Magnetohydrodynamic water-based hybrid nanofluid flow comprising diamond and copper nanoparticles on a stretching sheet with slips constraints

Humaira Yasmin; Laila A. AL-Essa; Showkat Ahmad Lone; Hussam Alrabaiah; Zehba Raizah; Anwar Saeed

doi:10.1515/phys-2024-0007

Article Open Access

Magnetohydrodynamic water-based hybrid nanofluid flow comprising diamond and copper nanoparticles on a stretching sheet with slips constraints

Humaira Yasmin , Laila A. AL-Essa , Showkat Ahmad Lone , Hussam Alrabaiah , Zehba Raizah and Anwar Saeed

Published/Copyright: April 3, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

Hybrid nanofluid problems are used for augmentation of thermal transportation in various industrial applications. Therefore, the present problem is studied for the heat and mass transportation features of hybrid nanofluid caused by extending surface along with porous media. In this investigation, the authors have emphasized to analyze hybrid nanofluid flow containing diamond and copper nanoparticles on an extending surface. Furthermore, the velocity, temperature, and concentration slip constraints are adopted to examine the flow of fluid. Heat source, chemical reactivity, thermal radiation, Brownian motion and effects are taken into consideration. Nonlinear modeled equations are converted into dimensionless through similarity variables. By adopting the homotopy analysis method, the resulting equations are simulated analytically. The impacts of various emerging factors on the flow profiles (i.e., velocities, temperature, concentration, skin frictions, local Nusselt number, and Sherwood number) are shown using Figures and Tables. The major key findings reveal that the hybrid nanofluid temperature is higher but the concentration is lower for a Brownian diffusivity parameter. Moreover, increment role of heat transport is achieved due to the increment in radiation factor, thermophoresis, Brownian motion factors, and Eckert number. It has also been observed that velocity in x-direction converges in the region − 0.8 ≤ ℏ f ≤ 0.5 , in y-direction velocity is convergent in the zone − 0.6 ≤ ℏ g ≤ 0.35 , while temperature converges in the region − 0.6 ≤ ℏ θ ≤ 0.4 and concentration converges in the region − 0.5 ≤ ℏ φ ≤ 0.4 .

Keywords: hybrid nanofluid flow; MHD; porous media; thermal radiation; heat source; stretching surface; slip conditions

Nomenclature

B 0: strength of magnetic field
C: concentration
C p: specific heat
D B: coefficient of Brownian diffusion
D T: thermophoretic coefficient
E: activation energy factor
E a: coefficient of activation energy
Ec: Eckert number
K: porosity factor
K ⁎: coefficient of porous media
K r: chemical reaction
k: thermal conductivity
k ⁎: mean absorption coefficient
k r: coefficient of chemical reaction
m: power index
M: magnetic factor
Nb: Brownian motion factor
Nt: thermophoresis factor
Pr: Prandtl number
P 1: first nanoparticle
P 2: second nanoparticle
Q: heat source factor
Q 0: heat source coefficient
q r: radiative heat flux
Sc: Schmidt number
T: temperature
u , v , w: velocity components
x , y , z: coordinates
μ: dynamic viscosity
ρ: density
σ 1: primary velocity slip constant
σ 2: secondary velocity slip constant
σ 3: temperature slip constant
σ 4: concentration slip constant
σ ⁎: Stefan–Boltzmann constant
ϕ 1: volume fraction of first nanoparticle
ϕ 2: volume fraction of second nanoparticle
β: temperature difference factor
δ: ratio factor
α 1: primary velocity slip factor
α 2: secondary velocity slip factor
α 3: temperature slip factor
α 4: concentration slip factor

1 Introduction

In the realm of advanced fluid dynamics and heat transfer research, the analysis of nanofluids has arose as a promising frontier with profound implications for various engineering applications. Magnetohydrodynamic (MHD) nanofluids, characterized by the synergistic coupling of magnetic field effects and nanoparticle dispersion within a fluid medium, have gathered substantial attention due to their potential to enhance thermal performance and control fluid flow behavior. This study delves into the complex dynamics of a MHD water-based hybrid nanofluid, composed of diamond and copper nanoparticles, as it flows over a stretching sheet; a scenario often encountered in industrial processes and engineering systems. Furthermore, the inclusion of slip boundary conditions adds an additional layer of complexity to the investigation, making this study a significant contribution to the evolving field of MHD nanofluid mechanics and its practical implications. In this context, the present work seeks to unravel the elaborated interaction of magnetic fields, nanoparticle additives, and slip effects on fluid flow and heat transfer characteristics, offering valuable insights for optimizing cooling systems, heat exchangers, and other thermal controlling applications at industrial level.

Nanofluid is a promising category of thermal transmission fluids that have been increasingly used in numerous areas, including aerospace engineering, automobile industry, and energy systems. The reason for their popularity is their unique properties that are not found in conventional fluids. Nanofluid is a mixture of a pure fluid and nanoparticles, which improves the thermal flow characteristics and thermal flow ability of fluid. Choi and Eastman [1] are the pioneers of the idea. Hashemi et al. [2] scrutinized thermal flow enhancement of a nanofluid flow in a conduit and observed that accumulation of nanoparticles significantly enhanced the coefficient of heat transference and improved the fluid’s overall thermal performance. Habib et al. [3] inspected MHD nanoparticles-based fluid flow subject to microorganisms and melting effects. Chu et al. [4] debated on the comparison for model-based study on MHD nanofluid flow amid two plates using the impact of nanoparticles shape. Eid and Nafe [5] examined the variation in thermal conductance and creation of heat for nanofluid MHD flow past a permeable sheet using slip constraints and found that growth in nanoparticles volume has augmented the thermal profiles while the velocity panels have weakened accordingly. Rajput et al. [6] explored numerically the nanoparticle’s radius impact on water-based nanofluid consisting of copper nanoparticles across an extending permeable surface with slip condition. Sarfraz and Khan [7] used the impacts of nanoparticles and dual diffusions on fluid flow induced by movable plate and observed that fluid velocity has declined with escalation in magnetic factor. Dawar et al. [8] examined non-homogenous MHD nanofluid flow model to simulate rotary thin inclined layer and concluded that augmentation in thermophoresis factor has weakened the concentration panels while thermal characteristics have augmented. Pasha et al. [9] studied statistically the entropy creation for nanofluid flow on permeable stretched sheet with impact of different shapes of nanoparticles and have deduced that the entropy of system has expanded for growth in nanoparticles concentration, Biot number, and radiation factor.

Hybrid nanofluids are mixing of multiple nanoparticles in pure fluid, which engrossed the attention of researchers. Such fluids have unique properties that are not present in single-nanoparticle nanofluids. For instance, hybrid nanofluids can have enhanced thermal, electrical conductivities, and heat transfer properties, making them useful in numerous applications like coolant of electric devices, nuclear reactor, and thermal exchangers. The behavior of hybrid nanofluids flow is affected by various features, including the size, shape, and concentration of nanoparticles, along with type and flow rate of pure fluid. Ojjela [10] investigated numerically the time-based silica–alumina hybrid nanofluid flow on an extended sheet using magnetic effects and has deduced that thermal distribution has augmented whereas velocity panels decreased with the increase in the concentration of nanoparticles. Sarfraz et al. [11] inspected thermal transference for radiative trihybrid nanofluid flow on an elongating surface and proved that shape of nanoparticles impacted the transportation of heat. Salahuddin et al. [12] debated on three-dimensional hybrid nanofluid flow at s stagnant point of wavy, heated, and stretched cylinder and observed that velocity panels have escalated at the beginning and end of the cylinder with the increase in nanoparticles but at the middle of the cylinder, the flow has declined. Zhang et al. [13] scrutinized nanoparticles flow near the elastic surface by employing nickel and tantalum nanoparticles with the effect of magnetic field. Sarfraz et al. [14] inspected stagnant point flow for hybrid nanofluid on spiraling and elongating sheet and noticed that fluid transportation of heat has augmented with the increase in the surface porosity. Alrabaiah et al. [15] deliberated on impact of gyrotactic microorganisms on hybrid nanofluid flow in the gap of cone and disk. Ouni et al. [16] debated on the thermal solar features for hybrid nanofluid flow in a solar collector by mixing gold and copper nanoparticles in engine oil. Barathi et al. [17] studied computationally the thermal transference for trihybrid EMHD nanofluid flow on a permeable surface. Pandey et al. [18] discussed the effect of iron oxide nanoparticles on the unsteady hybrid nanoliquid flow across a flat surface. Ahmed et al. [19] simulated computationally the hybrid nanofluid flow thorough annular sector of duct. Singh et al. [20] discussed thermal transference and impacts of thermally radiated trihybrid nanofluid flow on an elongating surface. Some valuable results are recently presented in previous studies [21,22,23].

The study of fluid’s behavior that is conducted electrically with impact of magnetic effects is called MHD. Recently, investigation of MHD fluid flow has gained immense consideration for its application like plasma propulsion, metallurgy, materials processing etc. In this literature review, we discuss recent advances in the field of MHD fluid flow. One of the fundamental problems in MHD fluid flow is the study of the permanency of the flow. The stability of MHD flow is an essential aspect of many industrial applications. Dey et al. [24] debated on the effects of zero mass flux and slip condition on the MHD nanoliquid flow across a plate surface. Goud Bejawada et al. [25] discussed radiative influences on MHD fluid flow on a sheet using the impact of a chemical reaction. Vishalakshi et al. [26] examined the MHD fluid flow on a shrinking and elongating permeable sheet subject to mass transportation and slip constraints and noted that the thermal and velocity distributions have augmented with the increase in the concentration of nanoparticles and radiative factor. Upreti et al. [27] considered the effects of radiations and heat sink/source on nanofluid flow over a permeable surface. Sharma et al. [28] inspected the convective MHD fluid flow on a gyrating disk and deduced velocity of fluid has opposed while thermal distribution has supported with progression in magnetic parameter. Raghunath et al. [29] studied the MHD fluid flow on a permeable surface using chemically reactive and radiation impacts and concluded that velocity panel was found to increase with the increase in the Soret and permeability factors while it weakened with augmentation in magnetic and inclined parameters. Many similar effects can be studied in previous literature [30–34].

Fluid flow over porous surfaces is a fundamental and ubiquitous phenomenon in many engineering and natural systems, such as ground water flow, geothermal energy extraction, oil reservoirs etc. Porous media, which are characterized by their porosity, permeability, and tortuosity, play a critical role in controlling fluid flow behavior. Porous media is commonly found in natural systems such as aquifers, soils, and rocks, and in engineered systems such as geothermal energy systems, catalytic reactors, and biomedical devices. When fluid flows over a porous surface, it can penetrate into the pores of the surface and interact with the material inside. Akbar Qureshi et al. [35] evaluated the impression of MHD flow of fluid on a permeable and orthogonal surface subject to effects of injection and small walled carbon nano tubes (SWCNTs) and inferred that the thermal panels enlarged with the increase in the porosity of the surface. Verma et al. [36] discussed the stability and comparative study of graphene oxide/water and graphene/water-based nanofluid flows across a permeable stretching surface. Raza et al. [37] scrutinized the hybrid nanoparticles flow on a porous sheet using the impacts of chemically reactive activated energy and magnetic field and detected that velocity panels have weakened for intensification in concentration of nanoparticles while thermal characteristics have increased. Krishna and Chamkha [38] inspected chemically reactive and Hall effects with slip constraints on MHD rotary flow over a porous infinite vertical sheet and have assumed velocity panels have diminished while thermal panels have increased with the increase in the porosity factor. Megahed and Abbas [39] discussed fluid flow on porous surface and highlighted that thermal panels have improved with progress in permeability factor while concentration characteristics have declined in this phenomenon. Pandey et al. [40] inspected free convective MHD flow of hybrid nanoparticles on a permeable surface subject to various slip constraints and have proved that velocity and concentration have weakened with expansion in natural convective factor.

Hybrid nanofluid has remarkable applications, while flowing across a stretching permeable surface, such as oil recovery, thermal transportation, automobiles, electronics equipment’s, vehicle engines, drug delivery, and biomedical fields. Therefore, the present problem is modeled based on the above applications of hybrid nanofluid flow, to evaluate the thermal characteristics of nanofluid consisting of diamond and copper nanoparticles across an extending surface. Furthermore, chemical reactivity, heat source, and thermal radiation effects along with activation energy features are taken into consideration. Thus, in this analysis, the authors are interested to answer the following research questions:

How do the slip parameter, ratio parameter, porosity parameter, and magnetic parameter affect the primary and secondary velocities?
How do the Eckert number, Brownian motion factor, heat generation factor, and thermal radiation factor affect the flow temperature?
How do the thermophoresis factor, activation energy factor, Brownian motion factor, and Schmidt number affect the flow concentration?
How do the embedded factors affect the skin frictions, local Nusselt number, and Sherwood number?

To answer the above questions, the present analysis is composed of subsequent sections. Section 2 presents the mathematical formulation fluid flow, which consists of diamond and copper nanoparticles. The solution of the transformed ODEs is tackled with the semi-analytical method called homotopy analysis method (HAM), which is described in Section 3. The convergence of the applied technique is shown in Section 4. Section 5 has two sub-sections. Sub-Section 5.1 presents the obtained results, and sub-Section 5.2 presents the physical discussion of the obtained results. The final concluding remarks are presented in Section 6.

2 Formulation of problem

Assume 3D flow of hybrid nanofluid (composed of diamond and copper nanoparticles while water is considered as pure fluid), over an elongating sheet with porous media. The surface stretches with velocity u w ( x ) = a x along x-direction and along y-direction with velocity v w ( y ) = b y , where a and b are fixed constants. The flow is considered to be magnetically influenced by applying the strength of magnetic field B 0 in a perpendicular direction (z-axis). In the context of porous medium flow, when viscous effects predominate and the flow is characterized by low Reynolds numbers, Darcy’s law is employed. The geometrical view of the considered stretching surface is shown in Figure 1. The concentration, velocity, and thermal slip constraints are adopted in this analysis. The flow of fluid is considered to be affected by various flow conditions. With above assumptions, the main equations are [41,42]:

(1) ∂ u ∂ x + ∂ v ∂ y + ∂ w ∂ z = 0 ,

(2) u ∂ u ∂ x + v ∂ u ∂ y + w ∂ u ∂ z = μ hnf ρ hnf ∂ 2 u ∂ z 2 − σ hnf ρ hnf B 0 2 + μ hnf ρ hnf 1 K ⁎ u ,

(3) u ∂ v ∂ x + w ∂ v ∂ z + v ∂ v ∂ y = μ hnf ρ hnf ∂ 2 v ∂ z 2 − σ hnf ρ hnf B 0 2 + μ hnf ρ hnf 1 K ⁎ v ,

(4) u ∂ T ∂ x + v ∂ T ∂ y + w ∂ T ∂ z = k hnf ( ρ C p ) hnf ∂ 2 T ∂ z 2 − 1 ( ρ C p ) hnf ∂ q r ∂ z − 1 ( ρ C p ) hnf Q 0 ( T − T ∞ ) + μ hnf ( ρ C p ) hnf ∂ u ∂ z 2 + ∂ v ∂ z 2 + ( ρ C p ) np ( ρ C p ) hnf D B δ C ∂ T ∂ z ∂ C ∂ z + D T T ∞ ∂ T ∂ z 2 ,

Figure 1

Geometrical view of the flow problem.

Following Cao et al. [43] and Song et al. [44], we have

(5) u ∂ C ∂ x + v ∂ C ∂ y + w ∂ C ∂ z = D B ∂ 2 C ∂ z 2 + δ C D T T ∞ ∂ 2 T ∂ z 2 − k r 2 ( C − C ∞ ) T T ∞ m e − E a k B T .

The boundary conditions are defined as follows:

(6) u = u w ( x ) + σ 1 ∂ u ∂ z , v = v w ( y ) + σ 2 ∂ v ∂ z , w = 0 , T = T w + σ 3 ∂ T ∂ z , C = C w + σ 4 ∂ C ∂ z at z = 0 , u → 0 , v → 0 , T → T ∞ , C → C ∞ as z → ∞ .

The value of q r is given as

(7) q r = − 4 σ ⁎ 3 k ⁎ ∂ T 4 ∂ z = − 16 σ ⁎ 3 k ⁎ ∂ 2 T ∂ z 2 .

Thus, Eq. (5) can be written as

(8) u ∂ T ∂ x + v ∂ T ∂ y + w ∂ T ∂ z = k hnf ( ρ C p ) hnf ∂ 2 T ∂ z 2 + 1 ( ρ C p ) hnf 16 σ ⁎ 3 k ⁎ ∂ 2 T ∂ z 2 − 1 ( ρ C p ) hnf Q 0 ( T − T ∞ ) + μ hnf ( ρ C p ) hnf ∂ v ∂ z 2 + ∂ u ∂ z 2 + ( ρ C p ) np ( ρ C p ) hnf D B δ C ∂ T ∂ z ∂ C ∂ z + D T T ∞ ∂ T ∂ z 2 ,

The value of μ hnf (dynamic viscosity) [45,46] is

(9) μ hnf = μ f ( 1 − ϕ 1 ) 2.5 ( 1 − ϕ 2 ) 2.5 ,

where ϕ 1 and ϕ 2 are, respectively, the volume fractions of first and second nanoparticles.

The density of the hybrid nanofluid is defined as [45,46]

(10) ρ hnf = ( ( 1 − ϕ 2 ) ( ρ f ( 1 − ϕ 1 ) + ρ P 1 ϕ 1 ) ) + ρ P 2 ϕ 2 ,

where P 1 and P 2 are the first and second nanoparticles.

The value of ( ρ C p ) hnf is given by [45,46]

(11) ( ρ C p ) hnf = ( ( 1 − ϕ 2 ) ( ( ρ C p ) f ( 1 − ϕ 1 ) + ϕ 1 ( ρ C p ) P 1 ) ) + ϕ 2 ( ρ C p ) P 2 .

The value of k hnf is given as [45,46]

(12) k hnf = k P 1 ϕ 1 + k P 2 ϕ 2 ϕ 1 + ϕ 2 + 2 k f + 2 ( k P 1 ϕ 1 + k P 1 ϕ 2 ) − 2 ( ϕ 1 + ϕ 2 ) k f k P 1 ϕ 1 + k P 1 ϕ 2 ϕ 1 + ϕ 2 + 2 k f − 2 ( k P 1 ϕ 1 + k P 1 ϕ 2 ) + ( ϕ 1 + ϕ 2 ) k f k f .

The value of σ hnf is given by [45,46]

(13) σ hnf = σ P 1 ϕ 1 + σ P 2 ϕ 2 ϕ 1 + ϕ 2 + 2 σ f + 2 ( σ P 1 ϕ 1 + σ P 1 ϕ 2 ) − 2 ( ϕ 1 + ϕ 2 ) σ f σ P 1 ϕ 1 + σ P 1 ϕ 2 ϕ 1 + ϕ 2 + 2 σ f − 2 ( σ P 1 ϕ 1 + σ P 1 ϕ 2 ) + ( ϕ 1 + ϕ 2 ) σ f σ f .

The thermophysical features of nanoparticles and water are defined in Table 1.

Table 1

Thermophysical features of nanoparticles and water [45,46]

Thermodynamic properties	Base fluid	Nanoparticles
Thermodynamic properties	Water	Diamond	Copper
ρ (kg m⁻³)	997.1	3100	8933
c _p (J kg⁻¹ K⁻¹)	4,179	516	385
k (Wm⁻¹ K⁻¹)	0.613	1,000	400
σ (Sm⁻¹)	0.05	34.84 × 10 6	5.96 × 10 7

To transform the above PDEs, variables (similarity) are defined as [41] follows:

(14) u = a x f ′ ( ζ ) , v = a y g ′ ( ζ ) , w = − a ν f ( g ( ζ ) + f ( ζ ) ) , φ ( ζ ) = C − C ∞ C w − C ∞ , θ ( ζ ) = T − T ∞ T w − T ∞ , ζ = z a ν f . .

The use of Eq. (14) converts the system of equation to

(15) μ hnf / μ f ρ hnf / ρ f f ‴ ( ζ ) + ( g ( ζ ) + f ( ζ ) ) f ″ ( ζ ) + f ′ 2 ( ζ ) − M σ hnf / σ f ρ hnf / ρ f f ′ ( ζ ) − K μ hnf / μ f ρ hnf / ρ f f ′ ( ζ ) = 0 ,

(16) μ hnf / μ f ρ hnf / ρ f g ‴ ( ζ ) + ( g ( ζ ) + f ( ζ ) ) g ″ ( ζ ) + g ′ 2 ( ζ ) − σ hnf / σ f ρ hnf / ρ f M g ′ ( ζ ) − μ hnf / μ f ρ hnf / ρ f K g ′ ( ζ ) = 0 ,

(17) k hnf / k f ( ρ C p ) hnf / ( ρ C p ) f + Rd ( ρ C p ) hnf / ( ρ C p ) f θ ″ ( ζ ) + Pr θ ′ ( ζ ) f ( ζ ) + g ( ζ ) + Pr Q ( ρ C p ) hnf / ( ρ C p ) f θ ( ζ ) + Pr Ec μ hnf / μ f ( ρ C p ) hnf / ( ρ C p ) f ( f ″ 2 ( ζ ) + g ″ 2 ( ζ ) ) + Pr ( ρ C p ) hnf / ( ρ C p ) f ( Nb θ ′ ( ζ ) φ ′ ( ζ ) + Nt θ ′ 2 ( ζ ) ) = 0 ,

(18) φ ″ ( ζ ) + Sc φ ′ ( ζ ) ( f ( ζ ) + g ( ζ ) ) + Nt Nb θ ″ ( ζ ) − K r Sc exp − E 1 + θ ( ζ ) β ( 1 + β θ ( ζ ) ) n φ ( ζ ) = 0 ,

with transformed constraints at boundary:

(19) f ( 0 ) = 0 , f ′ ( 0 ) = 1 + α 1 f ″ ( 0 ) , g ′ ( 0 ) = δ + α 2 g ″ ( 0 ) , g ( 0 ) = 0 , θ ( 0 ) = 1 + a 3 θ ′ ( 0 ) , φ ( 0 ) = 1 + a 4 φ ′ ( 0 ) , f ′ ( ∞ ) → 0 , g ′ ( ∞ ) → 0 , θ ( ∞ ) → 0 , φ ( ∞ ) → 0 .

where M = σ f B 0 2 ρ f a is the magnetic parameter, Pr = ( ρ C p ) f ν f k f is the Prandtl number, Ec = u w 2 ( C p ) f ( T w − T ∞ ) is the Eckert number, K r = k r a is the chemical reaction factor, Q = Q 0 a ( ρ C p ) f is the heat source factor, Nt = D T ( T w − T ∞ ) ( ρ C p ) np ν f T ∞ ( ρ C p ) f is the thermophoresis factor, Nb = ( C w − C ∞ ) D B ( ρ C p ) np ν f δ C ( ρ C p ) f is the Brownian motion factor, Sc = ν f D B is the Schmidt number, β = T w − T ∞ T ∞ is the temperature difference factor, E = E a T ∞ k B is the factor of activation energy, K = μ f a K ⁎ ρ f is the porosity factor, δ = b a is the ratio factor, α 1 = σ 1 a ν f is the primary velocity slip factor, α 2 = σ 2 a ν f is the secondary velocity slip factor, α 3 = σ 3 a ν f is the thermal slip factor, and α 4 = σ 4 a ν f is the mass slip factor.

The engineering quantities are demarcated as follows:

(20) Skin friction along x -axis C f x = τ w x ρ f u w 2 ,

(21) Skin friction along y -axis C fy = τ wy ρ f v w 2 ,

(22) Nusselt number Nu x = x q w k f ( T w − T ∞ ) ,

(23) Sherwood number Sh x = x q m D B ( C w − C ∞ ) ,

where

(24) τ w x = μ hnf ∂ u ∂ z z = 0 ,

(25) τ w y = μ hnf ∂ v ∂ z z = 0 ,

(26) q w = − k hnf ∂ T ∂ z z = 0 + q r ∣ z = 0 ,

(27) q m = − D B ∂ C ∂ z z = 0 .

Thus, we have

(28) Skin friction along x -direction Re x C f x = μ hnf μ f f ″ ( 0 ) ,

(29) Skin friction along y -direction Re x C f y = μ hnf μ f g ″ ( 0 ) .

(30) Nusselt number 1 Re x Nu x = − k hnf k f + Rd θ ′ ( 0 ) ,

(31) Sherwood number 1 Re x Sh x = − φ ′ ( 0 ) .

Here Re x = x u w ( x ) ν f and Re y = y v w ( y ) ν f are local Reynolds number.

3 HAM solution

The initial guesses are mathematically designated as follows:

(32) f 0 ( ζ ) = 1 1 + α 1 ( 1 − e − ζ ) , g 0 ( ζ ) = δ 1 + α 2 ( 1 − e − ζ ) , θ 0 ( ζ ) = 1 1 + α 3 ( 1 − e − ζ ) , φ 0 ( ζ ) = 1 1 + α 4 ( − e − ζ ) , ,

(33) L f ( ζ ) = f ‴ − f ′ , L g ( ζ ) = g ‴ − g ′ , L θ ( ζ ) = θ ″ − θ , L φ ( ζ ) = φ ″ − φ , ,

with properties

(34) L f ( χ 1 + χ 2 e − ζ + χ 3 e ζ ) = 0 , L g ( χ 4 + χ 5 e − ζ + χ 6 e ζ ) = 0 , L θ ( χ 7 e − ζ + χ 8 e ζ ) = 0 , L φ ( χ 9 e − ζ + χ 10 e ζ ) = 0 , .

where χ 1 − χ 10 are constants of general solution.

Moreover,

(35) ( 1 − ψ ) L f [ f ( ζ ; ψ ) − f 0 ( ζ ) ] = ψ ℏ f N f [ f ( ζ ; ψ ) , g ( ζ ; ψ ) ] ,

(36) ( 1 − ψ ) L g [ g ( ζ ; ψ ) − g 0 ( ζ ) ] = ψ ℏ g N g [ g ( ζ ; ψ ) , f ( ζ ; ψ ) ] ,

(37) ( 1 − ψ ) L θ [ θ ( ζ ; ψ ) − θ 0 ( ζ ) ] = ψ ℏ θ N θ [ θ ( ζ ; ψ ) , f ( ζ ; ψ ) , g ( ζ ; ψ ) , φ ( ζ ; ψ ) ] ,

(38) ( 1 − ψ ) L φ [ φ ( ζ ; ψ ) − φ 0 ( ζ ) ] = ψ ℏ φ N φ [ φ ( ζ ; ψ ) , f ( ζ ; ψ ) , g ( ζ ; ψ ) , θ ( ζ ; ψ ) ] ,

(39) f ( 0 ; ψ ) = 0 , f ′ ( 0 ; ψ ) = 1 + α 1 f ″ ( 0 ; ψ ) , g ′ ( 0 ; ψ ) = δ + α 2 g ″ ( 0 ; ψ ) , g ( 0 ; ψ ) = 0 , θ ( 0 ; ψ ) = 1 + a 3 θ ′ ( 0 ; ψ ) , φ ( 0 ; ψ ) = 1 + a 4 φ ′ ( 0 ; ψ ) , f ′ ( ∞ ; ψ ) = 0 , g ′ ( ∞ ; ψ ) = 0 , θ ( ∞ ; ψ ) = 0 , φ ( ∞ ; ψ ) = 0 .

(40) N f [ f ( ζ ; ψ ) , g ( ζ ; ψ ) ] = μ hnf / μ f μ hnf / μ f ∂ 3 f ( ζ ; ψ ) ∂ ζ 3 + ( f ( ζ ; ψ ) + g ( ζ ; ψ ) ) ∂ 2 f ( ζ ; ψ ) ∂ ζ 2 + ∂ f ( ζ ; ψ ) ∂ ζ 2 − M σ hnf / σ f μ hnf / μ f ∂ f ( ζ ; ψ ) ∂ ζ − μ hnf / μ f μ hnf / μ f K ∂ f ( ζ ; ψ ) ∂ ζ = 0 ,

(41) N g [ g ( ζ ; ψ ) , f ( ζ ; ψ ) ] = μ hnf / μ f μ hnf / μ f ∂ 3 g ( ζ ; ψ ) ∂ ζ 3 + ( g ( ζ ; ψ ) + f ( ζ ; ψ ) ) ∂ 2 g ( ζ ; ψ ) ∂ ζ 2 + ∂ g ( ζ ; ψ ) ∂ ζ 2 − σ hnf / σ f μ hnf / μ f M ∂ g ( ζ ; ψ ) ∂ ζ − μ hnf / μ f μ hnf / μ f K ∂ g ( ζ ; ψ ) ∂ ζ = 0 ,

(42) N θ [ θ ( ζ ; ψ ) , f ( ζ ; ψ ) , g ( ζ ; ψ ) , φ ( ζ ; ψ ) ] = k hnf / k f ( ρ C p ) hnf / ( ρ C p ) f + Rd ( ρ C p ) hnf / ( ρ C p ) μ f ∂ 2 θ ( ζ ; ψ ) ∂ ζ 2 + Pr ( f ( ζ ; ψ ) + g ( ζ ; ψ ) ) ∂ θ ( ζ ; ψ ) ∂ ζ + Pr Ec μ hnf / μ f ( ρ C p ) hnf / ( ρ C p ) f ∂ 2 f ( ζ ; ψ ) ∂ ζ 2 2 + ∂ 2 g ( ζ ; ψ ) ∂ ζ 2 2 + Q ( ρ C p ) hnf / ( ρ C p ) f θ ( ζ ; ψ ) + Pr ( ρ C p ) hnf / ( ρ C p ) f × Nb ∂ θ ( ζ ; ψ ) ∂ ζ ∂ φ ( ζ ; ψ ) ∂ ζ + Nt ∂ θ ( ζ ; ψ ) ∂ ζ 2 = 0 ,

(43) N φ [ φ ( ζ ; ψ ) , f ( ζ ; ψ ) , g ( ζ ; ψ ) , θ ( ζ ; ψ ) ] = ∂ 2 φ ( ζ ; ψ ) ∂ ζ 2 + Sc f ( ζ ; ψ ) + g ( ζ ; ψ ) ∂ φ ( ζ ; ψ ) ∂ ζ + Nt Nb ∂ 2 θ ( ζ ; ψ ) ∂ ζ 2 − Sc K r ( 1 + β θ ( ζ ; ψ ) ) n exp − E 1 + β θ ( ζ ; ψ ) φ ( ζ ; ψ ) = 0 .

For ψ = 0 and ψ = 1 , we have

(44) f ( ζ ; 0 ) = f 0 ( ζ ) , f ( ζ ; 1 ) = f ( ζ ) . g ( ζ ; 0 ) = g 0 ( ζ ) , g ( ζ ; 1 ) = g ( ζ ) . θ ( ζ ; 0 ) = θ 0 ( ζ ) , θ ( ζ ; 1 ) = θ ( ζ ) . φ ( ζ ; 0 ) = φ 0 ( ζ ) , φ ( ζ ; 1 ) = φ ( ζ ) .

By using the Taylor series expansion, we obtain

(45) f ( ζ ; ψ ) = f 0 ( ζ ) + ∑ m = 1 ∞ f m ( ζ ) ψ m ; f m ( ζ ) = 1 m ! ∂ m f ( ζ ; ψ ) ∂ ζ m ψ = 0 . g ( ζ ; ψ ) = g 0 ( ζ ) + ∑ m = 1 ∞ g m ( ζ ) ψ m ; g m ( ζ ) = 1 m ! ∂ m g ( ζ ; ψ ) ∂ ζ m ψ = 0 . θ ( ζ ; ψ ) = θ 0 ( ζ ) + ∑ m = 1 ∞ θ m ( ζ ) ψ m ; θ m ( ζ ) = 1 m ! ∂ m θ ( ζ ; ψ ) ∂ ζ m ψ = 0 . φ ( ζ ; ψ ) = φ 0 ( ζ ) + ∑ m = 1 ∞ φ m ( ζ ) ψ m ; φ m ( ζ ) = 1 m ! ∂ m φ ( ζ ; ψ ) ∂ ζ m ψ = 0 .

The mth-order deformation problem is defined as:

(46) L f [ f m ( ζ ) − p m f m − 1 ( ζ ) ] = ℏ f R m f ( ζ ) , L g [ g m ( ζ ) − p m g m − 1 ( ζ ) ] = ℏ g R m g ( ζ ) , L θ [ θ m ( ζ ) − p m θ m − 1 ( ζ ) ] = ℏ θ R m θ ( ζ ) , L φ [ φ m ( ζ ) − p m φ m − 1 ( ζ ) ] = ℏ φ R m φ ( ζ ) ,

(47) f m ( 0 ) = 0 , f m ′ ( 0 ) = 1 + α 1 f m ′ ′ ( 0 ) , g m ′ ( 0 ) = δ + α 2 g m ′ ′ ( 0 ) , g m ( 0 ) = 0 , θ m ( 0 ) = 1 + a 3 θ m ′ ( 0 ) , φ m ( 0 ) = 1 + a 4 φ m ′ ( 0 ) , f m ′ ( ∞ ) = 0 , g m ′ ( ∞ ) = 0 , θ m ( ∞ ) = 0 , φ m ( ∞ ) = 0 . ,

(48) R m f ( ζ ) = μ hnf / μ f ρ hnf / ρ f f m ′ ′ ′ ( ζ ) + ∑ n = 1 m − 1 f m − 1 − n ( ζ ) f ″ n ( ζ ) + ∑ n = 1 m − 1 g m − 1 − n ( ζ ) f n ′ ′ ( ζ ) + ∑ n = 1 m − 1 f m − 1 − n ′ ( ζ ) f n ′ ( ζ ) − M σ hnf / σ f ρ hnf / ρ f f m ′ ( ζ ) − μ hnf / μ f ρ hnf / ρ f K f ′ m ( ζ ) ,

(49) R m g ( ζ ) = μ hnf / μ f ρ hnf / ρ f g m ′ ′ ′ ( ζ ) + ∑ n = 1 m − 1 g m − 1 − n ( ζ ) g n ′ ′ ( ζ ) + ∑ n = 1 m − 1 f m − 1 − n ( ζ ) g n ′ ′ ( ζ ) + ∑ n = 1 m − 1 g m − 1 − n ′ ( ζ ) g n ′ ( ζ ) − M σ hnf / σ f ρ hnf / ρ f g m ′ ( ζ ) − μ hnf / μ f ρ hnf / ρ f K g m ′ ( ζ ) ,

(50) R m θ ( ζ ) = k hnf / k f ( ρ C p ) hnf / ( ρ C p ) f + Rd ( ρ C p ) hnf / ( ρ C p ) μ f θ ″ m ( ζ ) + Pr ∑ n = 1 m − 1 f m − 1 − n ( ζ ) θ ′ n ( ζ ) + ∑ n = 1 m − 1 g m − 1 − n ( ζ ) θ ′ n ( ζ ) + Pr Q ( ρ C p ) hnf / ( ρ C p ) f θ m ( ζ ) + Pr Ec μ hnf / μ f ( ρ C p ) hnf / ( ρ C p ) f ∑ n = 1 m − 1 f ″ m − 1 − n ( ζ ) f ″ n ( ζ ) + ∑ n = 1 m − 1 g ″ m − 1 − n ( ζ ) g ″ n ( ζ ) + Pr ( ρ C p ) hnf / ( ρ C p ) f Nb ∑ n = 1 m − 1 θ ′ m − 1 − n ( ζ ) φ ′ n ( ζ ) + Nt ∑ n = 1 m − 1 θ ′ m − 1 − n ( ζ ) θ ′ n ( ζ ) ,

(51) R m φ ( ζ ) = φ ″ m ( ζ ) + Sc ∑ n = 1 m − 1 f m − 1 − n ( ζ ) φ ′ n ( ζ ) + ∑ n = 1 m − 1 g m − 1 − n ( ζ ) θ ′ n ( ζ ) + Nt Nb θ ″ m ( ζ ) − Sc K r ( 1 + β θ m ( ζ ) ) n exp − E 1 + β θ m ( ζ ) φ m ( ζ ) ,

where

(52) p m = 0 , m ≤ 1 1 , m > 1 .

4 Convergence of HAM

Figure 2(a) and (b) depicts the convergence areas of different flow profiles of the modeled equations. HAM is a powerful method which can solve both linear and nonlinear differential equations. Also, HAM ensures the convergence area of the flow problem which validates the correctness of method. For the modeled equations, the convergence areas of the velocities, thermal and concentration panels are depicted in Figure 2(a) and (b). The convergence areas of the x-direction velocity is − 0.8 ≤ ℏ f ≤ 0.5 , y-direction velocity is − 0.6 ≤ ℏ g ≤ 0.35 , temperature is − 0.6 ≤ ℏ θ ≤ 0.4 , and concentration is − 0.5 ≤ ℏ φ ≤ 0.4 .

$Figure 2 (a) ℏ − \hslash - curves for f ″ ( 0 ) f^{\prime\prime} (0) and g ″ ( 0 ) g^{\prime\prime} (0) . (b) ℏ − \hslash - curves for θ ′ ( 0 ) \theta ^{\prime} (0) and φ ′ ( 0 ) \varphi ^{\prime} (0) .$

Figure 2

(a) ℏ − curves for f ″ ( 0 ) and g ″ ( 0 ) . (b) ℏ − curves for θ ′ ( 0 ) and φ ′ ( 0 ) .

5 Analysis and discussion of results

This paragraph shows the results and discussion of the present model. The flow fluid is considered on a stretching sheet. Figures 3–20 are displayed to see the influence effects of various factors on diverse flow distributions. The values of various factors are selected as ϕ 1 = ϕ 2 = 0.04 , M = 0.5 , Pr = 6.2 , Ec = 0.1 , K r = 0.5 , Q = 1.0 , Nt = 0.1 , Nb = 0.1 , Sc = 1.0 , β = 1.1 , E = 0.3 , K = 0.2 , α 1 = α 2 = α 3 = α 4 = 0.5 , and δ = 0.2 .

$Figure 3 Variation in velocity profile f ′ ( ζ ) f^{\prime} (\zeta ) via magnetic factor M M .$

Figure 3

Variation in velocity profile f ′ ( ζ ) via magnetic factor M .

$Figure 4 Variation in velocity profile g ′ ( ζ ) g^{\prime} (\zeta ) via magnetic factor M M .$

Figure 4

Variation in velocity profile g ′ ( ζ ) via magnetic factor M .

$Figure 5 Variation in velocity profile f ′ ( ζ ) f^{\prime} (\zeta ) via primary velocity slip factor α 1 {\alpha }_{1} .$

Figure 5

Variation in velocity profile f ′ ( ζ ) via primary velocity slip factor α 1 .

$Figure 6 Variation in velocity profile g ′ ( ζ ) g^{\prime} (\zeta ) via secondary velocity slip factor α 2 {\alpha }_{2} .$

Figure 6

Variation in velocity profile g ′ ( ζ ) via secondary velocity slip factor α 2 .

$Figure 7 Impact of variation in temperature profile θ ( ζ ) \theta (\zeta ) via thermal slip factor α 3 {\alpha }_{3} .$

Figure 7

Impact of variation in temperature profile θ ( ζ ) via thermal slip factor α 3 .

$Figure 8 Variation in concentration profile φ ( ζ ) \varphi (\zeta ) via concentration slip factor α 4 {\alpha }_{4} .$

Figure 8

Variation in concentration profile φ ( ζ ) via concentration slip factor α 4 .

$Figure 9 Variation in velocity profile f ′ ( ζ ) f^{\prime} (\zeta ) via porosity factor K K .$

Figure 9

Variation in velocity profile f ′ ( ζ ) via porosity factor K .

$Figure 10 Variation in velocity profile g ′ ( ζ ) g^{\prime} (\zeta ) via porosity factor K K .$

Figure 10

Variation in velocity profile g ′ ( ζ ) via porosity factor K .

$Figure 11 Variation in temperature profile θ ( ζ ) \theta (\zeta ) via Eckert number Ec \text{Ec} .$

Figure 11

Variation in temperature profile θ ( ζ ) via Eckert number Ec .

$Figure 12 Variation in temperature profile θ ( ζ ) \theta (\zeta ) via Brownian motion factor Nb \text{Nb} .$

Figure 12

Variation in temperature profile θ ( ζ ) via Brownian motion factor Nb .

$Figure 13 Variation in concentration φ ( ζ ) \varphi (\zeta ) via Brownian motion factor Nb \text{Nb} .$

Figure 13

Variation in concentration φ ( ζ ) via Brownian motion factor Nb .

$Figure 14 Variation in temperature profile θ ( ζ ) \theta (\zeta ) via thermophoresis factor Nt \text{Nt} .$

Figure 14

Variation in temperature profile θ ( ζ ) via thermophoresis factor Nt .

$Figure 15 Variation in concentration profile φ ( ζ ) \varphi (\zeta ) via thermophoresis factor Nt \text{Nt} .$

Figure 15

Variation in concentration profile φ ( ζ ) via thermophoresis factor Nt .

$Figure 16 Variation in temperature profile θ ( ζ ) \theta (\zeta ) via heat source factor Q Q .$

Figure 16

Variation in temperature profile θ ( ζ ) via heat source factor Q .

$Figure 17 Variation in temperature profile θ ( ζ ) \theta (\zeta ) via thermal radiation factor Rd \text{Rd} .$

Figure 17

Variation in temperature profile θ ( ζ ) via thermal radiation factor Rd .

$Figure 18 Variation in concentration profile φ ( ζ ) \varphi (\zeta ) via Schmidt number Sc \text{Sc} .$

Figure 18

Variation in concentration profile φ ( ζ ) via Schmidt number Sc .

$Figure 19 Variation in concentration profile φ ( ζ ) \varphi (\zeta ) via activation energy factor E E .$

Figure 19

Variation in concentration profile φ ( ζ ) via activation energy factor E .

$Figure 20 Variation in concentration profile φ ( ζ ) \varphi (\zeta ) via chemical reaction factor Kr \text{Kr} .$

Figure 20

Variation in concentration profile φ ( ζ ) via chemical reaction factor Kr .

5.1 Analysis of results

5.1.1 Discussion of results

Figures 3 and 4 demonstrate the impression of M on velocities f ′ ( ζ ) and g ′ ( ζ ) . Both profiles are reducing functions of magnetic factor. Actually, the increase in M generates Lorentz force which counterattacks the fluid’s motion. This opposing force is due to the higher skin friction at sheet’s surface. Therefore, the higher magnetic factor lessens the velocities in all directions. So, both f ′ ( ζ ) and g ′ ( ζ ) are the reducing functions of M . The results of the primary velocity slip factor α 1 and the secondary velocity slip factor α 2 on both f ′ ( ζ ) and g ′ ( ζ ) are described in Figures 5 and 6. It is detected that both f ′ ( ζ ) and g ′ ( ζ ) escalate with the increase in the primary velocity slip factor α 1 and the secondary velocity slip factor α 2 . Figures 7 and 8 explain the outcomes of the thermal slip factor σ 3 and mass slip factor α 4 , respectively, on θ ( ζ ) and φ ( ζ ) . It is observed that under the increment of thermal slip factor σ 3 and mass slip factor α 4 , both temperature profile and concentration profile have decreased, respectively. Figures 9 and 10 describe the fluctuation of f ′ ( ζ ) and g ′ ( ζ ) for the increase in the porosity parameter K . Figures 9 and 10 specify that f ′ ( ζ ) and g ′ ( ζ ) reduces due to the increase in the porosity parameter K . The relationship between Ec and θ ( ζ ) is exposed in Figure 11. It is shown in this figure that θ ( ζ ) is observed for increase in Ec. An increase in Eckert number amplifies the kinetic energy of the nanofluid, which is directly proportional to the temperature, leading to an elevation in the thermal properties of the nanofluid. Physically, an increase in Ec causes conversion of mechanical to thermal energy that leads to an escalation in thermal distribution. In Figure 12, the magnitude of θ ( ζ ) against greater values of Nb is discussed. It describes that with the increase in Brownian motion parameter Nb , hybrid nanofluid temperature profile θ ( ζ ) intensifies. Additionally, the random movement of molecules increases the kinetic energy of the system, leading to a strengthening of the thermal distribution. The variation in Nb on the concentration profile φ ( ζ ) is presented in Figure 13. The increase in Nb leads to the reduction in the hybrid nanofluid concentration distribution φ ( ζ ) . In physical point of view, the solutal distribution diminishes when the mass diffusion coefficient decreases and viscosity deteriorates with an increase in Nb. The alteration in θ ( ζ ) of fluid with respect to the increase in Nt is described in Figure 14. The temperature profile is amplifying for increase in thermophoresis factor Nt . Influence of thermophoresis factor Nt on the concentration profile φ ( ζ ) of fluid is displayed in Figure 15. For greater values of Nt , the value of φ ( ζ ) is lower. The features of the heat generation factor Q on θ ( ζ ) of hybrid nanofluid are examined in Figure 18. It is found that an increase in Q increased the hybrid nanofluid thermal distribution. The physical reason for increase in θ ( ζ ) is the increase in Q , which produces more energy. This justifies that temperature is higher for the greater values of Q . The fluctuation in hybrid nanofluid temperature profile θ ( ζ ) due to an increase in Rd is deliberated in Figure 17. Higher values of Rd amplified θ ( ζ ) . Actually, the thermal radiation is the measurement of the electromagnetic radiation emitted by a fluid element. This radiation is then transformed into thermal radiation that increases θ ( ζ ) . Influence of Sc on φ ( ζ ) is visualized in Figure 18. It is detected that φ ( ζ ) decreases due to an increase in Sc . Actually, Sc relates momentum diffusion to mass diffusion of fluid. When Sc increases, it indicates that the mass diffusivity is decreasing relative to momentum diffusivity; therefore, increase in Sc leads to a weakening in φ ( ζ ) due to the reduced mass diffusivity of the nanoparticles. The impression of activation energy factor E on the hybrid nanofluid concentration distribution φ ( ζ ) is explained in Figure 19. Increasing estimates of E amplifies φ ( ζ ) of the hybrid nanofluid. As E increases, the modified Arrhenius function diminishes in a physical sense. Therefore, the hybrid nanofluid concentration profile φ ( ζ ) shows an increasing role for the increase in E . Figure 20 displays the impacts of Kr on φ ( ζ ) . The declining role of φ ( ζ ) is examined for increase in Kr . In Table 2 and Figure 21, the current data are compared with that of Ramzan et al. [47]. The present results of the existing work show fine promise with the published data. Table 3 depicts the variation in skin friction coefficient μ hnf μ f f ″ ( 0 ) in x-direction, and skin friction coefficient μ hnf μ f g ″ ( 0 ) in y-direction for an increase in M , and porosity factor K . Both μ hnf μ f f ″ ( 0 ) and μ hnf μ f g ″ ( 0 ) are amplified for increasing values of M and K . Similarly, the physical aspects of radiation parameter Rd , Eckert number Ec , Brownian factor Nb , and thermophoresis factor Nt on the heat transport rate are deliberated in Table 4. It is found that the Improvement in radiation parameter Rd , Ec , Nb , and Nt led to improve the heat transport rate mechanism.

Table 2

Comparison of current data for −g″ (0) with work of Ramzan et al. [47]

δ	‒ g ″ ( 0 )
δ	Ramzan et al. [47]	Present results
0.0	0.0	0.0000000
0.1	0.073015	0.0730169
0.2	0.158223	0.1582240
0.3	0.254345	0.2543469
0.4	0.360590	0.3605902
0.5	0.476291	0.4762920

Figure 21

Comparison of current data for −g″ (0) with work of Ramzan et al. [47].

Table 3

Impacts of M and K on μ hnf μ f f ″ ( 0 ) and μ hnf μ f g ″ ( 0 )

M	K	μ hnf μ f f ″ ( 0 )	μ hnf μ f g ″ ( 0 )
0.4		1.578645	1.280633
0.6		1.653848	1.306885
0.8		1.726806	1.325946
	0.1	0.480742	0.478532
	0.2	0.508967	0.480652
	0.3	0.525809	0.497963

Table 4

Impacts of different factors on ‒ k hnf k f + Rd θ ′ ( 0 )

Rd	Ec	Nb	Nt	Q	‒ k hnf k f + Rd θ ′ ( 0 )
0.1					1.568757
0.2					1.608754
0.3					1.636709
	0.1				1.113578
	0.2				1.909757
	0.3				1.287537
		0.1			1.579986
		0.2			1.608864
		0.3			1.640875
			0.1		1.808646
			0.2		1.824575
			0.3		1.846906
				0.1	1.253689
				0.2	1.298764
				0.3	1.357896

6 Conclusion

The present problem determines the heat and mass transport characteristics across the stretching surface through the porous media. The velocity, thermal and concentration slip conditions are adopted in this analysis. HAM technique is employed for the simulation of highly nonlinear ODEs. In graphical form, the computation of how various physical parameters impact diverse flow profiles is presented. In this work, the significant key findings are examined and discussed, which include

The primary velocity profile declines due to magnetic field parameter, primary slip parameter, and porosity parameter.
Increment in magnetic field parameter, secondary velocity slip parameter, and porosity parameter led to amplifying the secondary velocity profile.
Thermal profile is weakened for upsurge in thermal slip factor but amplification in Eckert number, Brownian factor, heat generation factor, and thermal radiation factor amplified the thermal profile.
It is examined that the concentration distribution is greater for thermophoresis factor, and activation energy parameter. Further, reduction in concentration profile is observed for a solutal slip, Brownian, chemical reaction factors, and Schmidt number.
Heat transport rate shows increment behavior due to the increment in radiation parameter, Eckert number, Brownian motion, and thermophoresis factors.
With the enhancement of magnetic and porosity factors, the coefficients of skin friction in both x- and y-directions are amplified.
Further comparison of the existing work is also discussed and shows good agreements with the published results.
The present model can be modified by taking other types of fluid and coordinate system. Different types of physical effects and methodologies can be used to solve proposed model.
It has been observed that velocity in x-direction converges in the region − 0.8 ≤ ℏ f ≤ 0.5 , in y-direction velocity is convergent in the zone − 0.6 ≤ ℏ g ≤ 0.35 , while temperature converges in the region − 0.6 ≤ ℏ θ ≤ 0.4 , and concentration converges in the region − 0.5 ≤ ℏ φ ≤ 0.4 .
In future, the impacts of variable porous space will be incorporated in the proposed mathematical model to observe its impacts on velocity distribution and thermal transportation for hybrid nanofluid flow.

Funding information: Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R443), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, the Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. 6036).
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.
Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2023-12-22

Revised: 2024-02-02

Accepted: 2024-02-10

Published Online: 2024-04-03

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

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0007

Keywords for this article

hybrid nanofluid flow; MHD; porous media; thermal radiation; heat source; stretching surface; slip conditions

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