# Heat Transfer Analysis of the MHD Stagnation Point Flow of a Non-Newtonian Tangent Hyperbolic Hybrid Nanofluid past a Non-Isothermal Flat Plate with Thermal Radiation Effect

Authors: Abdullah Dawar; Saeed Islam; Ahmed Alshehri; Ebenezer Bonyah; Zahir Shah
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

Heat transfer phenomena are used in a variety of industries, including chemical devices, shipbuilding, power plants, electronic devices, and medicinal plants. Propylene glycol, engine oil, water, and ethylene glycol are common single-phase heat transfer liquids used in a variety of industries, including chemical process industries and thermal power plants. Therefore, the authors are interested in investigating the magnetohydrodynamic flow of a water-based hybrid nanofluid containing ferrous and graphene oxide nanoparticles past a flat plate. The stagnation points, as well as the effects of magnetic field and thermal radiation are taken into account in this analysis. The non-Newtonian tangent hyperbolic flow, which is laminar and incompressible, is also considered to investigate the non-Newtonian behavior of the hybrid nanofluid flow. The proposed model has been solved analytically with the help of HAM. The convergence of HAM is shown with the help of figure. The hydrothermal characteristics of hybrid nanofluid flow past a nonisothermal flat plate at a stagnation point are affected by the necessary parameters. The results show that the boosting volume fractions of the ferrous and graphene oxide nanoparticles have significantly reduced the velocity field, while the thermal field has increased with the augmenting volume fractions of the ferrous and graphene oxide nanoparticles. The increasing power-law index has augmented the viscosity of the non-Newtonian hybrid nanofluid flow due to which the velocity field escalated. However, this impact is opposite for the thermal field. Due to the direct relation between the Weissenberg number and relaxation time, the greater Weissenberg number has reduced the velocity profile, while increased the thermal field.


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## Body

## 1. Introduction

Because of various applications in healthcare and engineering, such as microelectronics, solar collectors, process industries, cancer therapy, heat exchangers, and power production, the mechanisms of heat exchange incorporating nanomaterials have piqued the interest of researchers. Regular liquids such as glycol mixtures, engine oil, and water had moderately poor thermal properties and inadequate capacity to attain higher thermal efficiency. The use of nanoparticles to develop the thermal conductivity of various cooling fluids is a contemporary method. Nowadays, temperature distribution plays an essential function in a variety of scientific and technical disciplines. Heat transfer phenomena have a wide range of applications in sectors, shipbuilding, electronic devices, power plants, medicinal, and chemical devices. To design heat exchangers and discover the optimal geometry, radiators, condensers, evaporators, and boilers, heat transfer analysis and the related cooling process become invaluable. Propylene glycol, engine oil, water, and ethylene glycol are common single-phase heat transfer liquids used in a variety of chemical process industries and thermal power plants. Due to its low thermal conductivity, the single-phase traditional liquids are acknowledged to have poor heat transmission ability. This improvement in working fluid heat transport is critical for achieving energy and cost reductions. In order to boost-up the thermal conductivity of the base fluids, many researchers have worked to resolve these issues and improve the thermal conductivity of the base fluids. Solid materials have higher thermal conductivity than those of liquids. As a result, dispersing microscopic solid particles into a base liquid is a novel technique to increase the thermal conductivity of the base fluids. Khan et al. [1] investigated the Casson nanofluid flow past a rotating disk. Shah et al. [2] addressed the applications of radius, heat flux, and mass flux of the water-based copper nanoparticles. Gul T et al. [3] investigated the flow of carbon nanotube nanofluid past a rotating cone and disk. Sowmya et al. [4] addressed the effects of convective condition and internal heat generation in a nanofluid flow past a porous fin. Ashraf et al. [5] analyzed the magnetohydrodynamic peristaltic flow of the blood-based magnetite nanoparticles. Dawar et al. [6] studied the unsteady flow of carbon nanotube nanofluid with the magnetic field impact. Rasool and Wakif [7] examined the electromagnetohydrodynamic second-grade nanofluid flow over a Riga plate. Alghamdi et al. [8] presented the magnetohydrodynamic flow of sodium alginate-based nanofluid bounded by slender surface with heat source impact. Rout et al. [9] analyzed the water- and kerosene-based nanofluid flow with viscous dissipation. Alshomrani and Gul [10] investigated the dissipative flow of water-based Al2O3 and Cu nanoparticles past a stretching cylinder with convective condition. Further related analyses can be studied in [11–14].Hybrid nanofluid is a class of nanofluids, developed by integrating a certain class of nanoparticles inside a functional fluid which has recently been used. Two different nanomaterials are suspended in a conventional fluid to create hybrid nanofluids. Hybrid nanofluids are widely used in a diversity of disciplines of engineering as well as refrigeration, space planes, biomedicals, machining coolant, motor cooling, heat pipe reduction in medicine, and high-performance boats. Jana et al. [15] investigated the conductive nanomaterials like copper and gold nanoparticles and their hybrids. Khashi’ie et al. [16] analyzed heat transfer of a magnetohydrodynamic flow of a water-based hybrid nanofluid comprehending Cu and Al2O3 nanoparticles. Their results show that the suction factor has a significant impact of heat transfer analysis. Additionally, they have computed the stability analysis as well. Nawaz and Nazir [17] studied the magnetohydrodynamic flow of an ethylene-based hybrid nanofluid flow containing MoS2 and SiO2 nanoparticles. They compared MoS2/ethylene-based and MoS2-SiO2/ethylene-based hybrid nanofluids. Their results showed that the thermal performance is greater for the MoS2-SiO2/ethylene-based as compared to MoS2/ethylene-based. Manjunatha et al. [18] presented the comparative analysis of the magnetohydrodynamic flows of Cu-H2O nanofluid and Cu-Al2O3/H2O hybrid nanofluid. They found that the nanoparticle volume fraction of the nanofluid and hybrid nanofluid has enhanced the velocity and thermal fields. Usman et al. [19] proposed the comparative analysis of the magnetohydrodynamic flow of a Cu/H2O nanofluid, Al2O3/H2O nanofluid, and Cu-Al2O3/H2O hybrid nanofluid. They claimed that the velocity fields of Cu-Al2O3/H2O have dominant role on other nanofluids; however, this impact is reverse for thermal profile. Iqbal et al. [20] offered the comparative analysis of magnetohydrodynamic flows of SiO2/H2O nanofluid and MoS2-SiO2/H2O hybrid nanofluid considering different shapes of the nanoparticles. Their results showed that the nanofluid has slower flow as compared to hybrid nanofluid. Additionally, the lower temperature is observed for brick-shaped nanoparticles of the nanofluid, while the blade-shaped nanoparticle of the hybrid nanofluid has extreme temperature. Ghadikolaei et al. [21] offered the comparative investigation of magnetohydrodynamic flow of Cu/H2O and hybrid nanofluid containing TiO2-Cu/H2O at a stagnation point. They also considered three different shapes of the nanoparticles named as platelets, bricks, and cylinders. It is clear from of this research that using platelet-shaped nanoparticles is more effective. Gul et al. [22] addressed the magnetohydrodynamic flow of hybrid nanofluid containing Cu and Fe3O4. Their results showed that the nanoparticle volume fractions of Cu and Fe3O4 have significantly improved the thermal transmission and velocity field. Alghamdi et al. [23] addressed the comparative analysis of the magnetohydrodynamic flows of blood-based Cu nanofluid and blood-based Cu-CuO hybrid nanofluid. It has been introduced that the hybrid nanofluid flow has more effective thermal conductivity in a contracting channels. Acharya [24] probed the application of solar energy toward a hybrid nanofluid flow containing alumina and copper nanoparticles. In another article, Acharya and Mabood [25] addressed the water-based hybrid nanofluid flow containing ferrous and graphene oxide nanoparticles. Thumma et al. [26] investigated the Cu-CuO nanoparticles past a porous extending surface. Acharya et al. [27, 28] analyzed the nanofluid and hybrid nanofluid flows under the impact of magnetic field.According to the authors’ knowledge, there is no study based on magnetohydrodynamic flow of water-based hybrid nanofluid containing ferrous and graphene oxide nanoparticles past a flat plate. The stagnation point along with the impacts of magnetic field and thermal radiation is taken in this consideration. The non-Newtonian tangent hyperbolic flow which is laminar and incompressible is also considered to investigate the non-Newtonian behavior of the hybrid nanofluid flow. The present analysis is composed of mathematical modeling which is shown in Section2. HAM solution and convergence of HAM are presented in Sections 3 and 4, respectively. Section 5 is composed of results and discussion. In the last, the concluding remarks are shown in Section 6.


## 2. Model Formulation

Consider the stagnation point flow of a water-based hybrid nanofluid containing graphene oxide (GO) and ferrous (Fe3O4) nanoparticles past a flat plate. The flat plate is chosen to be a nonisothermal. The non-Newtonian tangent hyperbolic model is taken to be laminar and incompressible. u and v are the velocity components which are considered along x- and y-directions, respectively. A magnetic field B=0,B0,0 is considered normal to the flow direction. The ambient velocity of the fluid flow along x-direction is uex=cx, where c is the positive constant. The wall temperature Twx=T∞+bx varies linearly along x-direction in which b is the positive constant and T∞ is the ambient temperature. Furthermore, the Hall current and thermal radiation effects are also considered. Following the above assumption, the leading equations are stated. Figure 1 shows the geometry of the hybrid nanofluid flow.
(1)∂u∂x+∂v∂y=0,u∂u∂x+v∂u∂y=ueduedx+μhnfρhnf1−n∂2u∂y2+2nΓ∂u∂y∂2u∂y2−σhnfB02ρhnfu−ue,u∂T∂x+v∂T∂y=αnf∂2T∂y2−1ρCphnf∂qr∂y+σhnfB02ρCphnfu2+μhnfρCphnf1−n∂u∂y2+nΓ2∂u∂y3.Figure 1
Geometry of the hybrid nanofluid flow.The relevance boundary conditions are defined as(2)u=0,v=0,T=Twaty=0,u⟶ue,T⟶T∞asy⟶∞.The radiative heat fluxqr is defined as
(3)qr=−4σ∗3k∗∂T4∂y.By using the Taylor series expansion,T4 can be written as
(4)T4≈4T∞3T−3T∞4.For the simulation of hybrid nanofluid flow, the thermophysical properties are defined as(5)μhnfμf=11−ϕ1−ϕ22.5,ρhnfρf=1−ϕ1−ϕ2+ρp1ϕ1+ρp2ϕ2ρf,ρCphnfρCp⌢f=1−ϕ1−ϕ2+ρCp⌢p1ϕ1+ρCpp2ϕ2ρCp⌢f,khnfkf=kp1ϕ1+kp2ϕ2/ϕ1+ϕ2+2kf+2kp1ϕ1+kp2ϕ2−2ϕ1+ϕ2kfkp1ϕ1+kp2ϕ2/ϕ1+ϕ2+2kf−kp1ϕ1+kp2ϕ2+ϕ1+ϕ2kf,σhnfσf=1+3σp1ϕ1+σp2ϕ2/σf−ϕ1+ϕ22+σp1ϕ1+σp2ϕ2/ϕ1+ϕ2σf−σp1ϕ1+σp2ϕ2/σf+ϕ1+ϕ2,where p1 and p2 represent Fe3O4 and GO nanoparticles, respectively, and ϕ1 and ϕ2 are the nanoparticle volume fractions of Fe3O4 and GO, respectively. The numerical values of the thermophysical properties are defined in Table 1.Table 1
The numerical values of the thermophysical properties of base fluids and nanoparticles [29].
Base fluids/nanoparticlesρ (kg.m‑3)Cp (J.kg−1.K−1)k [W.m−1.K−1]σ ((Ωm)−1)H2O99741800.60710.005Fe3O451806709.725000GO2250210025001×107The similarity transformations are defined as(6)u=cxϒ′ζ,u=−cνfϒζ,Θζ=T−T∞Tw−T∞,ζ=ycνf.Using the above similarity transformations, the leading equations are transformed as(7)11−ϕ1−ϕ22.51−n+nWeϒ″ϒ‴+1−ϕ1−ϕ2+ρp1ϕ1+ρp2ϕ2ρfϒϒ″+1−ϒ′2−M21+3σp1ϕ1+σp2ϕ2/σf−ϕ1+ϕ22+σp1ϕ1+σp2ϕ2/ϕ1+ϕ2σf−σp1ϕ1+σp2ϕ2/σf+ϕ1+ϕ2ϒ′−1=0,kp1ϕ1+p2ϕ2/ϕ1+ϕ2+2kf+2kp1ϕ1+kp2ϕ2−2ϕ1+ϕ2kfkp1ϕ1+kp2ϕ2/ϕ1+ϕ2+2kf−kp1ϕ1+kp2ϕ2+ϕ1+ϕ2kf+43RdΘ″+Pr1−ϕ1−ϕ2+ρCp⌢p1ϕ1+ρCpp2ϕ2ρCp⌢fϒΘ′−ϒ′Θ+EcPr1−ϕ1−ϕ22.51−nϒ″2+n2Weϒ″3+PrM2Ec1+3σp1ϕ1+σp2ϕ2/σf−ϕ1+ϕ22+σp1ϕ1+σp2ϕ2/ϕ1+ϕ2σf−σp1ϕ1+σp2ϕ2/σf+ϕ1+ϕ2ϒ′2=0,with boundary conditions
(8)ϒ0=0,ϒ′0=0,ϒ′∞=1,Θ0=1,Θ∞=0.The dimensionless parameters are defined as(9)Rd=4σ∗T∞3k∗kf,Pr=νfαf,M=σfB02ρfc,We=2cΓRex1/2,Rex=xuexνf,Ec=ue2CpfTw−T∞.Physical quantities of importance like skin frictionCfx and Nusselt number Nux are defined as
(10)Cfx=μhnfρhnfue21−n∂u∂y+nΓ2∂u∂y2y=0,Nux=−xTw−T∞khnfkf∂T∂y+4σ∗3k∗kf∂T4∂yy=0.Using the similarity transformations defined in equation (6), the above quantities are reduced to
(11)Rex1/2Cfx=11−ϕ1−ϕ22.51−ϕ1−ϕ2+ρp1ϕ1+ρp2ϕ2/ρf1−nϒ″0+nWe2ϒ″20,Rex−1/2Nux=−kp1ϕ1+p2ϕ2/ϕ1+ϕ2+2kf+2kp1ϕ1+kp2ϕ2−2ϕ1+ϕ2kfkp1ϕ1+kp2ϕ2/ϕ1+ϕ2+2kf−2kp1ϕ1+kp2ϕ2+ϕ1+ϕ2kf+43RdΘ′0.


## 3. HAM Solution

To attain the analytical solution of the proposed model along with the relevant boundary conditions, HAM method which was introduced by Liao [30] is applied. The initial guesses and linear operators are defined as
(12)ϒ0ζ=−1+ζ+e−ζ,Θ0ζ=e−ζ,Lϒ=ϒ‴−ϒ′,LΘ=Θ″−Θ,satisfying
(13)LϒR^1+R^2e−ζ+R^3eζ=0,LΘR^4e−ζ+R^5eζ=0,where R^1−R^5 are constants.


## 4. HAM Convergence

Homotopy analysis method guarantees the convergence analysis of the highly linear and nonlinear differential equations. The auxiliary parameterh insures the convergence area of the modeled problem. The convergence areas for velocity and temperature profiles are −1.0≤hϒ≤1.0 and −2.5≤hΘ≤1.5, respectively, as shown in Figure 2.Figure 2
h− curves for velocity and temperature profiles.


## 5. Results and Discussion

This part explains how the hydrothermal characteristics of hybrid nanofluid flow past a nonisothermal flat plate at a stagnation point are affected by the necessary parameters. To demonstrate physically accurate effects, thermal radiation and magnetic field are added. The hybrid nanofluid flow contains ferrous (Fe3O4) and graphene oxide (GO) nanoparticles and water (H2O) is used as base fluid. In the present analysis, the default values are considered as M=1.0, n=0.5, We=0.6, Pr=6.2, ϕ1=ϕ2=0.05, Ec=0.3, and Rd=0.7.The effects of the significant parameters onRexCfx and Nux/Rex are shown in Tables 2 and 3. The augmenting volume fraction of the Fe3O4 nanoparticles declines the surface drag force, while the augmenting volume fraction of the GO nanoparticles boosts up the skin friction coefficient. The augmenting impacts of Fe3O4 and GO nanoparticle volume fractions are found against heat transfer rate. The greater Weissenberg number We augments RexCfx; however, Nux/Rex reduces with the higher We. A similar impact of n is found for RexCfx and Nux/Rex. The greater magnetic parameter M augments RexCfx, while an opposite trend is observed for Nux/Rex. Also, the greater Eckert number Ec and thermal radiation parameter Rd have declining impacts on Nux/Rex. Figures 3 and 4 are displayed in order to clarify the variation in RexCfx and Nux/Rex via different embedded parameters. Figure 5(a) shows the impact of ϕ1 on ϒ′ζ when ϕ2=0.05. The augmenting ϕ1 declines ϒ′ζ. The increasing ϕ1 declines the boundary layer thickness, which consequently reduces ϒ′ζ. Figure 5(b) shows the streamline patterns for ϕ1 when ϕ2=0.05. Figure 6(a) shows the impact of ϕ2 on ϒ′ζ when ϕ1=0.05. A similar impact as of Fe3O4 nanoparticle is found here. Figure 6(b) shows the streamline patterns for ϕ2 when ϕ1=0.05. Figure 7(a) signifies the consequence of M on ϒ′ζ. The escalating magnetic parameter boosts up the velocity field. As the dynamic growth upsurges, the boundary layer of the velocity profile gets thinner, showing that the magnetic parameter augments the flow mobility near the heated plate. The present model is computed along with stagnation point flow, thus the augmenting impact of the magnetic parameter has been reported here. Figure 7(b) shows the streamline patterns for M when ϕ1=ϕ2=0.05. Figure 8(a) signposts the effect of We on ϒ′ζ. The augmenting We reduces ϒ′ζ. The maximum value of the parameter We increases ϒ′ζ, because We is directly related to the relaxation time Γ. The relaxation time of the examined non-Newtonian hybrid nanofluid has increased. As a result of this physical property, the water-based flow encounters extra barrier in developing easily across the flow boundary, lowering the hybrid nanofluid velocity. Figure 8(b) shows the streamline patterns for We when ϕ1=ϕ2=0.05. Figure 9(a) displays the effect of n on ϒ′ζ. The escalating n shows augmenting conduct against ϒ′ζ. The numerical value of the power-law index parameter is specified for two different fluids, namely, pseudoplastic n<1 and dilatant n>1. Physically, the escalating n interconnects an important augmentation in the viscosity of the non-Newtonian hybrid nanofluid flow. That is why the velocity boundary layer thickness is declined; as a result, ϒ′ζ is augmented. Figure 9(b) shows the streamline patterns for n when ϕ1=ϕ2=0.05. Figure 10 shows the effect of ϕ1 on Θζ when ϕ2=0.05. The increasing ϕ1 augments Θζ. Figure 11 shows the effect of volume fraction ϕ2 on Θζ when ϕ1=0.05. The increasing ϕ2 augments Θζ. Figure 12 shows the effect of power-law index n on Θζ when ϕ1=ϕ2=0.05. The rising n declines Θζ. The increasing n thickens the temperature boundary layer which diminishes the temperature of the hybrid nanofluid flow. Thus, a declining impact is found here. Figure 13 exhibits the effect of We on Θζ when ϕ1=ϕ2=0.05. The increasing We augments Θζ. The increasing We shows that the increased quantity of thermal energy provided to the nanofluidic system due to resistive nanofluid motion can explain this thermal behavior physically. Figure 14 displays the impact of M on Θζ when ϕ1=ϕ2=0.05. The augmenting M escalates Θζ of the hybrid nanofluid flow. Physically, as the magnetic parameter increases, the movement of particles of hybrid nanofluid escalates. Thus, both the thermal boundary and temperature of the hybrid nanofluid augment. Figure 15 designates the effect of Rd on Θζ when ϕ1=ϕ2=0.05. The increasing radiation parameter boosts up Θζ. Physically, the increasing radiation parameter augments the surface heat of the hybrid nanofluid flow which makes the hybrid nanofluid hotter. Thus, the escalating conduct is observed here. Figure 16 displays the effect of Eckert number Ec on Θζ when ϕ1=ϕ2=0.05. The increasing Eckert number augments Θζ. The link between kinetic energy and enthalpy in a flow is described by the Eckert number. It denotes the effort expended in converting kinetic energy to internal energy in the face of viscous fluid forces. An increase in the Eckert number implies that the fluid has a high kinetic energy; consequently, the intermolecular collisions take place which enhances the particles vibration. So, the increased molecule collisions increase heat dissipation in the boundary layer region, causing Θζ to climb.Table 2
Impacts ofϕ1, ϕ2, We, n, and M on RexCfx.
ϕ1ϕ2WenMRexCfx0.011.106390.021.104430.031.103810.011.050030.021.063740.031.077680.20.928560.30.956610.40.983080.21.090480.30.998560.40.898900.21.119860.31.142660.41.17459Table 3
Impacts ofϕ1, ϕ2, We, n, Ec, Rd, and M on 1/RexNux.
ϕ1ϕ2WenMEcRd1/RexNux0.010.427280.020.407960.030.385050.010.427480.020.408090.030.385150.20.608110.30.600890.40.593680.20.328740.30.372030.40.415320.20.322910.30.316720.40.308060.2-0.346870.3-0.779200.4-1.217530.20.246890.30.172590.40.08544Figure 3
Impacts ofϕ1, ϕ2, We, n, and M on RexCfx.Figure 4
(a) Impacts ofϕ1, ϕ2, We, and n on 1/RexNux. (b) Impacts of Ec, Rd, and M on 1/RexNux.
(a)(b)Figure 5
(a) Effect ofϕ1 on ϒ′ζ. (b) Streamline patterns for ϕ1.
(a)(b)Figure 6
(a) Effect ofϕ2 on ϒ′ζ. (b) Streamline patterns for ϕ2.
(a)(b)Figure 7
(a) Effect ofM on ϒ′ζ. (b) Streamline patterns for M.
(a)(b)Figure 8
(a) Effect ofWe on ϒ′ζ. (b) Streamline patterns for We.
(a)(b)Figure 9
(a) Effect ofn on ϒ′ζ. (b) Streamline patterns for n.
(a)(b)Figure 10
Effect ofϕ1 on Θζ.Figure 11
Effect ofϕ2 on Θζ.Figure 12
Effect ofn on Θζ.Figure 13
Effect ofWe on Θζ.Figure 14
Effect ofM on Θζ.Figure 15
Effect ofRd on Θζ.Figure 16
Effect ofEc on Θζ.


## 6. Conclusion

The magnetohydrodynamic flow of water-based hybrid nanofluid containing ferrous and graphene oxide nanoparticles past a flat plate has been studied in this article. The stagnation point along with the impacts of magnetic field and thermal radiation is taken in this consideration. The non-Newtonian tangent hyperbolic flow which is laminar and incompressible is also considered to investigate the non-Newtonian behavior of the hybrid nanofluid flow. The hydrothermal characteristics of the hybrid nanofluid flow past a nonisothermal flat plate at a stagnation point are affected by the necessary parameters. Key points of this analysis are as follows:(1)
The increasing volume fractions of the ferrous and graphene oxide nanoparticles have significantly reduced the velocity field, while the thermal field has increased with the augmenting volume fractions of the ferrous and graphene oxide nanoparticles(2)
The augmenting magnetic parameter has considerably enhanced the velocity and thermal fields(3)
Due to the direct relation between the Weissenberg number and relaxation time, the greater Weissenberg number has reduced the velocity profile, while increased the thermal field(4)
The increasing power-law index has augmented the viscosity of the non-Newtonian hybrid nanofluid flow due to which the velocity field escalated. However, this impact is opposite for the thermal field(5)
The augmenting Eckert number and thermal radiation parameter have increased the thermal field



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