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Numerical simulation of non-Newtonian hybrid nanofluid flow subject to a heterogeneous/homogeneous chemical reaction over a Riga surface

  • Muhammad Bilal EMAIL logo , Muhammad Bilal Riaz , Sanaa Ahmed Bajri , Adil Jhangeer EMAIL logo , Hamiden Abd El-Wahed Khalifa and Hijaz Ahmad
Published/Copyright: July 18, 2025
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Nanotechnology Reviews
From the journal Nanotechnology Reviews Volume 14 Issue 1

Abstract

Non-Newtonian hybrid nanofluids (NNHNFs) have a wide range of applications. Examples include optimized conveyance of heat, cooling, and maintenance in mechanically operated drug delivery, improved efficacy in microfluidic gadgets, advanced substance manufacturing, and energy-related uses such as energy storage and solar collector systems. For this purpose, the flow of NNHNFs past a porous Riga surface is examined. Two different types of NNHNFs known as SG (second-grade) and Walter’s B (WB) fluids have been considered. Molybdenum dioxide (MoS2) and single-walled carbon nanotube (SWCNT)-nanoparticles (NPs) are used in the base fluid sodium alginate (SA; C6H9NaO7) to prepare the hybrid nanofluid (HNF). The NNHNF flow is designed in the form of a non-linear system of partial differential equations, which are simplified to the dimensionless form of ordinary differential equations by using similarity transformation and then numerically handled through the BVP4C package. The numerical outcomes of the proposed model are compared with the published literature for validity purpose. The present results reveal higher similarity to the existing study. A stability analysis has also been performed to see which solution is stable in practice. From the graphical outcomes, it can be determined that the fluid temperature declines with the effect of MoS2 and SWCNT-NPs. Furthermore, with the increase of MoS2 and SWCNTs NPs in the SA-based HNF, the energy transfer rate enhances from 3.79 to 8.25% (in the case of SG hybrid nanoliquid), whereas in the case of WB HNF the energy transfer rate enhances from 3.88 to 9.86%.

Keywords: stability analysis; heterogeneous/homogeneous reaction; Walter B fluid; nanofluidics, efficiency; hybrid nanofluid; riga sheet

1 Introduction

The Riga surface is composed of an array of electrodes and fixed magnets aligned in a spanwise direction and affixed to a flat surface. This specific arrangement results in a decrease as one moves away from the plate in a direction perpendicular to its surface [1]. Nayak et al. [2] researched the characteristics of a Riga plate, considering various influencing factors. The investigation demonstrated that the boundary slip accelerated the fluid motion. Simultaneously, an elevated Weissenberg parameter was found to decrease. Shafiq et al. [3] examined the features near a permeable Riga surface. The analysis included the consideration of heat and mass phenomena. Eid [4] conducted a 3D flow study with copper nanoparticles (NPs) flowing through the Riga implantation plate. Prasannakumara et al. [5] explored the attributes of a hydromagnetic within a permeable medium. The analysis considered the wall boundary experiencing both linear deformations and featuring a quadratic surface temperature distribution. Allehiany et al. [6] evaluated the steady radiative fluid flow with the upshot of Lorentz force. Shah et al. [7] examined the hydromagnetic 2D flow across a flexible Riga wall situated within a absorbent medium. Khashiʼie et al. [8] illustrated the application of transpiration to attain stable solutions in counterflows, particularly in cases involving shrinking or opposing buoyancy. Alqahtani et al. [9] and Hua et al. [10] estimated the effects of a fluid constrained by a Riga surface. The research examined the flow characteristics above a Riga panel. Asogwa et al. [11] systematically explored the motion across the Riga surface, with a particular focus on time-dependent variations in concentration and temperature. Alhowaity et al. [12] analyzed a vertically convected Riga plate. It is worth noting that a Riga plate, in this context, refers to a surface-mounted electromagnetic actuator, featuring an array of alternating electrodes and fixed magnetization aligned spanwise. Nadeem et al. [13] assessed the characteristics of 2D nanofluid flow subject to radiation effects. The study focused on the 2D time-varying flow occurring over a stretchable/permeable vertical Riga plate. For recent and novel literature on the subject, readers can refer to Hamad et al. [14], Bilal et al. [15, 16], and Alqahtani et al. [17].

Currently, researchers have focused on enhancing the heat conduction properties of cooling liquids by incorporating suspensions of solid particles with nanometer dimensions into base fluids. Hybrid nanofluids (HNFs) are created by dispersing a synthesized nanocomposite material consisting of modified copper NPs [18]. Li et al. [19] deliberated the thermal characteristics of a Casson fluid flow. The nanofluid contains iron oxide nanocomposites, and it is presumed that the channel is non-uniform, intricate, undulating, and curved. Kavya et al. [20] investigated the dynamic variations in fluid momentum properties of an incompressible, laminar, and 2D pseudoplastic Williamson nanofluid. For this investigation, an HNF is employed consisting of a combination of Cu and MoS4 nanocomposite materials suspended in water. Khan et al. [21] estimated the flow features of HNFs, specifically those encompassing carbon nanotubes (CNTs) and ferrous oxide water between two parallel surfaces under the upshot of changing magnetization. Plant et al. [22] inspected the nanoliquid flow using a combination of experimental and mathematical approaches. The nanofluid, created by incorporating copper oxide nanostructures into a nanostructured aluminum oxide base, was tested within a permeable open-cell foam metal flow system. The permeable surface, composed of aluminum, featured a permeability of 0 . 92 × 1 0 − 9 m 2 . The investigations encompassed variable heat flux conditions, employing a blended nanofluid containing a 0.2% mass aqueous mixture of CuO NPs of size 29  ±  12 nm. Cimbala et al. [23] synthesized a hybrid nanoliquid by incorporating both ferric oxide and iron oxide NPs into a transformer lubricant. Employing dielectric relaxation spectroscopy, the study systematically investigated the insulating reaction across varying concentrations of C 60 − Fe 3 O 4 NPs within the frequency range of 0.10 MHz to 10 kHz. The findings highlighted the dominant influence of magnetic NPs. Some recent related results are reported by Adnan et al. [24], Alharbi et al. [25], and Ahmad et al. [26].

A heat source is a localized component within a fluid that discharges or produces heat energy. Since this heat source eliminates thermal energy, it can cause variations in pressure, temperature, and fluid circulation inside the system. Heat sources are commonly found in a variety of engineering uses, such as cooling systems, gasoline and diesel engines, and manufacturing operations. It is designed to assess these fluids according to power generation and element size, such as thermal exchangers and generators [27]. Mebarek-Oudina et al. [28] utilized the upshot of hydro-magnetic on the magnetized nanoliquid flow past a cylindrical tube to improve heat propagation. Farooq et al. [29] studied the phenomenon of bio-convection across a Carreau HNF flow subject to numerous thermal impacts. The flow was induced over a shrinking cylinder. Chamkha and Rashad [30] numerically analyzed the nanoliquid flow comprising solid NPs past an absorbent medium. Chamkha [31] assessed the fully developed non-Darcy mixed convective hydromagnetic fluid flow with heat source across a porous medium.

From the detailed overview of the literature, it can be determined that, so far, no one has reported non-Newtonian hybrid nanofluid (NNHNF) flow over a Riga sheet consisting of MoS2, single-walled CNTs (SWCNTs)-NPs, and SA (C6H9NaO7). For this purpose, the NNHNF flow across a porous surface of the Riga sheet is examined. Two dissimilar types of NNHNF have been assessed. MoS2 and SWCNT-NPs are used in the base fluid C6H9NaO7 to prepare the HNF. The NNHNF flow is designed in the form of non-linear partial differential equations (PDEs), which are simplified to the dimensionless form of ordinary differential equations (ODEs) by manipulation of similarity transformation and then numerically handled through the BVP4C package. The results of the proposed model are compared with the published literature for validity purpose.

2 Mathematical formulation

The steady and incompressible 2D NNHNF flow across the Riga surface is deliberated. The sheet stretches with velocity u w = a x , along the x-axis, where a is a constant, as exhibited in Figure 1(a) and (b). T ∞ and C ∞ are the ambient temperature and mass of the NNHNF, respectively. The surface of the Riga sheet is supposed to be under the upshot of electro-magnetic force F m = F m = exp π M 0 j 0 8 ρ n f − π c y [37].

Figure 1 Fluid flow graphical outline. (a) Riga sheet and (b) stretching sheet.
Figure 1

Fluid flow graphical outline. (a) Riga sheet and (b) stretching sheet.

A cubic autocatalytic reaction along with A* and B* of first order are as follows:

A ˜ + 2 B ˜ → 3 B ˜ , rate = k 1 C a C b 2 .

The single first-order reaction (isothermal) on the catalyst is signified as:

A ˜ + B ˜ → 3 B ˜ , rate = k 2 C a .

Here, C a and C b describe the volume of chemical species of A ˜ * and B ˜ , respectively, where k 1 and k 2 signify the constant rate. Furthermore, the flow equations are [32,33] as follows:

(1) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(2) u ∂ u ∂ x = μ hnf ρ hnf ∂ 2 u ∂ y 2 + π j 0 x M 0 8 ρ hnf exp − π c y − ν hnf K u − v ∂ u ∂ y ± 1 ρ hnf x υ hnf ∂ 3 u ∂ x ∂ y 2 − ∂ 3 u ∂ y 2 ∂ x + u ∂ 3 u ∂ x ∂ y 2 + v ∂ 3 u ∂ y 3 ,

(3) u ∂ T ∂ x + v ∂ T ∂ y = α hnf ∂ 2 T ∂ y 2 − 1 ( ρ C p ) hnf ∂ q r ∂ y + Q e ⁎ ( ρ C p ) hnf ( T − T ∞ ) exp − a υ f n y ,

(4) u ∂ C a ∂ x + v ∂ C a ∂ y = D A ∂ 2 C a ∂ x 2 + ∂ 2 C a ∂ y 2 − k 1 C a C b 2 ,

(5) u ∂ C b ∂ x + v ∂ C b ∂ y = D B ∂ 2 C b ∂ x 2 + ∂ 2 C b ∂ y 2 − k 1 C a C b 2 .

The boundary conditions (BCs) are as follows:

(6) u = λ u w , v = 0 , T = T w D A ∂ C a ∂ y = − D B ∂ C b ∂ y = k 2 C a at y = 0 u → 0 , ∂ u ∂ y → 0 , C a → C ∞ , C b → 0 , T → T ∞ as y → ∞ .

The following similarity variables are used to simplify equations (1)–(5) and (6) into a non-dimensional system of ODEs:

(7) u = x a f ′ , v = − a v f f , θ = T − T ∞ T w − T ∞ , φ = C − C ∞ C w − C ∞ , h ( η ) = C b C ∞ , g ( η ) = C a C ∞ , ψ = x a v f f , η = a ν f y .

Here, α 1 > 0 signifies the SG fluid and α 1 < 0 signifies the WB fluid, M 0 is the magnetization, Ea is the activation energy, λ is the sheet stretching and shrinking velocity, where λ < 0 for the shrinking case and λ > 0 for the stretching case, D A and D B are the species diffusion factors of A ˜ and B ˜ , respectively, j 0 is the applied electrode current density, K is the absorbent media permeability, and q r is the thermal radiation factor.

The heat radiation is expressed as follows [34]:

(8) q r = − 4 σ 3 k ∂ T 4 ∂ y .

By expanding T 4 , we obtain

T 4 = T ∞ 4 + 4 ( T − T ∞ ) T ∞ 3 + 6 ( T − T ∞ ) 2 T ∞ 2 + … .

Now, by removing higher order terms from equation (8), we obtain:

(9) q r = − 16 σ T ∞ 3 3 k ∂ T ∂ y .

By substituting equation (9) in (3), we obtain:

(10) u ∂ T ∂ x + v ∂ T ∂ y = k hnf ( ρ C p ) hnf + 16 σ T ∞ 3 3 k ∂ 2 T ∂ y 2 .

By incorporating equation (6) in equations (1)–(5), equation (5) becomes:

(11) f ‴ ℘ 1 ℘ 2 ± K 1 ℘ 2 [ 2 f ′ f ‴ − f f ′ ‴ − ( f ″ ) 2 ] + Q h ℘ 2 exp ( − η β 1 ) − λ 1 ℘ 1 ℘ 2 f ′ + f f ″ − ( f ′ ) 2 = 0 ,

(12) k h n f k f + 4 3 Rd 1 Pr ℘ 3 θ ″ + f θ ′ + Q e exp ( − n η ) = 0 ,

(13) h ″ + Sc D f h ′ + Sc D K g h 2 = 0 ,

(14) g ‴ + Sc f g ′ + Sc K g h 2 = 0 .

The BCs are as follows:

(15) f ( η ) = 0 , f ′ ( η ) = λ , θ ( η ) = 1 , h ′ ( η ) = − 1 D K s g ( 0 ) , g ′ ( 0 ) = K s g ( 0 ) at η = 0 f ′ ( η ) = 0 , θ ( η ) = 0 , h ( ∞ ) → 0 , g ( ∞ ) → 1 as η → ∞ .

The diffusion coefficient size are supposed to be the same, thus D A and D B are equal to ( D = 1 ) and g ( η ) + h ( η ) = 1 .

So,

(16) g ″ + Sc f g ′ − K Sc g ( 1 − g ) 2 = 0 .

And,

g ′ ( 0 ) = K s g ( 0 ) and g ( ∞ ) = 1 ,

where ℘ 1 , ℘ 2 and ℘ 3 are expressed as:

℘ 1 = 1 ( 1 − ϕ SWCNT s − ϕ MoS 2 ) 2.5 , ℘ 2 = ϕ SWCNTs ρ SWCNTs ρ bf + ϕ MoS 2 ρ MoS 2 ρ bf + ( 1 − ϕ SWCNTs − ϕ MoS 2 ) ,

℘ 3 = ϕ SWCNTs ( ρ C p ) SWCNTs ( ρ C p ) bf + ϕ MoS 2 ( ρ C p ) MoS 2 ( ρ C p ) bf + ( 1 − ϕ SWCNTs − ϕ MoS 2 ) .

In Table 1, C p is the capacity of heat, ρ is the density, σ is the electrical conductivity, and D f is the mass diffusivity. The thermochemical properties of the HNF ( ϕ 1 = ϕ SWCNTs , ϕ 2 = ϕ MoS 2 ) are the following [34]:

Table 1

Thermophysical properties of HNF [34]

Properties C p ( J kg − 1 K − 1 ) ρ ( kg m − 3 ) k ( kg ms − 3 K − 1 ) σ ( Ωm ) − 1
SA ( C 6 H 9 NaO 7 ) 4,175 989 0.6376 2.6 × 10 − 4
SWCNTs 425 2,600 6,600 5.96 × 10 7
MoS2 397.21 5.06 × 10 3 904.4 2.04 × 10 4

Viscosity ( μ ) :

μ hnf μ bf = 1 ( 1 − ϕ SWCNTs − ϕ MoS 2 ) 2.5 .

Density ( ρ ) :

ρ hnf ρ bf = ϕ SWCNTs ρ SWCNTs ρ bf + ϕ MoS 2 ρ MoS 2 ρ bf + ( 1 − ϕ SWCNTs − ϕ MoS 2 ) .

Thermal capacity ( ρ C p ) :

( ρ C p ) hnf ( ρ C p ) bf = ϕ SWCNTs ( ρ C p ) SWCNTs ( ρ C p ) bf + ϕ MoS 2 ( ρ C p ) MoS 2 ( ρ C p ) bf + ( 1 − ϕ SWCNTs − ϕ MoS 2 ) .

Thermal expansion ( ρ β T ) :

( ρ β T ) hnf ( ρ β T ) bf = ϕ SWCNTs ( ρ β T ) SWCNTs ( ρ β T ) bf + ϕ MoS 2 ( ρ β T ) MoS 2 ( ρ β T ) bf + ( 1 − ϕ SWCNTs − ϕ MoS 2 ) .

Thermal conductivity ( k ) :

k hnf k bf = ϕ SWCNTs k SWCNTs + ϕ MoS 2 k MoS 2 ϕ SWCNTs + ϕ MoS 2 + 2 k bf + 2 ( ϕ SWCNTs k SWCNTs + ϕ MoS 2 k MoS 2 ) − 2 ( ϕ SWCNTs + ϕ MoS 2 ) k bf ϕ SWCNTs k SWCNTs + ϕ MoS 2 k MoS 2 ϕ SWCNTs + ϕ MoS 2 + 2 k bf − 2 ( k SWCNTs ϕ SWCNTs + k MoS 2 ϕ MoS 2 ) + 2 ( ϕ SWCNTs + ϕ MoS 2 ) k bf .

Electrical conductivity ( σ ) :

σ hnf σ bf = ϕ SWCNTs σ SWCNTs + σ MoS 2 ϕ MoS 2 ϕ MoS 2 + ϕ SWCNTs + 2 σ bf − 2 ( ϕ SWCNTs + ϕ MoS 2 ) σ bf + 2 ( σ SWCNTs ϕ SWCNTs + ( ϕ MoS 2 ) σ MoS 2 ) ϕ SWCNTs σ SWCNTs + ϕ MoS 2 σ MoS 2 ϕ SWCNTs + ϕ MoS 2 + 2 σ bf + σ bf ( ϕ SWCNTs + ϕ MoS 2 ) − ( σ SWCNTs ϕ SWCNTs + ( ϕ MoS 2 ) σ MoS 2 ) .

Furthermore, the dimensionless numbers and parameters in equations (10)–(12) are itemized in Table 2.

Table 2

Non-dimensional physical quantities and their symbols

Parameters Symbols Expression
Viscoelastic factor K 1 K 1 = α 1 a ρ f ν f
WB fluid K 1 < 0 —
SG fluid K 1 > 0 —
Rate of chemical reaction R R = k r 2 c
Modified Hartmann number Q h Q h = π j 0 x M 0 8 ρ f a U w
Radiation factor Rd Rd = 4 σ T ∞ 3 k k f
Porosity term λ 1 λ 1 = ν f K a
Activation energy E E = E a κ T ∞
Schmidt number Sc Sc = υ f D f
Heat source/sink factor Q e Q e = Q e ⁎ l a ( ρ C p ) f
Reynolds number Re Re = x 2 a ν f
Magnitude and width of electrodes β 1 β 1 = π 2 ν f c 2 a
Prandtl number Pr Pr = ν f α f
Mass diffusion coefficient δ ⁎ δ ⁎ = D B D A
Homogeneous reaction factor K K = k 1 C 0 2 l U w
Heterogeneous reaction factor K s K s = k s D A v l U w
Schmidt number Sc Sc = ν D A

The engineering interest terms are as follows:

(17) Nu = − x ( k hnf + 16 σ T ∞ 3 / 3 k ) ( T w − T ∞ ) k f ∂ T ∂ y y = 0 , Sh = − x D f ( C w − C ∞ ) D f ∂ C ∂ y y = 0 , Cf = 1 U w 2 ρ f μ hnf ∂ u ∂ y ± α 1 v ∂ 2 u ∂ y 2 + ∂ 2 u ∂ x ∂ y u + 2 ∂ 2 u ∂ x ∂ y y = 0 .

By substituting equation (6) in (17), we obtain:

(18) Nu = − k hnf / k f + 4 3 Rd 1 θ ′ ( 0 ) Re , Sh Re = − φ ′ ( 0 ) , Cf = − 1 Re f ‴ ( 0 ) K 1 f ( 0 ) − 1 ℘ 1 ± 3 ( K 1 ) f ′ ( 0 ) f ″ ( 0 ) .

3 Stability analysis

Knowing that there are several solutions to a physical problem versus a single parameter, it is critical to determine the physical credibility of each solution for velocity and energy equations. The results of stability assessment could be used for engineering purposes. In this regard, we analyze the stability of the dual solutions. Therefore, an additional variable related to the initial value problem, which is suitable for the question: which solution will be derived in practice (physically realizable and stable). So, we revise equations (2), (3), (6), and (7) as follows (Weidman et al. [35] and Merkin [36]):

(19) ∂ u ∂ t + u ∂ u ∂ x = μ hnf ρ hnf ∂ 2 u ∂ y 2 + π j 0 x M 0 8 ρ hnf exp − π c y − ν hnf K u − v ∂ u ∂ y ± 1 ρ hnf x υ hnf ∂ 3 u ∂ x ∂ y 2 − ∂ 3 u ∂ y 2 ∂ x + u ∂ 3 u ∂ x ∂ y 2 + v ∂ 3 u ∂ y 3 ,

(20) ∂ T ∂ t + u ∂ T ∂ x + v ∂ T ∂ y = α hnf ∂ 2 T ∂ y 2 − 1 ( ρ C p ) hnf ∂ q r ∂ y + Q e ⁎ ( ρ C p ) hnf ( T − T ∞ ) exp − a υ f n y .

The following similarity variables are used to simplify equations (1) to (5) and (6) into a non-dimensional system of ODEs:

(21) u = x a f ′ ( η , τ ) , θ ( η , τ ) = T − T ∞ T w − T ∞ , v = − a v f f ( η , τ ) , φ ( η , τ ) = C − C ∞ C w − C ∞ , ψ = x a v f f ( η , τ ) , η = a ν f y , τ = a t .

Substituting equation (22) in equations (19) and (20), we obtain:

(22) 1 ℘ 1 ℘ 2 ∂ 3 f ∂ η 3 ± K 1 ℘ 2 2 ∂ f ∂ η ∂ 3 f ∂ η 3 − ∂ f ∂ η ∂ 4 f ∂ η 4 − ∂ 2 f ∂ η 2 2 + Q h ℘ 2 exp ( − η β 1 ) − λ 1 ℘ 1 ℘ 2 ∂ f ∂ η + ∂ f ∂ η ∂ 2 f ∂ η 2 − ∂ f ∂ η 2 − ∂ 2 f ∂ η ∂ τ = 0 ,

(23) k hnf k f + 4 3 Rd 1 Pr ℘ 3 ∂ 2 θ ∂ η 2 + ∂ f ∂ η ∂ θ ∂ η + Q e exp ( − n η ) − ∂ θ ∂ τ = 0 .

The BCs are as follows:

(24) f ( 0 , τ ) = 0 , θ ( 0 , τ ) = 1 , f ′ ( 0 , τ ) = λ at η = 0 θ ( η , τ ) = 0 , f ′ ( η , τ ) = 0 as η → ∞ .

To check the stability, f ( η ) = f 0 ( η ) & θ ( η ) = θ 0 ( η ) of equation (11) and (12) satisfying the basic model by inserting the following equations (Merrill et al. [37] and Weidman and Sprague [38]):

(25) f ( η , τ ) = f 0 ( η ) + e − γ τ F ( η ) & θ ( η τ ) = θ 0 ( η ) + e − γ τ H ( η ) .

Here, F ( η ) and H ( η ) are very small relative to f 0 ( η ) & θ 0 ( η ) and γ is a small eigenvalue, which we will calculate.

By substituting equation (26) in equations (24) and (25), we obtain:

(26) 1 ℘ 1 ℘ 2 ∂ 3 F ∂ η 3 ± K 1 ℘ 2 2 ∂ F ∂ η ∂ 3 F ∂ η 3 − ∂ F ∂ η ∂ 4 F ∂ η 4 − ∂ 2 F ∂ η 2 2 + Q h ℘ 2 exp ( − η β 1 ) − λ 1 ℘ 1 ℘ 2 ∂ F ∂ η + ∂ F ∂ η ∂ 2 F ∂ η 2 + F f ″ + ( γ − 2 f ′ 0 ) ∂ F ∂ η = 0 ,

(27) k hnf k f + 4 3 Rd 1 Pr ℘ 3 ∂ 2 H ∂ η 2 + ∂ F ∂ η ∂ H ∂ η + Q e exp ( − n η ) + γ H = 0 .

(28) F ( 0 ) = 0 , ∂ F ( 0 ) ∂ η = 0 , H ( 0 ) = 0 , at η = 0 ∂ F ( 0 ) ∂ η → 0 , H ( η ) → 0 , as η → ∞ .

Furthermore, the stability of the steady-state flow solutions can be obtained by fixing τ = 0 .

(29) 1 ℘ 1 ℘ 2 ( F ‴ 0 ) ± K 1 ℘ 2 [ 2 F ′ 0 F ‴ 0 − F ′ 0 F 0 i v − ( F ″ 0 ) 2 ] + Q h ℘ 2 exp ( − η β 1 ) − λ 1 ℘ 1 ℘ 2 F ′ 0 + F ′ 0 F ″ 0 + F 0 F ″ 0 + ( γ − 2 f ′ 0 ) F ′ 0 = 0 ,

(30) k hnf k f + 4 3 Rd 1 Pr ℘ 3 H ″ 0 + F ′ 0 H ′ 0 + Q e exp ( − n η ) + γ H 0 = 0 .

(31) F 0 ( 0 ) = 0 , F ′ 0 = 0 , H 0 ( 0 ) = 0 , at η = 0 F ′ 0 ( η ) → 0 , H 0 ( η ) → 0 , as η → ∞ .

To derive the eigenvalues, we relaxed F ′ 0 ( η ) → 0 , as η → ∞ (see Harris et al. [39]). Furthermore, the current study proved the existence of non-unique solutions in the HNF over a specific range of the shrinking factor, while a unique solution was discovered in the stretching case. The stability analysis revealed that the first solution had the smallest positive eigenvalue, implying that it was stable and feasible.

4 Numerical solution

“BVP4C” is employed to solve the simplified set of ODEs. “BVP” stands for Boundary Value Problem and “4C” usually denotes the four collocation points employed in the shooting method (SM). The SM is a numerical and iterative approach used for resolving the boundary value problems by converting them into IVPs.

The first important phase for the BVP4C is to convert equations (11), (12) and (16) into first order.

The simplified form of ODEs is as follows:

(32) f = $ 1 ( η ) , f ′ = $ 2 ( η ) , f ″ = $ 3 ( η ) , θ = $ 4 ( η ) , θ ′ = $ 5 ( η ) , g = $ 6 ( η ) , g ′ = $ 7 ( η ) .

By introducing equation (16) in (11), (12) and (16), we obtain:

(33) $ ′ 3 ( η ) ℘ 1 ℘ 2 ± K 1 ℘ 2 [ 2 $ 2 ( η ) $ ′ 3 ( η ) − $ 1 ( η ) $ ′ 3 ( η ) − ( $ 3 ( η ) ) 2 ] + Q 1 ℘ 2 exp ( − η β 1 ) − λ 1 ℘ 1 ℘ 2 $ 2 ( η ) + $ 1 ( η ) $ 2 ( η ) − ( $ 2 ( η ) ) 2 = 0 ,

(34) k nf k f + 4 3 Rd 1 1 Pr ℘ 3 $ ′ 5 ( η ) + $ 1 ( η ) $ 5 ( η ) = 0 ,

(35) $ ′ 7 + Sc $ 1 $ 7 − K Sc $ 6 ( 1 − $ 6 ) 2 = 0 .

The BCs are as follows:

(36) $ 1 ( η ) = 0 , $ 2 ( η ) = λ , $ 4 ( η ) = 1 , $ 7 ( η ) = K s $ 6 at η = 0 $ 2 ( η ) = 0 , $ 3 ( η ) = 0 , $ 4 ( η ) = 0 , $ 6 ( η ) = 1 as η → ∞ .

5 Results and discussion

In this section, the outcomes obtained through the BVP4c package are revealed through figures and tables. The solid lines show the results for the hybrid nanoliquid, whereas the dashes show results for the nanoliquid, and the values of viscoelastic factor are assumed to be K 1 = 0.5 and K 1 = − 0.5 , respectively. However, in some figures, the shrinking and stretching phenomenon is observed, in which λ < 0 presents the shrinking case (dashes lines), whereas λ > 0 presents the stretching case (solid lines) of sheet.

Figures 2–4 reveal the impact of the viscoelastic factor K 1 , Q 1 , and the porous surface factor λ 1 on f ′ ( η ) , respectively. Physically, K 1 plays an imperative role in fluid behavior, whether it is an SG or WB fluid. In Figure 2, it can be seen that the SG fluid has a higher fluid velocity than the WB fluid. Figure 3 illustrates the consequence of Q 1 on the fluid flow f ′ ( η ) . Physically, the effect of Q 1 produces the repellent force, which retards f ′ ( η ) . Figure 4 displays the upshot of a porous surface factor on the flow rate. Physically, the mounting values of λ 1 diminish the fluid velocity due to increased surface area and boundary layer formation. Actually, the pores enhance the surface area for the hybrid nanoliquid flow, so when the fluid passes through pores with a greater surface area, it causes friction and hence drops the fluid velocity. Similarly, as the fluid passes through pores, the slow-moving or stagnant fluid produces a boundary layer with the walls of pores, which act as a resistive force for the fluid flow. Figures 5–7 expose the effect of MoS2 and SWCNTs-NPs ( ϕ 1 , ϕ 2 ) , Rd , and Q e on the energy profile θ ( η ) , respectively. Figure 5 reveals the decline with the addition of MoS2 and SWCNTs-NPs ( ϕ 1 , ϕ 2 ) in the base fluid. Physically, the accumulation of Cu and Fe3O4-NPs in SA augments the convective heat transmission of the fluid, which results in lowering fluid temperature and more efficient cooling. Figure 6 displays that θ ( η ) is boosted by the influence of Rd . Thermal radiation is a type of heat propagation in which electro-magnetic waves emanating from heated objects are used. When a fluid is subjected to thermal radiation, the electro-magnetic waves are absorbed by the fluid, causing it to warm up. The fluid absorbed heat radiation and converted it into thermal energy. The heat source boosts the fluid temperature and works as a heating mediator for the fluid flow, as publicized in Figure 7. Figures 8–10 demonstrate the impact of Sc, K t, and K s on the mass profile g ( η ) , respectively. Figure 8 reveals that the intensifying upshot of Sc drops the mass curve. Physically, the molecular rate diffusion is contrarywise related to the Schmidt number. That is why, the mass curve falls with the upshot of Sc1. Figures 9 and 10 show that the influence of K t and K s declines the mass distribution. Homogeneous reactions encompass reactants in a single phase, whereas heterogeneous reactions incorporate reactants from various phases. When these reactions occur concurrently, the mass profile reduces due to conversations within the phases, resulting in a reduction in fluid concentration field. This is commonly observed in chemical reactors, during which reactions across different phases influence the entire concentration allocation. Figure 5(b) to 10(b) show the 3D view of energy and concentration profiles versus the varying impact of the physical parameters.

Figure 2 Viscoelastic factor K 1 {K}_{1} versus velocity profile f ′ ( η ) . f^{\prime} (\eta ).
Figure 2

Viscoelastic factor K 1 versus velocity profile f ′ ( η ) .

Figure 3 Modified Hartmann number Q 1 {Q}_{1} versus velocity profile f ′ ( η ) . f^{\prime} (\eta ).
Figure 3

Modified Hartmann number Q 1 versus velocity profile f ′ ( η ) .

Figure 4 Porous surface factor λ 1 {\lambda }_{1} versus velocity profile f ′ ( η ) . f^{\prime} (\eta ).
Figure 4

Porous surface factor λ 1 versus velocity profile f ′ ( η ) .

Figure 5 Hybrid NPs versus energy profile θ ( η ) . \theta (\eta ).
Figure 5

Hybrid NPs versus energy profile θ ( η ) .

Figure 6 Thermal radiation Rd \text{Rd} versus energy profile θ ( η ) . \theta (\eta ).
Figure 6

Thermal radiation Rd versus energy profile θ ( η ) .

Figure 7 Heat source/sink Q e {Q}_{\text{e}} versus energy profile θ ( η ) . \theta (\eta ).
Figure 7

Heat source/sink Q e versus energy profile θ ( η ) .

Figure 8 Schmidth number Sc versus mass profile g ( η ) g(\eta ) .
Figure 8

Schmidth number Sc versus mass profile g ( η ) .

Figure 9 Homogeneous reaction parameter K t versus mass profile g ( η ) g(\eta ) .
Figure 9

Homogeneous reaction parameter K t versus mass profile g ( η ) .

Figure 10 Heterogeneous reaction parameter K s versus mass profile g ( η ) g(\eta ) .
Figure 10

Heterogeneous reaction parameter K s versus mass profile g ( η ) .

6 Result validation

Table 3 presents the numerical assessment of the present results with the existing literature. It can be perceived that the present study’s results have the best similarity with the published work.

Table 3

Numerical result − θ ( 0 ) validation

Pr Madhukesh et al. [32] Abolbashari et al. [40] Das et al. [41] Present work
0.7 0.80876153 0.80853135 0.80875122 0.808761683
1.0 1.00000000 1.00000000 1.00000000 1.000000000
3.0 1.92357436 1.92368245 1.92357331 1.923574595
7.0 3.07314636 3.07225031 3.07314647 3.073146492
10 3.72067335 3.72067380 3.72067390 3.720673843

Table 4 shows the percentage ( % ) analysis for both types of fluids and also for the nanofluid and HNF versus γ 1 , B i , and ϕ . It can be seen that as compared to SG fluid, the WB fluid has greater ability to upsurge the energy transport rate. Furthermore, the rising quantity of MoS2 and SWCNTs-NPs in the SA (C6H9NaO7)-based nanofluid and an HNF, the energy transfer rate enhances from 2.59 to 5.23% (SG nanofluid), 3.79 to 8.25% (SG HNF), 2.79 to 5.75% (WB nanofluid), and 3.88 to 9.86% (WB HNF), respectively.

Table 4

Percentage ( % ) analysis for SG and WB fluids and also for the nanofluid and HNF versus γ 1 , B i , and ϕ

Parameters Values SG fluid (nanofluid) Cu/SG fluid (MoS2-SWCNTs) WB fluid (nanofluid) WB fluid (MoS2-SWCNTs)
θ ′ ( 0 ) (%) θ ′ ( 0 ) (%) θ ′ ( 0 ) (%) θ ′ ( 0 ) (%)
B i 0.2 0.070375 0.180371 0.580131 0.680230
0.3 0.110449 0.320245 0.580187 0.670286
0.4 0.151819 0.561827 0.621987 0.731886
γ 1 0.4 0.060765 0.170761 0.660955 0.770854
0.7 0.192779 0.392766 0.631013 0.741234
1.1 0.013379 0.443576 0.648985 0.758754
ϕ 1 = ϕ 2 0.02 2.593289 3.793477 2.794168 3.884267
0.03 3.793818 5.993404 4.894718 5.374817
0.04 5.253342 8.253230 5.754232 9.864331

7 Conclusions

We have examined the NNHNF flow across a porous surface of a Riga sheet. MoS2 and SWCNTs-NPs are used in the base fluid (C6H9NaO7) to prepare the HNF. The NNHNF flow is designed in the form of a non-linear system of PDEs, which are numerically handled through the BVP4C package. The numerical results of the proposed model are compared with the published literature for the limiting case. The basic deductions are as follows:

  • The fluid temperature drops with the variation of MoS2 and SWCNT-NPs.

  • The impact of thermal radiation Rd, Biot number B i , and heat source/sink Q e enhances the energy profile θ ( η ) .

  • The SG fluid has higher fluid velocity than the WB fluid against the variation of viscoelastic factor.

  • The effect of Hartmann number Q 1 and porous surface factor λ 1 declines the velocity profile f ′ ( η ) .

  • The energy transfer rate through the WB fluid enhances from 2.79 to 5.75% in the case of nanofluid and from 3.88 to 9.86% in the case of HNF.

  • With the increase of the quantity of MoS2 and SWCNTs-NPs in the SA (C6H9NaO7)-based nanofluid and HNF, the energy transfer rate enhances from 2.59 to 5.23% in the case of SG nanofluid and from 3.79 to 8.25% in the case of SG HNF.

Acknowledgments

This research is funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R527), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This article was produced with the financial support of the European Union under the REFRESH – Research Excellence For Region Sustainability and High-tech Industries project number CZ.10.03.01/00/22_003/0000048 via the Operational Programme Just Transition.

  1. Funding information: This research was funded by the Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R527), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This article was produced with the financial support of the European Union under the REFRESH – Research Excellence For Region Sustainability and High-tech Industries project number CZ.10.03.01/00/22_003/0000048 via the Operational Programme Just Transition.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2024-05-09
Revised: 2024-10-02
Accepted: 2025-06-26
Published Online: 2025-07-18

© 2025 the author(s), published by De Gruyter

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

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https://doi.org/10.1515/ntrev-2024-0133
Keywords for this article
stability analysis; heterogeneous/homogeneous reaction; Walter B fluid; nanofluidics, efficiency; hybrid nanofluid; riga sheet
Creative Commons
BY 4.0

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  2. MHD radiative mixed convective flow of a sodium alginate-based hybrid nanofluid over a convectively heated extending sheet with Joule heating
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  70. Retraction
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