# A Computational Model for the Radiated Kinetic Molecular Postulate of Fluid-Originated Nanomaterial Liquid Flow in the Induced Magnetic Flux Regime

Authors: Azad Hussain; Aysha Rehman; Sohail Nadeem; M. Riaz Khan; Alibek Issakhov
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The performance of mass transfer rate, friction drag, and heat transfer rate is illustrated in the boundary layer flow region via induced magnetic flux. In this recent analysis, the Buongiorno model is introduced to inspect the induced magnetic flux and radiative and convective kinetic molecular theory of liquid-initiated nanoliquid flow near the stagnant point. The energy equation is modified by radiation efficacy using the application of the Rosseland approximation. Through similarity variables, the available formulated partial differential equations are promoted into the nondimensional structure. The variation of the induced magnetic field near the wall goes up, and very far away, it decays when the size of the radiation characteristic ascends. The velocity amplitude expands by enlargement in the amount of the magnetic parameter, mixed convection, thermophoresis parameter, and fluid characteristic. The nanoparticle concentration reduces if the reciprocal of the magnetic Prandtl number expands. The temperature spectrum declines by enhancing the amount of the magnetic parameter. Drag friction decreases by the increment in the values of radiation and thermophoresis parameters. Heat transport rate increases when there is an increase in the values of Brownian and magnetic parameters. Mass transfer rate increases when there is incline in the values of the magnetic Prandtl and fluid parameter.


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## Body

## 1. Introduction

Improving the thermal efficiency of fluid flows under different conditions and applications has always been a famous research area. Besides, the significance of this issue because of the very wide range of applications in industries has made this area attractive to scientists and companies working in this field. To improve effectiveness, any solution proposed for this in any application can have different technical aspects that should be considered and investigated adequately [1]. For example, Ellahi et al. [2] explored the slip effect in the Newtonian fluid two-phase flow. Particles of the nanosized Hafnium are utilized in the base fluid. Two cases are discussed for fluid under consideration, namely, (i) phase of particles and (ii) fluid phase. Three forms of geometries are investigated in both cases. Relevant studies of nanofluids are discussed in recent articles [3–19].There are many engineering applications of the mixed convective boundary layer flow such as food processing, solidification system, and nuclear reactors. Convection also plays an important role in managing the production cycle such as medications and cosmetics. The transverse magnetic field that merged with the boundary layer-mixed convection flow towards an inclined plate with a wave is examined. The retardation inflow far from the magnetic field and leading-edge yield acceleration in the leading-edge close flow of the wavy sheet is observed by Wang and Chi-Chang [20]. Many researchers analyzed the regime of mixed convection flows in their articles [21–26].There are several uses of free convection in the presence of Lorentz forces, such as fire engineering and geophysics. Newly proposed nanotechnology is a new passive way to enhance heat transfer [27]. The induced magnetic field’s influence on temperature curves is represented by Ghosh et al. [28]. Vanita and Kumar [29] have examined transient flow towards a cone with the inclined magnetic field. Akbar et al. [30] investigated nanoparticle interaction for peristaltic flow in an asymmetric channel towards the magnetic field induced. Nadeem and Ijaz [31] explored the impulse of nanoparticles as a drug carrier to investigate theoretically stenosis arteries in the existence of the induced magnetic effect. Hayat et al. [32] observed second-grade nanoliquid flow with the induced magnetic field towards a stretched convectively heated surface. Rashid et al. [33] inspected the induced magnetic field effects of the Williamson peristaltic fluid flow in a curved channel.The radiation may be sunlight, infrared, or visible and the nature of the material emitted by such radiation depends on its exposure. Depending on how solar heat is collected and distributed or converted into solar electricity, a heat source and its systems are also categorized as either passive solar or active solar. Thermal radiations are defined as electromagnetic emissions from a sheet with a temperature greater than zero [34]. Viskant and Grosh [35] noted that when considering the power plants, hypersonic flights, cooling systems, and combustion chambers, radiations became an important part. Using Rosseland’s approximations, they addressed thermal radiation impacts for the flow of Falkner-Skan. Hayat et al. [36] portrayed the Ag-CuO/H2O rotating hybrid nanofluid flow in the existence of partial slip radiation impacts. Hussain et al. [37] noted the non-Newtonian fluid flow with radiation efficacy and time-dependent viscosity. Li et al. [38] examined the radiation impacts in the heat storage system by adding nanoparticles. Zeeshan et al. [39] investigated, due to entropy generation, the impacts of electromagnetohydrodynamics radiative diminishing internal energy of the pressure-driven dioxide-water titanium nanofluid flow.The non-Newtonian fluid is more naturalistic to consider because of the rheological characteristics of physiological and industrial fluids. There is no extensive model that can describe the moving structure of all fluids due to the complex behavior of non-Newtonian fluids. Thus, to study non-Newtonian fluid flow characteristics, numerous models have been developed. The Eyring–Powell model was obtained from a liquid molecular hypothesis. And the inclusion of additional analytical constants was further improved. It accurately reflects the essence of Newtonian for low and high shear values. For example, rubber melts, condensed liquids, toiletries, cosmetics, and vegetable products are included in these fluids [40]. Appropriate studies of Eyring–Powell fluid may be mentioned in these articles [41–45].This report is to narrate the specifications of radiative mass and heat transfer enhancement and flow analysis of the molecular kinetic theory of liquid-initiated boundary layer stagnation point nanofluid towards a vertical stretched surface. The non-Newtonian nanofluid model is manifested with the induced magnetic field, radiation efficacy, combined convection, Brownian, and thermophoresis diffusion. The flow field describes that, in the form of partial differential equations, the laws of conservation of momentum are considered. By reducing the number of independent variables by using the technique of similarity transformation, these coupled equations are then purified into the system of ordinary differential equations. The results are interpreted by the MATLAB bvp4c technique. Induced magnetic pattern near the wall decreases, and far away, it increases with an increase in the values of the reciprocal of the magnetic Prandtl number. The concentration curve enhances when the number of magnetic, stretching, and Prandtl characteristics incline. This study of nanofluid is mainly applied in heat transfer devices such as electrical cooling systems, radiators, and heat exchangers.


## 2. Mathematical Formulations

Consider the incompressible, steady, two-dimensional (2D) stagnant point flow under the assumption of the induced magnetic field in the molecular kinetic theory of liquid-initiated nanofluid and heat transport enhancement in the existence of combined convection and radiation towards a vertical stretched sheet as shown in Figure1. The surface stretching with velocity uwx=dx and ambient velocity is u∞x=bx while the origin is fixed at O; see Figure 1.Figure 1
Geometry of the problem.Taken the Cartesian coordinate structure, the velocity of liquid flow will change throughx- and y-axes in a way that x−axis is assumed vertically and y−axis is supposed horizontally. The fundamental form of the kinetic molecular postulate of liquid-originated nanofluid is [45](1)τ=μ+1δγ˙sinh−11c1γ˙A1,where(2)γ˙=trA1212,where δ, c1, μ,A1, and tr are fluid parameters, dynamic viscosity, first Rivlin-Ericksen tensor, and trace, respectively. Here, τ is the extra stress tensor and A1=gradvt+gradv. We consider the second-order approximation for sinh−1 function as(3)sinh−11c1γ˙≅γ˙c1−γ˙36c13,where1c1γ˙≪1.Then, equation (1) becomes(4)τ=μ+1δc1−γ˙26δc13A1.Under these premises, the governing equations of this particular investigation are as follows:(5)∂u∂x+∂v∂y=0,(6)∂H1∂x+∂H2∂y=0,(7)u∂u∂x+v∂u∂y−μ∞4πρfH1∂H1∂x+H2∂H1∂y=u∞du∞dx−μ∞H∞4πρfdH∞dx+ν+1δc1ρf∂2u∂y2−12δc13ρf∂u∂y2∂2u∂y2+g1−CβρfmρfT−T∞−gρp−ρfmρfC−C∞,(8)u∂H1∂x+v∂H1∂y−H1∂u∂x−H2∂u∂y=α1∗∂2H1∂y2,(9)u∂T∂x+v∂T∂y=α∂2T∂y2+ρcppρcpfDB∂C∂y∂T∂y+DTT∞∂T∂y2−1ρcpf∂qr∂y,(10)u∂C∂x+v∂C∂y=DB∂2C∂y2+DTT∞∂2T∂y2.The above equations narrate the viscosity coefficientν, where u,v and H1,H2 describe the velocity and magnetic field components along the x and y directions, respectively, whereas uwx=dx and H∞x=xH0 are the x velocity and y magnetic field at the edge of the boundary layer and H0 is the uniform value of the vertical magnetic field at infinity.The radiation heat flux is given by using Rosseland approximation:(11)qr=−4σ∗3k∗∂T4∂y,where σ∗,k∗ are the Stefan–Boltzmann and the mean absorption coefficient, respectively, whereas via extending T4 about T∞ in Taylor’s series and ignoring the larger terms,(12)T4≅4T∞3T−3T∞4.Substituting equations (11) and (12) into (9), the heat equation takes the form(13)u∂T∂x+v∂T∂y=kρcpf∂2T∂y2+ρcppρcpfDB∂C∂y∂T∂y+DTT∞∂T∂y2+16σ∗T∞33ρcpk∗∂2T∂y2.The relevant boundary conditions are(14)u=uwx=dx,v=0,∂H1∂y=0,H2=0,T⟶Tw,aty⟶0,(15)u=u∞x=bx,H1=H∞,T⟶T∞,C⟶Cw,aty⟶∞.Invoking similarity transformations are defined by(16)ψ=dνxFγ,γ=dνy,u=∂ψ∂y=dxFγ,v=−∂ψ∂x=−dνFγ,H1=H0xLh′γ,H2=−νdH0Lhγ,u=uwx=dx,H∞=H0xL,θγ=T−T∞Tw−T∞,φγ=C−C∞Cw−C∞.The magnetized pressure is described as(17)p=p+μH28π.Equations (5) and (6) are satisfied identically. Equations (7), (8), (10), and (13)–(15) reduce to(18)1+ε−εm1F″′+FF″−F′2+η2+βh′2−hh″−1+λ1θ−Nrφ=0,(19)α1h″′+Fh″−hF″=0,(20)1Pr1+43Rdθ″+θ′F+Nbφ′θ′+Ntθ′2=0,(21)φ″+LePrFφ′+NtNbθ″=0,with boundary conditions(22)F=0,F′=1,θ=1,φ=0,h=0,h″=0,atγ⟶0,(23)F′=η,θ=0,φ=1,h′=1,atγ⟶∞.Here, prime denotes derivative forγ and other dimensionless parameters are described as(24)λ1=1−CβρfmTw−T∞gd2xρf,Nr=ρp−ρfmCw−C∞gd2xρf,η=bd,Rd=4T∞3σ∗k∗kf,Nt=ρcppDTTw−T∞ρcpfνT∞,Pr=να,Nb=ρcppDBCw−C∞ρcpfν,Le=αDB,α=Kρcpf,ε=1δc1μ,m1=d3x22c12ν,ν=μρf,α1=α1∗ν,Rex1/2=uwxν,β=H02μ∞4d2πρf.Physical quantities are very valuable from an engineering point of view. These quantities reported the flow behavior which is defined by local Nusselt numberNux, skin friction Cf, and local Sherwood number Shx definitions as follows:(25)Cf=τwρuw2,Nux=xqwkTw−T∞,Shx=xqmDBCw−C∞,where τw represents the surface shear stress, qw denotes the surface heat flux, and qm presents the surface mass flux for fluid:(26)τw=μ+1δc1−16δc13∂u∂y2∂u∂yy=0,qw=−k∂T∂y−16σ∗T∞33k∗∂T∂yy=0,qm=−DB∂C∂yy=0.Using invoking transformation equation (16), the dimensionless local Nusselt number, skin friction, and the local Sherwood number become(27)CfRex1/2=1+εf″0−εm13f″03,NuxRex−1/2=−1+43Rdθ′0,ShxRex−1/2=−φ′0.


## 3. Results and Discussion

Coupled nonlinear differential equations (18)–(21) and their boundary conditions (22) and (23) are numerically worked out by employing the MATLAB scheme. This portion illustrates the impact of nondimensional sundry characteristics on induced magnetic, temperature, velocity, and concentration flow characteristics numerically and graphically. Figure 2 portrays the impact of η versus h′γ. It is noticeable that the induced magnetic spectrum falls when increasing in η. Figure 3 designates the effects of α1 on the induced magnetic pattern. Dual behavior has been seen for α1, near the wall, it is getting down, and far away, it moves upward. The field of h′γ rises by enhancing the amount of β in Figure 4. With larger values of mixed convection characteristic, the flow of induced magnetic expands as shown in Figure 5. Figures 6 and 7 demonstrate the effect of Brownian and thermophoresis diffusion on the induced magnetic field, respectively. When there is increase in the amount of Brownian diffusion, the induced magnetic field decreases, and the induced magnetic field increases by growing quantity of thermophoresis. Figure 8 shows the consequence of the Prandtl number on the induced magnetic curve, and profile falls by a bigger amount of Prandtl. The impacts of thermal radiation flux on the magnetic field are described in Figure 9. The variation of h′γ near the wall moves upward and very far away it moves downward by increasing the amount of thermal radiation. Figure 10 exhibits the outcome of the fluid parameter on h′γexpands in the values of ε which cause rise inh′γ. Figure 11 exhibits the deviation of β on velocity curve. When there is expansion in the amount of β, then there is increase in velocity flow. Figure 12 portrays the effect of λ1 on the flow of velocity. The diagram shows that the fluctuation shoots up by increment in the mixed convection parameter. Nanofluid behaviour is shown by Brownian and thermophoretic characteristics in in Figures 13 and 14. It is depicted that the contrary attitude showed variation in velocity decline for Nb and incline for Nt by enhancing the quantity of these parameters. Figure 15 investigates that the velocity graph expands if the fluid parameter ε rises. Figure 16 demonstrates that the field of velocity shrinks by rising the amount of the reciprocal of the magnetic Prandtl number. Figure 17 shows that when increasing stretching parameter, the field of F′γ enhances. Figures 18 and 19 show opposite behavior against the parameters of α1 and β, respectively. The temperature profile increases and decreases against the characteristics ofα1 and β. Figure 20 explores the impact of Lewis number on temperature distribution, θγ rises near the wall, and far away, it declines compared to Le. The temperature reduction against the Brownian motion diffusion is shown in Figure 21. Figure 22 describes that temperature sketch grows by inclining the quantity of thermophoresis diffusion. It is easily noticed that enlarging the amount of Prandtl number causes the figure of temperature decline in Figure 23. Figure 24 depicts the influence of η on the temperature field. The amplitude of temperature diminishes when the size of the stretching parameter boosts up. Figure 25 explores the influence of radiation characteristics on temperature portraits; field enhances by rising the amount of radiation parameter. Figure 26shows the influence of buoyancy ratio parameter on temperature distribution field increases by increasing the amount of buoyancy ratio parameter. The nanoparticle concentration field decreases if the reciprocal of the magnetic Prandtl number rises in Figure 27. Figure 28 exhibits the outcome of β on the concentration field. The figure of concentration inclines when the number of β grows. Figure 29 portrays the result of buoyancy ratio characteristics on nanoparticle concentration. It is easily observed that the field of concentration reduces if the number of Nr increases. Figures 30 and 31 define the matching outcomes on the concentration profile. When the size of Pr and η expands, the nanoparticle concentration grows. Streamlines graphs against the distinct amount ofα1andεare shown in Figures 32–35. Table 1 shows the impact of different characteristics on drag friction CfRex1/2. The Nusselt number for noticeable amounts of Rd, Nt, Nb, ε, λ1, m1, β, and α1 is analyzed and characterized in Table 2. Table 3 portrays the deviation of different amounts of parameters on the local Sherwood number.Figure 2
Outcomes ofη on h′γ.Figure 3
Impacts ofα1 on h′γ.Figure 4
Upshot ofβ on h′γ.Figure 5
Results ofλ1 on h′γ.Figure 6
Effects ofNb on h′γ.Figure 7
Consequences ofNt on h′γ.Figure 8
Upshot ofPr on h′γ.Figure 9
Consequence ofRd on h′γ.Figure 10
Upshot ofε on h′γ.Figure 11
Result ofβ on F′γ.Figure 12
Outcome ofλ1 on F′γ.Figure 13
Result ofNb on F′γ.Figure 14
Impact ofNt on F′γ.Figure 15
Upshot ofε on F′γ.Figure 16
Influence ofα1 on F′γ.Figure 17
Outcome ofη on F′γ.Figure 18
Upshot ofα1 on θγ.Figure 19
Result ofβ on θγ.Figure 20
Outcome ofLe on θγ.Figure 21
Influence ofNb on θγ.Figure 22
Deviation ofNt on θγ.Figure 23
Consequence ofPr on θγ.Figure 24
Influence ofη on θγ.Figure 25
Conclusion ofRd on θγ.Figure 26
Result ofNr on θγ.Figure 27
Outcome ofα1 on θγ.Figure 28
Upshot ofβ on θγ.Figure 29
Effect ofNr on ϕγ.Figure 30
Impact ofPr on ϕγ.Figure 31
Result ofPr on ϕγ.Figure 32
Flowlines forα1=0.5.Figure 33
Flowlines forα1=1.5.Figure 34
Flowlines forε=0.1.Figure 35
Flowlines forε=1.Table 1
Variation ofCfRex1/2 for distinct amounts of nondimensional parameters.
RdNtNbεPrλ1m1ηNrLeβα1CfRex1/20.10.10.1170.10.10.50.10.10.10.5−0.859740.3−0.870930.5−0.881480.10.20.1170.10.10.50.10.10.10.5−0.933030.3−1.005700.5−1.148350.10.10.2170.10.10.50.10.10.10.5−0.823440.3−0.811820.4−0.806350.10.10.11.270.10.10.50.10.10.10.5−1.432011.3−1.400421.4−1.370910.10.10.1170.20.10.50.10.10.10.5−0.832000.3−0.804320.4−0.776700.10.10.1170.10.20.50.10.10.10.5−0.842850.4−0.846590.6−0.850460.10.10.1170.10.10.50.10.10.20.5−0.666890.4−0.097880.60.683680.10.10.1170.10.10.50.10.10.10.2−0.840190.4−0.854000.6−0.86503Table 2
Variation ofNuxRex−1/2 for distinct amounts of nondimensional parameters.
RdNtNbεPrλ1m1ηNrLeβα1NuxRex−1/20.10.10.1170.10.10.50.10.10.10.53.682350.33.460530.53.257530.10.20.1170.10.10.50.10.10.10.52.104530.32.046080.51.915480.10.10.2170.10.10.50.10.10.10.52.218320.32.277280.42.338140.10.10.1190.10.10.50.10.10.10.52.46411122.86748153.226220.10.10.1170.10.20.50.10.10.10.52.158450.42.158090.62.157740.10.10.1170.10.10.50.10.10.20.52.181230.42.244570.62.324380.10.10.1170.10.10.50.10.10.10.22.160420.42.159190.62.15807Table 3
Variation ofShxRex−1/2 for distinct amounts of nondimensional parameters.
RdNtNbεPrλ1m1ηNrLeβα1ShxRex−1/20.10.10.1170.10.10.50.10.10.10.5−1.939210.3−2.207110.5−2.431440.10.20.1170.10.10.50.10.10.10.5−3.704630.3−5.348660.5−8.217110.10.10.2170.10.10.50.10.10.10.5−1.033880.3−0.733830.4−0.585750.10.10.1190.10.10.50.10.10.10.5−2.2149312−2.5793615−2.903860.10.10.1170.10.20.50.10.10.10.5−1.939060.4−1.938740.6−1.938420.10.10.1170.10.10.50.10.10.20.5−1.959580.4−2.016670.6−2.088580.10.10.1170.10.10.50.10.10.10.2−1.940830.4−1.939720.6−1.93872


## 4. Concluding Remarks

We studied the molecular theory of liquid-originated non-Newtonian nanofluids which are commonly used in heat transfer devices such as heat exchangers, engine oils, electrical cooling systems (such as flat plates), nuclear reactors, biomedicine lubricants, and radiators. The key points of observation in the recent analysis are as follows:(i)
Induced magnetic pattern near the wall declines, and far away, it inclines whenα1 intensifies. The variation of h′γ field near the wall goes up and very far away it decays when the size Rd ascends.(ii)
h′γ (nondimensionless induced magnetic function) falls, whereas η, Brownian diffusion Nb, and Prandtl number Pr rise. The field h′γ expands by enhancing the amount of magnetic parameter β, mixed convection λ1, thermophoresis parameter Nt, and fluid parameter ε. The variation of h′γ profile near the wall moves upward and very far away it moves down when the size Rd ascends.(iii)
The velocity amplitude expands by enlargement in the amount of magnetic parameterβ, mixed convection λ1, thermophoresis parameter Nt, fluid characteristic ε, and stretching parameter η.F′γ collapses by Brownian motion Nb and α1.(iv)
The temperature spectrum increases when the values ofα1, radiation parameterRd, and buoyancy ratioNr increases and decreases by Prandtl number, magnetic parameterβ, Brownian motion diffusionNb, and stretching parameterη. θγ(nondimensional temperature function) rises near the wall, and far away, it diminishes when there is increase in the values of Le.(v)
The nanoparticle concentration portrait reduces if the reciprocal of the magnetic Prandtl numberα1 and Nr rises. Concentration enlarges when the number of β, Pr, and η grows.(vi)
Drag friction decays by the inclination in the values ofRd, Nt, m1, and α1. When inclining the amount of Nb, ε, λ1, and β, drag force expands.(vii)
Heat transfer rates are increased when there is an increase in the values ofNb, Pr, andβand decrease when there is an increase in the values of Rd, Nt, m1, andα1.(viii)
Mass transfer rates diminish, forRd, Nt, Pr, and β, but increases by Nb, m1, and α1.



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