameter is consuming a noteworthy negative relationship with surface drag force and the volume fraction parameter of nanoparticle is consuming a positive affiliation with the same parameter. Table 3 revealed the fact that there is a negative association between Ec and heat transfer rate. And also, it is clear that the impact of M is same as Ec on the Nusselt number. Further, it is detected that there is a substantial positive affiliation between Sherwood number and Schmidt number (Table 4).

Validation
Validation is performed among our results and formerly published outcomes. We saw an acceptable concord (See Table 5).

Conclusion
Hybrid nanofluid, which is a combination of Propylene Glycol (PG) -Water (H 2 O) admixture and paraffin wax and sand, may be utilized as a standby for PG and H 2 O blend in solar thermal framework. With these things in mind, investigation is done on the dissipative flow propylene-glycol and water mixture based hybrid nanofluid by a sphere with chemical reaction and heat source parameters. MATLAB inbuilt simulator bvp4c is enforced to solve the transmuted classification of equations. Correlation coefficient is utilized to elucidate the impact of pertinent parameters on engineering parameters of concern, such as, surface friction factor. Additionally, we have verified the current results with the earlier outcomes and saw a decent concord. Primary conclusions of this study are displayed underneath: • Velocity minifies with bigger ϕ 1 .
• Buoyancy ratio parameter meliorates the fluid velocity. • minimizes the friction factor.
• Ec and H are useful to enhance the thermal boundary thickness. So, Nusselt number minifies with the raise in those parameters. • Sherwood Number is consuming a generous positive affiliation with Schmidt number.

Funding information:
The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest:
The authors state no conflict of interest.