# Stagnation Point Flow of EMHD Micropolar Nanofluid with Mixed Convection and Slip Boundary

Authors: Shahzada Muhammad Atif; Muhammad Abbas; Umair Rashid; Homan Emadifar
Journal: Complexity
 (2021)

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## Abstract

The aim of this numerical research is to study the stagnation point flow of the electrical magnetohydrodynamic micropolar nanofluid with slip conditions past a stretching sheet. The phenomenon of linear thermal radiation, Ohmic and internal heating, has also been considered in the energy equation. The modelled PDEs are converted into ODEs via similarity transformation, and converted ODEs are tackled via the shooting technique. The features of assorted parameters on the axial and angular velocities and energy and concentration fields are sketched. The numerical values of the Sherwood and Nusselt numbers have been computed numerically and displayed in the form of tables. Our analysis shows that the heat transfer rate is decreased as the thermal slip parameter and the diffusion slip parameter are enhanced. The present study illustrates that the energy and concentration distribution are decreased with each of the mass free convection parameter, stagnation parameter, and thermal free convection parameter.


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## Body

## 1. Introduction

The phenomenon of transfer of heat, which has significant application in many engineering and industry disciplines, is affected positively by the implementation of an appropriate magnetic field. The investigation of magnetohydrodynamic flow past a heated surface has gained significant attention due to its vast applications in engineering problems, i.e., magnetohydrodynamic power generators, petroleum industries, and crystal growth. Swedish scientist Alfven [1] was the first to introduce the magnetohydrodynamic fluid flow. He won the Nobel Prize in Physics for his work on MHD in 1970. He described the class of magnetohydrodynamic waves, which are now known as Alfven waves. Zheng et al. [2] reported the magnetohydrodynamic 2-D (dimensional) flow past a porous shrinking surface with slip conditions with conclusion that an acclivity in the shrinking parameter enhances the thermal boundary layer. By considering the slip effects in a porous medium, Ullah et al. [3] explored the magnetohydrodynamic (MHD) Casson fluid and noticed that boosting the unsteadiness parameter enhances the wall shear stress. Rahbari et al. [4] examined the magnetohydrodynamic Maxwell fluid flowing through parallel plates and determined that increase in the Deborah number increases the velocity. By investigating heat transfer in the magnetohydrodynamic flow past a radially shrinking/stretching sheet, Soid et al. [5] concluded that dual solutions exist only in case of suction and for small values of magnetic parameters. Decline in surface drag due to increment in the squeezed flow parameter in Carreau fluid with thermal radiation and magnetohydrodynamic effect past a sensor surface was reported by Atif et al. [6]. Transverse magnetohydrodynamic effect on nonlinear stretching sheet was ascertained by Ramana et al. [7]. They used modified Fourier flux law and concluded that the relaxation and retardation time have opposite effects on the thermal profile. For further studies, see [8, 9].Stagnation point flow is one of the fields in which scientists and engineers show keen interest. Some of the recent studies include the following: For MHD viscoelastic nanofluid, the dual solution of stagnation point past a porous stretching surface with radiation effect was reported by Juosh [10]. It was found that an acclivity in the Deborah number contributes to upsurge in the drag coefficient. Impact of MHD on stagnation point flow of a nanofluid with nonuniform thermal reservoir was analyzed by Rashid et al. [11]. Pal [12] put light on magnetohydrodynamic stagnation point flow with suction effect and reported that the Sherwood number was decreased as the Lewis number increased. Bioconvective stagnation point of Maxwell nanofluid flow past a convectively heated surface was reported by Abbasi et al. [13]. They observed that energy, concentration, and density profiles were higher for nonconvective surfaces than in convective heated surfaces. Lund et al. [14] performed the stability analysis and reported the dual solution of MHD stagnation point of Casson fluid. Their main observation was that the sign of the smallest eigenvalues shows that the first solution was stable. Effect of solar radiation on MHD stagnation point nanofluid flow was discussed by Ghasemia and Hatami [15] with a key finding that the energy profile is hiked as the Biot number is increased rapidly. The phenomena of thermal radiation have much significance in the transfer of heat and were discussed by many authors in the literature [16–19].Micropolar fluids can be characterized as fluids which exhibit the micro-rotational effects and micro-rotational inertia. Analysis of the micropolar fluids has been an active field of interest for many researchers. This class of fluids possesses certain simplicity and elegance in their mathematical formulation which should appeal to mathematicians. The micropolar fluids can support couple stress and body couples only. Physically, they may represent adequately the fluids consisting of dipole elements. Certain anisotropic fluids e.g., liquid crystals which are made up of dumbbell molecules are of this type. In fact, animal blood happens to fall in this category. Other polymeric fluids and fluids containing minute amounts of additives may be represented by the mathematical model underlying micropolar fluids. Eringen [20, 21], through his pioneering work, invited the attention of the researchers’ community in this interesting area of fluid dynamics. Sui et al. [22] investigated the nonlinear constitutive diffusion model in the micropolar fluid with the main finding that both the velocity and energy profiles are increased as the power exponent n is decreased from 1. Heat transfer of the free convective micropolar fluid with heat source past a shrinking sheet was noticed by Mishra et al. [23]. They observed that the fluid motion is declined as the heat generation coefficient is upsurged. Atif et al. [24] analyzed the bioconvective magnetohydrodynamic micropolar nanofluid with stratification and reported that the density distribution decreases as the density stratification and mixed number parameter are hiked. Micropolar nanofluid flow with nonlinear convection and multiple slip effects was examined by Zemedu and Ibrahim [25] with concluding remarks that boosting the solutal nonlinear convection parameter causes an increase in the velocity.The heat transfer in base fluids like mineral oils, water, and ethylene glycol is not as much effective as in nanofluids [26–31]. Nanofluids have the ability to improve the heat transfer properties. Their ability to move through capillaries and microchannels without making any blockage in flow makes them unique. By considering the induced magnetic field, Atif et al. [32] investigated the magnetohydrodynamic micropolar Carreau nanofluid and found that the angular velocity is increased rapidly as the magnetic Prandtl number increases. Three-dimensional Eyring–Powell nanofluid with Arrhenius energy was reported by Taseer et al. [33]. Khan [34] reported that nanoparticle dispersion reduces the Nusselt number in a partially heated vertical annulus. For a solutal-dominated regime, both Nusselt and Sherwood numbers declined for micropolar nanofluid as reported by Manaa et al. [35].In recent years, researchers have paid serious attention to electrical magnetohydrodynamics. Electrical magnetohydrodynamic stagnation point nanofluid with mixed convection and slip boundary over a stretching surface was scrutinized by Hsiao [36]. A major conclusion was that an acclivity in either the electrical or the magnetic parameter led to an upsurge in the temperature profile. Literature review indicates that the EMHD stagnation point micropolar nanofluid with mixed convection and slip boundary has not been investigated yet. In the present article, linear thermal radiation, Joule’s heating, and heat source have also been incorporated in the energy equation. In this study, four aspects have been focused. First, the heat and mass transfer of micropolar nanofluid are addressed. Second, the impact of thermal radiation and electrical magnetohydrodynamics on different profiles is examined. Third, the stagnation point flow is analyzed. Fourth, the analysis of the mixed convection and slip boundary conditions is performed. The arising ordinary differential equations for the problem are tackled through the shooting method. The influence of all the prominent parameters is examined numerically and displayed graphically.


## 2. Mathematical Model

An incompressible, 2D, mixed convection micropolar nanofluid flow over a stretching sheet with slip effects has been analyzed. By using Ohm’s law and Maxwell’s equations, the continuity equation, linear and angular momentum, and fluid energy and concentration equations have been formulated. Joule’s heating, thermal radiation, and heat source effects have also been considered in the energy equation.C,C∞,T, and T∞ denotes the surface concentration, ambient concentration surface temperature, and ambient temperature, respectively. The flow is assumed along the x−axis which is considered to be in the upward direction, whereas the y−axis is perpendicular to the sheet. A uniform magnetic field B0 has been implemented towards the y−axis as illustrated in Figure 1. It is also assumed that the magnetic Reynolds number is very small due to which induced magnetic number is ignored.Figure 1
Flow configuration.In the light of the above assumption, the governing equations of the modelled problem are as follows [36, 37]:(1)∂u∂x+∂v∂y=0,(2)u∂u∂x+v∂u∂y=U∞∂U∞∂x+ν+kρ∂2u∂y2+kρf∂N∂y+1ρσB02U−u+gxβtT−T∞+gxβcC−C∞+1ρσE0B0,(3)u∂N∂x+v∂N∂y=γρj∂2N∂y2−kρj2N+∂u∂y,(4)u∂T∂x+v∂T∂y=α∂2T∂y2−1ρCp∂qr∂y+σuB0−E02ρCp+τDB∂C∂y∂T∂y+DTT∞∂T∂y2+Q0ρCpT−T∞,(5)u∂C∂x+v∂C∂y=DB∂2C∂y2+DTT∞∂2T∂y2.The related BCs are as follows:(6)uw−cx=L∂u∂y,v=vw=ax,N=0,T=Tw+k1∂T∂y,C=Cw+k2∂C∂y,aty=0,u⟶0,N⟶0,T⟶T∞,C⟶C∞,asy⟶∞..In equation (3), the Rosseland radiative heat flux qr is given by qr=−4σ∗/3κ∗∂T4/∂y. The spin gradient viscosity is given by γ=μ+k/2j, where j=ν/a and k represents the microinertia density and vortex viscosity, respectively. For nondimensionlization, the following transformation [36] has been considered:(7)ψ=xaνfη,N=axaνgη,η=yaν,θη=T−T∞Tw−T∞,T=T∞+Axθη,ϕη=C−C∞Cw−C∞,C=C∞+Bxϕη..Continuity equation (1) is satisfied automatically, and equations (2)–(5) yield the following:(8)1+Kf″′−f′2+ff″+Kg′+S0−Mf′−1−E+Gtθ+Gcϕ=0,(9)1+K2g″−gf′+fg′−K2g+f″=0,(10)1+43Rdθ″+Prfθ′+λθ+Nbθ′ϕ′+Ntθ′2+MEcf′2+E2−2Ef′=0,(11)ϕ″+Scfϕ′+NtNbθ″=0.The associated boundary conditions in the dimensionless form are as follows:(12)f=S,f′=1+δ1f″,g=0,θ=1+δ2θ′,ϕ=1+δ3ϕ′atη=0,f′⟶0,g⟶0,θ⟶0,ϕ⟶0,asη⟶∞..Here,Pr=ν/α denotes the Prandtl number, S0=U∞2c+Lf″0a3/ν2/a2uw2 represents the stagnation parameter, Ec=a2x2/CpTw−T∞ denotes the Eckert number, Nb=τDBCw−C∞/ν denotes the Brownian motion parameter, K=k/μ represents the micropolar parameter, Gt=gxβtTw−T∞c+Lf″0a3/ν/a2Uw represents the thermal free convection parameter, M=σB02/aρ represents the magnetic number, E=E0/B0U represents the electric field parameter, Nt=τDTTw−T∞/νT∞ denotes the thermophoresis parameter, S=c/a+L/af″0a3/ν denotes the slip parameter, Sc=ν/DB denotes the Schmidt number, Gc=gxβcCw−C∞c+Lf″0a3/ν/a2Uw denotes the mass free convection parameter, Rd=4σ∗T∞3/kκ∗ denotes the thermal radiation parameter, λ=Q0/aρCp denotes the heat generation coefficient, δ1=La/ν represents the shear stress parameter, δ2=k1a/ν represents the temperature slip parameter, and δ3=k2a/ν represents the diffusion slip parameter, where k1 and k2 are the slip parameters associated with the reference temperature and concentration, respectively.


## 3. Quantities of Interest

The dimensionless Nusselt and dimensionless Sherwood numbers are the most concerning quantities in engineering and industries.(13)Nu=xqwkTw−T∞,Sh=xjwDBCw−C∞.In the nondimensional form, Nusselt and Sherwood numbers are given by(14)NuxRex−1/2=−1+43Rdθ′0,ShxRex−1/2=−ϕ′0,where Rex=ax2/ν.


## 4. Solution Methodology

The system of ODEs (8)–(11) along with BCs (12) is tackled numerically by the shooting technique. Now, we introduce ς1=f, ς2=f′, ς3=f″, ς4=g, ς5=g′, ς6=θ, ς7=θ′, ς8=ϕ, and ς9=ϕ′ as follows:(15)ς1′=ς2,ς2′=ς3,ς3′=11+Kς22−ς1ς3−S0−Kς5+Mς2−1−E−Gtς6−Gcς8,ς4′=ς5,ς5′=22+Kς2ς4−ς1ς5+K2ς4+ς3,ς6′=ς7,ς7′=−3Pr3+4Rdς1ς7+λς6+MEcς22+E2−2Eς2+Nbς7ς9+Ntς72,ς8′=ς9,ς9′=−Scς1ς9−NtNbς7′..The corresponding boundary conditions are(16)ς1=S,ς2=1+δ1ς3,ς4=0,ς6=1+δ2ς7,ς8=1+δ3ς9atη=0,ς2⟶0,ς4⟶0,ς6⟶0,ς8⟶0asη⟶∞..


### 4.1. Code Validation

For the verification of the correctness of the code, the results of the Nusselt and Sherwood numbers which were presented by Khan and Pop [38] and Hsiao [36] are successfully reproduced. Our simulations have a satisfactory agreement with the already published results of Khan and Pop [38] and Hsiao [36] in the literature which can be seen in Table 1.Table 1
Comparison of the presently computed values of−θ′0 and −ϕ′0.
−θ′0−ϕ′0NtNb[38][36]Present[38][36]Present0.10.10.95240.95240.9523712.12942.12942.1293560.20.69320.69320.6931732.27402.27402.2739560.30.52010.52010.5200812.52862.52872.5285420.40.40260.40260.4025842.57522.57522.7950410.50.32110.32110.3210593.03513.03523.0349790.10.20.50560.50560.5055802.38192.38192.3818400.30.25220.25210.2521562.41002.41002.4099910.40.11940.11940.1194062.39972.39972.3996250.50.05430.05420.0542542.38362.38362.383547


## 4.1. Code Validation

For the verification of the correctness of the code, the results of the Nusselt and Sherwood numbers which were presented by Khan and Pop [38] and Hsiao [36] are successfully reproduced. Our simulations have a satisfactory agreement with the already published results of Khan and Pop [38] and Hsiao [36] in the literature which can be seen in Table 1.Table 1
Comparison of the presently computed values of−θ′0 and −ϕ′0.
−θ′0−ϕ′0NtNb[38][36]Present[38][36]Present0.10.10.95240.95240.9523712.12942.12942.1293560.20.69320.69320.6931732.27402.27402.2739560.30.52010.52010.5200812.52862.52872.5285420.40.40260.40260.4025842.57522.57522.7950410.50.32110.32110.3210593.03513.03523.0349790.10.20.50560.50560.5055802.38192.38192.3818400.30.25220.25210.2521562.41002.41002.4099910.40.11940.11940.1194062.39972.39972.3996250.50.05430.05420.0542542.38362.38362.383547


## 5. Results and Discussion

Table2 is displayed to view the effect of the sundry parameters on the dimensionless Nusselt number and Sherwood number. It is observed that a boost in each of the electric parameter E, slip parameter S, micropolar parameter K, thermal radiation parameter Rd, the stagnation parameter S0, thermal free convection parameter Gt, diffusion slip parameter δ3, and mass free convection parameter Gc, causes an increase in Nusselt number, whereas it decreases for a boost in each of the heat generation coefficient λ, shear stress parameter δ1, magnetic parameter M, and the thermal slip parameter δ2. The Sherwood number is hiked as each of the material parameter K, magnetic parameter M, thermal free convection parameter Gt, velocity slip parameter S, electric parameter E, b mass free convection parameter Gc, stagnation parameter S0, and heat generation coefficient λ, and thermal radiation parameter Rd is boosted. However, it diminishes as the diffusion slip parameter δ3, thermal slip parameter δ2, and shear stress parameter δ1 are increased.Table 2
Numerical values of−θ′0 and −ϕ′0 with Pr=10 and Sc=10.
KMSERdλS0δ1δ2δ3GtGc−θ/0−ϕ/00.20.10.10.110.10.10.10.10.10.10.12.2086512.0246400.52.2328182.0364481.02.2619142.0508000.52.1845882.08473012.0806672.15017021.7392652.2728880.22.2437572.0419560.32.2758762.0580620.42.3055432.0731510.22.2183262.0249630.32.2262002.0256590.42.2322932.02672422.9841612.07117933.5926002.10646344.1122932.1330760.21.9426472.0964370.31.6427652.1763080.41.2975552.2668940.22.6423052.3092970.33.0861882.5798790.43.5350212.8353140.22.1504321.9937310.32.1019261.9681310.42.0606761.9464770.22.0409192.0403020.31.8935402.0547000.41.7636842.0678560.22.2895501.6286500.32.3450481.3620080.42.3854551.17030012.2748222.05730652.4885432.166622102.6707232.26340312.2518212.04598052.4059052.124113102.5492452.199092Figures2–14 are sketched to study the variations occurring due to dimensionless parameters in temperature distribution θη. For all graphs of the temperature dynamics, the values of Rd=1, K=Ec=Sc=0.2, Pr=10, and M=λ=Nt=Nb=S=E=S0=δ1=δ2=δ3=Gt=Gc=0.1. Figure 2 is prepared to visualize the fluctuation in the temperature distribution θη in response to the variation in the magnetic effect M. The temperature of the fluid is increased as M increases. This supports the general behaviour of the implementation of M. Resistance to the flow of the fluid is increased as M increases due to which θη is enhanced. The temperature distribution θη is diminished as the stagnation parameter S0 is enhanced. This effect is evident from Figures 3 and 4 demonstrated to view the effect of Pr on temperature distribution θη. These graphs indicate that upsurge in the Prandtl number Pr causes depreciation in the thermal profile. There is decline in the thermal conductivity of the fluid due to which θη is reduced. Fluctuation due to electrical parameter E in the temperature is divulged in Figure 5. An increment in E results in an enhancement in thermal profile. As the Lorentz force is associated with the magnetic and electric field, it leads to an increment in the resistance causing the energy distribution to enhance. The influence of the thermal radiation parameter Rd on temperature distribution θη is chalked out in Figure 6. These graphs reflect that an enhancement in Rd increases the energy profile. To visualize the behaviour of the thermophoresis parameter Nt on temperature, Figure 7 is sketched, which shows that there is an increment in θη as Nt is hiked. Physically, in thermophoresis, the particles apply force on the other particles due to which particles from the hotter region move towards the colder region. Larger values of Nt denotes more application of the force on the other particles and as a result, more fluid moves from the higher temperature region to the colder region. Figure 8 shows that θη is increased as the Brownian motion parameter Nb increases. Physically, the Brownian motion heats up the fluid and also aggravates the particles away from the fluid regime and therefore a decrement is seen in concentration profile. Figure 9 is illustrated to view the effect of heat generation coefficient λ on θη, which shows that θη is hiked for escalating values of λ. The viscous dissipation effect which is represented by the Eckert number Ec on energy field is analyzed in Figure 10. It is a number that represents the relation between the kinetic energy and the change in enthalpy. It is noticed that gradually boosting Ec leads to an increase in θη. Influence of the dimensionless slip parameter S on θη is presented in Figure 11, and it shows that an upsurge in S encourages the energy distribution θη to decline. The influence of the slip parameter δ2 which is associated with temperature on the temperature field is chalked out in Figure 12. The energy profile is found to be increasing as slip parameter δ2 goes up. The variation in the thermal profile due to the thermal free convection parameter Gt is shown in Figure 13. The energy distribution declined as the thermal free convection parameter Gt is hiked. Figure 14 depicts that the energy profile is decreases as Gc is boosted.Figure 2
Variation due toM in θη.Figure 3
Variation due toS0 in θη.Figure 4
Variation due toPr in θη.Figure 5
Variation due toE in θη.Figure 6
Variation due toRd in θη.Figure 7
Variation due toNt in θη.Figure 8
Variation due toNb in θη.Figure 9
Variation due toλ in θη.Figure 10
Variation due toEc in θη.Figure 11
Variation due toS in θη.Figure 12
Variation due toδ2 in θη.Figure 13
Variation due toGt in θη.Figure 14
Variation due toGc in θη.Figures15–23 have been outlined to study the fluctuations in the concentration field ϕη due to variation in the governing parameters. For all the graphical presentations of ϕη, we have considered Pr=10,Nb=0.3,Ec=Sc=K=0.2,andM=Rd=1,λ=Nt=S=E=S0=δ1=δ2=δ3=Gt=Gc=0.1. Figure 15 is given to study the impact of stagnation parameter S0 on ϕη. The concentration distribution ϕη is diminished as the stagnation point parameter is enhanced. Figure 16 represents the graph of ϕη for growing values of E. From these curves, it is clear that increasing electric parameter E diminishes ϕη. Figure 17 presents the role of Schmidt number Sc in the variation of ϕη. The concentration field ϕη is diminished as Sc is rapidly increased. The effect of Nt on dimensionless concentration field ϕη is reported in Figure 18. These graphs present that concentration field is enhanced as Nt is gradually increased. Figure 19 is presented to view the fluctuation in dimensionless ϕη caused by the increase in Nb. An increase in dimensionless parameter Nb causes a reduction in ϕη. The impact of the variation of the dimensionless velocity slip parameter S on the dimensionless ϕη is shown in Figure 20. These curves indicate that with an increment in the velocity slip parameter S, the dimensionless ϕη declined. Figure 21 depicts the graphs of the concentration profile for various values of the dimensionless diffusion slip parameter δ3. From these curves, it is noticed that an enhancement in the dimensionless diffusion slip parameter δ3 causes a decrease in ϕη. The fluctuation in the dimensionless concentration distribution due to the thermal free convection parameter Gt is shown in Figure 22. The concentration field is reduced as thermal free convection parameter Gt is increased. Figure 23 depicts that ϕη is decreased as the mass free convection parameter Gc is increased.Figure 15
Variation due toS0 in ϕη.Figure 16
Variation due toE in ϕη.Figure 17
Variation due toSc in ϕη.Figure 18
Variation due toNt in ϕη.Figure 19
Variation due toNb in ϕη.Figure 20
Variation due toS in ϕη.Figure 21
Variation due toδ3 in ϕη.Figure 22
Variation due toGt in ϕη.Figure 23
Variation due toGc in ϕη.


## 6. Concluding Remarks

In this study, two-dimensional free convection electrical magnetohydrodynamic micropolar nanofluid is analyzed. Some of the key observations are as follows:(i)
The energy field declined with an acclivity in the stagnation parameterS0, slip parameter S, thermal free convection parameter Gt, and thermal slip parameter δ2(ii)
The Nusselt number is escalated for the increasing values of slip parameterS, electric parameter E, stagnation parameter S0, and thermal radiation parameter Rd(iii)
The concentration field is diminished with an increase in stagnation parameterS0, electric parameter E, diffusion thermal slip parameter δ3, and slip parameter S(iv)
The Sherwood number is increased as slip parameterS, electric parameter E, heat generation coefficient λ, stagnation parameter S0, and mass free convection parameter Gc are increased



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