Dynamics of Nanoplatelets in Mixed Convective Radiative Flow of Hybridized Nanofluid Mobilized by Variable Thermal Conditions

The present mathematical model discloses the effects of Boussinesq and Rosseland approximations on unsteady 3D dynamics of water-driven hybridized nanomaterial with the movements of nanoplatelets (molybdenum disulfide, MoS 2 and graphene oxide, GO). Variable thermal conditions, namely, VST (variable surface temperature) and VHF (variable heat flux), are opted to provide temperature to the surface. MHD effects have also been used additionally to make the study more versatile. In order to transmute the transportation equations into nondimensionlized forms, similarity transformations have been adopted. The Keller-Box technique has been applied to obtain a numerical simulation of the modeled problem. The convergence of the solution for both VSTand VHF cases is presented via the grid independence tactic. Thermal setup against escalating choices of power indices and nonlinear thermal radiation parameter is discussed via graphical illustrations. The rate of heat transaction has been discussed with the growing choices of mixed convection, thermal radiation, and unsteady parameters through various tabular arrangements. It is observed through the present analysis that mixed convection parameter, radiation parameter, temperature maintaining indices ( r, s ) , and unsteady parameter magnify the rate of heat transference under the control of platelet-shaped nanoparticles.


Introduction
In the last few years, the topic of nano uids has become a major topic of research. Nano uid designates a liquid suspension containing particles of diameter less than 100 nm. Makinde and Aziz [1] inspected the e ect of convective-heating boundary constraints on nanoparticles. Mixed convection impact along with the steady dynamics of hybridized nano uid is explored by Waini et al. [2] along a vertical surface. Gul et al. [3] premeditated the hybridized nanomaterial stirring on a dispersion sheet with the impression of magnetic dipole. Moreover, Waini et al. [4] conferred the e ects of hybridized nanomaterial for heat transaction process for identical shear ow on a shrinking sheet. Forced convective heat transference of water conveying hybridized nano uid Al 2 O 3 and Al 2 O 3 − Cu with particle size 15 nm and 0.1% volume concentration was considered by Moghadassi et al. [5] and proposed that this model gives better results experimentally as compared to single phase tactic. Stagnation ow of TiO 2 − Cu/H 2 O hybridized nano uid over an elongating obstacle with the induced magnetic eld e ect is incorporated by Ghadikolaei et al. [6]. Here, it was noticed that hybridized nano uid proliferates heat transmission. Hybridized nanomaterial Ag − MoS 2 with base uid C 2 H 6 O 2 −H 2 O over a prolonging obstacle is incorporated by Ali et al. [7] by considering activation energy aspects. Here, enhancement of thermal transference was the core objective of the reported work. e thermal features of Williamson hybridized nanofluid (MoS 2 + ZnO) by considering engine oil as a base fluid are investigated by Yahya et al. [8] with the incorporation of heat source, thermal dissipation, and invariant magnetic field aspects. Heat transference in the stream of Maxwell hybridized (GO − MoS 2 ) nanofluid with the engine oil as a working liquid over a fluctuating vertical cylinder is addressed by Arif et al. [9]. e thermal performance of heat exchanger devices can be improved by the submersion of different shapes of nanoparticles into base fluids. e study of platelet-shaped nanoparticles (PbI 2 − PbMnI 2 ) surrounded by a polymer matrix is discussed by Savchuk et al. [10]. Possessions of different forms of nanoparticles such as blades, bricks, platelets, spherical, and cylindrical on the enactment of tube heat exchangers have been scrutinized by Elias et al. [11]. Anselmo et al. [12] investigated the biological utilization of platelet-shaped particles to target vascular injuries and summarized that platelets particles have an ability to interact with vascular injury sites and have capability to marinate the vascular walls. Platelets nanoparticles are facilitated by their flexibility, shape, and composite device interactions. C CuO/H 2 O forced convection nanofluid is explored under the inspiration of magnetic field by Sheikholeslami [13] and conferred the shapes effects of nanoparticles with the utilization of pedesis motion in the thermal stirring of nanofluids. Bahiraei and Monavari [14] glimpsed the hydro-thermal characteristics of microplate heat exchanger and studied nanoparticle shapes of blade, brick, platelet, and oblate spheroid. Here, it was deduced that platelet-shaped particles modify the performance of heat exchanger devices. Hayat et al. [15] examined the time-independent two-dimensional mixed-convection stream of micropolar fluid on nonlinear extendable device. Unsteady three-dimensional mixed-convection flow was considered by Hayat et al. [16] with the possessions of thermal conductivity and variable velocity on an exponentially elongated surface. Devi and Devi [17] numerically inspected the effects of Newtonian heating and Lorentz force on bidirectional movement of water-driven hybridized nanofluid with the thermal mixture of Cu and Al 2 O 3 nanoparticles. From this consideration, it is examined that hybridized nanofluid delivers the better rate of heat transference than conventional nanofluids. Khan et al. [18] portrayed the magnetowater nanofluid with nonlinear thermal-radiation effects in mixed-convection stirring. Here, characteristics of heat transference in manifestation of convective boundary conditions were explored. Elsaid and Abdel [19] elaborated the mixed convection aspects of hybridized nanofluid H 2 O − Cu/Al 2 O 3 with the effects of adjustable temperature and thermal radiative flux. Here, it was depicted that thermal radiation increases the rate of heat transference by 12%-22% according to the ratio of tiny particles within the base fluid. e collective implications of magnetohydrodynamic (MHD) flow, thermal radiation, and Lorentz force produce a vibrant role in the improvement of film blowing, wire coating, fiber spinning, electron ramifications, nuclear weapons, polarization process, heat-conduction process, petroleum reservoirs. Hayat et al. [20] also deliberated MHD three-dimensional nanofluid with the features of nonlinear thermal radiation and partial slip on a permeable extending surface. Mahanthesh et al. [21] scrutinized the combined features of radiation and nonlinear thermal convection in three-dimensional boundary layer stream of Newtonian fluid, numerically.
Khashi'ie et al. [22] scrutinized the heat transference and flow properties of hybrid Cu − Al 2 O 3 /water nanofluid with the aspects of suction, joule-heating, and MHD on an elongating obstacle. Here, 10% of AlO 3 volume fraction with 3% ≤ Cu ≤ 9% volume fraction was picked to scrutinize the thermal performance of the system. Tlili et al. [23] considered the three-dimensional MHD stirring of hybridized nanofluid with aluminum containment of AA7072 and AA7075 in methanol liquid above an irregular thickness surface with the slip effects. Here, the influence of Lorentz force was estimated much less for the stirring of hybridized nanofluids as equated to conventional nanofluids. Abbas et al. [24] numerically explained the stirring of hybridized nanofluids on a stagnation point flow with an inclined magnetic field over a nonlinear elongating cylinder by utilizing the theory of Xue and Yamada-Ota models. e deviation in the temperature variation at the geometric surface is benevolent for several engineering and industrial solicitations. Yacob et al. [25] investigated numerically the Falkner-Skan problem for moving and static devices in a nanofluid with prescribed surface heat flux. Prakash and Devi [26] depicted hybridized nanofluid stirring Al 2 O 3 /Cu water with the aspects of hydromagnetic Lorentz force and prescribed surface temperature over a slandering expandable sheet. Here, it was deduced that hybridized nanofluid has better efficiency than nanofluid. Waini et al. [27] incorporated the hybridized nanofluid stirring and heat transference on a thin needle with prescribed surface heat flux. Dynamics of Cu/H 2 O nanofluid with the consequences of prescribed surface temperature and prescribed heat flux have been scrutinized by Ahmed et al. [28]. Some recent novel contributions related with the applications of hybrid and nanofluids are made by researchers [29][30][31][32][33][34].
Under the circumstances of the above inclusive literature survey, it is perceived that much consideration has not been given to the stirring of hybridized nanofluid towards bidirectional elongating geometry. erefore, the main theme of the current investigation is to examine the aspects of platelet-shaped nanoparticles (molybdenum disulfide and graphene oxide) for radiative water-driven hybridized nanofluid flow towards an unsteady bidirectional elongating obstacle with variable thermal constraints. Furthermore, the action of Lorentz force is also incorporated in the mathematical model. Apposite mathematical relations/symbols have been acted to renovate the leading equations into a dimensionless system, and then, numerical simulation is made via the Keller-Box method [35][36][37]. Finally, the chief results obtained through the current mathematical study are presented via different graphs/tables.

Mathematical Formulation
Cartesian configuration (as shown in Figure 1) is adopted in order to illuminate the unsteady fluid model for the bidirectional stirring of water-driven hybridized nanofluid with nanoplatelets (molybdenum disulfide, MoS 2 and graphene oxide, GO). To incorporate the MHD (magnetohydrodynamics) process with variable strength B 0 , mathematical relation of Lorentz force is used, however, to examine the impact of thermal radiation mathematical relation of Rosseland approximation is carried. e tiny particles are deliberated in thermal equilibrium under the response of mixed convection. Flow is laminar and incompressible by virtue of no-slip phenomenon. e velocity u w � ax/1 − ct; a > 0, c > 0 is chosen along x-axis, and the velocity v w � by/1 − ct; b ≥ 0 is suggested along y-axis, but the space 0 < z < ∞ is shielded by the hybridized nanofluid. Two categories of thermal conditions, i.e., VHF (variable heat flux) and VST (variable surface temperature), are functionalized to provide nonuniform temperature distribution at the device. Table 1 is made to recapitulate the thermophysical characteristics of water H 2 O, molybdenum disulfide MoS 2 , and graphene oxide GO.
In the persistence of the above suppositions with boundary layer theory, the transport rheology is established as e thermal conditions and velocity field for equations (1)-(4) are conveyed as where g e represents the gravitational pull, T illustrates the temperature at the surface, (u, v, w) entitle the velocity components along x-, y-, and z-directions, respectively, (r, s) are power indices, time factor is suggested by t, T 0 and T 1 are dimensional constants, μ hnf is chosen to describe the effective viscosity of the hybridized mixture, k hnf is taken to mark the thermal conductivity of the hybridized mixture, ρ hnf is chosen to label the density of the hybridized mixture, α hnf is selected to state the thermal diffusivity of the hybridized mixture, β hnf is used to characterize the coefficient of thermal growth, C p hnf designates the specific heat capacity, and σ hnf is illustrated to express the stimulus of electrical conductivity of the hybridized nanofluid. e mathematical relations to familiarize the plateletshaped nanoparticles for the present evolution of hybridized nanomaterial are discussed as where quantities of molybdenum disulfide and graphene oxide tiny particles are expressed through ψ 1 and ψ 2 , respectively. e expression for radiative heat transaction is composed as (Ref. [21]) where k * is allotted for the mean-immersion coefficient, and σ * implied for the Stefan-Boltzmann constant. It is worthy to mention here that T 3 in equation (7) was prolonged about T ∞ in most literature related with thermal radiation; however, in the current scenario, this has been escaped to get more convincing and significant outcomes. us, the manifestation for thermal radiation in equation (7) is nonlinear, and it yields the effect of nonlinear thermal radiation for present flow exploration. e set of equations adopted to nondimensionalize the present mathematical formulation is written as x r y s ϕ(η).

(9)
With the incorporation of equations (8) and (9), principal equations become With boundary restrictions, e current model can be reduced to unidirectional flow by considering α � 0 and reduced to axisymmetric flow by adopting α � 1.0. Here, λ t � Gr/(Re x ) 2 illustrates the buoyancy factor, Gr � β f g e (T w − T ∞ )x 3 /ϑ 2 f represents the Grashof number, θ w � T w /T ∞ symbolized for the temperature constraint with T w > T ∞ , Hartman number is documented by M � (σ f /aρ f ) 1/2 B 0 , unsteadiness is stated by

S � c/a, stretching effect is expressed by α � b/a, Prandtl number is specified by Pr
the radiation factor, and (ε 1 , ε 2 , ε 3 , ε 4 ) are the relationships for current hybridized mixture and are elaborated as ε 1 � 1 + 37.1ψ 2 + 612.6ψ 2 2 1 + 37.1ψ 1 + 612.6ψ 2 e most captivating quantity for thermal processes is the local Nusselt number and is communicated in the form of Reynolds's number

Keller-Box Simulation
is numerical scheme introduced by Keller can be used in other physical models. is numerical technique has a quick convergence capability than other numerical techniques (such as shooting method, RK-method, and RKF-45 method). e Keller-Box method can be applied to solve those ODEs or PDEs whose linearization is possible. Keller-Box method has the following sequence of steps, and its coding procedure is described in Figure 2.
(i) Transformation process for second and higher-order differential equations into first-order differential equations (ii) Completion of discretization scheme using central difference approximations (iii) Transformation of the differential system into difference system of equations using the above discretization scheme (iv) Application of Newton Raphson method in order to linearize the difference equations system (v) Formation of the matrix-vector form of the system (vi) Solution of the matrix-vector form by using LUdecomposition tactic Assume the grid points [η 0 , η 1 , η 2 , . . . η N ], where η 0 � 0, η N � η max < ∞, and intermediate steps have not same length. To attain first approximation, we chose η 0 � 0, η ∞ � 15, n p � 600, h � η ∞ − η 0 /n p , and we got the required result, i.e., ε � 10 − 5 is achieved by varying the value of n p (the numbers of grid points) with the reduction in the value of h (step size).  Table 2 is obtained by selecting η 0 � 0, η ∞ � 15, n p � 600 in order to start the numerical solution with the defined accuracy, and then, the value of n p is increased uniformly up to thirty-six hundred. It is noticed in Table 2 that n p � 600 provides the stable solution for equations (10) and (11), whereas n p � 3000 provides the stable solution for equation (12) under the control of VST modulation. However, the solution is provided up to n p � 3600 to achieve the stability of the numerical solution. Table 3 explores the convergence analysis for present simulation under the regulation of the involved engineering parameters, i.e., α � S � M � 0.5, λ � 0.2, r � s � 1.0, ψ 1 � 0.01, ψ 2 � 0.02, R d � 0.4, θ w � 1.1, λ t � 0.2 for VHF case. e first row of Table 3 is obtained by selecting η 0 � 0, η ∞ � 15, n p � 600 in order to start the numerical solution with the defined accuracy, and then, the value of n p is increased gradually up to three thousand. It is noticed in Table 3 that n p � 600 provides the stable solution for equations (10) and (11), whereas n p � 2400 provides the stable solution for equation (12) under the control of VHF modulation. However, the solution is provided up to n p � 3000 to achieve the stability of the numerical solution.

Results and Discussion
e exposures of newly obtained involved parameters, i.e., temperature maintaining indices r& s and temperature ratio T w on thermal setups [θ(η) (for VST case) and ϕ(η) (for VHF case)] through present mathematical modeling are elaborated via Figures 3(a)-5(b). Figure 3(a) is used to predict the temperature fluctuation under VST modulation, whereas Figure 3(b) is plotted to predict the temperature fluctuation under VHF modulation for r � −2, −1, 0, 1, 2. It is concluded through Figure 3(a) that escalating value of r  reduces the temperature of the nanomaterial. For negative values of r, temperature of the material is observed very high as compared to the positive values of r. Moreover, the width of thermal layer is higher for consecutive negative amounts of r as compared to the consecutive positive amounts of r. e behaviour of the index r on thermal setup for VHF modulation is discussed via Figure 3(b). It is noticed through Figure 3(b) that temperature reduces with the higher values of r, but the temperature is detected higher for VHF case as compared to VST case for smaller values of r. Figure 4

Mathematical Problems in Engineering
For negative values of s, temperature of the material is observed very high as compared to the positive values of s. Moreover, the width of thermal layer is higher for consecutive negative amounts of s as compared to the consecutive positive amounts of s. e behaviour of the index s on thermal setup for VHF modulation is discussed via Figure 4(b). It is noticed through Figure 4(b) that temperature reduces with the higher values of s, but the temperature is detected higher for VHF case as compared to VST case for smaller values of r. e width of thermal layer in the case of s is much smaller than the width of thermal layer for the index r. Physically, the stretching ratio α is higher for x-direction as compared to y-direction. As r represents the index with respect to x-direction and s represents the index in the y-direction, so the above spectacular outcomes are obtained which are highly useful in many thermal engineering systems involved in industry and manufacturing. Figure 5(a) describes the impact of θ w on thermal setup for VST case, whereas Figure 5(b) illustrates the importance of θ w on thermal setup for VHF case. Positive tendencies in the temperature fluctuation for both VST and VHF cases are achieved with the escalation in the choice of θ w from 1.1 to 1.9. e worth of temperature is detected high for VST modulation as compared to VHF modulation. Table 4 enlightens the involvement of solid volume fractions ψ 1 and ψ 2 on local Nusselt numbers under the control of the involved engineering parameters, i.e., α � S � M � 0.5, λ � 0.2, r � s � 1.0, R d � 0.4, θ w � 1.1, λ t � 0.2 for both VST and VHF cases. From Table 4, it is established that increasing values of ψ 1 and ψ 2 , and local Nusselt numbers are enhanced in both VHF and VST cases. Table 5 discloses the impact of λ t (buoyancy factor), R d (radiation factor), and S (unsteady parameter) on Nusselt numbers under the influence of α � M � 0.5, λ � 0.2, r � s � 1.0, ψ 1 � 0.01, ψ 2 � 0.02, θ w � 1.1. It is noticed through Table 5 that increasing values of buoyancy factor, radiation factor, and unsteady parameter enhance the local Nusselt numbers for both VHF and VST cases.

Conclusions
is study gives the mathematical exploration for unsteady bidirectional dynamics of nanoplatelets (MoS 2 + GO) under variable thermal aspects. Nonlinear thermal radiation, magnetohydrodynamics, and mixed convection features have also been studied to make the investigation more impactful. Solution is made through Keller-Box technique, and significant observations are itemized as follows:    (i) Temperature profile is enhanced with the escalation of θ w , and it is reduced with appreciations of thermal indices (r, s) (ii) Nusselt number is observed higher for VHF case than VST case (iii) ermal performance of the system is enhanced with the suspension of nanoplatelets (molybdenum disulfide and graphene oxide) in water e outcomes of the present investigation are helpful in the development of solar collector, coating a sheet with hybrid nanomaterials, improvement of the thermal system used in industry, etc. In future, this work can be extended with non-Newtonian hybrid nanofluids and ternary hybrid nanofluids.
Data Availability e raw data supporting the conclusions of this article will be made available by the corresponding author without undue reservation.

Conflicts of Interest
e authors declare that there are no conflicts of interest.