Numerical investigation for rotating flow of MHD hybrid nanofluid with thermal radiation over a stretching sheet

This research investigates the heat and mass transfer in 3-D MHD radiative flow of water based hybrid nanofluid over an extending sheet by employing the strength of numerical computing based Lobatto IIIA method. Nanoparticles of aluminum oxide (Al2O3) and silver (Ag) are being used with water (H2O) as base fluid. By considering the heat transfer phenomenon due to thermal radiation effects. The physical flow problem is then modeled into set of PDEs, which are then transmuted into equivalent set of nonlinear ODEs by utilizing the appropriate similarity transformations. The system of ODEs is solved by the computational strength of Lobatto IIIA method to get the various graphical and numerical results for analyzing the impact of various physical constraints on velocity and thermal profiles. Additionally, the heat transfers and skin friction analysis for the fluid flow dynamics is also investigated. The relative errors up to the accuracy level of 1e-15, established the worth and reliability of the computational technique. It is observed that heat transfer rate increases with the increase in magnetic effect, Biot number and rotation parameter.

www.nature.com/scientificreports/ of fluids and judgment of their performance is still under evaluation phase. In recent past years, few researches have been conducted for the comparison of performance between nanofluid and hybrid nanofluids [24][25][26][27] . Magnetohydrodynamics (MHD) is the study where the magnetic field and the velocity field are coupled, given there is an electrically conducting fluid. The magnetic field can induce currents into such a moving fluid and this creates forces acting on the fluid and altering the magnetic field itself. Set of differential equations comprises of Navier-Stokes equations and Maxwell's equations describes the complete phenomenon of MHD. Kashi'ie et al. 28 numerically investigated the flow properties for the dynamics of fluidic system and phenomena of heat transfer for a MHD flow of hybrid nanofluid (Al 2 O 3 /H 2 O) due to stretching sheet while considering the joule heat effects. Osho et al. 29 discovered the flow characteristics of hybrid nanofluid (Al 2 O 3 -Zn/H 2 O) and noticed the significant effect of concentration of nanoparticles over the viscosity and specific heat of the flow. Aly et al. 30 theoretically and numerically studied the MHD stagnation point flow over stretching sheet of hybrid nanofluid with dissipation and slip effects and observed a relationship between MHD and rate of heat transfer. Aghahadi et al. 31 inspected the rheological performance of tungsten oxide-engine oil nanofluid at various concentration and temperature and found a linear relationship between applied shear stress and shear rate. Nagoor et al. 32 numerically explicated the influence of various physical constraints on velocity and temperature fields for Darcy-Forchheimer hybrid nanofluid in rotating frame by using Lobatto IIIA method. Huminic et al. 33,34 discussed heat transfer rate and entropy generation between ordinary and hybrid nanofluid in different physical situations. Saba et al. 35 numerically explored the phenomena of heat transfer for a hybrid nanofluid in an irregular channel with permeable walls. Furthermore, various effective results have been illustrated via plots. Oliverira et al. 36 experimentally studied an innovative method for addition of silver on the surface of diamond nanoparticle for the preparation of hybrid nanoparticles (Di-Ag). Different techniques including scanning electron microscopy (SEM) as well as X-ray diffraction (XRD) are executed to get required information about these hybrid nanoparticles. Lund et al. 37 examined the influence of different factors on the velocity and temperature profiles of a hybrid nanofluid (Cu-Al 2 O 3 /H 2 O) over stretched sheet under the effects of suction and viscous dissipation. Shahsavar et al. 38 inspected the impacts of concentration on entropy generation and heat transfer of non-Newtonian iron oxidebased hybrid nanofluid through concentric annulus. Iqbal et al. 39 inspected the Hall current effects on MHD flow of hybrid nanofluid in revolving channel under thermal radiations with different shapes of nanoparticles. During the recent past, many researchers investigated the heat transfer phenomenon in nanofluid flow [40][41][42][43][44][45][46][47][48] .
The inspiration behind this research work is above referred studies in which several researchers assumed various fluid with different types of nanoparticles and observed fascinating results for their thermal properties. A considerable research is being done about the numerical solution of the nanofluid flow problem [49][50][51] , but very few researchers tried to solve the hybrid nanofluid flow problem with novel numerical techniques. In this article, the authors investigate the problem of 3-D flow of MHD hybrid nanofluid over an extendable sheet in presence of thermal radiation. Main features of this study are as follows: • A novel scheme for 3-D MHD flow of hybrid nanofluid over an extendable sheet with thermal radiation effects has been modeled. System of PDEs expressing the flow model is then transmuted into the set of equivalent nonlinear ODEs while employing the appropriate mathematical transformations. • Detailed numerical study of the flow model is described by implementing the computational strength of Lobatto IIIA method with the aim to scan the influence of involved physical constraints on velocity and thermal fields. • To achieve the required solution of highly nonlinear ODEs, use of Lobatto IIIA technique in MATLAB software for this problem is an inventive work. Lobatto IIIA is the kind of bvp4c scheme depends on FDM. The strength of this technique is to solve the higher order nonlinear ODEs. • Detailed graphical and numerical explanation of result has also been presented, which evidently shows the variation of velocity and thermal fields on several constraints of interest.

Problem formulation
Consider the incompressible 3-D flow of hybrid nanofluid induced by a stretching and rotating effects with thermal convection and radiation along a sheet. The sheet is stretched through selected xy-coordinates system and nanofluid is assumed for z > 0 direction. Velocity components in x, y and z direction are denoted by u, v and w , Table 1. Improvement in heat transfer of fluids by using various nanoparticles. www.nature.com/scientificreports/ respectively. Figure 1 displays the schematic view of flow model in which Fig. 1a presents the geometry of the problem, Fig. 1b shows the microscopic view of surface and Fig. 1c depicts the structure of hybrid nanoparticles. T f and T denote the surface and fluid temperatures respectively, while the applied constant magnetic field acting in parallel direction to z-axis is represented by B 0 and h f is the coefficient of heat transfer. Hence, the balance of mass, balance of momentum and energy can be mathematically expressed as 52,53 : Corresponding boundary condition are: Mathematical relationships for various Thermophysical characteristics for hybrid nanofluids are 54 : (1) . The concentration of Al 2 O 3 and Ag nanoparticles are denoted by φ 1 and φ 2 respectively, whereas φ hnf is the total concentration of mix nanoparticles which can simply be calculated as (φ 1 + φ 2 ) . Values for density, thermal conductivity and specific heat of base fluids and nanoparticles are placed in Table 2, whereas Fig. 2 displays the well-known shapes of nanoparticles with numerical values of size, while ρ f , ρ s1 , ρ s2 represent the density of fluid, Al 2 O 3 particles and Ag particles, respectively. Thermal conductivity of Al 2 O 3 particles, Ag particles, base fluid and hybrid nanofluid is represented by k s1 , k s2 , k f and k hnf , respectively, C p s1 , C p s2 , C p f and C p hnf represent the specific heat of Al 2 O 3 particles, Ag particles, base fluid and hybrid fluid, respectively.
Dimensional form for coefficient of skin friction and Nusselt number can be written as 57 : To reduce the system of PDEs (1-4) into dimensionless set of ODEs, following mathematical transformations are introduced: Substituting above-mentioned transformations, the continuity equation is identically satisfied, while the Eqs.
(2-4) take the following form: whereas,  (6) are reduced to their dimensionless form as: In which Re x = u w x ν f . represents the Reynolds number.

Solution methodology
Transformed set of ODEs representing the flow problems given in Eqs. (8)- (12) are solved numerically by employing Lobatto IIIA technique in MATLAB software using bvp4c package as described in Fig. 3, while the detail information regarding the solution technique is available in 58,59 . The obtained graphical and numerical results portray the impact of all involved parameters on velocity as well as temperature fields. The convergence, stability and accuracy have been checked for solution and computation with the help of residual error for each case of all scenarios. Equations (8)- (12) are transformed to first order system of ODEs by the Lobatto IIIA technique.

Results and discussion
After solving the resultant set of ODEs, various forms of numerical with its graphical outcomes are obtained and displayed in Figs Table 3 to analyze the dynamic. Numerical simulation is performed for each scenario with four cases and observe their impact on the flow dynamics throughout in the presented study. Figure 4a-d show the influence of rotation parameter on f (η), f ′ (η), g(η) and θ(η), respectively, which depict that the rise the magnitude of results in the decline in velocity field and increase in temperature filed. In physical aspect, when the values of is larger, rotation rate gets higher than stretching rate. Therefore, higher values of results in extra resistance for the fluid, so the velocity component behaves as decreasing function of . This study reveals that plays an important role in the aeration of flow in y direction. It is due to the fact that higher values of correspond to higher oscillatory motion of fluid particles. As the sheet is stretched in x-direction and due to rotation effects, the fluid flows towards y direction. Figure 4e, f depict the variable behavior of g(η) and θ(η) against the various values of Pr. An increase in Pr upshots a decline in temperature due to weak thermal diffusivity, therefore temperature field acts as decreasing function of Pr. Figure 5a-d exposed the effects (11) (12)  Figure 6a-d demonstrate the impact of concentration of nanoparticles φ 2 on the f (η), f ′ (η), g(η) and θ (η) , respectively. Enhancement in f (η) and f ′ (η) whereas reduction in g(η) has been noticed for higher concentration of nanoparticles. Figure 6e, f represent the variation of g(η) and θ (η) against different values of γ . Higher values of γ give increase in heat transfer rate of flow. This is because γ depends on coefficient of heat transfer " h f " which has larger values for greater γ . Above discussion shows that the rate of heat transfer increases with the increase in Magnetic effect,iot number and rotation rate. It is also noticed that magnetic parameter M and rotation parameter have qualitatively same effect on velocity g(η).    Table 5. Best value for relative errors was observed for case 1 of scenario 3 in which relative errors up to 4.1871e-13 and 4.1871e-15 are observed for 1e-10 and 1e-12 convergence limits. Table 6 shows the number of evaluations for BCs, ODEs mesh points during computational process to achieve the targeted value of accuracy. It is seen from Tables 5 and 6 that for small convergence limit, the value of relative error is

Conclusions
In this study a numerical treatment for 3-D MHD flow of hybrid nanofluid over a stretchable sheet under the effects of thermal radiation has been conducted. Important findings of this research are listed as: In future one may explore the different characteristics of 3-D MHD flow of hybrid nanofluid with thermal radiation features through modern and advanced numerical computing skills based of artificial intelligence [60][61][62][63][64][65][66] .
Received: 14 June 2020; Accepted: 9 October 2020 Table 4. Mathematical data for Skin friction and Nusselt number. I  II  III  IV  I  II  III  IV  I  II  III  IV   I I  II  III  IV  I  II  III IV I  II  III  IV   1  www.nature.com/scientificreports/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.