# Analytic Simulation for Magnetohydrodynamic Unsteady Buongiorno Model Hybrid Nanofluid Flow over Stretching

Authors: Muhammad Jawad; Salah Boulaaras; Niaz Ali Shah; Rashid Jan; Sulima Ahmed Zubair
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this work, we inspect and analyze a two dimensional, unsteady mixed convectional hybrid nanofluid hydromagnetic flow (Al2O3-Cu/H2O) over a convectional heated an extending/contracting surface with the influence of thermal radiation. Hybrid nanofluid (Al2O3-Cu/H2O) flows with magnetohydrodynamic and heat source or sink. Brownian motion and thermophoresis were incorporated using the Buongiorno model. Hybrid nanofluid with vol. fraction range limited to 1.5% and within the higher temperature range of 50°C to 70°C is considered for thermal conductivity and viscosity analysis. The proposed model is then converted into ODEs through similarity transformation with the help of homotopy analysis. The effect of embedded input factors on the temperature, velocity, and concentration profiles is visually demonstrated and explained. The magnetic field has inverse impact on velocity and temperature profiles. Velocity profile increases for both mixed convection and buoyancy ratio parameters. It has been noticed that the temperature profile increases with thermal radiation. For increasing values of Lewis number, the concentration of hybrid nanoparticles is considerably lowered. Moreover, we observed an increase in the concentration of hybrid nanoparticles through a destructive chemical reaction, whereas a generative chemical reaction has the reverse effect. It has been proved that skin friction is increasing function of ε,Mand decreasing function of λ1,Nr. On the other hand, Nusselt number increased with the increase of R,Q,Nb,Nt while Sherwood number is decreased, with the increase of Nb,Nt,Le.


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## Body

## 1. Introduction

It is eminent that many studies have been carried out on two cases of nanoparticles floating in a base fluid known as “Hybrid Nanofluid” the forefront nanofluid. The key importance of the hybrid nanofluid is that by selecting the right mix of nanoparticles, favorable characteristics may be enhanced, and drawbacks can be mitigated owing to their interactive influence. It is reported that these hybrid nanofluids are new and have a number of application acoustics, defense, manufacturing, transportation, medical, microfluidics, and naval constructions. Nanofluid flow, in particular, is long familiar for its high heat conveyance when equated to regular fluid. The hybrid nanofluid is applied to raise it even further. Jamshed and Aziz [1] studied the Cattaneo-Christov heat flux impact and discovered that spherical shaped nanoparticles transmit heat at the fastest rate when compared to hexagon, platelet type nanoparticles. The authors in [2] examined squeezing flow in a hybrid base fluid with nanoparticles suspended in it. The hybrid nanofluid stagnation point flow past an extending sheet was investigated by the researchers [3]. The influence on flow between two riga-plates was investigated by Ahmed et al. [4]. The authors in [5] studied the impact of changing viscosity on a stretched sheet containing hybrid nanoparticles using the Runge–Kutta fourth order technique. The researchers in [6] interrogated the synthesis of hybrid nanofluid while in [7] examined the effects of a hybrid nanofluid of (Al2O3-Cu/H2O) on heat transmission.There are numerous numerical methods in the literature to inspect the boundary layer flow and heat transport of a hybrid nanofluid. These concepts were further explored by the researchers in their research [9] to conceptualize the main idea. The problem was then expanded to a three-dimensional flow capable to the Newtonian heating consideration by the authors [10]. In both the above studies, they obtained key results about the heat transfer. The difficulty of a hybrid nanofluid consisting of a rotating flow was described by Hayat and Nadeem [11]. Zainal et al. [12] inspected the unsteady stagnation point flow of a hybrid nanofluid past a convectional heated an extending/contracting surface while accounting for velocity slip’s influence on heat transfer. Using the Buongiorno model, Daniel et al. [13] investigated an unstable blended convectional electrical magnetohydrodynamic (MHD) flow and heat conveyance generated by nanofluid across a permeable stretched surface. The friction coefficient and wear volume were studied by Xie et al. [14] to determine the tribological properties of hybrid nanoparticles. Devi and Anjali [10] investigated the three-dimensional flow of (Cu-Al2O3/water) copper alumina/water hybrid nanofluid using the RK-Fehlberg integration technique.Thermal radiation’s effect on MHD blood flow and heat conveyance in a permeable capillary in extending motion was inspected. It has been examined that at high temperatures, heat transfer and thermal radiation are recognized to have a significant influence on numerous physiological processes, technological, and engineering industry equipment. Solar collector performance, plume dynamics, rocket propulsion, high-dose cancer therapy, fire propagation, material processing, and combustion systems are just a few examples. With the advancement of analytical and computational techniques, thermal convection flows with high radiative flux have received more attention recently. The effect of heat radiation on peristaltic transport of ionic nanoliquids in biomicrofluid channels had analyzed by Prakash et al. [15]. In context from melting heat transfer and thermal radiation, the researcher [16] proposed the dynamics of stagnation point flow of carbon nanotubes. To get series solution, the optimal homotopy technique was used. Muhammad et al. [17] make another notable attempt to address the key aspects of heat radiation and viscous dissipation impacts in a viscous nanofluid. Mixed convectional and slide impacts in viscous nanofluid across an extended sheet were described by Hsiao [18]. The solution was numerically defined using the convective boundary conditions [19]. The reader can further study about heat transfer through nanofluid flow in refs. [20–27]. The relevant study has been seen in [28–31].The main aim of this research is to see how magnetic fields, thermal radiation, and heat generation/absorption affect unstable hybrid nanofluid hydromagnetic flow (Al2O3-Cu/H2O) across a convectional heated an extending/contracting surface. The thermophoresis and Brownian motion characteristics were included in the Buongiorno model. To minimize the autonomous variables in governance equations resulting from mathematical modelling, a suitable collection of dimensionless variables is employed. By implementing the homotopy approach, an analytical solution has been calculated. Comparison between HAM and ND solve has been shown in Table 1.Table 1
Comparison of HAM with ND solve solutionϕ1=0.25,ϕ2=0.25,ε=0.3,Nr=1.0,λ1=0.2,Q=0.5,Pr=6.5,M=0.5,R=0.5,Sc=1.0,Nt=0.2,Nb=0.1.
ηfηf′ηθηΦηHAMND solveHAMND solveHAMND solveHAMND solve0000.024537600.024529780.4870.4870.550.550.50.034695680.034686590.040768650.040564210.508740010.508739860.612850070.612849991.00.060349580.060338850.204259120.204257420.512434320.512424150.551624020.551624161.50.123731580.123730980.217686350.217670850.527859020.527858990.845724170.845724062.00.115242380.115194270.227534760.227530870.537620420.537619970.854651070.854650992.50.285052170.285052140.108437050.108436040.547020620.547010151.046400871.046400153.00.370869180.370869080.107843040.107842910.490933520.490932951.044656011.044655923.50.392924870.392924670.020306970.020306910.475296520.475296051.035476181.035476084.00.409462080.409461790.023630140.023630010.488427520.488427021.038404471.038403964.50.415753080.415753050.041572260.041554230.496056020.496055951.039053271.039153065.00.424072580.424072390.042163020.042162710.397542620.397541871.057433071.057432985.50.448125380.448106150.043416040.043415810.355047320.355047111.058167371.058167066.00.452424140.452424090.045390120.045389970.346488620.346487951.009087971.009087066.50.473657470.473657290.046832010.046831920.357424720.357034150.942534570.942534067.00.495027980.495027960.047650240.047649810.375085820.375085650.927186080.927185947.50.503920980.503918960.048520790.048519720.387109020.387108750.858427010.85842689


## 2. Mathematical Formulation

Assume the unsteady hybrid nanofluid hydromagnetic flow (Al2O3-Cu/H2O) past a convectional heated an extending/contracting surface in the bearing of the magnetic field, chemical reaction, heat generation/absorption, and thermal radiation. The flow problem plot is depicted in Figure 1, where uwx,t=bx/1−ct, the extending/contracting is surface velocity, b indicates a constant that agree to extending b>0 and contractingb<0 instances, and c denotes the problem of unsteadiness. The velocity of the free stream is notified by uex,t=ax/1−ct, where a>0 is the strength of the stagnation flow. T1 and T0 are used to represent the ambient and reference temperature separately. Then, we consider that the bottom of the surface is heated by convectional from a hot fluid at a particular temperature Tfx,t=T1−T0ax2/2υ1−ct−3/2 and coefficient of heat transfer has noted by hf while the mass transfer coefficient is indicated by hs. The governing boundary layer equations may be recognized as [12, 13] based on all of the assumptions mentioned.
(1)∂u∂x+∂u∂y=0,(2)∂u∂t+u∂u∂x+v∂u∂y=∂ue∂t+μe∂ue∂x+μhnfρhnf∂2u∂y2+1ρhnf1−C∞βhnfρfT−T∞−ρp−ρfC−C∞g+σhnfβ02ρhnf1−ctu,(3)∂T∂t+u∂T∂x+v∂T∂y=khnfρcphnf∂2T∂y2+1ρcphnf∂qr∂y+τDB∂C∂y∂T∂y+DTΤ∞∂T∂y2+Q0ρcphnfT−T∞,Figure 1
Schematic diagram of the 2D flow past a stretching/shrinking sheet.Applying concept of Rosseland approximationqr is as follows:
(4)qr=−4σ13k∗∂T4∂y,where the parameters σ1 and k∗ are explained in nomenclature
(5)T4=4T04T−3T04.Further simplification leads us to the following:(6)qry=−16σ1T033k∗Tyy.Here, we rewrite Equation (3) as follows:
(7)∂T∂t+u∂T∂x+v∂T∂y=khnfρcphnf∂2T∂y2+1ρcphnf∂qr∂y++τDB∂C∂y∂T∂y+DTT∞∂T∂y2+Q0ρcphnfT−T∞,(8)∂C∂t+u∂C∂x+v∂C∂y=DB∂2C∂y2+DTT∞Tyy,where u denotes the factor of velocity in x-axis, v is the velocity factor in y-axis, μhnf is the Al2O3-Cu/H2O dynamic viscosity, and Db and Dt are the Brownian thermophoretic diffusion terms while ρhnf is the density of Al2O3-Cu/H2O, T is the Al2O3-Cu/H2O temperature, khnf is the thermal/heat conductivity of Al2O3-Cu/H2O, and ρcphnf is the Al2O3-Cu/H2O heat capacity. The boundary conditions, as well as the velocity partial slip, are set to
(9)u=uwx,t+H1υ∂u∂y,v=0,−khnfTy=hfTf−T,DbCy=hsCf−Caty=0,u⟶uex,t,T⟶T∞,C⟶C∞aty⟶∞,whereH1=H1−ct1/2 is the velocity slip term, in which H stands for the starting merit of the velocity slip term. Table 2 lists the thermophysical characteristics of copper (Cu), as well as aluminium oxide (Al2O3) and water (H2O) nanoparticles. The thermophysical characteristics of hybrid nanofluid are shown in Table 3. The solid volume fraction of nanoparticles is described as ϕ, where ρf denotes the density, ρs is the density of the hybrid nanoparticle, cp is the constant pressure of heat capacity, kf represents the thermal conductivity of H2O, and ks is the thermal conductivity of the hybrid nanoparticle.Table 2
Cu thermophysical properties along with Al2O3 and H2O [12].
Thermophysical propertiesCuAl2O3H2OkW/mK400400.613ρkg/m3893339709971cpJ/kgK3857654179β×1051/K1.670.8521Table 3
Applied models for thermophysical properties of the hybrid nanofluid [12].
PropertyHybrid nanofluidViscosityμhnf=1/1−ϕhnf2.5Densityρhnf=1−ϕhnfρf+ϕ1ρs1+ϕ2ρs2Thermal capacityρcphnf=1−ϕhnfρcpf+ϕ1ρcps1+ϕ2ρcps2Thermal conductivitykhnf/kf=ϕ1ks1+ϕ2ks2/ϕhnf+2kf+2ϕ1ks1+ϕ2ks2−2ϕhnfkf/ϕ1ks1+ϕ2ks2/ϕhnf+2kf−2ϕ1ks1+ϕ2ks2−ϕhnfkfElectrical conductivitywhereσhnf/σnf=σs2+σnf−2ϕ2σnf−σs2/σs2+2σnf+ϕ2σnf−σs2σnf/σf=σs1+2σf−2ϕ1σf−σs1/σs1+2σf+ϕ1σf−σs1Thermal expansionρβhnf=1−ϕ21−ϕ1+1−ϕ2ϕ1ρβs1/ρβf+ϕ2ρβs2/ρβfThe follower similarity transformations are provided in order to represent the governed Equations (1), (2), (7), and (8) concerning the BCs (9) in a much easy way [12].
(10)ψ=aυ1−ct1/2xfη,θη=T−T∞Tf−T∞,Φη=C−C∞Cf−C∞η=aυ1−ct1/2y,where ψ is the stream function that can be specified as u=∂ψ/∂y and v=−∂ψ/∂x and η is the similarity variable. Thus, we attain
(11)u=ax1−ctf′η,v=−aυ1−ct1/2fη.Equations (2), (7), and (8) are converted into the following set of nonlinear similarity ODEs by using the similarity variables (10) and (11) in light of the aforementioned relationships.
(12)μhnf/μfρhnf/ρff‴+2ff″−f′2+1−εf′−12ηf″−1−λβhnfβθ−Nrβhnf/βϕ+σhnf/σfρhnf/ρfMf′=0,(13)1Pr1ρcphnf/ρcpfkhnfkf+43Rθ″+fθ′−2f′θ+ε2ηθ′+3θ+Nbθ′Φ′+Ntθ′2+Qρcpfρcphnfθ=0,(14)Φ″+ScfΦ′−Scε2ηΦ′+3Φ+NtNbθ″=0.Now the starting and final constraints (9) are also changed into the following:
(15)f0=0,f′0=λ+γf″0,−khnfkfθ′0=Bi1−θ0,Φ′0=−Nd1−Φ0,f′η⟶1,θη⟶0,Φη⟶0,whileη⟶∞,where ε,M,R,Pr,Nr,Bi,Rex,λ,Q,λ1,Sc,Nt,andNb are unsteadiness parameter, magnetic parameter, thermal radiation parameter, Prandtl number, buoyancy ratio parameter, Biot number, local Reynolds number in x-axis, ratio of velocity parameter, heat source/sink parameter, mixed convection parameter, Lewis number, thermophoretic parameter, and Brownian motion parameter.
(16)ε=ca,M=σβ02ρfa,γ=Haυ1/2,Bi=hfkfυ1−cta,Rex=uexυ,R=4σ∗T∞33k∗kfQ=Q01−ct2ax,Pr=υα,Nr=ρp−ρfΔCρfβ1−C∞ΔT,λ=ba,λ1=GrRex2,Sc=υDB,Nt=τDTTs−T0νfT0,Nb=τDBCw−C∞νf.


### 2.1. Physical Quantities of Interest

For above model local Nusselt numberNux, the skin friction coefficient cf and Sherwood number are clear as follows:
(17)cf=τwρfue2,Nux=xqwkfTf−T∞,Shx=xqmDBCf−C∞,where
(18)τw=μhnf∂u∂yy=0,qw=−khnf∂T∂y−qry=0,qm=−DbCyy=0.In dimensional form, we have from above as(19)Rex1/2Cf=μhnfμff″0,Rex−1/2Nux=−khnfkf1+43Rθ′0,Rex−1/2Sh=−ϕ′0.


### 2.2. Idea of HAM for the Model

Here, we used HAM to solve Equations (12)–(14) using boundary conditions (15). The Mathematica software is utilized for this, and the proposed model can be solved using HAM in the following manner:
(20)Lf⌢f⌢=f‴⌢−f⌢′,Lθ⌢θ⌢=θ⌢″−θ⌢,LΦ⌢Φ⌢=Φ⌢″−Φ⌢,where operators are specified as Lf⌢ and Lθ⌢; moreover, we have
(21)Lf⌢γ1+γ2e−η+γ3eη=0,Lθ⌢γ4e−η+γ5eη=0,Lϕ⌢γ6e−η+γ7eη=0.We also define the operators, such that(22)Nf⌢f⌢η;ζ,θ⌢η;ζ,Φ⌢η;ζ=μhnf/μfρhnf/ρff⌢ηηη+2f⌢f⌢ηη−f⌢η2+1−εf⌢η−12ηf⌢ηη−1−λβhnfβθ⌢−Nrβhnf/βΦ⌢+σhnf/σfρhnf/ρfMf⌢η,Nθ⌢f⌢η;ζ,θ⌢η;ζ=1Pr1ρcphnf/ρcpfkhnfkf+43Rθ⌢ηη+f⌢θ⌢η−2f⌢ηθ⌢+ε2ηθ⌢η+3θ⌢+τhnfτ/υfNbθ′⌢Φ′⌢+Ntθ′⌢2+Qρcpfρcphnfθ⌢,NΦ⌢Φ⌢η;ζ,f⌢η;ζ,θ⌢η;ζ=Φ⌢ηη+Scf⌢Φ⌢η−Scf⌢ηΦ⌢−Scε2ηΦ⌢η+3Φ⌢+NtNbθ″⌢.For Equations (12)–(14), the 0th-order system is shown as follows:
(23)1−ζLf⌢f⌢η;ζ−f⌢0η=pℏf⌢N;f⌢f⌢η;ζ,1−ζLθ⌢θ⌢η;ζ−θ⌢0η=pℏθ⌢Nθ⌢θ⌢η;ζ,f⌢η;ζ,1−ζLΦ⌢Φ⌢η;ζ−Φ⌢0η=pℏΦ⌢NΦ⌢Φ⌢η;ζ,f⌢η;ζ,θ⌢η;ζ,with the BCs are
(24)f⌢η;ζη=0=0,∂f⌢η;ζ∂ηη=0=λ+γ∂2f⌢η;ζ∂η2η=0,f0=0,knfkf∂θ⌢η;ζ∂ηη=0=−Bi1−θ⌢0,∂Φ⌢η;ζ∂ηη=0=−Nd1−Φ⌢0∂f⌢η;ζ∂ηη=∞⟶1,θ⌢η;ζη=∞⟶0,Φ⌢η;ζη=∞⟶0.While the embedding constraint isζ∈0,1, to regulate for the solution convergence ℏf⌢, ℏθ⌢, and ℏΦ⌢ are used. At specific values ζ=0andζ=1, the following is obtained:
(25)f⌢η;1=f⌢η,θ⌢η;1=θ⌢η,Φ⌢η;1=Φ⌢η.Expand thef⌢η;ζ, θ⌢η;ζ, and Φ⌢η;ζ through Taylor’s series for ζ=0(26)f⌢η;ζ=f⌢0η+∑n=1∞f⌢nηζn,θ⌢η;ζ=θ⌢0η+∑n=1∞θ⌢nηζn,Φ⌢η;ζ=Φ⌢0η+∑n=1∞Φ⌢nηζn,f⌢nη=1n!∂f⌢η;ζ∂ηp=0,θ⌢nη=1n!∂θ⌢η;ζ∂ηp=0,Φ⌢nη=1n!∂Φ⌢η;ζ∂ηp=0.While BCs are
(27)f⌢0=0,f′⌢0=λ+γf″⌢0,khnfkfθ′⌢0=−Bi1−θ⌢0,Φ′⌢0=−Nd1−Φ⌢0,f′⌢η⟶0,θ⌢η⟶0,Φ⌢η⟶0η⟶∞.Rnf⌢η=μhnf/μfρhnf/ρff⌢n−1‴+2∑j=0w−1f⌢w−1−jf″⌢j−f′⌢n−12+1−εf⌢n−1′−12ηf⌢n−1″−1+σhnf/σfρhnf/ρfMf⌢n−1′−λθ⌢n−1−NrΦ⌢n−1,Rnθ⌢η=1Pr1ρcphnf/ρcpfkhnfkf+43Rθ″⌢n−1+∑j=0w−1θ⌢w−1−j′f∧j−2∑j=0w−1θ⌢w−1−jf⌢j′+ε2ηθ⌢n−1′+3θ⌢n−1+τhnfτ/υfNb∑j=0w−1θ⌢w−1−j′Φ′⌢j+Ntθ′⌢n−12+Qρcpfρcphnfθ⌢n−1,RnΦ⌢η=Φ⌢n−1″+Sc∑j=0w−1f⌢w−1−jΦ⌢j′−Sc∑j=0w−1f⌢w−1−j′Φ⌢j+NtNbθ⌢n−1″+Scε2ηΦ⌢n−1′+3Φ⌢n−1.While
(28)χn=0,ifn≤1,1,ififn>1.


## 2.1. Physical Quantities of Interest

For above model local Nusselt numberNux, the skin friction coefficient cf and Sherwood number are clear as follows:
(17)cf=τwρfue2,Nux=xqwkfTf−T∞,Shx=xqmDBCf−C∞,where
(18)τw=μhnf∂u∂yy=0,qw=−khnf∂T∂y−qry=0,qm=−DbCyy=0.In dimensional form, we have from above as(19)Rex1/2Cf=μhnfμff″0,Rex−1/2Nux=−khnfkf1+43Rθ′0,Rex−1/2Sh=−ϕ′0.


## 2.2. Idea of HAM for the Model

Here, we used HAM to solve Equations (12)–(14) using boundary conditions (15). The Mathematica software is utilized for this, and the proposed model can be solved using HAM in the following manner:
(20)Lf⌢f⌢=f‴⌢−f⌢′,Lθ⌢θ⌢=θ⌢″−θ⌢,LΦ⌢Φ⌢=Φ⌢″−Φ⌢,where operators are specified as Lf⌢ and Lθ⌢; moreover, we have
(21)Lf⌢γ1+γ2e−η+γ3eη=0,Lθ⌢γ4e−η+γ5eη=0,Lϕ⌢γ6e−η+γ7eη=0.We also define the operators, such that(22)Nf⌢f⌢η;ζ,θ⌢η;ζ,Φ⌢η;ζ=μhnf/μfρhnf/ρff⌢ηηη+2f⌢f⌢ηη−f⌢η2+1−εf⌢η−12ηf⌢ηη−1−λβhnfβθ⌢−Nrβhnf/βΦ⌢+σhnf/σfρhnf/ρfMf⌢η,Nθ⌢f⌢η;ζ,θ⌢η;ζ=1Pr1ρcphnf/ρcpfkhnfkf+43Rθ⌢ηη+f⌢θ⌢η−2f⌢ηθ⌢+ε2ηθ⌢η+3θ⌢+τhnfτ/υfNbθ′⌢Φ′⌢+Ntθ′⌢2+Qρcpfρcphnfθ⌢,NΦ⌢Φ⌢η;ζ,f⌢η;ζ,θ⌢η;ζ=Φ⌢ηη+Scf⌢Φ⌢η−Scf⌢ηΦ⌢−Scε2ηΦ⌢η+3Φ⌢+NtNbθ″⌢.For Equations (12)–(14), the 0th-order system is shown as follows:
(23)1−ζLf⌢f⌢η;ζ−f⌢0η=pℏf⌢N;f⌢f⌢η;ζ,1−ζLθ⌢θ⌢η;ζ−θ⌢0η=pℏθ⌢Nθ⌢θ⌢η;ζ,f⌢η;ζ,1−ζLΦ⌢Φ⌢η;ζ−Φ⌢0η=pℏΦ⌢NΦ⌢Φ⌢η;ζ,f⌢η;ζ,θ⌢η;ζ,with the BCs are
(24)f⌢η;ζη=0=0,∂f⌢η;ζ∂ηη=0=λ+γ∂2f⌢η;ζ∂η2η=0,f0=0,knfkf∂θ⌢η;ζ∂ηη=0=−Bi1−θ⌢0,∂Φ⌢η;ζ∂ηη=0=−Nd1−Φ⌢0∂f⌢η;ζ∂ηη=∞⟶1,θ⌢η;ζη=∞⟶0,Φ⌢η;ζη=∞⟶0.While the embedding constraint isζ∈0,1, to regulate for the solution convergence ℏf⌢, ℏθ⌢, and ℏΦ⌢ are used. At specific values ζ=0andζ=1, the following is obtained:
(25)f⌢η;1=f⌢η,θ⌢η;1=θ⌢η,Φ⌢η;1=Φ⌢η.Expand thef⌢η;ζ, θ⌢η;ζ, and Φ⌢η;ζ through Taylor’s series for ζ=0(26)f⌢η;ζ=f⌢0η+∑n=1∞f⌢nηζn,θ⌢η;ζ=θ⌢0η+∑n=1∞θ⌢nηζn,Φ⌢η;ζ=Φ⌢0η+∑n=1∞Φ⌢nηζn,f⌢nη=1n!∂f⌢η;ζ∂ηp=0,θ⌢nη=1n!∂θ⌢η;ζ∂ηp=0,Φ⌢nη=1n!∂Φ⌢η;ζ∂ηp=0.While BCs are
(27)f⌢0=0,f′⌢0=λ+γf″⌢0,khnfkfθ′⌢0=−Bi1−θ⌢0,Φ′⌢0=−Nd1−Φ⌢0,f′⌢η⟶0,θ⌢η⟶0,Φ⌢η⟶0η⟶∞.Rnf⌢η=μhnf/μfρhnf/ρff⌢n−1‴+2∑j=0w−1f⌢w−1−jf″⌢j−f′⌢n−12+1−εf⌢n−1′−12ηf⌢n−1″−1+σhnf/σfρhnf/ρfMf⌢n−1′−λθ⌢n−1−NrΦ⌢n−1,Rnθ⌢η=1Pr1ρcphnf/ρcpfkhnfkf+43Rθ″⌢n−1+∑j=0w−1θ⌢w−1−j′f∧j−2∑j=0w−1θ⌢w−1−jf⌢j′+ε2ηθ⌢n−1′+3θ⌢n−1+τhnfτ/υfNb∑j=0w−1θ⌢w−1−j′Φ′⌢j+Ntθ′⌢n−12+Qρcpfρcphnfθ⌢n−1,RnΦ⌢η=Φ⌢n−1″+Sc∑j=0w−1f⌢w−1−jΦ⌢j′−Sc∑j=0w−1f⌢w−1−j′Φ⌢j+NtNbθ⌢n−1″+Scε2ηΦ⌢n−1′+3Φ⌢n−1.While
(28)χn=0,ifn≤1,1,ififn>1.


## 3. Outcomes with Discussion

In the following discussion section, the outcomes of various parametric quantity on velocity, concentration profiles, and temperature are shown graphically and discussed. These parameters are unsteadiness parameter, buoyancy ratio parameter, magnetic parameter, thermal radiation parameter, Lewis number, mixed convection parameter, Prandtl number and thermophorotic parameter, and Brownian motion parameter. The geometry of the problem is shown in Figure1. It can be seen that when the magnetic parameter M assigns maximum values, the velocity distribution retains its declining tendency as shown in Figure 2. There is a periodic oscillation velocity with diminishing amplitude while the magnetic force produces a resistive nature force known as the Lorentz force which controls the flow of fluid particles. As a result, to demonstrate the value of reducing the velocity of moving particles, the effect of the unsteadiness parameter on ε the nanofluid velocity profile has seen in Figure 3. Because of the acceleration situation, ε>0 brings down rate of fluid flow and a narrower momentum boundary barrier thickness; the behavior occurs. The velocity profile reduces for the higher acceleration.Figure 2
Illustration of the effect ofM on f′η.Figure 3
Representation of the effect ofε on f′η.Figure4 illustrates the influence of the buoyancy ratio parameter on velocity drawings and also increasing levels of Nr; there is a noticeable increase in velocity. This increase in velocity drawings is more noticeable in stable condition than in the unsteadness situation. Figure 5 shows the effect of λ1 on f′η. From Figure 5, it can be expected that f′η has greater values of λ1. Physically, this is because of the larger values of buoyancy force. The impact of heat generation/absorption on the temperature is plotted in Figure 6. The temperature of the fluid and the thickness of the thermal boundary level are both increased when a heat source Q>0 is used. The heat sink Q<0, on the other hand, lowers the fluid’s temperature and thins the thermal boundary level thickness.Figure 4
Illustration of the effect ofNr on f′η.Figure 5
Illustration of the effect ofλ1 on f′η.Figure 6
Illustration of the effect ofQ on θη.To denoteQ=0, the lack of heat generation or absorption, the increase in thermal radiation raises the temperature but lowers the concentration profiles, as seen in Figure 7. This is because an increase in thermal radiation gives to a greater extent heat to the hybrid nanofluid, resulting in a rise in temperature and the thickness of the thermal boundary layer. Physically, radiative aspect creates the Brownian motion of minute ingredients faster than normal; thus, random migrated particles strike with one another, and the caused frictional energy transforms it to thermal energy. The effect from the Prandtl number along the temperature profile has seen in Figure 8. By increasing the Prandtl number, the fluid’s thermal conductivity decreases. This fact is due to inverse relation of Pr with thermal diffusivity, and it is well-known fact that the fluid with higher values of Pr has weaker thermal diffusion so that the temperature declines. Figures 9 and 10 demonstrate the consequence of the thermophoresis parametric quantity Nt along the thermal profile. When the thermophoresis parametric quantity is increased which enhance the behavior of the system, thermophoresis effect is associated with movement of nanoparticles from a hot wall to a cold wall, and since it is generated by temperature gradients, it creates a fast flow away from the moving plate; thus, more fluid is heated away from the surface, and this leads to an increase in the temperature within the thermal boundary layer. Figure 10 demonstrates the consequence of the Brownian diffusion parameter Nb on the thermal profile. Physically, the imperfect nature of the Brownian motion parametric quantity enables heating the physical setup. This heating causes nanoparticles to be transferred from the cooler stretched sheet area to the quiescent fluid zone. Because the particles migrate from a high up to a low concentration part, an increase in the Brownian motion parameter Nb and Φη leads decrease in the fluid concentration gradient, as seen in Figure 11.Figure 7
Representation of the effect ofR on θη.Figure 8
Graphical view to show the effect ofPr on θη.Figure 9
Illustration of the effect ofNt on θη.Figure 10
Graphical view of the effect ofNb on θη.Figure 11
Representation of the effect ofNb on Φη.The Brownian motion is the movement of fluid particles from higher to lower concentration so increases in the thermophoresis parameterNt produce a huge rise in the concentration dispersion of the fluid flow Φη, which meet to zero at the boundary level, as shown in Figure 12. A minor change in the thermophoresis parameter causes the fluid particles to move quickly, releasing surplus heat energy and causing a huge rise in the concentration dispersion. As a result, with an increase in the value of Nt, Figure 12 indicates a considerable rise in the concentration dispersion. Figure 13 also shows the influence of the Schmidt number on the concentration dispersion. It is noticed that for larger values of Sc, the fluid concentration decreases. This fact is quite similar with that of Prandtl number impact on temperature. Actually, Sc has inverse relation with mass diffusivity; therefore, larger values of Sc correspond thinner concentration boundary layer.Figure 12
Illustration of the effect ofNt on Φη.Figure 13
Representation of the effect ofSc on Φη.


## 4. Discussion on Tabulated Results

Table4 shows that Cf is increased when the values of ε,M are increased. The Cf is decreased, when the values of λ1,Nr are increased. Table 5 shows that Nux is increased when the values of R,Q,Nb,Nt are increased. The Nux is decreased, when the values of Pr are increased. Table 6 shows that Sh is decreased, when the values of Nb,Nt,Le are increased. Table 1 shows the very excellent agreement of the HAM and ND solve solution by computer-based package Mathematica 11.1.0.Table 4
Representation of the influence of different physical factors on skin frictionRex1/2Cf=μhnf/μff″0.
λ1NrMεμhnf/μff″00.30.30.50.31.67352370.51.43218031.01.23075390. 32.23167390.52.13176931.02.10380330.51.31560631.01.43671091.51.48373330.32.13968530.52.19463790.72.2453103Table 5
Illustration of the influence of different physical factors over Nusselt numberRex−1/2Nux=−khnf/kf1+4/3Rθ′0.
RQPrNtNbε−khnf/kf1+4/3Rθ′00.8-0.57.50.70.50.31.83639041.01.72143181.21.3274134-0.51.3435906-1.01.1643153-1.51.03661737.51.34531398.51.64370589.51.84678930.21.23278110.41.13643830.61.09536470.31.47353330.71.37945310.91.23934070.31.61572030.51.72403670.71.8090526Table 6
Illustration of the influence of different physical factors over Sherwood numberRes−1/2Shx=−ϕ′0..
NtNbLe−ϕ′00.20.32.02.14937310.42.17306910.62.23934140.32.35187010.72.45041030.92.49937982.02.56230873.32.65739194.02.6935286


## 5. Conclusions

In this research work, the uniform mixed convectional with the combined consequences of thermal radiation and heat generation/absorption along the magnetohydrodynamic (MHD) flow of hybrid nanofluid via a convectional heated the stretching/shrinking surface is studied. Brownian motion and thermophoresis were incorporated using the Buongiorno model. We came at the following key conclusions based on our findings with various factors in this study:(i)
Increasing the combined convection and buoyancy ratio parameters improved nanofluid flow properties(ii)
It is well established that an increase in the unsteadiness and magnetic parameters result in a decrease in the velocity profile(iii)
Consequently, a rise in the thermophorotic parameter, radiation parameter, or Brownian motion parameter means an increase in the temperature profile(iv)
It is noticed that the increase of the Prandtl number decreases the temperature profile(v)
Increases in the values of the thermophoretic parameter and Brownian motion parameter have an inverse influence on the concentration profile(vi)
TheΦη displays lessening tendency for rising values of Le(vii)
Skin friction is augmented when the values ofε,M are improved. When the values of λ1,Nr is enlarged, Cf is reduced(viii)
Nusselt number is improved when the valuesR,Q,Nb,Nt are increased. The Nux is reduced, when the values of Pr are augmented(ix)
Sherwood number is declined, when the values ofNb,Nt,Le are enlarged



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