# Thermal Radiation Effects on Unsteady Stagnation Point Nanofluid Flow in View of Convective Boundary Conditions

Authors: Haroon Ur Rasheed; Saeed Islam; Zeeshan Khan; Sayer O. Alharbi; Waqar A Khan; Waris Khan; Ilyas Khan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The present communication particularizes nonlinear convective non-Newtonian stagnation point flow and heat transference effects in stretchable flow of nanofluid. Magnetohydromagnetic steady viscous flow of nanofluid is examined. Heat transfer attributes of nanofluids are addressed via a numerical algorithm. Conductivity and diffusivity characteristics of fluid are depending on temperature and concentration and furthermore, on mass conservation, momentum, energy, and concentration yield partial differential equations (PDEs). The boundary layer flow concept pioneered by Prandtl has been employed to simplify the nonlinear constitutive flow laws which are then changed to ordinary differential equations. A built-in bvp4c algorithm in Mathematica software yields convergent outcomes of nonlinear (ODEs) systems. A comprehensive analysis has been made elucidating the physical significance of various governing parameters effects presented graphically. Additionally, the flow nature was confirmed versus streamlines.


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## Body

## 1. Introduction

The low thermal conductivity features of various non-Newtonian fluids have serious problems with heat transport mechanisms in different scientific and practical engineering applications. Several attempts have been made by researchers and engineers to address the low thermal behavior of the base fluids. Different tools and mechanisms have been adopted for the enhancement of thermal conductivity issues. Out of these suggested mechanisms, incorporating small nanosized metallic particles into base fluid yields potential results. The word nanofluid was adopted by Choi and Eastman [1]. Khan et al. [2] discussed fluid thermal conductivity features by utilizing nanofluid for the development of heat transportation analysis for various applications in different heat exchanger equipment. There are many important applications and narrative characteristics of nanofluids in various applications in engineering and industrial units. These nanofluid applications include heat exchange processes such as chillers, nuclear power stations, car engines, microelectronics, fuel cells, coolant, and many more engineering utilization as discussed by Turkyilmazoglu [3]. The nanofluid has significant advantages and applications in manufacturing industries. A great number of mathematicians, researchers, and scientists addressed such utilizations of nanofluid for development in [4–10]. For the first time, Hiemenz [11] conducted extensive research and discussed the exact solution to stagnation point flow by Navier–Stokes’s equation. Furthermore, stagnation point flow was investigated, which can be steady or unsteady, normal or oblique, two or more dimensional, onward or opposite, and viscous or inviscid. Khan et al. [12] elucidated different aspects of unsteady non-Newtonian stagnation point nanofluid flow and multiple solutions analyzed and furthermore, added the enhancement of thermal behavior of the base fluid by incorporating the nanofluid. A potential investigation has been made to analyze three-dimensional stagnation point hybrid nanofluid flow over a slip surface by Abbas et al. [13]. Serious attention has been given in recent times by researchers, engineers, and mathematicians to explore non-Newtonian stagnation point flow nanofluid flow [14–17]. The viscoelastic nanofluids have a variety of practical usage in daily life and large-scale applications. Such important utilizations of non-Newtonian fluids are foods, multiphase mixtures molecular weight systems, and various solutions. Abbas et al. [18] studied stagnation point MHD flow and thermal radiation effects on micropolar flow past through a round cylinder with sinusoidal radius variation. Furthermore, porosity influence and velocity slip jump are evaluated. Akbar et al. [19] explored hydromagnetic convective condition along with slip effects on stagnation point nanofluid by incorporation of CNT particles past a stretchable surface, and thermal radiations have been analyzed. Moreover, the convective boundary conditions with slip flow and incorporated CNT nanofluid vied in a stretchable plane surface. Those materials showing non-Newtonian fluid flow features contain the natural products, food products, polymer melts and solutions, biological fluids, multiphase mixtures, and personal care products including investigated toiletries and cosmetics and by Ijaz et al. [20]. The non-Newtonian fluids have a great number of significant utilizations in industrial units, medical technologies, and engineering application. The expanding uses of these nanofluids describe petroleum drilling, steel substances, blasting of glasses, and extrusion of polymers. Some of the polymers employed in telecommunication appliances, agriculture, and medical are analyzed by Hussain et al. [21]. Muhammad et al. [22] investigated nonlinear thermal radiation in three-dimensional Eyring–Powell nanofluid flow with activation energy. Hussain and Muhammad [23] discussed convective heat transfer at the boundaries is incorporated by employing effective thermal conductivity of nanoliquid. Muhammad et al. [24] studied Carreau nanofluid under the influence of slip over a wedge with motile microorganisms and thermal radiation effects. Besides, these many scientists carried to analyze the stagnation point flow and other non-Newtonian fluids with aspect to the following citations, provided a detailed view of flows [25–33].


## 2. Mathematical Formulation

Here, we formulate unsteady non-Newtonian stagnation point stretchable flow of incompressible viscous nanofluid along with convection boundary conditions. Lower surface of sheet heated convectively fluid having temperatureTf and heat transportation transfer coefficient hf and the ambient fluid having temperature T∞ and concentration C∞, respectively. The stretchable sheet velocity at the surface is defined by the relation uwx=axn, where a is the fixed positive constant. Magnetic field source is adequate with strength B0 positioned in +veydirection normal to stretchable surface for magnetization. The influence of induced magnetic properties is small with respect to the applied one and is neglected. Furthermore, there is a thermal equilibrium to base fluid and suspended nanoparticles. Flow diagram and coordinate axes are shown in Figure 1. Boundary layer governs flow expressions byMass conservation expression:(1)∂u∂x+∂υ∂y=0.Momentum expression:(2)u∂u∂x+v∂u∂y=υ∂2u∂y2+U∞∂U∞∂x+σB02ρfU∞−u.Energy expression:(3)u∂T∂x+v∂T∂y=α∂2T∂y2+ΓDB∂C∂y∂T∂y+DTT∞∂T∂y2.Concentration of nanoparticles:(4)u∂C∂x+v∂C∂y=DB∂2C∂y2+DTT∞∂2T∂y2.Subject to flow conditions are as follows:(5)u=uwx=ax,v=0,−k∂T∂y=hfTf−T,DB∂C∂y+DTT∞∂T∂y=0,fory=0,u⟶U∞=bx,v=0,T⟶T∞,C⟶C∞,fory=∞.Herein,uandv velocity components are in coordinates axes, along and normal to flow directionThe kinematics viscosity isυ, inside boundary layer temperature is T, nanoparticles effective heat capacity is ρcp, thermal capacity of fluid is ρcf, fluid density is ρ, Γ=ρcp/ρcf, and T∞ is the ambient temperature away from stretchable surface.Introducing dimensionless parameters,(6)η=aυy,ψ=aυxfη,θη=T−T∞Tf−T∞,ϕη=C−C∞C∞.Here, the stream functionψx,y is defined by(7)u=∂ψ∂y,v=−∂ψ∂x.Equation (1) is justified automatically, and equations (2)–(4) take the following forms by employing equation (6):(8)f‴+ff″−f′2+A2+MA−f′=0,(9)1Prθ″+fθ′+Nbϕ′θ′+Ntθ′=0,(10)ϕ″+LePrfϕ′+NtNbθ″=0.Transformed conditions:(11)f0=0,f′0=1,θ′0=Biθ0−1,Nbϕ′0+Ntθ′0=0,f′∞=A,θ∞=0,ϕ∞=0.Variables seeming in constitutive flow laws (8)–(11) are defined as follows:M=σB02/ρfa, the magnetic parameter; Le=α/DB, Lewis number; Pr=υ/α, Prandtl number; A=b/a, velocity ratio parameter; Nb=ρcpDBC∞/ρcfυ, Brownian motion parameter; Nt=ρcpDTTf−T∞/ρcfυT∞, thermophoresis parameter; and Bi=υ/ahf/k, Biot number, respectively.Expressions of physical quantitiesCfx,Nux are defined by(12)Cfx=τwρuw2,Nux=xqwkTf−T∞.(13)τw=μ∂u∂y|aty=0,qw=−k∂T∂y|aty=0.By substituting (13) in (12), we have(14)Rex0.5Cfx=−f″0Rex−0.5Nux=−θ′0.Figure 1
Flow geometry.


## 3. Numerical Solution and Convergence Analysis

For numerical solution, the 1st order differential equations are found from equations (8)–(10) by introducing the following transformations relations. Let the transformations variables be defined by(15)fη=x1,f′η=x2,f″η=x3,θη=x4,θ′η=x5,ϕη=x6,ϕ′η=x7.Following are the first-order differential equations generated:(16)x1′=x2,x2′=x3,x3′=x22−x1x3−A2−MA−x2,x4′=x5,x5′=−Prx1x5+Nbx7x5+Ntx52,x6′=x7,x7′=−LePrx1x7+NtNbx5′.The initial boundary conditions are(17)x10=0,x20=1,x50=Bix40−1,Nbx70+Ntx50=0x2∞=A,x4∞=0,x6∞=0.A suitable step sizeΔη=0.001 is selected for the entire numerical computation work. A possible convergence limit has been adopted as 10−6. Furthermore, the boundary conditions given in (11) value at infinity is chosen as η∞=4; after selecting the appropriate value for η∞, all numerical solutions converge to it and are satisfied correctly. Finally, residual error analysis has been evaluated. Dwindles in error perceive for higher order deformations as shown in Figure 2. Furthermore, the algorithm of bvp4c routine calculation is shown in Figure 3.Figure 2
Total residual error via order of deformations.Figure 3
Algorithm of bvp4c routine in Mathematica software.


## 4. Results and Discussion

The constative flow laws of nonlinear differential systems consisting of (8)–(10) subjected to specified boundary are solved numerically by employing the bvp4c algorithm. The attributes of affective pertinent flow factors on velocity field, thermal field, and nanoparticles volume are evaluated graphically and discussed in Figures 4–25.Figure 4
f′η versus AA>1.Figure 5
f′η versus AA<1.Figure 6
f′η versus M.Figure 7
θη versus Pr.Figure 8
θη versus Nt.Figure 9
θη versus Bi.Figure 10
θη versus A.Figure 11
ϕη versus Nb.Figure 12
ϕη versus Nt.Figure 13
ϕη versus Pr.Figure 14
ϕη versus Le.Figure 15
ϕη versus Bi.Figure 16
ϕη versus A.Figure 17
Variation in−f″0 via AandM.Figure 18
Variation in−f″0 via MandA.Figure 19
Variation in−θ′0 via Bi andNt.Figure 20
Variation in−θ′0 via Bi andA.Figure 21
Variation in−θ′0 via PrandNt.Figure 22
Streamlines forM=1.0.Figure 23
Streamlines forM=10.0.Figure 24
Streamlines forA=1.0.Figure 25
Streamlines forA=10.0.Figures4 and 5 depict variations in f′η velocity subjected to velocity ratio parameter A. These figures unveil enhancement in boundary layer thickness for higher A, perceive when A>1, and diminish the hydromagnetic boundary layer thickness for lower values of A when A<1. In reality, stretching velocity is less than the free stream velocity, due to which the ratio of stretching velocity to free stream velocity is less than unity; consequently, decreasing force vanishes, and clearly, f′η profile is augmented. Attributes of M are shown in Figure 6. Herein, we noticed f′η is lower subjected to increment in magnetic field. Such scenario found due to higher M implies strong resistance force came into existence called Lorentz force. In consequence, dwindles in velocity f′η profile perceived. The contribution of the Prandtl number Pr on thermal field θη is evaluated as shown in Figure 7. The thermal diffusivity diminishes when Pr boosts through larger values. Hence, θη decays. Consequently, thermal boundary layer thickness escalates. Figure 8 explains variations in θη thermal field curves for the thermophoresis Nt parameter. This figure unveils enhancement in θη when Nt is enlarged. In fact, nanoparticles attain extra heat subject to the Nt factor. In consequence, θη upsurges. The temperature field θη curves for the Biot number Bi also called convective heating are shown in Figure 9. One can clearly perceive that θη is a growing function of Bi. In reality, heat transference at surface boosts through larger Bi. Consequently, θη upsurges. Hence, thermal boundary layer thickness escalates. Variations in temperature field θη against A are interprets as shown in Figure 10. Clearly, larger velocity ratio factor diminishes thermal boundary layer thickness.Impact of concentrationϕη profile against the Nb parameter is portrayed as shown in Figure 11. This figure witnesses enhancement in ϕη. Clearly, ϕη is an augmenting function of the Brownian motion parameter. In reality, one can observe heat transference boosts through larger Nb. Consequently, ϕη augments. Figure 12 discloses contribution of the Nt parameter on ϕη profile. Clearly, dwindles ϕη subject to increment in Nt. Figure 13 shows the effects of Prandtl number Pr on concentration gradient. As the influence of the Pr parameter increases, nanoparticles diffuse outside which reduced the concentration at the surface. The contribution of Lewis number Le on the concentration profile is shown in Figure 14. An increasing value of the Le parameter shows a poor Nb diffusion coefficient as a result of small penetration depth observed for ϕη and displays decreasing trends in the concentration profile. The contribution of Boit number Bi parameter via ϕη is shown in Figure 15. Clearly, larger estimations correspond to upsurges in the ϕη profile. Figure 16 unveils A velocity ratio factor influences on ϕη. It is viewed that the concentration ϕη profile is a decreasing function of A. Such scenario is noticed because augmentation in A yields reduction in fluid thermal conductivity. Consequently, ϕη vanishes. Figures 17 and 18 emphasize A,M,Nt, and Bi influences on −f″0. Here, skin friction diminishes for AandM are augmented. However, opposite appearances noticed for Ntand Bi enlarged. The attributes of Bi,Nt,A, and Pr interpreted on −θ′0 are shown in Figures 19–21. These figures confirm that −θ′0 Nusselt number upsurges subject to augmented in Bi,Nt,A, and Pr parameters. Figures 22–25 explain the impression of magnetic and velocity ratio parameters on streamlines.


## 5. Conclusions

Here, we formulated hydrodynamics stagnation point nanofluid flow over a surface with the magnetic effect and convective boundary condition. Furthermore, heat transportation features subject to viscous dissipation has been discussed. We witnessed the following are the noteworthy points from aforementioned investigation:We found lower−f″0 when A velocity ratio factor incremented and −f″0 upsurges for a higher magnetic field valueAn augmentation inNb and Nt yields an opposite report for the solutal ϕη boundary layerLarger velocity parameterA and Boit number Bi yields −θ′0 escalatesAn increment in thermal boundary is noticed for higherNt and diminishes for larger Pr valuesBoth parametersNb and Nt yield higher thermal θη fieldThe solutalϕη boundary layer thickness is near stretching sheet negative and zero elsewhere



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