# Numerical Solutions of Micropolar Nanofluid over an Inclined Surface Using Keller Box Analysis

Authors: Khuram Rafique; Hammad Alotaibi; Taher A. Nofal; Muhammad Imran Anwar; Masnita Misiran; Ilyas Khan
Journal: Journal of Mathematics
 (2020)

---

## Abstract

The Brownian motion and thermophoretic impacts attained a noticeable intention of the recent researchers because these factors trigger the thermal conductivity of the nanofluid. In this study, we focus on radiation and Soret effects on a slanted stretchable sheet. Buongiorno’s model is taken into account with Brownian motion and thermophoretic effects. Compatible transformations are implemented to attain the nonlinear differential equation from the boundary value PDE’s. The physical quantities of practical interest are treated by graphically as well as numerically. For numerical results, the Keller box technique is applied. The numerical outcomes through tabulated magnitudes performed a good settlement with already existing results. Energy transfer rate against involved factor exhibited via graphs. Energy and mass transport rates enhance against increment in Soret factor while skin friction diminishes. Moreover, Nusselt number and Sherwood number decrease on improving inclination while skin friction increases.


---

## Body

## 1. Introduction

Due to the ideal potential of heat and mass exchange impacts, the nanofluids have pulled in consideration of analysts worldwide. These fluids are the mixture of nano-sized particles along with base fluids. The main purpose to mix the nanoparticles into base fluids is to enhance the thermal conductivity. Brownian motion and thermophoretic effects are two principal ideas for an abnormal upgrade of thermal conductivity. This model is becoming more widely used since it empowers us to efficiently explore different applications in the marvels of science and technology. Recently, the flow of the Casson nanofluid over an inclined surface was discussed numerically by Rafique et al. [1]. Moreover, the flow of Casson nanofluid on a slanted rotating disk was probed by Saeed et al. [2]. In addition, Rani and Govindarajan [3] investigated the nanofluid flow through an inclined plate numerically. Nowadays, many investigators discussed the flow of nanofluid by considering different geometries [4–10].Thermal radiations are an active part of the research which is very valuable in the latest technology, cancer cure, missiles, and nanotechnology. Pal and Roy [11] discussed the numerical impact of thermal radiation on the flow of nanofluid along the sheet. More recently, Ghadikolaei et al. [12] scrutinized Casson nanofluid flow along permeable slanted surface via a famous numerical technique. Saidulu et al. [13] considered an exponentially slanted sheet for examining the radiation influences on nanofluid flow. For further details about the literature on the flow of nanofluid by considering different geometries, see [14–24].The behavior of the flow of non-Newtonian fluid is a study of keen interest of scholars and practical significance. There is a little regular and mechanical utilization of such fluids, for occasion volcanic magma, molten polymers, penetrating mud, oils, certain paints, liquid suspensions, food products, and cosmetic and numerous others. In the literature, there exist numerous numerical models with various constitutive conditions including a distinctive set of exact parameters. The micropolar liquid model is suitable for exotic oils, animal blood, fluid crystals with rigid atoms, certain organic liquids, and colloidal or suspensions solutions. The micromotion of liquid constituents, spin inertia, and the influences of the couple stresses are very noteworthy in micropolar liquids. Eringen [25] offered the philosophy of micropolar fluids based on constitution equations. Uddin [26] studied the variable electrical conductivity on the flow of micropolar liquid. Recently, Shamshuddin and Thumma discussed the flow of micropolar fluid on an inclined sheet numerically [27]. Nandhini and Ramya [28] studied the flow of micropolar fluid by incorporating the radiation effects. For some recent investigations on micropolar fluids with different impacts, see [29–34].This work focuses on establishing the basis for a novel study on thermal investigation in solar magnetohydrodynamic nanotechnology. The approach empowers us with great flexibility to dissect the Brownian motion and thermophoretic impact on the flow of micropolar nanofluid with thermal radiations and Soret impacts on the inclined geometry. The physical quantities of practical interest such as energy exchange, velocity, skin friction, and concentration species are elaborated in graphical and tabulated form. It is worth stating there is no investigation assumed for the study under concern and all the results are new and contrast to the current outcomes with already available literature [35]. In this article, we employed the Keller box technique for numerical results. Moreover, the current results can be obtained via another numerical technique, but the Keller box method is more easier to program, friendly, and flexible. For complete detail about Keller box scheme, see [36].


## 2. Problem Formulation

Our main aim in this analysis is to examine the radiations and Soret impacts on MHD flow of micropolar nanofluid. A uniform external magnetic field of strengthB0 is considered perpendicular to the inclined surface in this study, while induced magnetic field is neglected [37]. The slanted sheet here is stretched with a stretching rate “a” due to which the flow is generated. Moreover, ω is the angle taken with the vertical direction of the stretching sheet. Brownian motion and thermophoretic impacts are considered. In addition, suction or injection impacts on heat and species exchange rates are discussed via graphs.The Lorentz force in momentum equation is expressed byJ×B whereJ represents the current density,B denotes the total magnetic field:B=B0+b,B0 denotes the applied magnetic field,b is the induced magnetic field,J=σE+V×B (Ohm’s law),σ is the electrical conductivity,E=0 is the polarization charge effect,J=σV×B,J×B=σV×B×B (cross product with B),J×B=σV⋅BB−B⋅BV (vector identity),Using Afify [38] ×B=σB02u.The governing equations for the study under concern are(1)∂u∂x+∂v∂y=0,(2)u∂u∂x+v∂u∂y=μ+K1∗ρ∂2u∂y2+K1∗ρ∂N∗∂y+gβtT−T∞−βcC−C∞cosω−σB02ρu,(3)u∂N∗∂x+v∂N∗∂y=γ∗j∗ρ∂2N∗∂y2−K1∗j∗ρ2N∗+∂u∂y,(4)u∂T∂x+v∂T∂y=α∂2T∂y2−1ρcf∂qr∂y+τDB∂C∂y∂T∂y+DTT∞∂T∂y2,(5)u∂C∂x+v∂C∂y=DB∂2C∂y2+DTKTT∞∂2T∂y2.The Rosseland approximation is described by(6)qr=−4σ∗3k∗∂T4∂y.By employing Taylor expansion, approximate value ofT4 is given as(7)T4≅4T∞3T−3T∞4.By employing (6) and (7), equation (4) becomes(8)u∂T∂x+v∂T∂y=α+16σ∗T∞33k∗ρcf∂2T∂y2+τDB∂C∂y∂T∂y+DTT∞∂T∂y2.The subjected boundary conditions are(9)u=uwx=ax,v=Vw,T=Tw,N∗=−m0∂u∂y,C=Cw,aty=0,u⟶u∞=0,v⟶0,T⟶T∞,N∗⟶0,C⟶C∞,aty⟶∞.The stream functionψ=ψx,y for the concerned study are in the following form:(10)u=∂ψ∂y,v=−∂ψ∂x.The similarity transformations are defined as(11)u=axf′η,v=−avfη,η=yav,N∗=axmam+1xm−12vhx,θη=T−T∞Tw−T∞,ϕη=C−C∞Cw−C∞.The converted form of equations (2)–(5) given by employing equation (8) is(12)1+Kf″′+ff″−f′2+Kh′+Grxθ+Gcxϕcosω−Mf′=0,1+K2h″+fh′−f′h−K2h+f″=0,PrNθ″+fθ′+Nbϕ′θ′+Ntθ′2=0,ϕ″+Lefϕ′+SrLeθ″=0,where(13)M=σB02aρ=νDB,Pr=να,Nb=τDBCw−C∞ν,Nt=τDtTw−T∞νT∞,Grx=gβtTw−T∞x−1a2,Rex=uwxν,Gcx=gβcCw−C∞x−1a2,PrN=1Pr1+43N,Sr=DTKTTw−T∞νT∞Cw−C∞.where in order to make local Grashof number and local modified Grashof number free from x, the coefficient of thermal expansion βt and coefficient of concentration expansion βc are proportional to x1. Hence, we assume that (see references [39–41])(14)βt=nx1,βc=n1x1.where n and n1 are the constants; thus Grx and Gcx become(15)Gr=gnTw−T∞a2,Gc=gn1Cw−C∞a2.The corresponding boundary conditions are(16)fη=S,f′η=1,hη=0,θη=1,ϕη=1,atη=0,f′η⟶0,hη⟶0,θη⟶0,ϕη⟶0,asη⟶∞.The related expressions for the skin friction coefficientCfx0=f″0, the reduced Sherwood number −ϕ′0, and the reduced Nusselt number −θ′0 are demarcated as(17)−θ′0=Nux1+4/3NRex,−ϕ′0=ShxRex,Cfx=CfRex.


## 3. Results and Discussion

The objective behind this section is to elaborate on the numerical results along with graphically results via tables and graphs. In order to check the validity of our numerical values with already published results (Table1), we managed the deficiency of Gr, Gc, M, N, Sr,K,andS, taking factor Pr=Le=10, with ω = 90°. The variations in −θ′0, −ϕ′0∞, and Cfx0 for physical factors Nb, Nt, M, N, Gr, Gc, ω, Pr,Le, Sr,S, and K are presented in Table 2. Table 2 demonstrates that the energy flux slows down by improving the Brownian motion effect. Moreover, the mass flux increases with the effect of the Brownian factor for large values. Moreover, the exchange rate of heat reduces with improving Brownian motion factor, and mass exchanges enlarge on enhancing the thermophoretic impacts. Physically, the enhancement in the Brownian motion causes the boundary layer to get thicker. Besides, skin friction shows a direct relation with Brownian motion thermophoretic effects. The magnetic effect factor boosts the skin friction against improving magnetic field magnitude. Consequently, the fluid applies a drag force on the solid boundary layer. In addition, the Nusselt number along with Sherwood number and skin friction reduces for increasing the suction parameter. The Soret effect improves the mass exchange and reduces the skin friction on enhancing the Soret impact on the mass flow.Table 1
Contrast of−θ′0 and −ϕ′0 against ω = 90°, M, K, Gc, N, S,Sr, and Gr = 0, with Pr = Le = 10.
NbNtKhan and Pop [35]Present results−θ′0−ϕ′0−θ′0−ϕ′00.10.10.95242.12940.95242.12940.20.20.36542.51520.36542.51520.30.30.13552.60880.13552.60880.40.40.04952.60380.04952.60380.50.50.01792.57310.01792.5731Table 2
Values of−θ′0, −ϕ′0, and Cfx0.
NbNtPrLeMKGrGcSrNSω−θ′0−ϕ′0Cfx00.10.16.55.00.11.00.11.11.01.00.545°0.41020.84260.62110.50.16.55.00.11.00.11.11.01.00.545°0.28231.12780.65080.10.56.55.00.11.00.11.11.01.00.545°0.24211.01430.68050.10.110.05.00.11.00.11.11.01.00.545°0.37281.20440.61290.10.16.510.00.11.00.11.11.01.00.545°0.40601.25670.64120.10.16.55.01.01.00.11.11.01.00.545°0.36680.74441.14700.10.16.55.00.13.00.11.11.01.00.545°0.42020.87150.99260.10.16.55.00.11.01.01.11.01.00.545°0.42850.87110.31540.10.16.55.00.11.00.12.01.01.00.545°0.43930.90020.22860.10.16.55.00.11.00.11.12.01.00.545°0.47831.10290.43210.10.16.55.00.11.00.11.11.03.00.545°0.39290.51000.64440.10.16.55.00.11.00.11.11.01.00.745°0.24610.83470.57480.10.16.55.00.11.00.11.11.01.00.560°0.39710.81520.7807Figures1–8 are exhibited to demonstrate the performance of incorporated constraints that impact on the velocity sketch. Figure 1 indicates that the impact of the magnetic field creates resistance in the path of fluid flow because of the Lorentz force which reduces the velocity profile. Moreover, the angular velocity presents opposite behavior against the magnetic field factor in Figure 2. Figure 3 shows that the velocity contour upsurges against material factor K. Physically, increment in factor K declines the viscosity and upturns the velocity. On the contrary, against a higher magnitude of the factor K, the angular velocity upturns (Figure 4). This demonstration corresponds with the outcomes of Rafique et al. [42]. Figure 5 represents as we enhance, the buoyancy force the velocity increases. Besides, the velocity profile shows the direct relation with Gc in Figure 6. In addition, the velocity field and inclination parameter correspond to an inverse relation drawn in Figure 7. Physically, by considering ω=0, the gravitational force reaches its maximum value. On the contrary, in the case of ω=90°, the sheet will be in horizontal position, and that is why the power of the bouncy forces declines which is the reason behind the reduction in the velocity profile. Moreover, in Figure 8, the velocity profile enhances against the higher values of S.Figure 1
Velocity profile for several values ofM.Figure 2
Angular velocity for several values ofM.Figure 3
Velocity profile for several values ofK.Figure 4
Angular velocity profile for several values ofK.Figure 5
Velocity profile for several values ofGr.Figure 6
Velocity profile for several values ofGc.Figure 7
Velocity profile for several values ofω.Figure 8
Velocity profile for several values ofS.


### 3.1. Temperature

The impacts of incorporated factors in our current study against temperature profile are demonstrated in Figures9–12. The temperature profile boosts against a stronger magnetic field because the boundary layer thickness upsurge corresponds to a higher magnetic effect (Figure 9). The Prandtl number triggers the temperature profile shown in Figure 10. The thermal boundary layer becomes thinner, and the thermal diffusivity becomes weaker on improving the Prandtl number steadily. The large values of the radiation effect accelerate the temperature profile as shown in Figure 11. Physically, the conductive heat exchange is greater than the radiative heat exchange, which causes reduction in boundary layer thickness and buoyancy force. The recovered outcome is the affirmative proof of the relation qr=−4σ∗/3k∗∂T4/∂y. The temperature profile relates directly proportional to the Brownian motion effect as shown in Figure 12. Physically, the Brownian motion constraint improves the boundary layer heat which leads to rises in the fluid temperature. The variation between reference temperature and wall temperature is enhanced by growing the thermophoresis influence which corresponds the enhancement in temperature profile (Figure 13).Figure 9
Temperature profile for several values ofM.Figure 10
Temperature profile for several values ofPr.Figure 11
Temperature profile for several values ofN.Figure 12
Temperature profile for several values ofNb.Figure 13
Temperature profile for several values ofNt.


### 3.2. Concentration

Figures14 to 18 indicate concentration profiles against different incorporated parameters. The concentration profile improves on strengthening the magnetic impact as shown in Figure 14. An increment in Brownian motion parameter declines the concentration profile and the boundary layer thickness (Figure 15). Figure 16 presents more nanoparticles that pass away from the hot surface on enhancing the thermophoretic effects which cause the improvement in the concentration contour. The boundary layer viscosity decreases against the Lewis number which relates to a drop in the concentration profile (Figure 17). The concentration profile corresponds to a direct variation with the Soret factor drawn in Figure 18.Figure 14
Concentration profile for several values ofM.Figure 15
Concentration profile for several values ofNb.Figure 16
Concentration profile for several values ofNt.Figure 17
Concentration profile for several values ofLe.Figure 18
Concentration profile for several values ofSr.


### 3.3. Heat and Mass Exchange

In order to check the behavior of dimensionless heat and mass exchange rates at the wall along with skin friction against the involved parameters, i.e.,Nb and ω, Figures 19 to 21 are drawn. Figure 19 reveals the impact of the Brownian motion factor against different inclination parameters. It is noted that −θ′0 inversely relates the Brownian motion factor and inclination factor. As we improve the inclination and the Brownian motion factor, the heat transfer rate decreases. The same impacts of the Brownian motion factor along with the inclination parameter on the mass exchange rate have been noticed in Figure 20. However, Cfx0 enhances for higher magnitudes of Brownian motion and inclination depicted in Figure 21. Figures 22 to 24 reveal the variations in −θ′0,−ϕ′0, and Cfx0 versus Nt and ω. On the contrary, heat exchange declines on the higher values of thermophoretic effect and inclination (Figure 22). Moreover, the mass exchange rate shows similar behavior like heat transfer rate against altered values of inclination and thermophoresis impacts in Figure 23. In addition, skin friction is improved by enhancing the magnitudes of inclination and thermophoretic impacts (Figure 24).Figure 19
Variations of reduced Nusselt number withNb for different values of ω.Figure 20
Variations of reduced Sherwood number withNb for different values of ω.Figure 21
Variations skin friction withNb for different values of ω.Figure 22
Variations of reduced Nusselt number withNt for different values of ω.Figure 23
Variations of reduced Sherwood number withNt for different values of ω.Figure 24
Variations skin friction withNt for different values of ω.


## 3.1. Temperature

The impacts of incorporated factors in our current study against temperature profile are demonstrated in Figures9–12. The temperature profile boosts against a stronger magnetic field because the boundary layer thickness upsurge corresponds to a higher magnetic effect (Figure 9). The Prandtl number triggers the temperature profile shown in Figure 10. The thermal boundary layer becomes thinner, and the thermal diffusivity becomes weaker on improving the Prandtl number steadily. The large values of the radiation effect accelerate the temperature profile as shown in Figure 11. Physically, the conductive heat exchange is greater than the radiative heat exchange, which causes reduction in boundary layer thickness and buoyancy force. The recovered outcome is the affirmative proof of the relation qr=−4σ∗/3k∗∂T4/∂y. The temperature profile relates directly proportional to the Brownian motion effect as shown in Figure 12. Physically, the Brownian motion constraint improves the boundary layer heat which leads to rises in the fluid temperature. The variation between reference temperature and wall temperature is enhanced by growing the thermophoresis influence which corresponds the enhancement in temperature profile (Figure 13).Figure 9
Temperature profile for several values ofM.Figure 10
Temperature profile for several values ofPr.Figure 11
Temperature profile for several values ofN.Figure 12
Temperature profile for several values ofNb.Figure 13
Temperature profile for several values ofNt.


## 3.2. Concentration

Figures14 to 18 indicate concentration profiles against different incorporated parameters. The concentration profile improves on strengthening the magnetic impact as shown in Figure 14. An increment in Brownian motion parameter declines the concentration profile and the boundary layer thickness (Figure 15). Figure 16 presents more nanoparticles that pass away from the hot surface on enhancing the thermophoretic effects which cause the improvement in the concentration contour. The boundary layer viscosity decreases against the Lewis number which relates to a drop in the concentration profile (Figure 17). The concentration profile corresponds to a direct variation with the Soret factor drawn in Figure 18.Figure 14
Concentration profile for several values ofM.Figure 15
Concentration profile for several values ofNb.Figure 16
Concentration profile for several values ofNt.Figure 17
Concentration profile for several values ofLe.Figure 18
Concentration profile for several values ofSr.


## 3.3. Heat and Mass Exchange

In order to check the behavior of dimensionless heat and mass exchange rates at the wall along with skin friction against the involved parameters, i.e.,Nb and ω, Figures 19 to 21 are drawn. Figure 19 reveals the impact of the Brownian motion factor against different inclination parameters. It is noted that −θ′0 inversely relates the Brownian motion factor and inclination factor. As we improve the inclination and the Brownian motion factor, the heat transfer rate decreases. The same impacts of the Brownian motion factor along with the inclination parameter on the mass exchange rate have been noticed in Figure 20. However, Cfx0 enhances for higher magnitudes of Brownian motion and inclination depicted in Figure 21. Figures 22 to 24 reveal the variations in −θ′0,−ϕ′0, and Cfx0 versus Nt and ω. On the contrary, heat exchange declines on the higher values of thermophoretic effect and inclination (Figure 22). Moreover, the mass exchange rate shows similar behavior like heat transfer rate against altered values of inclination and thermophoresis impacts in Figure 23. In addition, skin friction is improved by enhancing the magnitudes of inclination and thermophoretic impacts (Figure 24).Figure 19
Variations of reduced Nusselt number withNb for different values of ω.Figure 20
Variations of reduced Sherwood number withNb for different values of ω.Figure 21
Variations skin friction withNb for different values of ω.Figure 22
Variations of reduced Nusselt number withNt for different values of ω.Figure 23
Variations of reduced Sherwood number withNt for different values of ω.Figure 24
Variations skin friction withNt for different values of ω.


## 4. Conclusions

The article presents numerical simulations of the micropolar-type nanofluid flow over a slanted surface. Similarity results for velocity, temperature, and concentration are recovered through the Keller box technique. The statistical outcomes are compared with the already available literature. In research field, some notable points from this study are as follows:(i)
The irregular movement of particles boosts with the increment in Brownian motion; and as a result energy and species exchange rate diminish, whereas the skin friction is enhanced.(ii)
The temperature field is more influenced by increasing thermal radiation.(iii)
The concentration profile enhances with the growing magnitude of Soret impact.(iv)
The boundary layer viscosity decreases against Lewis number which relates to a drop in the concentration profile(v)
The conductive heat exchange is greater than the radiative heat exchange, which causes reduction in boundary layer thickness and buoyancy force.



---

*Source: 6617652-2020-12-14.xml*