Impact of inclined magnetic field on non-orthogonal stagnation point flow of CNT-water through stret

Abstract

Keywords: Oblique Stagnation Point; Nanofluid; Stretching Surface; Non-Linear Thermal Radiation; Inclined Magnetic Field; Porous Medium; Heat Transfer

Introduction

The study of point of stagnation has been attracting numerous researchers throughout the history of fluid dynamics by reason of its practical applications for example,

and cooling electronic equipment. When a fluid flow interacts with a solid wall, the fluid velocity at such point is relatively zero, which is known as point of stagnation. Less focus

*Corresponding author. *E-mail address: Driouich.cedoc@gmail.com This paper was recommended for publication in revised form by Ahmet Selim Dalkılıç Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

has been given to oblique point of stagnation compared to orthogonal stagnation point flows which have been studied by many authors. Wang [1] investigated numerically the unsteady flow of three-dimensional non-orthogonal point of stagnation using the fifth-order RK Fehlberg algorithm. The oblique Axially symmetric point of stagnation flow of a fluid supposed viscous on a circular cylinder by means of coupled ordinary differential equations has been studied by Weidman et al. [2]. The 2D oblique steady flow at the point of stagnation of a viscous incompressible fluid over a stretching surface has been discussed by Reza et al. [3], By varying the stretching parameter of the surface, they found that the ratio of heat exchange at the stretched surface may be enhanced or decreased. Labropulu et al. [4] numerically investigated the oblique flow at the point of stagnation of a fluid considered to be non-Newtonian on a stretched sheet with heat exchange using a quasi-linearization technique, they conclude that the an upsurge in the fluid elasticity lead to enhance the velocity and reduce the temperature. Sarkar et al. [5] studied the slip impacts on the oblique point of flow stagnation over a rough surface of viscous fluid. Awan et al. [6] investigated the unsteady non-orthogonal point of flow stagnation of second-grade fluid across a stretched sheet taking into account the slip effects. The magnetohydrodynamics (MHD) is the discipline of the dynamics of magnetic fields in an electrically conductive fluid, has been conducted by many scientists and engineers [7–12]. Singh et al. [13] examined the steady two-dimensional, incompressible and viscous fluid flow at oblique point of stagnation hitting over a stretching sheet in the presence of magnetic field and radiative heat. Borrelli et al. [14] examined the 2-D steady flow of a electrically conducting Newtonian fluid at an oblique stagnation point with effect of a uniform external electromagnetic field. In the presence of a uniformly applied magnetic field, an unsteady oblique point of flow stagnation of a viscous fluid over an oscillating flat plate has been investigated by Javed et al. [15], the equations are solved using Keller-box method and they found that because of the oscillating plate and magnetic field, the velocity profile of the Newtonian fluid raised in the non-orthogonal region of the stagnation point only. Awan et al. [16] studied numerically the magnetohydrodynamic impacts of an incompressible two-dimensional unsteady Jeffrey fluid model at an oblique point of flow stagnation driven by an oscillating stretching sheet. Mohamed et al. [17] numerically examined the magnetohydrodynamic (MHD) point of flow stagnation over a stretching sheet of blood based Casson ferrofluid with Newtonian heating and using the model of Tiwari and Das nanofluid. The Fe3O4 cobalt ferrite CoFe2O4 ferroparticles and ferrite was used for the study. As a result, they found that the stretching parameter play an important role to reduce the difference in surface temperature between cobalt, ferrite blood based Casson ferrofluid and blood based fluid. Countless applications have been provided to the field of nanofluids ranging from biomedical to chip cooling [18–20].

They are known to significantly increase heat conductivity [21]. The heat exchange efficiency of carbon nanotubes (CNTs) has been proven to be extremely high, in addition to conventional nanofluids that comprise 1-100 nm nanoparticles for heat transfer enhancement reasons. These tubes with diameter measured in nanometer are made from rolled-up graphene sheets. Due to their high thermal conductivities, carbon nanotubes (CNTs) represent a great opportunity for usage as nanofluid additives. SWCNTs are carbon nanotubes with a single surface, whereas MWCNTs have many surfaces. SWCNTs are known to be more effective than MWCNTs because they tend to lose some of their electronic characteristics, whereas SWCNTs are utilized specifically for electronic applications. Although, the benefit of MWCNTs over SWCNTs since they are more rigid, simple and affordable to create on a wide scale. A considerable amount of studies on the nanofluid with magnetohydrodynamics for instance [22–26]. The inclined hydromagnetic flow of nanofluid under thermal radiation close to the point of stagnation through a stretched sheet, and varying viscosity of the nanofluid has been studied by Khan et al. [27]. Mehmood et al. [28] analyzed the oblique point of flow stagnation of an Ethylene- Glycol and water based nanofluid with heat exchange across a stretched sheet, it was found that, in comparison to Ethylene-based nanofluids, water-based nanofluids exhibit local heat flux. Khan et al. [29] studied the oscillatory non-orthogonal point of flow stagnation of MHD nanofluid over a flat plate considering the slip effect, examining three distinct nanoparticles Titania, Copper and Alumina while utilizing water as the working fluid, they observed that Cu nanoparticles, when compared to other nanoparticles examined, can increase the magnetic susceptibility of nanofluids. Nadeem et al. [30] examined the unsteady MHD oblique point of flow stagnation over an oscillatory stretching/shrinking surface of a nanofluid with suction effect. Rizwana et al. [31] numerically studied the MHD oblique point of flow stagnation through a Riga plat of copper-water nanofluid, they conclude that the Riga parameter and Casson parameter increase the velocity field. Ghasemi et al. [32] studied the effects of solar radiation on two-dimensional flow at point of stagnation of MHD nanofluid on a stretched sheet under thermal radiation, Ohmic heating and viscous dissipation. The governing equation was solved using the differential quadrature method (DQM), it was found that the temperature profiles enhanced by decreasing the Prandtl number and enhancing the solar radiation parameter. Zainal et al. [33] used the Bvp4c approach to solve the two-dimensional unsteady flow of MHD hybrid nanofluid (Al2O3-Cu/H2O) at point of stagnation across an exponentially permeable stretching/shrinking sheet. They concluded that raising the suction parameter and the MHD parameter increases the skin friction coefficient and ratio of heat exchange of the hybrid nanofluid. Nandi et al. [34] explored a computational and statistical study of the MHD flow at the point of stagnation of Fe3O4/Cu/Ag-CH3OH nanofluid on a heated permeable stretched sheet. With the impact of viscous dissipation, activation energy, velocity

slip, concentration slip, thermophoresis and chemical reaction. The method of Runge-Kutta-Fehlberg was used for the numerical approach. They concluded that the fluid velocity is always lower for Fe3O4-CH3OH and Cu-CH3OH nanofluids as compared to Ag-CH3OH nanofluid. The analytical or numerical solutions in the above studies are examined using the approximation of Rosseland for the thermal radiation, which is the emission of electromagnetic radiations in all directions by a heated substance, since the flow temperature difference is very small, they described T4 as a linear function of T. Ghaffari et al. [35] explored the nonlinear radiation impact of viscoelastic fluid on 2-D non-orthogonal point of flow stagnation in a impermeable media; the results of the study are obtained using Iterative Scheme of Chebyshev Spectral Newton they found that small value of surface heating and radiation parameter can control the width of thermal limit layer. Khan et al. [36] investigated the impact of nonlinear thermal radiation on the MHD point of flow stagnation of two Burgers’ nanofluids driven by cylinder stretching the solution have been obtained using HAM. The nonlinear radiative heat effects on a point of flow stagnation of MHD Casson nanofluid with boundary conditions of Thompson and Troian slip have been studied by Akaje et al. [37]. Abbasi et al. [38] consider the axisymmetric rotational point of flow stagnation of hybrid nanofluid with effects of activation energy and nonlinear thermal radiation and considering both type of CNTs nanoparticle and water as base fluid, they found that increasing activation energy and nanoparticle volume fraction reduce the concentration of nanoparticles. Motivated by the above studies, the effects of inclined magnetic field on oblique point of flow stagnation of carbon nanotubes over a stretching surface through permeable medium with nonlinear thermal radiation is examined. The differential partial equations (PDEs) are changed, by applying suitable transformation to the dimensionless ordinary differential equations (ODEs). The bvp4c approach has been used to solve numerically the resulting equations. In an attempt to observe the effect of diverse physical factors on the normal velocity, tangential velocity and temperature, the results were examined graphically. To our best knowledge, no such works have been published which discuss the present problem.

Mathematical Formulation

The 2-D stationary flow of nanofluid at nonorthogo-nal point of stagnation over a stretched sheet with thermal radiation considered nonlinear. The flow is subjected to a tilted magnetic field B0 with constant intensity and makes an angle ζ with the y-axis. The induced magnetic field is ignored since it is assumed to be significantly small compared to the external magnetic field. Two identical and opposed forces are employed along the x-axis with velocity u = ex so that the surface is stretched while maintaining the center fixed. We take Tw the surface temperature andT∞ denote the ambient temperature, as appear in Figure 1.

Under the above assumptions and considering the fluid in a permeable media, the fluid flow equations can be expressed as [39–41]: (1)

(4) where qr is the thermal radiation and is stated as [40,42,43]: (5) The appropriate boundary conditions are, (6) where and are normal and tangential velocity in x and y-axis respectively. Simplification of Mathematical Formulation Introducing these quantities,

(19) with C1, C2 are constant depend on integration. The converted boundary conditions are:

(20) The integration constants C1 and C2 obtained using the boundary conditions (20) at infinity, f '(∞) = γ2. An examination of the layer boundary equation (17) divulges that f(y) behaves as f(y) = γ2 y + A as y → ∞ where A is a constant depending on the displacement of boundary layer. Setting the limit as y → ∞ and using the condition g'(∞) = γ1 we get,

(12) Substitution of equation (12) into equations (9), (10) and (11), differentiate partially equations and making to eliminate the pressure term,

(15) We quest for solution of equations (13) and (14) as [40]: (16) where f(y) and g(y) refers to the component normal and tangential of the flow respectively. Using equation (16) in equation (13) and (14), after integration, we find the corresponding differential ordinary equations:

ity parameter, obliqueness of the flow, parameter of stretching ratio, Prandtl number, parameter of thermal radiation and the heating parameter. Introducing, (24) Equation (22) becomes:

we use [0,5] as the range of this work instead of [0,∞[. The procedure is explained as follows. Let:

The thermo-physical properties for nanofluid are given as [44]:

(33) Physical Quantities A mathematical expression for the shear stress τw, local heat flux qw, the local Nusselt number Nux and the skin friction coefficient Cf can be expressed as [40,42]: (27)

The procedure of the numerical method used is depicted in Figure 2. Comparisons of Numerical Results Table 1 shows the comparison of our work with Mandal et al. [40] and Ramzan et al. [42] for f ''(0), h'(0) in the absence of M, ϕ and Ω. It is observed that the accuracy of the method used for our model is quite excellent in agreement with the previous data published in the literature. Table 2 present the validation of our work with Labropulu et al. [4] for θ'(0) in the absence of M, ϕ, Ω and Nr, with Pr = 1. A perfect correlation between the values is found.

Numerical Method The corresponding differential partial equations (PDE’s) are changed to system of ordinary equations (ODEs) (21), (23) and (25) along with the relevant boundary conditions (20) using suitable transformation, then solved using the Bvp4c approach, It’s a code of finite difference that employ formula of the three-stage Lobatto IIIa, which is a collocation that offers a C1-continuous solution fourth-order accurate in the interval of integration uniformly. The residual of the continuous solution is used for mesh selection and error control. However, this approach is an expanded form of the Runge-Kutta method. This solver has been widely used by other researchers to solve the boundary value problem. In this case, the value y = y∞ = 4 and 5 was used. Because no significant difference in the results for y > 5. In other words,

Table 1. Comparison of outcome for f ''(0) and h'(0) when M = Ω = ϕ = 0 Present results f''(0)

Table 2. Comparison of results for −θ'(0) when M = Ω =ϕ = Nr = 0 and Pr = 1 γ2

Results And Discussion

In order to understand the quantitative effects of the differents flow parameters such as magnetic parameter M, angle of inclination ζ, stretching ratio parameter γ2, permeability parameter Ω, volume fraction ϕ, on normal velocity, tangential velocity and temperature, we have illustrated Figs. 3-20 for both SWCNTs and MWCNTs. The values of the fixed parameters for the graphical results and the values of Nusselt number and skin friction, are chosen as M = 0.5, ζ = π/2, Ω = 0.5, γ1 = 1, γ2 = 0.3, A = 0.5, ϕ = 0.03, Pr = 6.7, θw = Nr = 0.5. Further, the Thermophysical properties consist of density, heat capacitance and thermal conductivity of considered base fluid water, SWCNTs and MWCNTs are mentioned in Tables 3.

Effect of Magnetic Parameter (M) From Figure 3 we observed a decrease in the normal velocity profiles with increasing the values of magnetic parameter M. This can be explained by the formation of the force of Lorentz which was liable for slowing down the fluid motion and causing the reduction of normal velocity profiles. The influence of magnetic parameter is discussed through Figure 4. An increase in tangential velocity profiles was observed with increasing values of M near the sheet, while opposite behavior is observed away from the sheet. This can be explained by the formation of the force of Lorentz which was liable for slowing down the fluid motion and causing the reduction of tangential velocity profiles when moving away from the sheet, whereas it increases near the sheet. Tables 5 and 6 presents the opposite nature of the propagation of the values of skin friction coefficient for normal (CfN) and tangential (CfT) components respectively of both SWCNT and MWCNT, it is noticed that the normal component of skin friction is a decreasing functions of magnetic parameter, whereas the contrary is observed for the tangential component of skin friction. One may realize from these tables that MWCNT gives maximum results for tangentiel skin friction (CfT) comparing with SWCNT causing a higher drag effect on the surface. In addition, the tangentiel friction factor increase rate of SWCNT is higher than that MWCNT, which is estimated to be 11.62% and 11.34% for SWCNT and MWCNT, respectively. While an opposite behavior is seen for the normal friction factor decrease rate which evaluated as 35.6% and 36.6% for SWCNT and MWCNT, respectively.

Table 3. Physical properties of base fluid and nanoparticles [40] Physical properties

Mwcnt

Figure 5 reveals the effects of magnetic parameter M on temperature profiles. It is clearly shown that the temperature of the fluid increase with an increase in the magnetic parameter. In addition, the increase of the temperature field is due to the improvement of the Lorentz force, which dissipates more thermal energy in the flowing fluid. That is, thicker thermal boundary layer. Table 4 shows the effect of magnetic parameter on local Nusselt number. It is evident from Table 4 that the local Nusselt number decreases with an increase in the magnetic field. It can be seen from this Table that SWCNT give the best results compared to MWCNT. This means that the increase in heat transfer is more significant for SWCNT. The Nusselt number reduction rate for SWCNT is 3.98%, while it is 4.01% for MWCNT. The cooling improvement found will be effective for MWCNT.

Figure 5. Temperature distribution for variation in M. Effect of Inclination Angle of Magnetic Field (ζ) From Figure 6 we observed a decrease in the normal velocity profiles with increasing the values of angle of inclination ζ. This can be explained by the formation of the force of Lorentz which was liable for slowing down the fluid motion and causing the reduction of normal velocity profiles, further, Since sin(ζ) is an increasing function when 0 < ζ < 90, the strength of magnetic field also increases, and reduces the boundary layer thickness. The same effect is observed by Ref. [41]. Tables 5 and 6 show the contrary nature of the distribution of skin friction coefficient values for the normal (CfN) and tangential (CfT) components of both SWCNT and MWCNT. It is noted that the normal component of skin friction is a decreasing function of angle of inclination of magnetic field. These Tables show that MWCNT produces the best results for the tangentiel skin friction (CfT) as compared to SWCNT, resulting in a larger drag impact on the surface. Furthermore, the tangentiel friction factor reduction rate of SWCNT is greater than that of MWCNT, which is predicted to be 0.54% for SWCNT and 0.52% for MWCNT, respectively. The normal friction factor reduction rate, which was 1.35% and 1.40% for SWCNT and MWCNT, respectively, exhibited the opposite tendency. Figure 7 describes the impact of angle of inclination ζ on tangential velocity profiles. The plots show that an increase of ζ increases the strength of magnetic field, and reduces the boundary layer thickness, which decreases the tangential velocity. One may realize from these Figs. that MWCNTs gives maximum results comparing with SWCNTs this is due to the low density of MWCNT. The temperature distribution for different values of angle of inclination of external magnetic field ζ is given in Figure 8. It is observed that the temperature distribution increases with an increase in the values of angle of inclination that strengthens the applied magnetic.

Effect of Stretching Ratio Parameter (γ2) The behavior of the stretching ratio parameter can be described in Figure 9. It is perceived that the normal velocity profiles increases with an increase in the stretching ratio parameter. The stretching rate of the sheet has the potential to increase fluid velocity. As the stretching parameter value increases, the fluid velocity becomes greater than the free stream and the momentum boundary layer becomes thicker. The tangential velocity for different values of stretching ratio parameter is plotted in Figure 10. We observe that the tangential velocity increase by increasing the value of γ2. Table 5 and 6 reveal the variation of the normal and tangentiel friction factor (CfN), (CfT) with the stretching ration parameter γ2. From these tables, we observe the increasing nature of (CfN) and (CfT) as a function of γ2. The same results are verified for both types SWCNT and MWCNT, although the amplification rate value of (CfN) found to be larger than (CfT) , which is 36.53% for SWCNT, 36.51% for MWCNT and 5.18% for SWCNT, 5.14% for MWCNT, respectively. Figure 11 elaborates the effect of the parameter of stretching ratio γ2 on temperature. It is seen that temperature distribution decreases with an increase in the stretching ratio parameter. The stretching ratio parameter has the potential to reduce fluid temperature. As the stretching parameter value increases, the thermal boundary layer gets thinner, which reduce the temperature of the fluid. The local Nusselt number for diverse valeus of stretching ratio parameter for both SWCNT and MWCNT is computed on Table 4. It is perceived that an upsurge in the stretching ratio parameter lead to increase Nu. Similarly, it can be noticed that the heat transport is lower for MWCNT nanofluid. Therefore, the cooling procedure obviously remains high for SWCNT. The percentage of reduction of the local Nusselt number for MWCNT is 4.08%, whereas it is 4.19% for SWCNT.

Thus, in order to achieve optimal velocity of the system, we need to keep Ω within a certain range, otherwise the drag will be too higher for the fluid movement. Morever, the tangentiel friction factor increase rate of SWCNT is higher than that MWCNT, which is estimated to be 9.60% and 8.61% for SWCNT and MWCNT, respectively. For the normal friction factor decrease rate was evaluated as 28.75% and 27.91% for SWCNT and MWCNT, respectively. Figure 14 describes the infleunce of Ω on temperature distribution. It is readily apparent that temperature increase with an increase in the values of the permeability parameter which make the thermal boundary layer thicker, this is evident from the fact that the porous medium opposes fluid movement and offers resistance to flow. Table 4 express the changes in local Nusselt number under the positive change in permeability parameter Ω. It depicts a decrese in Nu with the increase in Ω for both SWCNT and MWCNT. The values of Nu for MWCNT

Figure 11. Temperature distribution for variation in γ2. Effect of Permeability Parameter (Ω) Figure 12 depicts the variation of permeability parameter with the normal velocity. We remark that the increase of Ω follow by the decrease of normal velocity, that’s return to the fact that the permeability parameter is inversely proportional to the permeability of the medium, causes a high fluid flow restriction, which results in the retardation of the fluid, and leads to decelerate the fluid in the boundary layer. Figure 13 gives account of variation of permeability parameter Ω over the tangential velocity. We notice that the tangential velocity decreases with increasing the values of permeability parameter. In addition, it is also noted that the normal component of skin friction (CfN) is a decreasing functions of permeabilty parameter, while opposite behavior is seen for the tangential component of skin friction (CfT). MWCNTs perform the highest tangential skin friction (CfT) compared to SWCNTs, resulting in a higher drag effect on the surface.

are slightly lower than that of SWCNT. The reduction rate of Nu in MWCNT is 2.84%, while it is 2.97% in SWCNT composition. Effect of Volume Fraction of Nanoparticle (ϕ) The variation of nanoparticle volume fraction is discussed in Figure 15. It is observed that nanoparticles with high concentration possess higher values of normal velocity. Figure 16 graphically illustrates the influence of volume fraction ϕ on tangential velocity. We analyzed that when increasing the value of volume fraction the tangential velocity increase. Table 4 shows that with positive inputs of , the normal component of skin friction (CfN) decreases, while the tangentiel component of skin friction (CfT) tend to increase. Although similar results are observed for both SWCNT and MWCNT. The rate of decrease in (CfN) evaluated for the SWCNT is 5.24%, while it is 4.11% for the MWCNT.

The (CfT) increase rate estimated for the SWCNT is 6.47%, while it is 7.12% for the MWCNT. The influence of volume fraction on temperature is explored in Figure 17. We analyzed that the temperature distribution increase with an increase volume fraction of nanoparticle. Physically the thermal conductivity of nanofluids is enhanced due to a rise in the volume fraction of nanoparticles, which make the temperature profile rises. Table 4 shows that local Nusselt number increases due to the positive changes in ϕ. It is also vital to mention that Nu is better for SWCNT compared to MWCNT. This means that the procedures of cooling our system will happen quickly considering SWCNT. The increase percentage of Nu for SWCNT is 6.59%, while it is 6.45% for MWCNT.

Effect of Radiation Parameter (Nr) The temperature distribution for diverse values of radiation parameter Nr is given in Figure 18. It is observed that temperature increase with an increase of Nr. In fact, The radiation parameter is the ratio of conductive heat exchange to radiative heat exchange, the increase of the conduction effect inside the thermal limit layer due to the increase of Nr which leads to enhance the temperature. The results qualitatively agree with the report from the literature [45].As a result, utilizing the thermal radiation mechanism, one may simply regulate the rate of heat transfer characteristics. The local Nusselt number for variation in radiation parameter Nr is presented in table 4. The Nusselt number increase due to Nr improvment for both SWCNT and MWCNT. Here, the maximum values is achieved by the SWCNT compared to MWCNT. The increasing rate of local Nusselt number for SWCNT is about 2.04%, while for the MWCNT is 2.10%. Effect of Heating Parameter (θw) Figure 19 describes the impact of the heating parameter θw on temperature distribution. It is revealed that the temperature increase with an increase of θw. This is due to the increase of thermal boundary layer, that is improves the internal heat production which inspires the heat transfer and reinforces the increase of the temperature field. This observation and result are in good agreement with other published article Ref. [46]. Table 4 shows the local Nusselt number for a variety of θw inputs. In both SWCNT and MWCNT, Nu is amplified due to the increase in θw, although a lower impact is seen for the MWCNT. Therefore, For SWCNT, the increasing rate of Nu is 16.43%, while it is 16.81% for the MWCNT.

Figure 20. Temperature distribution for variation in Pr. Effect of Prandtl Parameter (Pr) The temperature distribution for changing values of Prandtl number is ploted in Figure 20. As noticed. The Prandtl number is, in fact, is the ratio of momentum diffusivity to thermal diffusivity. [47]; for larger Prandtl number, the thermal diffusivity becomes smaller and causing a decrease in the boundary layer thickness, which reduces the temperature. From this results, it can be seen that SWCNTs have a higher temperature than MWCNTs due to their high thermal condictivity. Table 4 is showing a clear elevation in local Nusselt number with increasing Prandtl number for both SWCNT and MWCNT. It is also evident from the table that Nu is more advanced for the SWCNT compared with MWCNT. The amplification rate of Nu for SWCNT and MWCNT is 47.71% and 47.58%, respectively.

Table 5. Calculation table of normal component of skin friction coefficient. M

Mwcnt

Table 6. Calculation table of tangential component of skin friction coefficient M

Conclusion

The aim of this paper was to investigate the impact of an inclined magnetic field on oblique stagnation point flow of carbon nanotubes in water as base fluid with nonlinear thermal radiation through a stretching surface in a permeable medium. The objectives were to study the normal velocity, tangential velocity, temperature profiles for both SWCNTs and MWCNTs, for the evaluation of the effect of various physical factors such as the magnetic parameter, angle of inclination of external magnetic field, permeability parameter, volume fraction of nanoparticle and stretching ration parameter. In this regard, we proposed the governing equations, which are, the momentum equation, and the energy equation. With the help of a suitable transformation the dimensionless differential partial equations (PDE’s) have been transformed into differential ordinary equations (ODE’s). Then, the equations have been solved numerically using bvp4c approach. The numerical outcomes are provided in graphical and tabular forms and the conclusions are as follows: • The increasing of permeability parameter, magnetic parameter and angle of inclination reduces the normal and tangential velocity while it enhances the temperature. • The addition of nanoparticles upsurge the normal velocity and temperature while decreasing the tangential velocity. • The temperature distribution is enhanced when radiation parameter and heating parameter are increased. • The normal and tangential velocities are higher for MWCNT compared to SWCNT, the contrary of this trend is found for temperature distribution. • The normal component of skin friction coefficient values is larger for MWCNT compared to SWCNT as a function of decreasing M, ζ, Ω and ϕ. • The tangential component of skin friction coefficient values is larger for MWCNT compared to SWCNT as a function of increasing M, ζ, Ω and ϕ. • Nusselt number rises with rise in radiation parameter, heating parameter and Prandtl number, but decrease with magnetic parameter, permeability parameter, and becomes higher for SWCNT than for MWCNT.

Skin friction coefficient Reynolds number Constants Displacement coefficient Dimensionless normal velocity Dimensionless tangential velocity

Greek symbols μ Dynamic viscosity α Thermal diffusivity, m2 / s θ Dimensionless temperature ρ Density, kg/m3 θw Heating parameter γ1 Obliqueness of the flow γ2 Stretching parameter Φ Particle volume fraction τw Wall shear stress Ω Permeability parameter σ* Stefan–Boltzmann constant, W / m2.K4 ψ Stream function v Kinematic viscosity, m2 / s Subscripts nf Indicate nanofluid f Indicate fluid s Indicate nanoparticle

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

Specific heat, J/K Temperature of the ambient, K Mean absorption coefficient, m-1 Temperature of the surface, K Thermal conductivity, W / m.K Permeability of porous medium Radiation parameter Magnetic parameter Radiative heat flux, W / m2 Prandtl number

References

1. k Thermal conductivity, W / m.K K Permeability of porous medium [1] Wang CY. The unsteady oblique stagnation point Nr Radiation parameter flow. Phys Fluids. 1985;28:2046−2049. [CrossRef] M Magnetic parameter [2] Weidman PD, Putkaradze V. Axisymmetric stagna- qr Radiative heat flux, W / m2 tion flow obliquely impinging on a circular cylinder. Pr Prandtl number Eur J Mech B Fluids. 2003;22:123−131. [CrossRef] 128 J Ther Eng, Vol. 10, No. 1, pp. 115−129, January, 2024
2. Reza M, Gupta AS. Steady two-dimensional oblique stagnation-point flow towards a stretching surface. unsteady oblique stagnation point based Jeffrey Fluid Dyn Res. 2005;37:334−340. [CrossRef] nanofluid flow over an oscillatory stretching sheet.
3. Labropulu F, Li D, Pop I. Non-orthogonal stagna- Adv Mech Eng. 2020;12:1−13. [CrossRef] tion-point flow towards a stretching surface in a [17] Mohamed MKA, Yasin SHM, Salleh MZ, Alkasasbeh non-Newtonian fluid with heat transfer. Int J Therm HT. MHD Stagnation Point Flow and Heat Transfer Sci. 2010;49:1042−1050. [CrossRef] Over a Stretching Sheet in a Blood-Based Casson
4. Sarkar S, Sahoo B. Analysis of oblique stagnation Ferrofluid With Newtonian Heating. J Adv Res Fluid point flow over a rough surface. J Math Anal Appl. Mech Therm Sci. 2021;82:1−11. [CrossRef] 2020;490:124208. [CrossRef] [18] Asirvatham LG. Nanofluid heat transfer and appli-
5. Awan AU, Aziz M, Ullah N. Thermal analysis cations. J Therm Eng. 2015;1:113−115. of oblique stagnation point flow with slippage [19] Ravisankar R, Venkatachalapathy VSK, Alagumurthi on second-order fluid. J Therm Anal Calorim. N. Application of nanotechnology to improve the 2022;147:3839−3851. [CrossRef] performance of tractor radiator using Cu-water
6. Bhuvaneswari M, Eswaramoorthi S, Sivasankaran S, nanofluid. J Therm Eng. 2018;4:2188−2200. [CrossRef] Hussein AK. Cross-diffusion effects on MHD mixed [20] Madani K, Maad RB, Abidi-Saad A. Numerical convection over a stretching surface in a porous investigation of cooling a ribbed microchannel medium with chemical reaction and convective con- using nanofluid. J Therm Eng. 2018;4:2408−2422. dition. Eng Trans. 2019;67:3−19. [CrossRef]
7. Maatki C, Ghachem K, Kolsi L, Hussein AK, Mohamed B, Aissia HB. Inclination effects of mag- thermal instability in a low prandtl number nano- netic field direction in 3D double-diffusive natural fluid layer in a porous medium. J Appl Fluid Mech. convection. Appl Math Comput. 2016;273:178−189. 2015;8:265−272. [CrossRef] [CrossRef] [22] Ali B, Hussain S, Nie Y, Habib D, Hussein AK.
8. Mansour MA, Rashad AM, Mallikarjuna B, Finite element investigation of Dufour and Soret Hussein AK, Aichouni M, Kolsi L. MHD mixed impacts on MHD rotating flow of Oldroyd-B nano- bioconvection in a square porous cavity filled by fluid over a stretching sheet with double diffusion gyrotactic microorganisms. Int J Heat Technol. Cattaneo Christov heat flux model. Powder Technol. 2019;37:433−445. [CrossRef] 2021;377:439−452. [CrossRef]
9. Ahmed SE, Hussein AK, Mohammed HA, Kayode I, Kolsi L, Zhang X, et al. Viscous dissipation and Talebizadehsardari P. MHD mixed convection due radiation effects on MHD natural convection in a to a rotating circular cylinder in a trapezoidal enclo- square enclosure filled with a porous medium. Nucl sure filled with a nanofluid saturated with a porous Eng Des. 2014;266:34−42. [CrossRef] media. Int J Mech Sci. 2020;181. [CrossRef]
10. Hussein AK, Ashorynejad HR, Sivasankaran S, Kolsi L, Kayode I, Sivanandam S, et al. Modeling Hybrid nanofluids: Significance of gravity modula- of MHD natural convection in a square enclosure tion, heat source/sink, and magnetohydrodynamic having an adiabatic square shaped body using on dynamics of micropolar fluid over an inclined Lattice Boltzmann Method. Alexandria Eng J. surface via finite element simulation. Appl Math 2016;55:203−214. [CrossRef] Comput. 2022;419:126878. [CrossRef]
11. Pakdee W, Yuvakanit B, Hussein AK. Numerical analysis on the two-dimensional unsteady magneto- A, Malekshah EH, Kanna PR et al. Three- hydrodynamic compressible flow through a porous dimensional investigation of the effects of external medium. J Appl Fluid Mech. 2017;10:1153−1159. magnetic field inclination on laminar natural con- [CrossRef] vection heat transfer in CNT-water nanofluid filled
12. Singh P, Sinha D. MHD oblique stagnation-point cavity. J Mol Liq. 2018;252:454−468. [CrossRef] flow towards a stretching Sheet With Heat transfer. [26] Ashraf MZ, Rehman SU, Farid S, Hussein AK, Int J Appl Math Mech. 2010;6:94−111. Ali B, Shah NA, et al. Insight into Significance
13. Borrelli A, Giantesio G, Patria MC. MHD oblique of Bioconvection on MHD Tangent Hyperbolic stagnation-point flow of a Newtonian fluid. Z Nanofluid Flow of Irregular Thickness across a Angew Math Phys. 2012;63:271−294. [CrossRef] Slender Elastic Surface. Mathematics. 2022;10:1−17.
14. Javed T, Ghaffari A, Ahmad H. Numerical study [CrossRef] of unsteady MHD oblique stagnation point flow [27] Khan WA, Makinde OD, Khan ZH. Non-aligned and heat transfer due to an oscillating stream. MHD stagnation point flow of variable viscosity Thermophys Aeromech. 2016;23:383−391. nanofluids past a stretching sheet with radiative heat. [CrossRef] Int J Heat Mass Transf. 2016;96:525−534. [CrossRef] J Ther Eng, Vol. 10, No. 1, pp. 115−129, January, 2024 129
15. Mehmood R, Nadeem S, Sher Akbar N. Non-aligned Ethylene-Glycol 30% based stagnation point fluid thermal radiation and activation energy features in over a stretching surface with hematite nano parti- axisymmetric rotational stagnation point flow of cles. J Appl Fluid Mech. 2016;9:1359−1366. [CrossRef] hybrid nanofluid. Int Commun Heat Mass Transf.
16. Khan AU, Nadeem S, Hussain ST. Phase flow study 2021;126:105335. [CrossRef] of MHD nanofluid with slip effects on oscillatory [39] Devi R, Poply V, Manimala M. Effect Of Aligned oblique stagnation point flow in view of inclined Magnetic Field And Inclined Outer Velocity In magnetic field. J Mol Liq. 2016;224:1210−1219. Casson Fluid Flow Over A Stretching Sheet With [CrossRef] Heat Source. J Therm Eng. 2021;7:823−844. [CrossRef]
17. S Nadeem M, KhanArif R, Khan U. MHD oblique stagnation point flow of nanofluid over an oscilla- numerical simulation of mixed convective and arbi- tory stretching/shrinking sheet: existence of dual trarily oblique radiative stagnation point slip flow solutions. Phys Scr. 2019;94:075204. [CrossRef] of a CNT-water MHD nanofluid. J Therm Anal
18. Rizwana R, hussain A, Nadeem S. Series solution Calorim. 2021;143:1901−1916. [CrossRef] of unsteady MHD oblique stagnation point flow [41] Sulochana C, Sandeep N, Sugunamma V, Rushi of copper-water nanofluid flow towards Riga plate. Kumar B. Aligned magnetic field and cross-dif- Heliyon. 2020;6. [CrossRef] fusion effects of a nanofluid over an exponentially
19. Ghasemi SE, Hatami M. Solar radiation effects on stretching surface in porous medium. Appl Nanosci. MHD stagnation point flow and heat transfer of a 2016;6:737−746. [CrossRef] nanofluid over a stretching sheet. Case Stud Therm [42] Ramzan M, Shahmir N, Alotaibi H, Ali H, Ghazwani Eng. 2021;25:100898. [CrossRef] S, Muhammad T. Thermal performance comparative
20. Zainal NA, Nazar R, Naganthran K, Pop I. Unsteady analysis of nanofluid flows at an oblique stagnation MHD stagnation point flow induced by expo- point considering Xue model: a solar application. J nentially permeable stretching/shrinking sheet Comput Des Eng. 2022;9:201−215. [CrossRef] of hybrid nanofluid. Eng Sci Technol an Int J. [43] Mallikarjuna B, Rashad AM, Hussein AK, Hariprasad 2021;24:1201−1210. [CrossRef] Raju S. Transpiration and Thermophoresis Effects
21. Nandi S, Kumbhakar B, Sarkar S. MHD stagnation on Non-Darcy Convective Flow Past a Rotating Cone with Thermal Radiation. Arab J Sci Eng. point flow of Fe3O4/Cu/Ag-CH3OH nanofluid 2016;41:4691−4700. [CrossRef] along a convectively heated stretching sheet with
22. Nadeem S, Mehmood R, Akbar NS. Oblique stag- partial slip and activation energy: Numerical and nation point flow of a casson-nano fluid towards statistical approach. Int Commun Heat Mass Transf. a stretching surface with heat transfer. J Comput 2022;130:105791. [CrossRef] Theor Nanosci. 2014;11:1422−1432. [CrossRef]
23. Agbaje TM, Mondal S, Makukula ZG, Motsa SS, Radiation on Non-Newtonian Fluid in the Region Sibanda P. A new numerical approach to MHD of Oblique Stagnation Point Flow in a Porous stagnation point flow and heat transfer towards a Medium: A Numerical Study. Transp Porous Media. stretching sheet. Ain Shams Eng J. 2016;9:233−243. 2016;113:245−266. [CrossRef] [CrossRef]
24. Kumar A, Sugunamma V, Sandeep N. Impact of magnetized Burgers' nanofluid subject to ther- of Non-linear Radiation on MHD Non-aligned mal radiation. Appl Nanosci. 2020;10:5233−5246. Stagnation Point Flow of Micropolar Fluid over a [CrossRef] Convective Surface. J Non-Equilibrium Thermodyn.
25. Akaje TW, Olajuwon BI. Impacts of Nonlinear 2018;43:327−345. [CrossRef] Thermal Radiation on a Stagnation Point of an [47] Hayat T, Shafiq A, Alsaedi A, Asghar S. Effect of Aligned MHD Casson Nanofluid Flow with inclined magnetic field in flow of third grade fluid Thompson and Troian Slip Boundary Condition. with variable thermal conductivity. AIP Adv. 2015;5. 2021;1:1−15. [CrossRef]

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