Numerical study of hybridized Williamson nanofluid flow with TC4 and Nichrome over an extending surface

Asmat Ullah Yahya; Imran Siddique; Nadeem Salamat; Hijaz Ahmad; Muhammad Rafiq; Sameh Askar; Sohaib Abdal

doi:10.1515/phys-2022-0246

Article Open Access

Numerical study of hybridized Williamson nanofluid flow with TC4 and Nichrome over an extending surface

Asmat Ullah Yahya , Imran Siddique , Nadeem Salamat , Hijaz Ahmad , Muhammad Rafiq , Sameh Askar and Sohaib Abdal

Published/Copyright: June 19, 2023

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From the journal Open Physics Volume 21 Issue 1

Abstract

Enhancement in thermal distribution of Williamson hybrid nanofluid flow is articulated in this research. Nichrome and TC4 nanoparticles are homogenously diffused in the water, which is the base fluid. An elongating surface holds the flow and thermal transition phenomenon in the existence of uniform sources of magnetic field and heat radiation. The boundary of wall obeys a suction and slip condition. The formulation for physical conservation laws is made as a system of partial differential equations. For the solution purpose, their boundary-value problem is transmuted into the ordinary differential form. Then, Matlab code involving Runge–Kutta procedure is run to compute the variation in velocity as well as temperature profiles under impacts of the controlling factors. The comparative computations are made for two cases: nanofluids ( TC 4 + water ) and hybrid nanofluids ( TC 4 , Nichrome + water ) . The heat for that hybrid nanofluid case is larger than that for the nanofluids. The velocity curve is decreased against increasing magnetic field strength and Williamson parameter. Enhancement in thermal distribution is observed with increasing concentration ϕ 2 of Nichrome.

Keywords: Williamson fluid; hybrid nanofluid; magnetohydrodynamic; Runge–Kutta method

1 Introduction

Nanosized particles are utilized to enhance heat energy conservation, increase the effectiveness of medicines, and improve equipment in medical and engineering fields. Also, when nanosized metallic particles are used in fluid, it change its characteristics that are beneficial in fluid mechanics. Normally, these nanosized particles are constructed by carbides, oxides, metals, etc. By using the nanoparticles, Hazarika et al. [1] discussed the thermophoresis process in the nanofluid. Berrehal et al. [2] predicted the shape impacts of nanoparticles on stretching sheet. On the unsteady fluid, the effects of nanoparticle were investigated by Tlili et al. [3]. Zahmatkesh et al. [4] analyzed the thermal system by using the nanoparticle impacts. The study of radiative nanofluid stream in the presence of nanoparticles was predicted by Acharya et al. [5]. The investigation of non-isothermal system containing nanoparticles was described by Abbas et al. [6]. He et al. [7] investigated the theory of harvesting of energy with carbon nanotube-embedded boundary-layer. Gorji et al. [8] showed how they used X-ray diffraction technique to examine the surface of an IC chip without causing any damage.

The fluids that are independent of stress or having constant viscosity are classified as non-Newtonian fluids. Also, Newton’s laws of viscosity are not applicable for non-Newtonian fluids. At present these fluids are widely used in engineering, medical fields, and daily life. Non-Newtonian fluids are most commonly encountered as melted butter, ketchup, shampoo, corn starch, apple juice, and starch suspensions. Williamson fluid is an important non-Newtonian fluid, which was introduced by Williamson in 1929 [9]. This fluid is used in different areas such as fluid film condensation process, emulsion coating on photographic films, and behavior of pseudo-plastic fluid that is extensively used in industrial applications. Yahya et al. [10] scrutinized the enhancement of thermal application of Williamson hybrid nanofluids using different nanoparticles. The study of Williamson nanofluid with inclined magnetic field was examined by Abdelmalek et al. [11]. Wang et al. [12] deliberated the flow of Williamson nanofluid through elastic surface with non-uniform thickness. The impacts on pour stretching sheet with the stream of Williamson nanofluid were depicted by Li et al. [13]. Zhou et al. [14] analyzed the effects of non-linear Williamson nanofluid on gyrotactic microorganisms.

In recent, decades, many researchers are interested to arrange the hybrid nanofluids. Different investigations have revealed the limitations, applications of single-type nanoparticles, and the attributes of their colloidal mixtures. Al 2 O 3 –Cu , Al–Zn , TiO 2 + CuO , etc. are the examples of hybrid nanofluids with some base fluids. Zubair et al. [15] scrutinized the heat conductivity of the Williamson hybrid nanofluid with radiation effects. The study of thermal and heat transformation in Williamson hybrid nanofluid with chemical reaction was depicted by Nazir et al. [16]. Eshgarf et al. [17] analyzed the features, formation, and stability analysis of hybrid nanofluids to optimization of energy usage. By using the hybrid nanofluids, Sathyamurthy et al. [18] investigated the phenomenon of cooling in photovoltaic penal. The analysis of thermodynamic process in different cycle driven by using hybrid nanofluids in solar collector tubes was carried out by Abid et al. [19].

The study of behavior and characteristics of magnetic produced by electrically conducting liquids is known as magnetohydrodynamics (MHD). Alfven [20] was the first who investigated this concept in 1942 and received Noble Prize in 1970. Later on, many researchers put their contribution and use it in many fields of engineering and medical science, such as cooling in refrigerators and electronic devices, medicines, and automobiles. Alblawi et al. [21] analyzed the implications of MHD in carbon nanotubes using casson nanofluid. In a cylindrical geometry, MHD effects on carreau nanofluid along different heat source were deliberated by Sabu et al. [22]. Rasheed et al. [23] studied the analytical study of nanofluid flow over convective boundaries with MHD and chemical impacts. The numerical investigation of nanofluid flow on stretched sheet using MHD to be effective was carried out by Makkar et al. [24]. Jimoh et al. [25] described the reacting effects of MHD at different thermal conductivity of nanofluid flow with radiation impacts. The study of MHD hybrid nanofluid motion over straining cylinder with non-uniform heat flux was depicted by Ali et al. [26].

This work is perceived to examine the enhancement in heat transfer Nichrome and TC4 that are emulgated homogenously in a bulk volume of water. This hybrid nanofluid and heat transition flow across a elongating surface is formulated on the basis of the Williamson fluid model and the Tiweri-Das model. It is presumed that thermal distribution is enhanced and that the application of the results can be helpful in heat exchangers and electronics. This study can further be discussed for a variety of nanofluids to analyze the impacts of thermal properties of materials that are useful for heat transportation phenomena.

2 Mathematical formulation

Williamson nanofluid is presumed to flow in two-dimensions across an elongated surface (Figure 1). The liquid is regarded incompressible, and the stream is convinced to be linear as a result of the dominance of viscous forces inside the boundary-layer flow. The fluid attributes are recognized not to modify with time once the sheet is prolonged in the +ve x -direction with a non-uniform speed.

U w ( x , t ) = c x ,

where c signifies the initial straining rate The temperature of an insulated wall is T w ( x , t ) = T ∞ + ( c x ) , and this is supposed to be specified at x = 0 ; for simplicity, here T w and T ∞ depict the heat of the exterior and surroundings, respectively. A slipping exterior is presumed, and a temperature distribution is adapted to the sheet. In the ordinary direction of flow, a consistent magnetic field of strength B ( t ) = B 0 is initiated.

Figure 1

Flowchart.

2.1 Suppositions and conditions of the model

The preceding presumptions and situations are implemented to the problem formulation:

two-dimensional laminar stable flow,
varying heat capacity,
permeable elongating sheet,
non-Newtonian Williamson nanoparticles,
MHD,
boundary-layer approximation,
heat radiation,
nanoparticles’ shape factor,
Tiwari and Das model,
convective and slip boundary situations.

The computational representation of the Williamson liquid stress tensor is presented by:

(1) S ∗ = − p I + τ i j ,

where

(2) τ i j = μ ∞ + ( μ 0 − μ ∞ ) ( 1 − ϕ γ ˜ ) A 1 .

The governing equations of stream for a viscous Williamson nanofluid were developed and modified using standard boundary-layer estimations, heat radiation, and heat-reliant thermal conductivity.

(3) ∂ v 1 ∂ x + ∂ v 2 ∂ y = 0 ,

(4) v 1 ∂ v 1 ∂ x + v 2 ∂ v 1 ∂ y = μ h n f ρ h n f ∂ 2 v 1 ∂ y 2 + 2 Γ μ h n f ρ h n f ∂ v 1 ∂ y ∂ 2 v 1 ∂ y 2 − σ h n f B t 2 v 1 ρ h n f ,

(5) v 1 ∂ T ∂ x + v 2 ∂ T ∂ y = 1 ( ρ C p ) h n f ∂ ∂ y k h n f ∗ ( T ) ∂ T ∂ y − 1 ( ρ C p ) h n f ∂ q r ∂ y .

With supportive boundary condition,

(6) v 1 ( x , 0 ) = U w = W ∗ μ h n f ∂ v 1 ∂ y , v 2 ( x , 0 ) = V w , − k h n f ∂ T ∂ y = h f ( T w − T ) , as y = 0 , v 1 → 0 , T → T ∞ , as y → ∞ .

Magnetic force term

F = q E + q v ∗ B ,

(7) η = c ν f y , u = c x f ′ ( η ) , v = − c ν f f ( η ) , θ ( η ) = T − T ∞ T w − T ∞ , k h n f ∗ ( T ) = 1 + ε T − T ∞ T w − T ∞ , ∂ q r ∂ y = − 16 T ∞ 3 σ ∗ 3 k ′ k ∂ 2 T ∂ y 2 .

From the aforementioned literature review, the basic thermo-physical characteristics of nanofluids are given in Table 1. The thermo-physical characteristics of water taken as base fluid are given in Table 2. Transformed ordinary differential equations are as follows:

(8) f ‴ − A 1 A 2 ( f ′ 2 − f f ″ ) + λ ( f ″ f ‴ ) − A 1 A 3 M f ′ = 0 ,

(9) θ ″ ( 1 + ε θ + 1 A 5 Pr Rd ) + ε θ ′ 2 + Pr A 4 A 5 ( f θ ′ − f ′ θ ) = 0 ,

(10) f ( η ) = S , f ′ ( η ) = 1 + β A 1 f ″ ( η ) , k h n f k f θ ′ ( η ) = B i ( 1 − θ ( η ) ) , at η = 0 , f ′ ( η ) → 0 , θ ( η ) → 0 , as η → ∞ ,

where M = σ h n f B t 2 c ρ h n f , β = W ∗ c ν f μ h n f , Bi = h f k h n f ν f c , λ = Γ x 2 c 3 ν f , Pr = ν f ( ρ C p ) f κ f , Rd = 16 T ∞ 3 σ ∗ 3 k ′ k ν f ( ρ C p ) f , We = − V w 1 c ν f .

Table 1

Thermo-physical properties

Physical properties	TC 4	Nichrome	Water
ρ ( kg ⋅ m − 3 )	4,420	8,314	997.1
C p ( J ( kg ⋅ ∘ k ) )	610	460	4,179
κ ( W ( m ⋅ ∘ k ) )	5.8	13	0.613

Table 2

Thermal characteristics

Properties	Nanofluid	Hybrid nanofluid
μ (viscosity)	μ n f = μ f ( 1 − Φ ) 2.5	μ h n f = μ f ( 1 − Φ 1 ) 2.5 ( 1 − Φ 2 ) 2.5
ρ (density)	ρ n f = ρ f ( 1 − Φ ) + Φ ρ s ρ f	ρ h n f = ρ f ( 1 − Φ 2 ) ( 1 − Φ 1 ) + Φ 1 ρ s 1 ρ f + Φ 2 ρ s 2
ρ C p (heat capacity)	( ρ C p ) n f = ( ρ C p ) f ( 1 − Φ ) + Φ ( ρ C p ) s ( ρ C p ) f	( ρ C p ) h n f = ( ρ C p ) f ( 1 − Φ 2 ) ( 1 − Φ 1 ) + Φ 1 ( ρ C p ) s 1 ( ρ C p ) f + Φ 2 ( ρ C p ) s 2
κ (thermal conductivity)	κ n f κ f = κ s + ( s f − 1 ) κ f − ( s f − 1 ) Φ ( κ f − κ s ) κ s + ( s f − 1 ) κ f + Φ ( κ f − κ s )	κ h n f κ b f = κ s 2 + ( s f − 1 ) κ b f − ( s f − 1 ) Φ 2 ( κ b f − κ s 2 ) κ s 2 + ( s f − 1 ) κ b f + Φ ( κ b f − κ s 2 )
		where κ b f κ f = κ s 1 + ( s f − 1 ) κ f − ( s f − 1 ) Φ 1 ( κ f − κ s 1 ) κ s 1 + ( s f − 1 ) κ f + Φ ( κ f − κ s 1 )
σ (electrical conductivity)	σ n f σ f = 1 + 3 ( σ − 1 ) Φ ( σ + 2 ) − ( σ − 1 ) Φ	σ h n f σ b f = 1 + 3 Φ ( σ 1 Φ 1 + σ 2 Φ 2 − σ b f ( Φ 1 + Φ 2 ) ) ( σ 1 Φ 1 + σ 2 Φ 2 + 2 Φ σ b f ) − Φ σ b f ( ( σ 1 Φ 1 + σ 2 Φ 2 ) − σ b f ( Φ 1 + Φ 2 ) )

Also,

A 1 = ( 1 − Φ 1 ) 2.5 ( 1 − Φ 2 ) 2.5 ( 1 − Φ 2 ) ( 1 − Φ 1 ) + Φ 1 ρ s 1 ρ f + Φ 2 ρ s 2 ρ f , A 2 = ( 1 − Φ 2 ) ( 1 − Φ 1 ) + Φ 1 ρ s 1 ρ f + Φ 2 ρ s 2 ρ f , A 3 = ( 1 − Φ 2 ) ( 1 − Φ 1 ) + Φ 1 ( ρ C p ) s 1 ( ρ C p ) f + Φ 2 ( ρ C p ) s 2 ( ρ C p ) f , A 4 = [ ( 1 − Φ 1 ) 2.5 ( 1 − Φ 2 ) 2.5 ] , k h n f k f = κ s 2 + ( s f − 1 ) κ b f − ( s f − 1 ) Φ 2 ( κ b f − κ s 2 ) κ s 2 + ( s f − 1 ) κ b f + Φ 2 ( κ b f − κ s 2 ) . κ s 1 + ( s f − 1 ) κ f − ( s f − 1 ) Φ 1 ( κ f − κ s 1 ) κ s 1 + ( s f − 1 ) κ f + Φ 1 ( κ f − κ s 1 ) .

Physical quantities are

(11) C f = τ w ρ f U w 2 , Nu = x q w k f ( T w − T ∞ ) , τ w = μ h n f ( 1 + Γ ) ∂ u ∂ y , q w = k h n f ∂ T ∂ y , at y = 0 .

Using similarity transformation, we obtain

(12) C f x Re x 1 ∕ 2 = 1 + λ 2 f ″ ( 0 ) A 4 f ″ ( 0 ) , Nu x Re x − 1 ∕ 2 = − [ k h n f ( 1 + Rd ) θ ′ ( 0 ) ] k f .

3 Numerical scheme

The two-point boundary-value problem, governing the heat and mass transfer problem is highly non-linear. These equations are difficult to yield analytical solution. Numerical and semi-analytical methods [27–37] are suggested for such complex problems. However, semi-analytical methods such as the homotopy perturbation method are used for this purpose [38–41]. Numerical procedure is mostly used for solution of such problems because of easily available computer and variant numerical techniques. Runge–Kutta method is a very effective method [42,43]. The non-linear non-dimensional revised problem and the boundary instances (8) were rectified using the MATLAB built-in code RK-4 strategy and a non-linear shooting methodology. The shooting tactic incorporates the first-order ordinary differential equations (ODEs) and starting instances employing the RK-4 procedure, and deficiency constraints are modified using the shooting technique until the scheme validates the required accuracy. The presence of exponential convergence is affirmed for η max = 8 . All numerical values acquired in this situation are restricted to a 1 0 − 5 range. The framework of higher-order differential equations (DEs) is transformed into a structure of first-order basic differential equations using the factors described below.

s 1 ′ = s 2 , s 2 ′ = s 3 , s 3 ′ = A 1 A 2 ( s 2 − s 1 s 3 ) − λ ( s 3 s 3 ′ ) + A 1 A 3 M s 2 , s 4 ′ = s 5 , s 5 ′ 1 + ε s 4 + 1 A 5 Pr Rd = − ε s 5 2 − Pr A 4 A 5 ( s 1 s 5 − s 2 s 4 ) .

In addition to the boundary situations:

s 1 = S , s 2 = 1 + β A 1 s 3 , k h n f k f s 5 = B i ( 1 − s 4 ) , at η = 0 , s 2 → 0 , s 4 → 0 as η → ∞ .

4 Results and discussion

The preceding description is used to attain numerical findings from the solutions of Eqs (8)–(10). The influence of the physical factors on the velocity and temperature is noted. The numerical computations are carried out for TC4 and TC4 + Nichrome water-based nanofluids. The graphical attitude of TC4-water nanofluid is depicted in green in the color graphs, whereas the tendency of TC4 + Nichrome–water is exhibited in red.

Figure 2 depicts the implications of the magnetic factor M and Williamson factor λ on the velocity distribution f ′ ( η ) . With a large magnetic factor M , the lowering trend of the fluid velocity is seen, leading to a reduction in the depth of the momentum boundary-layer. Actually, Lorentz force emerges when a typically imposed magnetic field reacts with an electric field to retard the boundary-layer thickness. When the intensity of the imposed magnetic force increases, so does the Lorentz force, which works in the reverse way of fluid movement, and thus, the induced fluid friction diminishes the depth of the velocity boundary-layer. The reduction in flow velocity is accompanied by a rise in, as a consequence wherein the momentum boundary-layer depth reduces. Moreover, the fluid’s velocity is delayed as a response to the Williamson parameter. This is owing to the belief that the Williamson factor is the combination of relaxation and deceleration time.

$Figure 2 Velocity f ′ ( η ) f^{\prime} \left(\eta ) fluctuation with M M and λ \lambda .$

Figure 2

Velocity f ′ ( η ) fluctuation with M and λ .

Figure 3 displays the effects of varying the suction factor S and velocity slip factor β on the flow velocity f ′ ( η ) . The velocity f ′ ( η ) diminishes as the suction parameter value increases. The velocity slip factor β exhibits a declining pattern in the velocity distribution; this could be due to an increment in slip impact that retards the liquid motion that depresses the fluid.

$Figure 3 Velocity f ′ ( η ) f^{\prime} \left(\eta ) fluctuation with S S and β \beta .$

Figure 3

Velocity f ′ ( η ) fluctuation with S and β .

Figure 4 portrays the impact of nanoparticle concentration ϕ 2 on liquid velocity f ′ ( η ) and temperature θ ( η ) when ϕ 1 = 0.0 for TC4 and ϕ 1 = 0.01 for TC4 + Nichrome. By boosting the valuation of ϕ 2 , the velocity is reduced, so is the energy of the boundary-layer depth. Larger nanoparticles thickened the fluid, enabling the momentum boundary-layer to collapse. The amount of nanoparticles enhances the heat conductivity of nanoparticles. As a consequence of the rise in heat capacity, the energy boundary-layer revealed a decreasing tendency. Higher heat conductance of nanofluids, on the other hand, has a beneficial impact on fluid temperature since it improves with higher nanoparticle concentration.

$Figure 4 Impact of ϕ 2 {\phi }_{2} on velocity f ′ ( η ) f^{\prime} \left(\eta ) and temperature θ ( η ) \theta \left(\eta ) .$

Figure 4

Impact of ϕ 2 on velocity f ′ ( η ) and temperature θ ( η ) .

In Figure 5, the heat variation of nanoparticles increases with an increase in the intensity of factor M , causing the heat boundary-layer to expand. This is also revealed that the variable M is inversely associated with the thickness of the nanofluid, and thus, boosting M diminishes velocity when the temperature of the liquid surges. As the Williamson factor λ increases, the heat boundary-layer extends, allowing the temperature to elevate leading to a rise in the elasticity stress factor.

$Figure 5 Temperature θ ( η ) \theta \left(\eta ) fluctuation with M M and λ \lambda .$

Figure 5

Temperature θ ( η ) fluctuation with M and λ .

Figure 6 explores the effects of the heat radiation factor Rd on the temperature dispersion θ ( η ) of Williamson nanofluids, indicating that the heat of the nanofluid improves with larger values of Rd = 0.1, 0.2, 0.3, 0.4, and 0.5. The depth of the heat boundary-layer improves as the temperature increases. Similarly, as ε values rise, the temperature dispersion rises. The temperature of the nanoparticles is also shown to expand as the Biot number increases. The smaller the Biot number, the greater the conductivity inside the surface, so the higher the Biot number, the greater the extreme conductivity at the exterior level. Because of the enhanced heat energy in nanoparticles, the temperature keeps rising. A rise in Bi promotes more heat transmission from the barrier to the liquid, thus increasing the conductivity of the boundary-layer.

$Figure 6 Temperature θ ( η ) \theta \left(\eta ) fluctuation with Rd, ε \varepsilon , and Bi.$

Figure 6

Temperature θ ( η ) fluctuation with Rd, ε , and Bi.

Table 3 portrays that as the values of magnetic parameter M and suction factor S enhance, the skin friction factor − f ″ ( 0 ) boosts for both TC4 nanofluid and TC4 + Nichrome nanofluids, whereas it declines for upsurging values of the Williamson parameter λ and velocity slip parameter β . In Table 4, the Nusselt number − θ ′ ( 0 ) increases for increasing values of Rd and Bi in both TC4 nanofluid and TC4 + Nichrome nanofluid cases; however, the Nusselt number − θ ′ ( 0 ) diminishes for upsurging values of ε .

Table 3

Results for skin friction factor − f ″ ( 0 )

M	λ	S	β	TC4	TC4 + Nichrome
0.2	0.1	0.1	0.3	0.8038	0.8561
0.4				0.8659	0.9103
0.6				0.9198	0.9582
0.6	0.1			0.9198	0.9582
	0.3			0.8701	0.9027
	0.5			0.8118	0.8372
	0.1	0.1		0.9198	0.9582
		0.3		0.9668	1.0155
		0.5		1.0151	1.0742
		0.1	0.1	1.1828	1.2520
			0.3	0.9198	0.9582
			0.5	0.7951	0.7838

Table 4

Results for Nusselt number − θ ′ ( 0 )

Rd	ε	Bi	TC4	TC4 + Nichrome
0.1	0.1	0.2	0.2196	0.2051
0.3			0.2500	0.2328
0.5			0.2794	0.2595
0.3	0.1		0.2500	0.2328
	3.1		0.2471	0.2298
	6.1		0.2440	0.2264
	0.1	0.1	0.1329	0.1243
		0.2	0.2500	0.2328
		0.3	0.3539	0.3285

5 Conclusion

The two-dimensional MHD movement of the Williamson nanofluid was scrutinized with the influence of heat capacity varying from heat to heat radiation over the flexible surface. The complex equations that control velocity and heat, known as non-linear partial differential equations, are simplified into simpler equations called ordinary differential equations. This simplification is achieved by using a suitable transformation method called similarity transformation. The RK-4 approach was used to generate the computational model solution. The significance of the non-dimensional speed and heat gradient implications of the numerous physical factors under discussion is depicted graphically. For every value of the individual governing factors, the coefficient of skin friction as well as the Nusselt number is presented in a tabulated form. Following a comprehensive analysis, we came to the preceding conclusion:

An elevation in the Williamson factor, suction parameter, and nanoparticle concentration leads to a reduction in the velocity distribution.
Nanoparticles are mostly engaged in fluids to enhance thermal characteristics. As an outcome, rising the density of nanoparticles boosts the temperature of the nanoparticles and thus the depth of the heat boundary-layer.
TC4-Nichrome-water-based nanofluid is considered to be a better heat conductor than TC4-water-based nanofluid.
Boosting the magnetic factor declines the depth of the velocity boundary-layer while improving the temperature distribution.
An upsurge in the Williamson, thermal radiation factors, Biot number, and ε leads to an elevation in the temperature gradient.
Skin friction improves with rising magnetic and suction factors for both TC4 and TC4-Nichrome nanofluids, but decreases with higher Williamson and velocity slip parameters.
Heat transfer efficiency dropped as the quantity of the parameter ε expanded, but it expanded with the radiation parameter and Biot number boosted in both instances.

Acknowledgments

This research was supported by Research Supporting Project number (RSP2023R167), King Saud University, Riyadh, Saudi Arabia.

Funding information: This project was funded by King Saud University, Riyadh, Saudi Arabia.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

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Received: 2023-03-09

Revised: 2023-04-20

Accepted: 2023-04-28

Published Online: 2023-06-19

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2022-0246

Keywords for this article

Williamson fluid; hybrid nanofluid; magnetohydrodynamic; Runge–Kutta method

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