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Abstract

The current investigation explores the effect of activation energy on the MHD radiative Williamson nanofluid flow across a wedge using heat generation and binary chemical reactivity. The flow model consists of partial differential equations (PDEs) by transforming them into ordinary differential equations (ODEs). Numerical computations have been carried out through the bvp4c MATLAB package. The most effective solutions for flow profiles have been displayed through graphs, while the numeric solutions for the drag friction, heat, and mass transport have been displayed via tables. Numerical findings demonstrate that the temperature field is accelerated by the increase in radiation parameter. In addition, it is intriguing to discover that the concentration boundary layer thickness improves as the activation energy increases. A fundamental study further reveals that the local skin friction coefficient is a rising function of thermal and concentration Grashof numbers. Moreover, it is concluded that the enhanced Brownian motion, thermophoresis, and Eckert number decline the heat transfer rate.

Keywords

Chemical reaction Williamson nanofluid porous medium thermal radiation energy generation activation energy

Introduction

Heat and mass movement of liquids are assured in a variety of thermal technology uses such as crude oil refineries, heat exchangers, geothermal processes, and isolation processes with a defence system. The theory of boundary layers is crucial to many branches of engineering and practical problems. This theory is most useful when trying to calculate the drag caused by skin friction for a body moving in an aircraft, turbine blade, and many more. Elbashbeshy and Dimian¹ inspected computationally the impact of radiated viscous fluid flow past a wedge and exposed that the thermal distribution is an enhancing function of radiation parameter. The non-Newtonian fluid due to a saturated porosity wedge with a vertical sheet was conducted by Hayat et al.² Habib et al.³ developed the mathematical model for the MHD stagnant flow of Prandtl fluid past a stretching/shrinking wedge under the sway of activation energy. The numerous investigations for the transport of various viscous fluids over a stretching wedge are detailed in the literatures.^4–7

The pseudoplastic behavior of many substances may be characterized by the Williamson fluid model, which is based on a class of non-Newtonian fluids. Pseudoplastic fluids are the most well-known example of the liquids experienced in the study of non-Newtonian fluids; they are encountered in the production of emulsion sheets like imaging films, polymeric sheets extraction, the transport of blood and plasma, etc. Hamid and Khan⁸ explored the two-layered heat transport of Williamson fluid over a wedge. Hussain et al.⁹ scrutinized the significance of dissipation and internal heat source for MHD Williamson fluid flow on a permeable wedge. Further, they revealed that the Nusselt number rises as the heat generation escalates. Extensive scientific and theoretical investigations on non-Newtonian fluids made available in the literatures.^10–12

Magnetohydrodynamic (MHD) fluid flow on a stretching porous wedge is the subject of several researchers with relevance in a broad of technological fields including oil beds, MHD energy generation, plasmas, nuclear reactor cooling operations, and heat exchangers. Kameswaran et al.¹³ investigated the hydromagnetic nano-liquid motion past a stretched sheet under the impact of the chemical reaction. Goqo et al.¹⁴ analyzed the entropy production on MHD nanofluid flow on a porous wedge through the sway of thermally radiated effects. The MHD flow with chemical reactivity on the Cross nanoparticles flow using a wedge was investigated by Haq et al.¹⁵ Hussain et al.¹⁶ analyzed the heat transfer of the MHD flow of non-Newtonian fluid past a wedge geometry in the presence of radiation and convective boundary conditions.

As a form of energy transmission, thermal radiation has several potential applications, most notably in the solar sector. A material medium is unnecessary for this kind of heat transmission. Numerous academics have taken notice of heat radiation impact, most likely as a result of its extensive use in polymer production processes. Ullah et al.¹⁷ discovered the mixed convective movement of radiative nanofluid by utilizing the Keller box scheme in the presence. Swain et al.¹⁸ examined the heat generation effect on the MHD flow of Maxwell fluid on a porous stretched sheet with thermal radiation. Recently, various academics^19,20 have described the heat transfer phenomenon with energy generation and thermal radiation effects.

Nanofluids have been demonstrated to enhance heat transmission in a variety of industrial settings, including nuclear reactors, semiconductors, medical diagnostics, and food processing. Buongiorno²¹ explored the essential applications and models for studying nanofluids including Brownian motion and thermophoresis effects. The physical characteristics of Williamson nanofluid flow past a Wedge were studied by Hashim et al.²² Amar et al.²³ discovered the MHD fluid flow with radiation effect and wedge emerged due to a spongy medium. The Analysis of non-Newtonian nanofluid past an expanding surface with activation energy and chemical reaction was performed by Shamshuddin and Mabood.²⁴ Moreover, some studies on nanofluids with various effects are referred to in the literatures.^25–31 The activation energy is the additional energy supplied to initiate an advantageous work done. In chemical engineering, water emulsion, oil reservoir, geothermal, and mechanical engineering, activation energy accompanying binary chemical reactions plays an important role. Many fields, including material design, geothermal storage, atomic reaction cooling, and thermal oil recovery, have shown interest in the development of mass gradient studies with activation energy. The naturally convective fluid flow through a porous area with activation energy was originally studied by Bestman.³² Azam and Abbas³³ have recently described the effect of Arrhenius activation energy on the radiated flow of a nanofluid past a melted wedge. Ali et al.^34,35 have presented the MHD Falkner-Skan motion of non-Newtonian nanofluids past a wedge under radiation and activation energy.

In the present analysis, the MHD Williamson nanofluid flow due to a porous wedge with heat generation, activation energy, and binary chemical reaction is considered. In addition, the effects of thermal radiation and viscous dissipation are taken into account. The partial differential equations (PDEs) that are transmuted have been altered into nonlinear ODEs with the assistance of appropriate transformation. The bvp4c MATLAB package is implemented to fix the system of ODE. Further, the consequences of numerous physical quantities for the flow quantities are discussed and elaborated in detail. It is crucial to note that the proposed investigation has broad applications in many disciplines including engineering such as solar power, biomedicine, power storage, cancer treatment, microelectronics, thermal engineering, and several other areas.

How did the nanoparticles affect the mixed convection flow?

How does the Lorentz force affect the velocity of Williamson fluid?

What is the significant of velocity Williamson nanofluid against numerous variables?

What is the mechanism of Williamson nanofluid temperature through wedge flow parameters?

Mathematical formulation

We assumed the magneto-Williamson nanofluid flow on a wedge through a porous surface under the activation energy effect. Also, the thermal radiative, viscous dissipation, and heat source are taken into consideration. The movement of nanoparticle effects is observed by using Buongiorno’s model. The coordinate system and flow model considered in the current study is presented in Figure 1. The coordinates ( $x$ , $y$ ) have been assumed for the modelling of the study where $x$ -axis is considered with the wedge in fluid flow and $y$ -axis is normal to it with surface and ambient velocities as $u_{w}$ , and $U$ . A constant magnetic field $B_{o}$ isused in y-direction and vertical to the flow direction. The uniform and ambient temperatures as well as concentrations are $T_{w}$ , $T_{\infty}$ , and $C_{w}$ and $C_{\infty}$ respectively. $Ω = β π$ is the angle of a wedge with $β = \frac{2 m}{m + 1}$ is the Hartree pressure gradient parameter. Under these assumptions, the governing equations are²³:

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,

(1)

\begin{matrix} u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = U \frac{\partial U}{\partial y} + \sqrt{2} υ Γ \frac{\partial u}{\partial y} \frac{\partial^{2} u}{\partial y^{2}} + g β_{T} (T - T_{\infty}) \\ + g β_{C} (C - C_{\infty}) + (\frac{σ B_{°}^{2} u}{ρ} + \frac{υ}{k_{1}}) (U - u), \end{matrix}

(2)

\begin{matrix} u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = α \frac{\partial^{2} T}{\partial y^{2}} + τ [D_{B} \frac{\partial C}{\partial y} \frac{\partial T}{\partial y} + \frac{D_{T}}{T_{\infty}} {(\frac{\partial T}{\partial y})}^{2}] + \frac{Q_{°}}{ρ C_{p}} (T - T_{\infty}) \\ + \frac{υ}{C_{p}} {(\frac{\partial u}{\partial y})}^{2} - \frac{1}{ρ C_{p}} \frac{\partial q_{r}}{\partial y} + \frac{σ B_{°}^{2}}{ρ C_{p}} (U - u)^{2}, \end{matrix}

(3)

\begin{matrix} u \frac{\partial C}{\partial x} + v \frac{\partial C}{\partial y} = D_{B} \frac{\partial^{2} C}{\partial y^{2}} + \frac{D_{T}}{T_{\infty}} \frac{\partial^{2} T}{\partial y^{2}} \\ - K_{r}^{2} (C - C_{\infty}) {(\frac{T}{T_{\infty}})}^{n} \exp (\frac{- E_{a}}{kT}) . \end{matrix}

(4)

The associated boundary conditions are:

\begin{matrix} u = u_{w} = - λ U, v = v_{w}, - k \frac{\partial T}{\partial y} = δ_{1} (T_{w} - T), \\ - D_{B} \frac{\partial C}{\partial y} = δ_{2} (C_{w} - C), at y = 0 . \end{matrix}

u \to U, T \to T_{\infty}, C \to C_{\infty}, as y \to \infty .

(5)

Here, $λ$ is noted in three cases like as $λ < 0$ referred as stretching wedge, $λ > 0$ denoted as contracting wedge, $λ = 0$ indicates that fixed wedge. Here, u and v are velocity works along with $x$ and $y$ directions, $ρ$ – is the density, $k_{1}$ – porosity coefficient of the porous medium, $α$ – is the thermal diffusivity, $q_{r}$ – is the radiative heat flux, $Q_{°}$ – is the heat source/sink coefficient, $K_{r}$ – is the rate of chemical reaction respectively.

Figure 1.

Geometrical view of flow problem.

To alter governing equations into non-dimensional form we establish the following suitable transformation:

\begin{matrix} ζ = y \sqrt{\frac{(m + 1) U_{\infty}}{2 υ}} x^{\frac{(m - 1)}{2}}, ψ = \sqrt{(\frac{2 υ U_{\infty}}{1 + m})} x^{\frac{(m + 1)}{2}} F (ζ), \\ u = \frac{\partial ψ}{\partial y}, v = - \frac{\partial ψ}{\partial x}, ϑ (ζ) = \frac{T - T_{\infty}}{T_{w} - T_{\infty}}, Φ (ζ) = \frac{C - C_{\infty}}{C_{w} - C_{\infty}} . \end{matrix}

(6)

Using equation (6) in equations (2)–(4) with boundary conditions equation (5), we have

\begin{matrix} F ″' + F F ″ + We F ″ F ″' + \frac{1}{(m + 1)} (M + K) (1 - F') \\ + β (1 - F^{' 2}) + Gr ϑ + Gc Φ = 0, \end{matrix}

(7)

\begin{matrix} (1 + \frac{4}{3} Rd) ϑ ″ + \Pr (ϑ' φ' Nb + F ϑ' + {ϑ'}^{2} Nt \\ + \frac{1}{(m + 1)} MEc (1 - F^{' 2})) + Q ϑ + Ec {F ″}^{2} = 0, \end{matrix}

(8)

\begin{matrix} Φ ″ + ScF Φ' + \frac{Nt}{Nb} ϑ ″ - \frac{1}{(m + 1)} Sc γ {(1 + α_{1} ϑ)}^{n} \\ \exp [\frac{- E_{1}}{(1 + α_{1} ϑ)}] Φ = 0 . \end{matrix}

(9)

The transformed boundary conditions are as:

\begin{matrix} F (0) = S, F' (0) = - λ, ϑ' (0) = - B i_{1} + B i_{1} (ϑ (0)), \\ Φ' (0) = - B i_{2} + B i_{2} (Φ (0)) \end{matrix}} at ζ = 0 .

F' (\infty) \to 1, ϑ (\infty) \to 0, Φ (\infty) \to 0, as ζ \to \infty .

(10)

where $β = \frac{2 m}{m + 1}$ (Hartree pressure gradient parameter), $Ω = β π$ (total wedge angle), $M = \frac{σ B_{°}^{2} x^{1 - m}}{ρ U_{\infty}}$ (magnetic field), $We = \sqrt{\frac{Γ^{2} (m + 1) U_{\infty}^{3} x^{3 m - 1}}{υ}}$ (Williamson parameter), $K = \frac{2 υ x^{1 - m}}{k_{1} U_{\infty}}$ (porosity parameter), $Nb = \frac{τ D_{B} (C_{w} - C_{\infty})}{υ}$ (Brownian motion), $Rd = \frac{4 σ^{*} T_{\infty}^{3}}{k^{*} k}$ (radiation parameter), $Ec = \frac{U^{2}}{C_{P} (T_{w} - T_{\infty})}$ (Eckert number), $Q = \frac{2 Q_{°} x}{ρ C_{P} (m + 1) U}$ (heat generation), $Nt = \frac{τ D_{T} (T_{w} - T_{\infty})}{T_{\infty} υ}$ (thermophoresis parameter), $\Pr = \frac{υ}{α}$ (Prandtl number), $Sc = \frac{υ}{D_{B}}$ (Schmidt number), $γ = \frac{2 k_{r}^{2} x^{1 - m}}{U_{\infty}}$ (chemical reaction parameter), $E_{1} = \frac{E_{a}}{k T_{\infty}}$ (activation energy), and $α_{1} = \frac{(T_{w} - T_{\infty})}{T_{\infty}}$ (temperature difference).

In engineering interest, the physical interpretation of friction factor, Nusselt number, and Sherwood number are stated as:

\begin{matrix} C f_{x} {(R e_{x})}^{0.5} = 2 {(\frac{m + 1}{2})}^{0.5} f (0), \frac{N u_{x}}{{(R e_{x})}^{0.5}} = - {(\frac{m + 1}{2})}^{0.5} θ' (0), \\ and \frac{S h_{x}}{{(R e_{x})}^{0.5}} = - {(\frac{m + 1}{2})}^{0.5} ϕ' (0) . \end{matrix}

(11)

Numerical solution

The transmuted equations (7)–(9) with the boundary conditions (10) are utilized numerically via a shooting method with the bvp4c MATLAB package. In this method, the higher-order nonlinear ODEs are altered into first-order ODEs by considered variables. The considered variables are $F = T_{1}$ , $F' = T_{2}$ , $F ″ = T_{3}$ , $ϑ = T_{4}$ , $ϑ^{'} = T_{5}$ , $ϕ = T_{6}$ , $ϕ^{'} = T_{7}$ . Then the first-order ODEs are written in the below form:

\begin{matrix} F = T_{1}, \\ F' = T_{2}, \\ F ″ = T_{3}, \end{matrix}

\begin{matrix} T_{3}' = \frac{1}{(1 + We T_{3})} \\ [- T_{1} T_{3} - (M + K) (1 - T_{2}) \frac{1}{(1 + m)} - β (1 - {(T_{2})}^{2}) - Gr T_{4} - Gc T_{6}] \\ ϑ = T_{4}, \\ ϑ' = T_{5}, \end{matrix}

\begin{matrix} T_{5}' = \frac{\Pr}{[(1 + \frac{4}{3} Rd)]} [\begin{matrix} - T_{1} T_{5} - Nb T_{7} T_{5} - Nt {(T_{5})}^{2} - Q T_{4} \\ - Ec {(T_{3})}^{2} - \frac{1}{(1 + m)} MEc (1 - {(T_{2})}^{2}) \end{matrix}], \\ ϕ = T_{6}, \\ ϕ' = T_{7}, \end{matrix}

T_{7}' = - Sc T_{1} T_{7} - \frac{Nt}{Nb} T_{5}' + γ Sc {(1 + α_{1} T_{4})}^{n} \exp [\frac{- E}{1 + α_{1} T_{4}}] T_{6},

Along with boundary conditions

\begin{matrix} T_{1} (0) = S, T_{2} (0) = - λ, T_{5} (0) = - B i_{1} (1 - T_{4} (0)), \\ T_{7} (0) = - B i_{1} (1 - T_{6} (0)) at ζ = 0, \end{matrix}

T_{2} (\infty) = 1, T_{4} (\infty) = 0, T_{6} (\infty) = 0 as ζ \to \infty .

Results and discussion

The significance of the problem was empirically determined by utilizing numerical findings of the velocity, friction factor, temperature, heat transfer rate, concentration, and mass transfer rate for the pertinent crucial parameters, which were revealed via graphical and tabular representations. The contribution of numerous flow parameters, including the magnetic field $(M)$ , Hartree pressure gradient parameter $(β)$ , porosity parameter ( $K$ ), Williamson parameter $(We)$ , and moving wedge parameter $(λ)$ , on the flow velocity field is depicted in Figure 2(a)–(e). As observed in Figure 2(a), the velocity distribution drops as the magnetic field $M$ increases. Physically, the utilization of a magnetic field to an electrically leading fluid provides the resistive force known as the Lorentz force, which leads to lower velocity. In Figure 2(b), the Hartree pressure gradient parameter for a significant consequence with the fluid velocity distribution. Rising of $β$ is shown to enhance the velocity profile. The velocity plot for different values of the porosity parameter is demonstrated in Figure 2(c). The findings indicate that an upsurge in $K$ diminishes the velocity distribution. Figure 2(d) and (e) describe the effect of the Williamson parameter and the movable wedge parameter on the velocity distribution. The distribution of velocity has been demonstrated to decrease with increasing $We$ and $λ$ .

Figure 2.

(a–e) $F' (ζ)$ versus $M, β, K, We$ , and $λ$ .

The magnetic field influence on the thermal distribution is exhibited in Figure 3(a). As an increase $M$ declines the thermal distribution. As noticed in Figure 3(b) and (c), the temperature profile improves with an increase in thermal radiation and heat source parameters. Physically, this is because the boundary layer of the fluid becomes hotter and thicker when thermal radiation and energy generation are introduced into it. In Figure 3(d), the impact of Eckert number $Ec$ on the thermal profile is illustrated. It is noticed that the higher $Ec$ elevates the temperature. In terms of physics, an increased Eckert number drives up fluid temperature by increasing the internal source of energy. The Brownian movement effect of Williamson nanofluid on the thermal profile is presented in Figure 3(e). Physically, the Brownian motion force acts on the surrounding particles to drive them from heat to cold zones. Thus, increased Brownian motion input leads to a greater temperature.

Figure 3.

(a–e) $ϑ (ζ)$ versus $M, Rd, Q, Ec$ , and $Nb$ .

The influence of the temperature difference parameter $(α_{1})$ , activation energy parameter $(E_{1})$ , and chemical reaction parameter $(γ)$ on the concentration profile is viewed in Figure 4(a)–(c). Figure 4(a) and (c) present the behavior of temperature ratio and chemical reaction parameters on concentration profile. The concentration profile exhibits a declining pattern when $α_{1}$ and $γ$ gets raised. The influence of activation energy $(E_{1})$ on the concentration field is demonstrated in Figure 4(b). It noted that with a rise of $E_{1}$ , escalations the concentration field. Further, Figures 5(a)–(b) and 6(a)–(b) depict the Contour and streamline plot for numerous physical parameters.

Figure 4.

(a–c) $Φ (ζ)$ versus $α_{1}, E_{1}$ and $γ$ .

Figure 5.

(a) Contour plot for numerous values of $M = 0.1$ and $K = 0.2$ , (b) contour plot for numerous values of $β = 1.0$ and $Gr = 0.1$ .

Figure 6.

(a) Streamlinesfor different values of $M = 0.1$ and $K = 0.2$ , (b) stream line for different values of $β = 1.0$ and $Gr = 0.1$ .

Table 1 shows a limiting case comparison of $- ϑ' (0)$ . These numerical results also support the reliability of the currently used approach to solving the problem. Table 2 describes the consequences of the Williamson number $(We)$ , porosity parameter $(K)$ , Hartree pressure gradient parameter $(β)$ , thermal and concentration Grashof numbers on the drag friction coefficient. From the table, it is revealed that the friction factor increases by climbing the amount of $β$ , $K$ , $Gr,$ and $Gc$ . But a reverse trend can be seen in the case of increased $We$ . The sway of influential parameters such as Eckert number $Ec$ , chemical reaction parameter $(γ)$ , Radiation parameter $(Rd)$ , Prandtl number $(\Pr)$ , thermophoresis, and Brownian movement parameters on Nusselt number is presented via Table 3. It is confirmed that the Nusselt number is enhanced with the enhancement of $Rd$ , $\Pr$ whereas the opposite behavior is observed with increased $Ec$ , $Nt$ , and $Nb$ . Table 4 demonstrates the variations of the Sherwood number for the Schmidt number $(Sc)$ , thermophoresis parameter $(Nt)$ , Brownian motion $(Nb)$ , temperature ratio parameter $(α_{1})$ , activation energy $(E_{1})$ , and Chemical reaction $(γ)$ . As the increased values of $Nb$ , $Nt$ , $α_{1},$ and $γ$ decrease the Sherwood number and it strengthens due to an augment of $Sc$ and $E_{1}$ .

Table 1.

Comparison values of $- ϑ' (0)$ for various values of $\Pr$ when Bi→ 0, $Nb \to 0$ , $We = 0, Gr = 0, Gc = 0, β = 0, M = 0, Sc = 0, α_{1} = 0, E_{1} = 0, and γ = 0$ .

$- ϑ' (0)$
$\Pr$	Ahmed et al.³⁶	Current result
0.7	0.44391	0.443912
1.0	0.58197	0.581977
1.5	0.76028	0.760280
2.0	0.91115	0.911151

Table 2.

Numerical values of ${Re}_{x}^{1 / 2} C f_{x}$ for various parameter values.

We	K	$β$	Gr	Gc	${Re}_{x}^{1 / 2} C f_{x}$
0.1					1.1296
0.2					1.0646
0.3					1.0129
0.5					0.9343
	0.1				0.7472
	0.3				0.7918
	0.5				0.8320
	0.7				0.8686
		0.7			0.9156
		0.9			0.9596
		1.1			1.0001
		1.3			1.0377
			0.1		1.0728
			0.2		1.0961
			0.3		1.1188
			0.4		1.1412
				0.3	1.1064
				0.6	1.1556
				0.9	1.2035
				1.0	1.2192

Table 3.

Numerical values of ${Re}_{x}^{- 1 / 2} N u_{x}$ for various parameter values.

Ec	$γ$	$N t$	Nb	Rd	Pr	$R e_{x}^{- 1 / 2} N u_{x}$
0.1						0.1738
1.0						0.1141
1.3						0.0545
1.7						0.0051
	0.1					0.0102
	0.2					0.0387
	0.3					0.2456
	0.4					0.2897
		0.01				0.0421
		0.02				0.0408
		0.03				0.0394
		0.04				0.0381
			0.1			0.0656
			0.3			0.0617
			0.5			0.0579
			0.6			0.0540
				0.1		0.1738
				0.2		0.1758
				0.3		0.1766
				0.4		0.1770
					1.1	0.6512
					1.2	0.7187
					1.3	0.7877
					1.4	0.8586

Table 4.

Numerical values of ${Re}_{x}^{- 1 / 2} S h_{x}$ for various parameter values.

Sc	Nb	$Nt$	$α_{1}$	$E_{1}$	$γ$	${Re}_{x}^{- 1 / 2} S h_{x}$
1.1						1.2473
1.3						1.3084
1.5						1.3704
1.7						1.4335
	0.1					0.5903
	0.2					0.5898
	0.3					0.5768
	0.4					0.5667
		0.1				0.9989
		0.2				0.9765
		0.3				0.9543
		0.4				0.9323
			0.1			2.9014
			0.3			2.8924
			0.5			2.8855
			0.6			2.8654
				0.1		2.8528
				0.2		2.8707
				0.3		2.8855
				0.4		2.9056
					1.1	2.9272
					1.2	2.9190
					1.3	2.9115
					1.4	2.9047

Conclusion

In this study we explore the magneto-Williamson nanofluid flow past a wedge in a porous medium under the influence of heat generation, activation energy, and binary chemical reaction is considered. In addition, the effects of thermal radiation and viscous dissipation are taken into account. Numerical execution of the R-K method through the bvp4c MATLAB package has enabled the parametric investigation to be accomplished. It appears that this work’s findings are promising and they may be used to improve heat transfer in large-scale technological operations. The important conclusions of this study are listed as:

The rate of heat and mass transport diminishes with enrichment in thermophoresis and Brownian motion parameters.

The Hartree pressure gradient parameter and Williamson parameter diminish the velocity field.

An increased activation energy upsurges the concentration profile whereas reverse performance is observed in the case of rising amounts of temperature ratio and chemical reaction parameters.

The friction factor is enhanced with an increment in thermal and concentration Grashof numbers.

As the enhanced thermal radiation and Eckert number boost the temperature field.

Footnotes

Appendix

Handling Editor: Chenhui Liang

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Anwar Saeed

Availability of data and materials

All data used in this manuscript have been presented within the article.

References

Elbashbeshy

EMA

Dimian

. Effect of radiation on the flow and heat transfer over a wedge with variable viscosity. Appl Math Comput 2002; 132(2–3): 445–454.

Hayat

Hussain

Nadeem

, et al. Falkner–Skan wedge flow of a power-law fluid with mixed convection and porous medium. Comput Fluids 2011; 49(1): 22–28.

Habib

Salamat

Abdal

SHS

, et al. Numerical investigation for MHD Prandtlnanofluid transportation due to a moving wedge: Keller box approach. Int Commun Heat Mass Transf 2022; 135: 106141.

Zheng

Zhang

, et al. MHD mixed convective heat transfer over a permeable stretching wedge with thermal radiation and ohmic heating. Chem Eng Sci 2012; 78: 1–8.

Raju

CSK

Sandeep

. Nonlinear radiativemagnetohydrodynamic Falkner-Skan flow of Casson fluid over a wedge. Alex Eng J 2016; 55(3): 2045–2054.

Rana

Gupta

. Numerical and sensitivity computations of three-dimensional flow and heat transfer of nanoliquid over a wedge using modified Buongiorno model. Comput Math Appl 2021; 101: 51–62.

Khatun

Islam

. Influence of magnetic field and heat generation/absorption on Unsteady MHD Convective Flow along a permeable stretching/shrinking wedge with thermophoresis and variable fluid properties. Int J Thermofluids 2022; 16: 100204.

Hamid

Khan

. Numerical simulation for heat transfer performance in unsteady flow of Williamson fluid driven by a wedge-geometry. Results Phys 2018; 9: 479–485.

Hussain

Jahan

Ranjha

, et al. Suction/blowing impact on magneto-hydrodynamic mixed convection flow of Williamson fluid through stretching porous wedge with viscous dissipation and internal heat generation/absorption. Results Eng 2022; 16: 100709.

10.

Ramamoorthy

Pallavarapu

. Radiation and Hall effects on a 3D flow of MHD Williamson fluid over a stretchable surface. Heat Transfer 2020; 49(8): 4410–4426.

11.

Mulinti

Pallavarapu

. Influence of thermal radiation and viscous dissipation on MHD flow of UCM fluid over a porous stretching sheet with higher order chemical reaction. Spec Top Rev Porous Media 2021; 12(4), 33.

12.

Hussain

Ranjha

Anwar

, et al. Eyring-Powell model flow near a convectively heated porous wedge with chemical reaction effects. J Taiwan Inst Chem Eng 2022; 139: 104510.

13.

Kameswaran

Narayana

Sibanda

, et al. Hydromagneticnanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects. Int J Heat Mass Transfer 2012; 55(25–26): 7587–7595.

14.

Goqo

Oloniiju

Mondal

, et al. Entropy generation in MHD radiative viscous nanofluid flow over a porous wedge using the bivariate spectral quasi-linearization method. Case Stud Therm Eng 2018; 12: 774–788.

15.

Haq

Shahzad

Khan

, et al. Characteristics of chemical processes and heat source/sink with wedge geometry. Case Stud Therm Eng 2019; 14: 100432.

16.

Hussain

Ghaffar

Ali

, et al. MHD thermal boundary layer flow of a Casson fluid over a penetrable stretching wedge in the existence of nonlinear radiation and convective boundary condition. Alex Eng J 2021; 60: 5473–5483.

17.

Ullah

Shafie

Khan

, et al. Brownian diffusion and thermophoresis mechanisms in Casson fluid over a moving wedge. Results Phys 2018; 9: 183–194.

18.

Swain

Ibrahim

Dharmaiah

, et al. Numerical study of nanoparticles aggregation on radiative 3D flow of maxwell fluid over a permeable stretching surface with thermal radiation and heat source/sink. Results Eng 2023; 19: 101208.

19.

Khan

Azam

Alshomrani

. Effects of melting and heat generation/absorption on unsteady Falkner-Skan flow of Carreaunanofluid over a wedge. Int J Heat Mass Transfer 2017; 110: 437–446.

20.

Vinodkumar Reddy

Lakshminarayana

. Higher order chemical reaction and radiation effects on magnetohydrodynamic flow of a Maxwell nanofluid with Cattaneo–Christov heat flux model over a stretching sheet in a porous medium. J Fluids Eng 2022; 144(4): 041204.

21.

Buongiorno

. Convective transport in nanofluids. J Heat Transfer 2006; 128(3): 240–250.

22.

Hashim

Hamid

. Numerical investigation on time-dependent flow of Williamson nanofluid along with heat and mass transfer characteristics past a wedge geometry. Int J Heat Mass Transfer 2018; 118: 480–491.

23.

Amar

Kishan

. The influence of radiation on MHD boundary layer flow past a nano fluid wedge embedded in porous media. Partial Differ Equ Appl Math 2021; 4: 100082.

24.

Shamshuddin

Mabood

. A numerical model for analysis of binary chemical reaction and activation energy of thermo solutal micropolar nanofluid flow through permeable stretching sheet: nanoparticle study. Phys Scr 2021; 96: 075206.

25.

Ibrahim

Lorenzini

Vijaya Kumar

, et al. Influence of chemical reaction and heat source on dissipative MHD mixed convection flow of a Casson nanofluid over a nonlinear permeable stretching sheet. Int J Heat Mass Transfer 2017; 111: 346–355.

26.

Hamid

Hashim

. Impacts of binary chemical reaction with activation energy on unsteady flow of magneto-Williamson nanofluid. J Mol Liq 2018; 262: 435–442.

27.

Ram

Shamshuddin

Spandana

. Numerical simulation of stagnation point flow in magneto micropolar fluid over a stretchable surface under influence of activation energy and bilateral reaction. Int Commun Heat Mass Transfer 2021; 129: 105679.

28.

Ramzan

Ullah

, et al. A numerical treatment of radiative nanofluid 3D flow containing gyrotactic microorganism with anisotropic slip, binary chemical reaction and activation energy. Sci Rep 2017; 7(1): 17008.

29.

Shamshuddin

Asogwa

Ferdows

. Thermo-solutal migrating heat producing and radiative casson nanofluid flow via bidirectional stretching surface in the presence of bilateral reactions. Numer Heat Tr Part A Appl 2023: 1–20.

30.

Reddy

RCS

Reddy

. A comparative analysis of unsteady and steady Buongiorno’s Williamson nanoliquid flow over a wedge with slip effects. Chin J Chem Eng 2020; 28(7): 1767–1777.

31.

Reddy

Lakshminarayana

. MHD radiative flow of Williamson nanofluid with Cattaneo-Christov model over a stretching sheet through a porous medium in the presence of chemical reaction and suction/injection. J Porous Media 2022; 25(12): 1–15.

32.

Bestman

. Natural convection boundary layer with suction and mass transfer in a porous medium. Int J Energy Res 1990; 14(4): 389–396.

33.

Azam

Abbas

. Recent progress in Arrhenius activation energy for radiative heat transport of cross nanofluid over a melting wedge. Propuls Power Res 2021; 10(4): 383–395.

34.

Ali

Hussain

Nie

, et al. Buoyancy effetcs on falknerskan flow of a Maxwell nanofluid fluid with activation energy past a wedge: Finite element approach. Chin J Phys 2020; 68: 368–380.

35.

Ali

Liu

, et al. A comparative study of unsteady MHD Falkner–Skan wedge flow for non-Newtonian nanofluids considering thermal radiation and activation energy. Chin J Phys 2022; 77: 1625–1638.

36.

Ahmed

Khan

Irfan

, et al. Transient MHD flow of Maxwell nanofluid subject to non-linear thermal radiation and convective heat transport. Appl Nanosci 2020; 10: 5361–5373.

Magneto-Williamson nanofluid flow past a wedge with activation energy: Buongiorno model

Abstract

Keywords

Introduction

Mathematical formulation

Numerical solution

Results and discussion

Conclusion

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iD

Availability of data and materials

References