# MHD Mixed Convection Nanofluid Flow over Convectively Heated Nonlinear due to an Extending Surface with Soret Effect

Authors: Jamel Bouslimi; M. A. Abdelhafez; A. M. Abd-Alla; S. M. Abo-Dahab; K. H. Mahmoud
Journal: Complexity
 (2021)

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## Abstract

The aim of this paper is to investigate the flow of MHD mixed convection nanofluid flow under nonlinear heated due to an extending surface. The transfer of heat in nanofluid subject to a magnetic field and boundary conditions of convective is studied to obtain the physical meaning of the convection phenomenon. The governing partial differential equations (PDEs) of the boundary layer are reduced to ordinary differential equations (ODEs) considering a technique of the transformation of similarity. The transformed equations are solved numerically considering the technique of an efficient numerical shooting applying the Runge–Kutta technique scheme from the fourth-fifth order. The results corresponding to the dimensionless speed, temperature, concentration profiles, and the Nusselt number reduced, and the Sherwood numbers are presented by figures to display the physical meaning of the phenomena. A comparison has been made between the obtained results with the previous results obtained by others and agrees with them if the new parameters vanish. The results obtained indicate the impacts of the nondimensional governing parameters, namely, magnetic field parameterM, Soret number Sr, heat source λ, thermal buoyancy parameterλT , and solutal buoyancy parameterλC on the flow, temperature, and concentration profiles being discussed and presented graphically.


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## Body

## 1. Introduction

Nanofluids are suspended particles of the fluid. They have particles with a nanometer size, and they have a less uniform dispersion in the rigid particles. Nanofluids have crucial usages in science and technology, marine engineering, and applications in the field of industry such as plastic, polymer industries, cancer home therapy, and building sciences. They flow through transferring vertical plane plate also having enormous applications in the field of aerosols engineering, aerodynamics, and civil engineering and because of this reason, the researchers are likely to investigate this field. Nonlinear warm air radiation and chemical reaction effects on MHD 3D Casson fluid movement in the porous medium were considered by Sulochana et al. [1]. Wahiduzzaman et al. [2] investigated MHD Casson fluid movement going through a nonisothermal porous linearly stretching sheet. Ramreddy et al. [3] analyzed the Soret effect on mixed convection flow in a nanofluid under convective boundary condition radiation and the Soret effects of MHD nanofluid moving freely from a moving vertical moving plate in the porous medium were discussed by Raju et al. [4]. The mixed convective flow of Maxwell nanofluid goes beyond an absorbent vertical stretched surface. An optimal solution is considered by Ramzan et al. [5]. Stagnation electrical MHD nanofluid varied convection with slip boundary on a stretching sheet was discussed by Hsiao [6]. Rafique et al. [7] have obtained the numerical solution of casson nanofluid streams over a nonlinear sloping surface through Soret and Dufour effects by the Keller–Box method. Mkhatshwa et al. [8] investigated the MHD mixed convective nanofluid flow about a vertical thin cylinder using the interfering multidomain spectral collocation approach. Huang [9] analyzed the effect of non-Darcy and MHD on free convection of non-Newtonian fluids over a vertical plate which can be infiltrated in a pored medium with Soret/Dufour effects and thermal radiation.Mahanthesh et al. [10] expanded this work by considering nonlinear radiative heat transfer in MHD three-dimensional water stream based nanofluid above a nonlinearly stretching sheet with convective boundary condition. Sivaraj and Sheremet [11] investigated the MHD accepted convection and entropy generation of ferrofluids in a void with a nonuniformly heated horizontal plate. Abad et al. [12] discussed the analysis of transfers in a reaction to the flow of hybrid nanofluid around the body discussed a trick embedded in porous media using a synthetic neural network and a special swarm improvement. Siavashi [13] studied the nanofluid and porous fins influence on natural thermal convection and entropy generation of flow within a cavity. Bilal [14] discussed the partial flow of EMHD nanoparticles with nonlight thermal radiation and sliding effects. Alomar [15] studied the analysis of the effects of unbalanced and nonbloody heat flow on natural load in a square enclosure with a hot plate in L shape. Dogonchi and Hashim [16] investigated the transfer of heat through a natural load Fe304-water nanofluid to neutralize between a wavelength circular cylinder and a rhombus cylinder. Hsiao analyzed the micropolar nanoparticle flow containing MHD and Mercury dissipation effects towards an extended sheet with a multimedia feature [17]. Dinarv et al. [18] studied Buongiorno’s model for the double flux slump point of a nanofluid effect of the binary base. Kefayati and Tang [19] studied the MHD mixed convection of viscoplastic fluids in different aspect rates of a lid-driven cavity using LBM. Ramzan et al. [20] discussed the mixed flux of second grade nanofluid with lipid boundary conditions: an optimal solution. Kumar et al. [21] studied Bejan’s heat and mass lines ideation of multiforce influence on convection in a pored enclosure. Aurangzaib et al. [22] investigated the influence of a fractional slide on an unstable MHD mixed convection stagnation-point flow of a micropolar fluid towards a porous contraction plate. Krishna and Chamkha [23] considered the hall and ionic slide effects on the rotation flow of MHD of the water flow of the coherent fluid through the porous medium. Zaidi and Mohyud-Din [24] are the founders of the analysis of wall jet flow for Soret, Dufour, and chemical response effects in the presence of MHD with uniform suction/injection. Machireddy and Naramgari [25] examined the heat and mass transfers in radiative MHD Carreau fluid by cross-diffusion. Alsabery et al. [26] examined the fluid structure reaction in natural load heat transfer in an implicit space with a flexible oscillating fin and partial heating. Maleki et al. [27] presented the investigation of the heat transfer and nanofluid flow over a pored plate with radiation and slip boundary conditions. Do et al. [28] studied the case of Navier when slipping at the time Darcy–Forchheimer nanofluid relies on using a spectral relaxation method. Beg [29] discussed the mixed radioactive dual magnetic flow of acid with traces of the number of Biot and Richardson’s effects. Srinivasacharya and Jagadeeshwar [30] investigated the seismic flow on a porous leaf extending significantly with acidic thermal boundary conditions.Chemi et al. [31] researched the critical impedance load of double-wall carbon nanotubes using nonlocal theoretical elasticity. Timesli [32] studied the twisting analysis of double-wall carbon nanotubes embedded in a flexible Kerr medium under axial pressure using the nonlocal Donnell shell theory. Mirjavadi et al. [33] studied the analysis of fine nonlinear forced vibrations of two-phase magnetic flexible nanoparticles under elliptical type force. Some related investigations can be found in the article [34].In this investigation, the researchers aim at presenting the influence of the prominent Soret impact on mixed convection heat and mass transfer in the border layer region of a semi-infinite vertical flat plate in a nanofluid, under the circulatory motion of the boundary conditions. The consequences of nondimensional governing parameters, namely, the volume fraction of nanoparticles, magnetic field parameter, radiation parameter, Soret number, buoyancy parameter, and porosity parameter on the flow, temperature, and concentration profiles are discussed and presented graphically. Also, the friction aspect and Nusselt and Sherwood numbers are deliberated and demonstrated in tabular form for two nanofluids separately. Some new contributions near to the present article are discussed in [35]–[44].


## 2. Construction of the Problem

Let us consider two-dimensional constant MHD nanofluid flows in the regiony>0 along nonlinearly stretched sheet assuming Cartesian coordinates x,y chosen in which x−axis is measured along with the sheet whereas is y−axis normal to it under the influence of heat absorption/generation, magnetic field, suction, and injection parameters. The stretched sheet is assumed to have general power-law surface velocity distribution uwx=axn where a>0, n≥0 are constants. We consider that the flow is concerned with a variable magnetic field of strengthBx=B0xn−1/2. The induced magnetic field is neglected whereas the electric field is absent by assuming the low magnetic Reynolds number. The temperature of the sheet is defined by a function Twx=T∞+Axn where A>0 is constant, T∞ is the ambient fluid temperature, and C∞ is the ambient nanoparticles concentration. The coordinate system and physical flow are shown in Figure 1.Figure 1
Diagrammatic of the problem.The equation of rheological state for the anisotropic flow of a Casson fluid [1, 2]:(1)τij=2µB+Pz2πeij,π>πc,2µB+Pz2πeij,π<πc,where π=eijeij,(2)∂u∂x+∂v∂y=0,(3)u∂u∂x+v∂u∂y=υ1+1β∂2u∂y2+gαTT−T∞+αCC−C∞−σB2xρfu,(4)u∂T∂x+v∂T∂y=α∂2T∂y2+Q0ρcpT−T∞+τDB∂T∂y∂C∂y+DTT∞∂T∂y2+νcp1+1β∂u∂y2,(5)u∂C∂x+v∂C∂y=DB∂2C∂y2+KTDTT∞∂2T∂y2,where u and v are the velocity components along the x− and y−directions respectively, υ is the kinematic viscosity, β=μB2πc/Py is Casson fluid parameter,σ is the electrical conductivity, ρ is the fluid density, the thermal diffusivity is denoted by α, DB and DT are the coefficients of the thermophoresis and Brownian diffusions, Q0 is the dimensional heat generation or absorption coefficient, cp is the specific heat, τ=ρcp/ρcf indicates the ratio of effective heat capacity of the nanoparticle material to the effective heat capacity of the fluid, g is the acceleration due to gravity, αT is the thermal expansion coefficient, αC is the concentration expansion coefficient, and KT is the thermal diffusion ratio. The boundary conditions in the present problem are(6)u=uwx=axn,v=vw,T=Twx=T∞+Axn,DB∂C∂y+DTT∞∂T∂yaty=0,u⟶0,T⟶T∞,C⟶C∞asy⟶∞.The nondimensional transformations are taken as follows:(7)u=axnf′η,v=−axn−1/2υan+12fη+n−12ηf′η,η=aυxn−1/2y,θη=T−T∞Tw−T∞,ϕη=C−C∞Cw−C∞,(8)1+1βf‴η−nf′2η+n+12fηf″η−Mf′η+λTθ+λCϕ=0,(9)1Prθ″+n+12fθ′−nf′θ+λθ+Nbθ′ϕ′+Ntθ′2+1+1βEcf″2=0,(10)φ″+n+12Lefϕ′+LeSrθ″=0.The boundary conditions are(11)f0=fw,f′0=1,θ0=1,Nbϕ′0+Ntθ′0=0,atη=0,f′∞⟶0,θ∞⟶0,ϕ∞⟶0,asη⟶∞,,where Ec is Eckert number, fwis absorption/intromission variable fw>0 for absorption and fw<0 for intromission), M is the attractive field variable, Nb is the Brownian motion variable, Nt is the thermophoresis parameter, Pr is the Prandtl number, Le is the Lewis number, λ is the birthplace of heat λ>0 or sink parameterλ<0, Sr is Soret number, λT is thermal buoyancy parameter, and λC is solutal buoyancy variable. These are specified as follows:(12)M=σB2axn−1ρf,K=υaxn−1k,λT=GrxRex=αTga2xn−1xnTw−T∞,λC=GcxRex=αCga2xn−1xnCw−C∞,Grx=αTgTw−T∞x3υ2,Gcx=αCgCw−C∞x3υ2,Rex=uwxxυ=axn+1υ,Pr=υα,λ=Q0ρcp,Nb=τDBCw−C∞υ,Nt=τDTTw−T∞T∞υ,Ec=uw2Tw−T∞cp,Le=υDB,Sr=KTDTTw−T∞υT∞Cw−C∞.The amounts of practical interest include the coefficient of skin abrasionCf, regional Nusselt volume Nux, and local Sherwood volume Shx specified as follows:(13)Cfx=τwρuw2,Nux=xqwαTw−T∞,Shx=xqmDBCw−C∞,where τw is the stress of wall clipping; αf is the thermal conductivity of the nanofluid; and qw,qm are the wall heat and mass influx. These are given by(14)τw=μB1+1β∂u∂yy=0,qw=−αf∂T∂yy=0,qm=−DB∂C∂yy=0.Using equations (7) and (14) in equation (13), one obtains(15)Rex1/2Cfx=1+1βf′0,Rex−1/2Nux=−θ′0,Rex−1/2Shx=−ϕ′0,where Rex=uwx/υ is the local Reynolds number.


## 3. Numerical Results and Discussion

The orders of not linear normal distinctive equation (3) for (5) with the limit situation (6) are detached in terms of numbers by the Rung–Kutta method with MATLAB package. The outcome displays show the influences of the nondimensional controlling variable, which is magnetic area variable M, Soret number Sr, birthplace of heat λ, thermic buoyancy parameter λT, and solutal buoyancy parameter λC on the flow, heat, and concentricity profiles are shown and illustrated diagrammatically. The agent of rubbing and Nusselt and Sherwood numbers is discussed and given in tabular form for two nanofluids individually. The numeral outcomes are as follows:(16)Pr=0.7,Le=0.4,Nb=Nt=0.3,Ec=0.5,λ=0.1,β=0.5,Fw=0.2,n=2.Figures2–4 show that the difference of speed f′η, heat θη, and concentricity ϕη with respect to η -axis for various amounts of λC. There is an observation of the velocity decrease along with the increase of λC, whereas there is an increase of both the heat and concentricity profiles along with the increase of λC for magnetic field parameter M=1,2. This causes development in both the thermic stretched sheet thickness and velocity and concentration sheet thickness. From these figures, it is observed that the increase of magnetic area parameter leads to the increase of velocity. It is also interesting to note from Figure 3 that temperature rises as M increases; Figure 4 also shows the fall of concentricity while the magnetic area parameter increases.Figure 2
Variation of velocityf′ concerning η with various values of λc (M = 1ــــ, M = 2 ---).Figure 3
Variation of temperatureθ concerning η with various values of λc (M = 1ــــ, M = 2 ---).Figure 4
Variation concentrationϕ concerning η with various values of λc (M = 1ــــ, M = 2 ---).Figures5–7 display variance of speed f′η, heat θη, and concentration ϕη with respect to η-axis for various amounts of thermic buoyancy parameter λT. The increase of velocity along with the increase of thermic buoyancy parameter λ_T is noticed, whereas the heat and centricity profiles reduce with the growing flux of magnetic area parameter. This causes development in both the thermic stretched sheet thickness and velocity and concentration sheet thickness.Figure 5
Variation of velocityf′ concerning η with various values of λT (M = 1ــــ, M = 2 ---).Figure 6
Variation of temperatureθ concerning η with various values of λT (M = 1ــــ, M = 2 ---).Figure 7
Distinction of concentrationϕ concerning η with various values of λT (M = 1ــــ, M = 2 ---).Figures8–10 show the alteration velocity f′η, temperature θη , and concentricity ϕη with respect to η-axis for various numbers of thermal buoyancy parameter λT. It is noticed that the velocity value increases along with thermal buoyancy parameter λT, whereas the temperature and the concentricity profiles fall with the increase of thermal buoyancy parameter λT of the flux for the Solutal buoyancy parameterλC=0.5,1. This causes to make the thickness of both the thermic stretched sheet and velocity and concentricity sheet better.Figure 8
Diversity of velocityf′ concerning η with various values of λT (λc=0.5ــــ, λc=1---).Figure 9
Variation of temperatureθ concerning η with various values of λT (λc=0.5ــــ, λc=1---).Figure 10
Variation of concentrationϕ concerning η with various values of λT (λc=0.5ــــ, λc=1---).Figures11–13 plot the velocity variance f′η, temperature θη, and concentration ϕη with respect to η-axis for various numbers of magnetic area parameter M. It is watched that, by the growing of magnetic area parameter, the velocity value grows too. However, the increase of the magnetic field area causes a reduction in the profiles of temperature and concentricity. This leads to developing the thermic stretched sheet thickness along with velocity and concentration sheet thickness.Figure 11
Variation of velocityf′concerning η with various values of M.Figure 12
Variation of temperatureθ concerning η with various values of M.Figure 13
Variation of concentrationϕ concerning η with various values of M.Figures14–16 clear the different values of velocity f′η, temperature θη, and concentration ϕη with respect to η-axis for changeable values of magnetic field parameter. By observation, it is clear that the increasing positive values of the magnetic area cause the reduction of velocity which also grows along with the reduction of negative values of a magnetic area, while it increases with decreasing of negative values of the magnetic field, though the temperature increases with decreasing the negative values of the magnetic field. Also, there is no effect of the positive values of magnetic field on the temperature, and the concentration outlines increase with increasing the negative values of the magnetic field parameter, while as well it is not affected by positive values of magnetic field on the concentration. This causes thermal stretched sheet thickness to get better along with velocity and concentration sheet thickness.Figure 14
Variation of velocityf′ concerning η with various values of M.Figure 15
Variation of temperatureθ concerning η with various values of M.Figure 16
Variation of concentrationϕ concerning η with various values of M.Figures17–19 illustrate the difference of velocity f′η, temperature θη , and concentricity ϕη with respect to η-axis for different values of Soret number Sr. The growth of profiles of both the velocity and concentricity is noted along with the increase of Soret number for n=0.5,2, while the temperature profiles go down with the increase of Soret number of the flow for n=0.5,2. This leads to making the thermal stretched sheet thickness better along with velocity and concentration sheet thickness.Figure 17
Variation of velocityf′ concerning η with various values of Sr (n = 2 __, n = 0.5 ---).Figure 18
Variation of temperatureθ concerning η with various values of Sr (n = 2 __, n = 0.5 ---).Figure 19
Variation of concentrationϕ concerning η with various values of Sr (n = 2 __, n = 0.5 ---).Figures20–22 show the variance of velocity f′η, temperature θη, and concentricity ϕη with respect to η-axis for different values of Soret number Sr. It is watched that the increasing Soret number causes the increase of velocity and the decrease of temperature; however, both of them are not affected by Lewis number Le on the velocity and temperature, while the concentration profiles increase with increasing Soret number of the flow, but they are not affected by Lewis number on the concentration. These causes developing the thermal stretched sheet thickness along with velocity and concentration sheet thickness.Figure 20
Variation of velocityf′ concerning η with various values of Srand Le.Figure 21
Variation of temperatureθ concerning η with various values of Sr and Le.Figure 22
Variation of concentrationϕ concerning η with various values of Sr and Le.Figure23 plots the diversity of Nusselt number NuxRex1/2 with respect to Solutal buoyancy parameter λC-axis for various numbers of thermal buoyancy parameter λT and magnetic field M. Through observation, it is clear that the Nusselt number grows up along with the increasing thermic buoyancy parameter but falls with the increasing magnetic area.Figure 23
Variation ofNuxRex−1/2 concerning λc with various values of λT and M.Figure24 displays the difference of the Nusselt number NuxRex1/2 with respect to Lewis number Le-axis for different values of Soret number Sr and thermal buoyancy parameter λT. It is noticed that the increase of both the Sort number and thermic buoyancy parameter causes the Nusselt number to increase as well.Figure 24
Variation ofNuxRex−1/2 concerning Lewith number various values of λT and Sr.Figure25 clears the variance of the Nusselt number NuxRex−1/2 with respect to λc-axis for different values of the Soret number Sr and thermal buoyancy parameter λT. There is a notice that the Nusselt number grows along with the increase of both the Soret number and thermic buoyancy parameter.Figure 25
Variation ofShxRex−1/2 concerning λT with various values of λc and M.Figure26 shows the variation of the Sherwood number ShxRex−1/2 with respect to solutal buoyancy parameter λC-axis for various numbers of thermal buoyancy parameter λT and magnetic field M. It is clear that the Sherwood number goes up with the progress of the thermic buoyancy parameter but it goes down with the progress of the magnetic field.Figure 26
Variation ofShxRex−1/2 concerning Lewith various values of λT and Sr.Table1 displays a comparison for the heat transfer rate at the sheet for different values of n,β,andM, when Ec=fw=λ=0,Nt=Nb=0.5,Pr=7. It is strongly obvious that the present study has original values near the results obtained by the others [9].Table 1
Heat transfer rate at the sheet for numerous values ofn,β,andM, when Ec=fw=λ=0,Nt=Nb=0.5,Pr=7.
ConstantsRef. [9]Present worknβM−θ′0−θ′010.632.691562.6915810.632.513542.5135510.632.315542.3155920.233.411433.4113220.633.310403.3103021.033.263193.2631130.604.188304.1882930.614.148204.1482230.634.082194.08222Table2 shows the heat transfer rate at the sheet for numerous values of Sr,λT,λC,β,fw,andM when Nt=Nb=0.5,Pr=7,n=1,Ec=0.5. It is clear that the external parameters impact the heat transfer rate comparing with the consequences obtained in the previous works.Table 2
Heat transfer rate at the sheet for numerous values ofSr,λT,λC,β,fw,andM when Nt=Nb=0.5,Pr=7,n=1,Ec=0.5.
λTλCSrMβfw−θ′00.10.10.230.60.20.110490.20.044390.10.20.095180.40.064950.10.10.269660.30.003260.211.3386920.5887130.20.594670.40.266450.6−0.40.366490.40.13465


## 4. Conclusion

In this research, the authors analyzed the impact of magnetic fieldM, Soret number Sr, heat source λ, thermal buoyancy parameter λT, and solutal buoyancy parameter λC of a nanofluid flow over convectively nonlinear heated due to an extending surface. The partial differential equations that dominate flow are transformed into normal changed-in-variables equations using the similarity transform and then solved numerically. The impacts of nondimensional ruling parameters specifically the Soret number of nanoparticles, magnetic field parameter, thermal buoyancy parameter λT, Lewith number, and λC on the flow, speed, temperature, and concentration profiles are discussed and presented graphically. The conclusions can be drawn as follows:(1)
The distribution of nanoparticle concentration is an increasing and decreasing function of the magnetic field and Soret number.(2)
Speed distribution decreases by increasing the values of the Soret number with growing values of the magnetic field.(3)
Temperature and concentration decrease with the increase of the values of thermal buoyancy parameterλT and solutal buoyancy parameter λC.(4)
Nusselt and Sherwood local numbers are greater for the incremental values of Soret number Sr and solutal buoyancy parameterλC.For future work, we can use other influences of a nanofluid flow over convectively heated nonlinear due to an extending surface.



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