# Convective Heat Transport in Bidirectional Water Driven Hybrid Nanofluid Using Blade Shaped Cadmium Telluride and Graphite Nanoparticles under Electromagnetohydrodynamics Process

Authors: Iftikhar Ahmad; Qazi Zan-Ul-Abadin; Muhammad Faisal; K. Loganathan; Tariq Javed; Ngawang Namgyel
Journal: Journal of Mathematics
 (2022)

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## Abstract

The recent progress in nanotechnology provides the concept of hybrid-class nano-fluids having advanced thermal features comparing to regular nanofluids. The idea of a hybrid nanofluid has motivated many researchers because of its convincible performance in thermal systems. The novel theme of the present effort is to scrutinize the consequences of convective heat transfer in bidirectional water driven hybridclass nanofluid involving blade shaped cadmium tellurideCdTe and graphite C nanoparticles with electromagnetohydrodynamics (EMHD) process. The transport equations representing the aforementioned topic are firstly nondimensionalized by using scaling-group transformation and then tackled by the Keller-box method, numerically. The significant results for pertinent parameters have been simulated and presented graphically as well as in tabular forms. Coefficients of drag forces are diminished with the more loadings of cadmium telluride and graphite nanoparticles and opposite results are noticed in the case of the Nusselt number. Heat transport has been improved significantly with more loadings of nanoparticles from 1 wt% to 10 wt%. A comparison benchmark for a limited version of the investigation is made with the previously published data.


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## Body

## 1. Introduction

The electromagnetohydrodynamics (EMHD) process augments the level of heating/cooling in thermal systems with the flow of hybrid-class nanofluid. The incorporation of magnetized nanoparticles in the fluid increases the performance and strengthens the process of heat transportation. Convective heat transference via stretchy barrier recently has got the attention of many scientific experts due to its extensive coverage in modern technologies, especially in energy storage devices, exchange, and recovery of thermal heat in controlled reactors. The thermal conductivity of conventionally heat transfer fluids plays a significant role on the thermal transport coefficient between the transport medium and the available surface. Das et al. [1] disclosed the convective heat transferal in the flow mechanism of an incompressible fluid with the consequences of second order velocity slip and induced magnetic field via H2O−EG driven nanofluid comprising TiO2 tiny-particles on impermeable elongating sheet. Jusoh et al. [2] discussed the heat transport mechanisms in boundary-layer flow of magneto-nano-fluid that comprises of Fe3O4, Cu, Al2O3, and TiO2 tiny-particles with water as base fluid over a bidirectional extending sheet, here, copper Cu nanoparticles are found clearly decent in thermal-conductivity. However, higher meditation of tiny-particles leads to weakening the heat transaction rate. Irfan et al. [3] considered theoretically the unsteady bio-magnetic nanofluid flow in a porous domain with a thermal link of heat production/consumption across elongating obstacle. The collective effects of EMHD, thermal radiation, Joule heating, and thermal radiation on the steady two-dimensional flow of nanofluid due to nonlinear elongating obstacle are theoretically investigated by Daniel et al. [4]. Here, it is perceived that the electric field is observed to be dominant over the magnetic field, and nonlinear thermal radiation aggregates the temperature profile. Saeed et al. [5] explored the hybrid-class nanofluid flow over a stretchy obstacle by considering a couple of stress nanofluid with solid materials of CNTs under the thermal link of Newtonian heating and viscous-dissipation. Here, the hybrid-class nanofluid is organized due to the deferment of solid tiny-particles i.e., SWCNTs and MWCNTs in pure blood with the function of EMHD. Here, the influence of the embedded constraints on hybrid-class nanofluid flow are discussed and designated that hybrid-class nanofluids are the best thermal-conductor and play significant role in the circulation of blood in human body. Shahzad et al. [6] analyzed MHD rotating flow of Fe3O4 tiny-particles through stretchable obstacle with the manifestation of heat transfer. Here, Coriolis as well as centripetal forces taken into account. Iftikhar et al. [7] carried out effects of MHD, heat convection and different shapes of tiny-particles on stirrings of Cu−H2O nanofluid and SiO2−Cu/H2O hybrid-class nanofluid containing relaxation and contraction phenomenas. Here, it is revealed that diminution is detected in the size of bolus in case of Cu−H2O, while escalation is noticed in the size of bolus for hybrid tiny-particles. Unsteady EMHD stagnated dynamics of hybrid-class nanofluid Al2O3−Cu/H2O towards a shrinking obstacle is reported by Zainal et al. [8]. Kashi’ie et al. [9] analyzed the consequences of different tiny-particles shape factors, EMHD and radiation parameters for Cu−Al2O3/H2O flow towards the shrinking Riga obstacle. Zainal et al. [10] also investigated mathematically heat transference possessions in MHD flow contacting bidirectional extending sheet of hybrid-class nanofluid and revealed that the rate of heat transference is fortified by the intensity of suction consideration, while the presence of heat generation condensed the performance of heat transaction rate. Liu and Andersson [11] analyzed the heat transport in viscous fluid driven by bidirectional extending device with the aspects of variable thermal conditions and internal heat generation.Nanofluids have novel properties that make them potentially advantageous in many applications of heat transport phenomenon including microelectronics, thermal management, fuel cells, hybrid-power engines, heat exchangers, domestic refrigerator, chillers, grinding and machining, and temperature reduction in the boiler flue gas. These fluids incorporate the thermal-conductivity and the convective-heat transport coefficient related to the host fluid/liquid. Pourmehran et al. [12] examined the dynamics of MHD convective nanofluid by considering the buoyancy effect and thermal radiations over a vertical elongating obstacle with three different base fluids and four different tiny-particles. Ahmad et al. [13] addressed numerically heat transaction of dynamics of MHD 3D magnetic nanofluid over a bidirectional stretching sheet with water as base fluid including Fe3O4 tiny-particles. Nayak et al. [14] investigated the stimulus of thermal radiation as well as transverse magnetic field on bidirectional free convective dynamics of nanofluid touching the linear elongating barrier. Here, it is established that the manifestation of the magnetic field reduces the fluid dynamics, whereas it augments the fluid-temperature, and the rate of heat transport diminishes by increasing thermal radiation. Most recent investigations on heat transportation in the dynamics of magneto nanofluid over a bidirectional stretchy sheet have been reported by some experts in (References [15–17]).The advanced class of nanofluid is termed as hybrid nanofluid. Hybrid-class nanofluids for convection heat transfer deliberate the maximization phenomenon of heat transference with the addition of tiny-particles into conventional fluids. It explores thermophysical properties, preparation procedures, and heat transfer features with mathematical models and core applications with implementation challenges of hybrid-class nanofluids. Waini et al. [18] incorporated uniform flow with heat transaction in hybrid-class nanofluid by considering copper Cu and alumina Al2O3 tiny-particles with water as host fluid over a permeable biaxial stretching device. Here, it is observed that dual-solutions occur upto definite choices of suction and stretching parameters. In order to examine the firmness of dual-solutions, temporal-stability scrutiny is presented in the published report. Shoaib et al. [19] scrutinized numerically the heat and mass transference of MHD three-dimensional dynamics of hybrid-class nanofluid Al2O3−Ag−H2O with the consequences of thermal radiations over an enlarging obstacle. Shah et al. [20] numerically investigated the hybrid-class nanofluid Fe3O4−MWCNT−H2O damped in permeable media with Lorentz force effect and free convection interaction. Wanatasanapan et al. [21] experimentally inspected the consequences of TiO2−Al2O3 tiny-particles fraternization ratio on dynamic viscosity, thermal-conductivity, and rheology properties of TiO2−Al2O3/water hybrid-class nanofluid at a temperature between 30° to 70°C. Almeshaal et al. [22] analyzed natural convection dynamics inside a T-shaped enclosure with water based CNT−Al2O3 hybrid-class nanofluid. Here, scrutinization about the consequences of size of solid-particles, fraction of CNT particles, volumetric concentration of solid particles, and Rayleigh number in heat transference process is presented. Some additional considerations about water based hybrid nano-fluids and their modeling are highlighted by some experts in [23–27]. Few of scholar-works also emphasized the impression of tiny-particles shapes on the thermophysical possessions which reveal the thermal augmentation of the nanofluids. Shahsavar Goldanlou et al. [28] glimpsed the turbulent forced-convection heat transport in Fe3O4−CNT/H2O hybrid-class nanofluid through a heat exchanger fortified with blade-shaped turbulators. Furthermore, consequences of different velocities 6000<Re<8000 and tiny-particles volume concentration 0.1vol%<φ<0.9vol% are studied in the published communication. Implications of dual-stretching and suction on 3D dynamics of water driven nanofluid with several shapes and densities are considered by Elnaqeeb et al. [29]. Venkateswarlu and Satya Narayana [30] disclosed the momentous impacts of viscous-dissipation and variable viscosity on porous enlarging sheet via the manifestation of Cu−Al2O3/H2O hybrid-class nanofluid and radiative heating. The uniqueness of this discussion is to present “variable viscosity along with natural convection in momentum equation and viscous dissipation in the energy equation.” Sheikh et al. [31] used the fractional model employing generalized Fourier’s law for MHD dynamics of Casson fluid with cadmium telluride CdTe tiny-particles. Shah et al. [32] presented a wide description of hybrid-class nanofluid, their features, preparations methods, and the starring role of hybrid-class nano-fluids in either direct or indirect energy harvesting system. Ali et al. [33] discussed various applications of advanced energy storage materials used in solar energy, in buildings, electronic cooling, waste heat recovery, and photovoltaic thermal systems. Recent advancements regarding hybrid-class nanofluids are discussed in (References [34–37]).The foremost purpose of the present exploration is to provide a scientific contribution in literature by considering the consequence of convective heat transport in bidirectional water driven hybrid-class nanofluid using blade shaped nanoparticles cadmium tellurideCdTe and graphite C under the electromagnetohydrodynamics process. Cadmium telluride is a crystalline compound and has a lot of applications in semiconducting materials science e.g., manufacturing of solar PV cell. On the other hand, graphite is a crystalline carbon found in igneous and metamorphic rocks. Graphite cleaves easily with light pressure and possesses low specific gravity. Graphite has also a variety of applications in this modern world, e.g., manufacturing of writing materials and batteries, stabilization of nuclear reactors, and helpful in the production of iron, glass, and steel. These nanoparticles have a high tolerance of heat and so are suitable for nanoemulsion. A mathematical formulation of these nanoparticles has been presented in the present analysis. Transports equations are initially modeled as PDEs and then transformed into nonlinear ODEs before computational inspection is made via the Keller-box procedure discussed in [38–41]. Numerical results of this contribution are helpful in heating/cooling processes, nano-drug delivery, power generation, and cancer therapy. Some other recent research studies are seen in the References [42–52].


## 2. Mathematical Formulation

Consider an unsteady bidirectional electromagnetohydrodynamics (EMHD) flow of hybrid-class nanofluidCdTe−C/H2O touching a stretching obstacle with the incorporation of convective heat transport in xy- plane. It is supposed that the device is stretched in both x- and y- directions with velocities uwx,t=ax/1−ct;a>0,c≥0 and vwy,t=by/1−ct;b≥0, respectively. However, the region covered by the hybrid-class CdTe−C/H2O nanofluid is above the xy- plane. Velocity components in x,y and z directions are signified by u,v and w, respectively. The symbol Tw=T∞+a0xrys/1−ct designates the surface temperature with thermal indices r,s, dimensional constant a0 and ambient (surrounding) temperature T∞. Nonuniform magnetic field possessing strength B0=b0/1−ct with initial magnetic strength b0 acted parallel to z- direction, and electric field E0=e0/1−ct3/2 with an initial electrical field e0 is applied in the direction of stretching. Figure 1 describes the graphical abstract of the present investigation. The boundary-layer balances of mass, momentum, and energy with the electromagnetohydrodynamics process are conveyed as follows (References [4, 5, 41]):(1)∂u∂x+∂v∂y+∂w∂z=0,(2)u∂u∂x+v∂u∂y+w∂u∂z+∂u∂t=μhnfρhnf∂2u∂z2+σhnfρhnfE0B0−B02u,(3)u∂v∂x+v∂v∂y+w∂v∂z+∂v∂t=μhnfρhnf∂2v∂z2+σhnfρhnfE0B0−B02v,(4)u∂T∂x+v∂T∂y+w∂T∂z+∂T∂t=αhnf∂2T∂z2.Figure 1
Schematic abstract of the physical model.The convective heat transfer and bidirectional stretching conditions are as follows:(5)z=0:u=ax1−ct,v=by1−ct,w=0,−kf∂T∂z=htTw−T,z⟶∞:u⟶0,v⟶0,T⟶T∞,Here, μhnf deliberates the dynamic viscosity of hybrid-class nanofluid, ρhnf represents the density of hybrid-class nanofluid, αhnf=khnf/ρCphnf (ratio of thermal-conductivity to heat capacity) in terms of the thermal diffusivity of hybrid-class nanofluid, and ht=h0/1−ct illustrates the convective heat transference coefficient with h0 as initial heat convection. In order to incorporate the impact of blade-shaped nanoparticles the thermophysical properties of the tiny-sized elements and based fluid (i.e., water) are given in Table 1, whereas the following mathematical relations are introduced to connect the thermophysical properties of solid-particles with working-liquid as follows (References [9, 41, 48]):(6)ρhnf=ψ1ρp1+ψ2ρp2+1−ψ1−ψ2ρf,(7)ρCphnf=ψ1ρCpp1+ψ2ρCpp2+1−ψ1−ψ2ρCpf,(8)khnfkbf=kp2+7.26kbf+7.26ψ2kp2−kbfkp2+7.26kbf−ψ2kp2−kbf,(9)kbfkf=kp1+7.26kf+7.26ψ1kp1−kfkp1+7.26kf−ψ1kp1−kf,(10)σhnfσbf=1+3σp2/σbf−1ψ2σp2/σbf+2−σp2/σbf−1ψ2,(11)σbfσf=1+3σp1/σf−1ψ1σp1/σf+2−σp1/σf−1ψ1,(12)μhnfμbf=1+14.6ψ2+123.3ψ22,(13)μbfμf=1+14.6ψ1+123.3ψ12,Here, quantities of cadmium telluride and graphite are articulated through ψ1 and ψ2, respectively. The subscripts f,bf,hnf,p1 and p2 denote fluid, based fluid (i.e., water), hybrid-class nanofluid, first type of nanoparticles (i.e., cadmium telluride), and second type of nanoparticles (i.e., graphite).Table 1
Thermophysical measures of cadmium telluride and graphite nanoparticles with water as working fluid (references [41, 50, 52]).
PropertiesBase fluidNanoparticlesH2OPr=6.2CdTeψ1Cψ2Densitykgm−3997.158552060Thermal conductivityWm−1K−10.6137.525Specific heatJkg−1K−14179209.0720Electrical conductivitySm−10.057.0 × 10−75.0 × 102Now, the following scaling group of transformations is introduced to articulate the transport equations (1)–(5) into dimensionless forms (References [16, 41]).(14)u=ax1−ctf′η,v=ay1−ctg′η,w=−aϑf1−ctfη+gη,Tx,y,z,t=T∞+Twx,y,t−T∞θη,η=aϑf1−ctz,Here, θη represents the dimensionless temperature, f′η denotes the dimensionless velocity along x- direction, g′η illustrates the dimensionless velocity along y- direction, ϑf=μf/ρf labels the kinematic viscosity, and η is the similarity variable that connects the dimensional quantities to nondimensional quantities. Outcomes of equation (14) with the aids of equations (6)–(13) are communicated in the following equations as follows:(15)ε1f‴−f′2+f+gf″−Sf′+η2f″−ε2M2f′+ε2ExM2=0,(16)ε1g‴−g′2+f+gg″−Sg′+η2g″−ε2M2g′+ε2EyM2=0,(17)ε3θ″+Prf+gθ′−rf′+sg′θ−Sθ+η2θ′=0.Subject to the boundary conditions,(18)η=0:fη+gη=0,f′η=1,g′η=α,θ′η=−Bi1−θη,η⟶∞:f′η⟶0,g′η⟶0,θη⟶0,Here, Bi=h0/kfϑf/a indicates the Biot number, M=σf/aρf1/2b0 illustrates the Hartmann number and S=c/a specifies the unsteady factor, elongation ratio is conveyed by α=b/a (This parameter is very crucial to maintain the bidirectional flow (Liu and Andersson [11]). We are concentrating to choose the range 0 < α < 1 because for α > 1, the axes x and y are interchanged and for α = 1, the stretching rate becomes alike in both directions; thus, the flow profile turns out to be axisymmetric. Also, considering the limiting situation α ⟶ 0, the flow becomes unidirectional. Wang [46] reported that without any loss of generality, the condition f(0) + g0 = 0 can be restored by f(0) = 0 and g0=0, Prandtl factor is identified by Pr=ϑf/αf, Ex=e0/b0ax describes the parameter for electric-field effect along x- direction, Ey=e0/b0ay illustrates the parameter representing the electric-field effect along y- direction and ε1,ε2,ε3 are the terms/relations interrelated with the involvement of nano-composites and these relations are mathematically conveyed as follows:(19)ε1=1+14.6ψ2+123.3ψ221+14.6ψ1+123.3ψ12ψ1ρp1/ρf+ψ2ρp2/ρf+1−ψ1−ψ2,(20)ε2=1+3σp2/σbf−1ψ2/σp2/σbf+2−σp2/σbf−1ψ21+3σp1/σf−1ψ1/σp1/σf+2−σp1/σf−1ψ1ψ1ρp1/ρf+ψ2ρp2/ρf+1−ψ1−ψ2,(21)ε3=kp2+7.26kbf+7.26ψ2kp2−kbf/kp2+7.26kbf−ψ2kp2−kbfkp1+7.26kf+7.26ψ1kp1−kf/kp1+7.26kf−ψ1kp1−kfψ1ρCpp1/ρCpf+ψ2ρCpp2/ρCpf+1−ψ1−ψ2.The rate of heat transport and coefficients of drag forces in the bidirectional flow of hybrid-class nanofluid are calculated from the following mathematical relations in the form of Reynold’s numberRex=xuw/ϑf as follows:(22)Rex1/2Cfx=1+14.6ψ1+123.3ψ121+14.6ψ2+123.3ψ22f″0,Rex1/2Cfy=α−3/21+14.6ψ1+123.3ψ121+14.6ψ2+123.3ψ22g″0,Rex−1/2Nu=−xkhnf∂T/∂zz=0kfTw−T∞=−khnfkfθ′0.


## 3. Numerical Simulation

The numerical approach that is accurate, completely efficient, and stable for the solution of mathematical problems is termed as Keller-box method and is used here to compute the numerical solution of the investigated model. In this technique, first of all, nonlinear governing equations are transformed into a first order system. After this, using central difference, the first order system is approximated. Since the system of equations is nonlinear, Newton’s method is linked here to linearize the system. Tridiagonal matrix situation is used to transmute the linearized-system into matrix-vector form. To solve the matrix-vector problem LU-decomposition scheme is used and an unknown vector is obtained for further simulation.


### 3.1. Convergence Analysis

Using the aforementioned steps of numerical simulation, Table2 portrays the convergence of the formulated problem. Here, the initial-value of η (i.e. η0) is chosen exactly zero, and then it is augmented regularly to attain the final-value of η (i.e. η∞=20). The further augmentation in the value of η∞ did not disturb the solution. The 1st approximation of numerical inspection is acquired by selecting np=500 (i.e. number of grid points) with step-size h=η∞−η0/np and then the value of np is augmented slowly upto 2500 in edict to accomplish the stable and convergent solution with 10−6 as the desired accuracy. It is concluded via Table 2 that fifteen hundred grid points are sufficient to acquire the convergent-value of f″0, twenty-five hundred grid points are compulsory to get the convergent-amount of g″0 and θ′0 for convective heat transference under the values of involved parameters α=S=M=0.5,Ex=0.3,Ey=0.2,Bi=0.3,r=s=1.0,ψ1=3wt%,ψ2=2wt%. These values of np that deliver the convergent-solution are then further used for the simulation of heat transport.Table 2
Convergence table for convective heat transfer case under the influence ofα=S=M=0.5,Ex=0.3,Ey=0.2,Bi=0.3,r=s=1.0,ψ1=3wt%,ψ2=2wt%.
np−f″0−g″0−θ′05000.9481770.4109850.21563210000.9481740.4109810.2156315000.9481730.4109810.21562920000.9481730.410980.21562925000.9481730.410980.215629


## 3.1. Convergence Analysis

Using the aforementioned steps of numerical simulation, Table2 portrays the convergence of the formulated problem. Here, the initial-value of η (i.e. η0) is chosen exactly zero, and then it is augmented regularly to attain the final-value of η (i.e. η∞=20). The further augmentation in the value of η∞ did not disturb the solution. The 1st approximation of numerical inspection is acquired by selecting np=500 (i.e. number of grid points) with step-size h=η∞−η0/np and then the value of np is augmented slowly upto 2500 in edict to accomplish the stable and convergent solution with 10−6 as the desired accuracy. It is concluded via Table 2 that fifteen hundred grid points are sufficient to acquire the convergent-value of f″0, twenty-five hundred grid points are compulsory to get the convergent-amount of g″0 and θ′0 for convective heat transference under the values of involved parameters α=S=M=0.5,Ex=0.3,Ey=0.2,Bi=0.3,r=s=1.0,ψ1=3wt%,ψ2=2wt%. These values of np that deliver the convergent-solution are then further used for the simulation of heat transport.Table 2
Convergence table for convective heat transfer case under the influence ofα=S=M=0.5,Ex=0.3,Ey=0.2,Bi=0.3,r=s=1.0,ψ1=3wt%,ψ2=2wt%.
np−f″0−g″0−θ′05000.9481770.4109850.21563210000.9481740.4109810.2156315000.9481730.4109810.21562920000.9481730.410980.21562925000.9481730.410980.215629


## 4. Results and Discussion

In this work, inspection on hybrid-class nanofluidCdTe−C/H2O flow with convective heat transport is perceived from graphical results via Figures 2(a)–9(h) by observing the consequences of unsteadiness parameter S, elongation ratio α , Hartmann number M, electric field parameters Ex,Ey, thermal indices r,s, Biot number Bi, load of first nanoparticle ψ1 , and load of second nanoparticle ψ2. The impact of an unsteady factor S on hybrid-class nanofluid CdTe−C/H2O flow on velocity component f′η, velocity component g′η , and thermal setup θη for convective heat transference is discussed through Figures 2(a)–2(c), respectively. It is noticed through Figures 2(a) and 2(b) that the advancement in the adoption of unsteady factor reduces velocity in the starting region and after sometimes escalation is noticed in dimensionless velocities. Thermal setup θη for convective heat transport (CHT) is sketched via Figure 2(c) and progression in the choice of unsteady factor reduces the thermal impact. Physically, the unsteady factor is the ratio of dimensional constant c to the expansion rate a , and thus the escalation in the choice of S produces a lower expansion rate and enhances the impact of the unsteadiness number. Thus, the nano-composite velocity in the present scenario is reduced. Additionally, the thickness of the boundary-stream is also condensed with the enhancement in the choice of unsteady factor. Figures 3(a)–3(c) are enlisted to visualize the provocation of elongation ratio α of hybrid-class nanofluid CdTe−C/H2O flow on velocity component f′η, velocity component g′η , and thermal setup θη for CHT distribution.Figure 2
The results of an unsteady factorS on f′η via panel (a), on g′η via panel (b), and on θη via panel (c).
(a)(b)(c)Figure 3
The results of elongation ratioα on f′η via panel (a), on g′η via panel (b), and on θη via panel (c).
(a)(b)(c)Figure 4
The results of hartmann factorM on f′η via panel (a) and on g′η via panel (b).
(a)(b)Figure 5
The result of electric fieldEx on f′η via panel (a) and the result of Ey on g′η via panel (b).
(a)(b)Figure 6
The result of temperature controlling indexr on θη via panel (a) and the results of s on θη via panel (b).
(a)(b)Figure 7
The results of Biot numberBi on thermal setup θη.Figure 8
The computation of Nusslet number with the pairs (a):ψ1 (cadmium telluride) versus S and (b): ψ2 (graphite) versus S.
(a)(b)Figure 9
The calculations of skin-friction coefficients for graphite and cadmium telluride nanocomposites with the pairs (a and b):ψ2 versus Ex; (c and d): ψ2 versus Ey; (e and f): ψ1 versus Ex; (g and h): ψ1 versus Ey.
(a)(b)(c)(d)(e)(f)(g)(h)Figure3(a) portrays that velocity is diminished with the enhancement of elongation ratio α, whereas Figure 3(b) envisages that the velocity of hybrid-class nanofluid CdTe−C/H2O is improved due to enhancement in the physical range of α and Figure 3(c) depicts that thermal fluctuation is lessened by the augmentation in the range of expansion ratio. As alpha α is the ratio of expansion rate along y- direction to the expansion rate along x- direction. The expansion rate along y- direction dominates the expansion rate along x- direction with the variation in the selection of alpha α which is noticed as the main cause of the thermal reduction. The influence of the Hartmann factor M on the unsteady EMHD flow over a bidirectionally expanding barrier in a hybrid-class CdTe−C/H2O nanofluid is exhibited through Figures 4(a) and 4(b). Hartmann number is the ratio of magnetic force to viscous force. The solution is displayed in Figures 4(a) and 4(b) proves that the inclusion of M eventually increases velocity, dramatically, but this explosion in velocity is seen much superior in Figure 4(b) compared to Figure 4(a). Lorentz force is a resistive force which is instigated by a magnetic field in the dynamics of nanoparticles. Whenever the magnetic field is pragmatic, the resistive force acts to fluid particles, and thus, fluid velocities escalate. Figures 5(a) and 5(b) sketched the effects of electric field constraints Ex,Ey of hybrid-class nanofluid CdTe−C/H2O flow on velocity components f′η and g′η, respectively. It is portrayed through Figures 5(a) and 5(b) that velocity of hybrid-class CdTe−C/H2O nanofluid is enhanced by escalating the choices of electric field constraints Ex&Ey, but it is seen through Figure 5(b) that escalation in velocity is much dominant then Figure 5(a).The implication of heat distributing indicesr and s on thermal setup are conferred via Figures 6(a) and 6(b) for CHT environment. It is noticed through Figures 6(a) and 6(b) that intensifying choices of r and s diminish the thermal setup for heating/cooling mechanisms. Physically, r and s are involved in the energy equation that is the main cause of temperature reduction. Furthermore, negative choices of r and s yield more temperature dissemination than the positive choices of r and s. Likewise, the temperature distribution is activated higher for r<−2, whereas it is triggered higher for all choices of the index s. It is remarkable that the present results for the bidirectional widening barrier show a more prominent influence of r than of s (i.e. temperature distribution along yz plane). The influence of the Biot number Bi on thermal setup θη of hybrid-class nanofluid CdTe−C/H2O is investigated via Figure 7. Biot number is used for heat transference calculations, which is ratio of thermal resistances inside of the body and at the surface of body. It is identified via Figure 7 that thermal fluctuation is increased with the augmentation of the Biot number. Problems concerning small Biot numbers are thermally simple because of uniform temperature fields inside the body. Here, values of Bi are restricted between 0.1−0.9 (i.e. small values). Biot number much larger than one specifies additional difficult problems due to the nonuniformity of temperature fields within the flow regime. Values of Biot number smaller than 1/ten imply that the heat conduction inside the body is considerable faster than heat convection away from its surface.Figure8(a) designates the consequences of reduced Nusslet number for cadmium telluride ψ1 tiny particles with several choices of un-steadiness. It is demonstrated that a higher percentage amount of cadmium telluride nano-composites develops the rate of heat transaction for values of un-steadiness. As the Nusselt number is the ratio of convective to conductive heat transference through the boundary, therefore, augmenting values of S the heat conduction process enhances. Figure 8(b) explores the implications of reduced Nusslet number for graphite ψ2 tiny-particles with various values of unsteadiness. The rate of heat transaction is extremely improved with higher percentage amount of graphite tiny-particles. Moreover, the Nusslet number is also enhanced with the various percentage amounts of graphite nano-composites.Figures9(a)–9(h) explores the consequences of skin-friction coefficients for graphite and cadmium telluride nanocomposites with the pairs Figures 9(a) and 9(b): ψ2 versus Ex; Figures 9(c) and 9(d): ψ2 versus Ey; Figures 9(e) and 9(f): ψ1 versus Ex and Figures 9(g) and 9(h): ψ1 versus Ey. Skin friction is generally expressed in terms of Reynolds number which is the ratio between inertial to viscous forces. Skin friction drag is the component of aerodynamic drag that is a resilient force employed on an object moving in a fluid. Figures 9(a) and 9(b) associates the stimulus of electric field parameter Ex on skin-friction coefficients Cfx&Cfy with different percentage amounts of graphite (i.e., ψ2) when the percentage amount of cadmium telluride (i.e., ψ1) is kept constant (i.e., zero wt%). Escalation in the choice of Ex from 0.1 to 0.7 enhances the skin-friction coefficient Cfx because of the higher rate of electric field parameter Ex (electric field along x- direction), but skin-friction coefficient Cfy diminishes by enhancing the values of electric field parameter Ex and this reduction is not so much clear because the total impact of Ex is along x- direction. Figures 9(c) and 9(d) deduced the consequences of electric field parameter Ey on skin-friction coefficients Cfx&Cfy with different percentage amounts of graphite (i.e., ψ2) when the percentage amount of cadmium telluride (i.e., ψ1) is kept constant (i.e., zero wt%). Escalation in the choice of Ey from 0.1 to 0.7 lessens the skin-friction coefficient Cfx because of a higher rate of electric field parameter Ey (electric field along y- direction), but skin-friction coefficient Cfy augments by enhancing the values of electric field parameter Ey (electric field along y-direction). Figures 9 and 9(f) highlights the influence of Ex from 0.1 to 0.7 on skin-friction coefficients Cfx and Cfy with different percentage amounts of cadmium telluride (i.e., ψ1) when the percentage amount of graphite (i.e., ψ2) is kept constant (i.e., zero wt%). It is deduced that escalation in the choice of Ex heightens the worth of Cfx and reduces the net worth of Cfy (drag-force along y- direction). Moreover, the enhancement in the value of Ex increases the flow rate of the hybrid-class nanofluid at the surface and hence Cfx is improved and Cfy diminishes. Figures 9(g) and 9(h) depict the impact of Ey from 0.1 to 0.7 on skin-friction coefficients Cfx and Cfy with different percentage amounts of cadmium telluride (i.e., ψ1) when the percentage amount of graphite (i.e., ψ2) is kept constant (i.e., zero wt%). It is presumed that escalation in the choice of Ey heightens the worth of Cfx and reduces the net worth of Cfy. Moreover, the enhancement in the value of Ey develops the flow rate of the hybrid-class nanofluid at the surface and hence Cfx is improved and Cfy diminishes. A comparison benchmark with the results from the published paper [11] is made via Table 3 to validate the whole present investigation. In the published paper, Runge–Kutta method (RKM) were used and here we used the Keller-box method (KBM).Table 3
Comparisons of−f″0 and −g″0 for various choices of α when M=S=Ex=Ey=ψ1=ψ2=0.0,ε1=ε2=1.0.
α−f″0−g″0Present (KBM)Published [11] (RKM)Present (KBM)Published [11] (RKM)0.0011000.251.0488114151.0488130.1945639950.1945650.501.0930956791.0930960.4652052980.4652060.751.1344867221.1344860.7946190980.7946191.001.1737220291.1737211.1737220291.173721


## 5. Conclusion

In this report, we numerically estimated the convective heat transport in the unsteady 3D dynamics of hybrid-class water based electromagnetohydrodynamics nanofluid with the movement of blade-shaped nanoparticles, namely, cadmium telluride and graphite. The present report has a variety of applications in thermal engineering, biomechanics, and medical science. The core outcomes of the analysis are as follows:(i)
Thermal and momentum profiles are reduced with the higher estimation of unsteady factor(ii)
Velocity alongy- direction is improved with the appreciation of expansion ratio and electromagnetohydrodynamics constraints, whereas velocity along x- direction is reduced with the positive tendency of expansion ratio, and the opposite trend is observed for electromagnetohydrodynamics parameters(iii)
Thermal setup is condensed with the increasing values of expansion ratio and thermal indices, whereas opposite inference is detected for the case of Biot number(iv)
The Nusselt number is improved with more loadings of cadmium telluride and graphite nanoparticles, and an opposite trend is noticed for the coefficient of drag forces(v)
Rate of heat transport is upgraded with the inclusion of blade shaped nanoparticles



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