An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion

Sridevi Dandu; Venkata Ramana Murthy Chitrapu; Raghunath Kodi

doi:10.1515/phys-2024-0043

Article Open Access

An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion

Sridevi Dandu , Venkata Ramana Murthy Chitrapu and Raghunath Kodi

Published/Copyright: June 14, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

This article investigates the unsteady mixed convention two-dimensional flow of magnetohydrodynamic Casson hybrid nanofluids (alumina oxide and titanium oxide nanoparticles with base fluid water) flow through porous media over a linearly stretched sheet. We analyzed the heat and mass transfer in mixed convection, thermal radiation, variable thermal conductivity, variable mass diffusivity, and chemical reaction in the presence of thermophoresis and Brownian motion. A system of partial differential equations is reduced to a solvable system of ordinary differential equations by applying a suitable similarity transformation. We used the Runga–Kutta method along with the shooting procedure to solve the flow, heat, and mass transfer equations along with boundary conditions. The results obtained from MATLAB codes are compared with previously published results of the same type in a limiting case. The results of the velocity, temperature, and concentration profile of the hybrid nanofluid for varying different flow parameters are obtained in the form of graphs, while the rate of shear stress, rate of heat, and mass transfer are expressed in tables. We noticed that velocity and temperature diminish as an unsteady parameter increases; however, the reverse trend was observed in the nanoparticle concentration profile. With an increase in the thermal radiation parameter, the resultant velocity and temperature profile improves, while the concentration of nanoparticle profiles decreases. The velocity and temperature increase with higher Brownian motion, while the velocity increases and temperature decreases with higher thermophoresis.

Keywords: Brownian motion; thermophoresis; thermal radiation; porous media; chemical reaction

1 Introduction

In nature, some non-Newtonian fluids behave like elastic solid, which means that no flow occurs with low shear stress. Casson fluid is one of these fluids. This fluid has distinct features and has recently gained popularity. Casson fluid model was introduced by Casson in 1959 for the prediction of the flow behavior of pigment-oil suspensions. So, for the flow to occur, the shear stress magnitude of Casson fluid must be greater than the yield shear stress; otherwise, the fluid acts as a rigid body. This type of fluid can be classified as purely viscous with high viscosity. The Casson model is based on a structure model that describes the interaction of solid and liquid phases in a two-phase suspension. Some well-known Casson fluids include jelly, tomato sauce, honey, soup, and concentrated fruit juices. Human blood can also be treated as Casson fluid since it contains protein, fibrinogen, and globulin in aqueous base plasma and human red blood cells. In the earlier studies on Casson fluid, Boyd et al. [1] discussed the steady and oscillatory flow of blood by taking into account Casson fluid, whereas Fredrickson [2] investigated its steady flow in a tube. The peristaltic flow of a Casson fluid in a two-dimensional channel is described by Mernonr et al. [3]. Kumar et al. [4], Gowda et al. [5], Kumar et al. [6], Zeeshan et al. [7], Sanni et al. [8], and Anwar et al. [9] studied computational analysis of induced magnetohydrodynamic (MHD) non-Newtonian nanofluid flow over a nonlinear stretching sheet. Awan et al. [10] discussed the significance of magnetic fields and the Darcy–Forchheimer law on the dynamics of Casson–Sutterby nanofluid subjected to a stretching circular cylinder.

The combined effects of magnetic field and thermal radiation are encountered in different technological applications such as purification of nuclear reactor coolers, molten metals, metal casting, geothermal energy extraction, and so on. Prathiba and Lakshmi [11] discussed the effects of radiation and convection on three-dimensional (3D) Casson hybrid nanofluid flow in the presence of suction over a rotary sheet. Olkha and Choudhary [12] reviewed a numerical simulation of MHD radiative Casson nanofluid flow over a nonlinearly heated stretching sheet embedded in a porous medium. Sravan Kumar et al. [13] investigated the impact of Lorentz force and viscous dissipation on unsteady nanofluid convection flow over an exponentially moving vertical plate. Sharma et al. [14] reported the buoyancy effects of unsteady MHD chemically reacting and rotating fluid flow over a plate in a porous medium. Saeed et al. [15] reviewed the Darcy–Forchheimer couple stress hybrid nanofluids flow with varied fluid properties. Gul et al. [16] discussed irreversibility analysis of the couple stress hybrid nanofluid flow in the presence of the electromagnetic field.

Although it is well recognized that nanofluids have many potential applications for improving thermal efficiency, research is still being conducted to identify new types of nanofluids for improved performance. Hybrid nanofluids are a new and advanced type of nanofluid that has higher thermal efficiency than conventional nanofluids. Hybrid nanofluids are formed by dispersing two or more types of nanoparticles in a basic fluid medium. This results in a homogeneous mixture that has physicochemical properties of different substances that could hardly be speculated to exist in an individual substance. Hybrid nanofluids have received much attention owing to their significant ability to improve heat transfer. They have been found to be potentially applicable in various areas such as generator cooling, lubrication, solar heating, welding, electronic cooling, cooling and heating in buildings, vehicle thermal management, space air-crafts, bio-medical and drug reduction, and so on. Pal and Mandal [17] studied stability analysis and implications of Darcy magnetic-radiative hybrid reactive nanofluid heat transfer on a shrinkable surface with Ohmic heating. Gopinath and Dulal [18] expressed dual solutions of radiative Ag-MoS₂/water hybrid nanofluid flow with varying viscosity and thermal conductivity along an exponentially shrinking porous Riga surface: Stability and entropy generation analysis. Dulal and Gopinath [19] discussed the effects of the aligned magnetic field on heat transfer of water-based carbon nanotubes nanofluid over a stretching sheet with homogeneous–heterogeneous reactions. Jakeer et al. [20] scrutinized the electromagnetohydrodynamic (EMHD) ternary hybrid nanofluid flow on a stretchable surface. Numerical analysis is conducted on Darcy-Forchheimer flow with thermal radiation effect using the BVP4C solver in MATLAB to obtain numerical solutions for the problem. Alfwzan et al. [21] discussed the EMHD nanofluid flow with hybrid nanoparticles and activation energy phenomena. The results can be used to manage bleeding by changing the magnetic field in the drug infusion mechanism.

The study of MHD flow of non-Newtonian fluids in a porous medium has attracted the attention of many researchers. Of course, it is due to the fact that such phenomena are mostly found in the optimization of metal and metal alloy solidification processes, the geothermal source investigation, and nuclear fuel debris treatment. However, non-Newtonian fluids are subtle compared to Newtonian fluids. Indeed, the resulting equations of non-Newtonian fluids produce highly nonlinear differential equations that are usually difficult to solve. These equations add further complexities when MHD flows in a porous space have been taken into account. MHD flows of non-Newtonian fluids in a porous medium have numerous applications in irrigation problems, heat-storage beds, biological systems, petroleum processing, textile, paper, and polymer composite industries. Numerous studies have been conducted on various aspects of MHD flows of non-Newtonian fluid flows passing through a porous medium. One may refer to some recent investigations. Humane et al. [22] studied the effects of chemical reactions and thermal radiation on MHDs flow of Casson–Williamson nanofluid over a porous stretched surface. Umair et al. [23] reviewed radiative mixed convective flow induced by hybrid nanofluid over a porous vertical cylinder in porous media with irregular heat sink/source. Ravikumar et al. [24] studied the effects of Joule heating and reaction mechanisms on coupled stress fluid flow with peristalsis in the presence of a porous material through an inclined channel. Ramachandra Reddy et al. [25] reviewed the effects of Hall current, activation energy, and diffusion temperature on MHD Darcy–Forchheimer Casson nanofluid flow in the presence of Brownian motion and thermophoresis. Yasmin et al. [26] discussed HAM simulation for bioconvective MHD flow of Walters-B fluid containing nanoparticles and microorganisms past a stretching sheet with velocity slip and convective conditions. Khan et al. [27] reviewed the numerical analysis of the bioconvection-MHD flow of Williamson nanofluid with gyrotactic microbes and thermal radiation: a new iterative method. Zhang et al. [28] possessed analyses of electrokinetic energy conversion for periodic EMHD nanofluid through the rectangular microchannel under the Hall effects. Raghunath et al. [29] analyzed the effects of diffusion, thermodynamics, and chemical reactions on MHD Jeffrey nanofluid over an inclined vertical plate in the presence of radiation absorption and a constant heat source. Zhang et al. [30] investigated the effect of chemical reactions on MHD Newtonian fluid flow on a vertical plate in a porous medium in conjunction with thermal radiation. Ali et al. [31] discussed heat transfer analysis of the MHD stagnation-point flow of third-grade fluid across a porous sheet with thermal radiation effect: an algorithmic approach. Tarakaramu et al. [32] demonstrated thermal radiation and heat generation for three-dimensional Casson fluid motion using a porous stretching surface with variable thermal conductivity. Aruna et al. [33] examined an unsteady MHD flow of a second‐grade fluid passing through a porous medium in the presence of radiation absorption, resulting in Hall and ion slip effects. Akbar et al. [34] discussed a computational analysis of the thermal characteristics of magnetized hybrid Casson nanofluid flow in a converging–diverging channel with radiative heat transfer. Sedki [35] studied the effect of thermal radiation and chemical reaction on MHD mixed convective heat and mass transfer in nanofluid flow due to nonlinear stretching surface through a porous medium. Saeed et al. [36] discussed a mixed convective flow of an MHD Casson fluid through a permeable stretching sheet with first-order chemical reaction. Ismail et al. [37] demonstrated thermal instability of tri-hybrid Casson nanofluid with thermal radiation saturated porous medium in different enclosures.

We extend the work of Santhi et al. [38] by including mixed convection and porous media to examine the effects of heat radiation and chemical reactions on an unsteady MHD-free convective hybrid nanofluid flow via pores in the medium and a vertical stretching sheet in the presence of thermophoresis and Brownian motion. The governing equations are transformed via the use of a similarity transformation, and the resulting dimensionless equations are solved numerically by the use of the Runge–Kutta fourth directive method with the shooting approach application. The impacts of a variety of dissimilar governing factors on the momentum, temperature, concentration, skin friction factor, Nusselt numeral, and Sherwood digit are illustrated in the sculptures and tablelands, and they are explained in great depth.

2 Description of the problem

The unsteady, laminar, two-dimensional, maximum heat transfer boundary layer that transfers heat and mass from a TiO₂/Al₂O₃‐water-dependent hybrid nanoliquid flow via a stretched sheet with thermophoresis and Brownian motion effects was considered. This flow is shown in Figure 1. It is necessary to take the x-axis along the stretching surface and in the direction of flow, while the y-axis is measured in a direction that is normal to it. The speed of the stretched sheet is denoted by the equation U _w(x,t). A magnetic field of intensity B ₀ is applied consistently to the plate in a normal direction. T _w and C _w represent the homogeneous temperature and concentration of the sheet surface. Additionally, T _∞ and C _∞ represent the temperature and concentration of the natural fluid, respectively, on the sheet surface. Table 1 shows the nanoliquid’s thermophysical characteristics. These equations are based on the presumptions that were previously presented by Santhi et al. [38]

(1) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(2) ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y = − 1 ρ hnf ∂ p ∂ x + μ hnf 1 + 1 γ ∂ 2 u ∂ y 2 − σ hnf B 2 ( t ) ρ hnf u + g [ β T ( T − T ∞ ) + β c ( C − C ∞ ) ] − ϑ f k u ,

(3) ∂ T ∂ t + u ∂ T ∂ x + v ∂ T ∂ y = k hnf ( ρ c p ) hnf ∂ 2 T ∂ y 2 − 1 ( ρ c p ) hnf ∂ q r ∂ y + τ hnf D B ∂ C ∂ y ∂ T ∂ y + D T T ∞ ∂ 2 T ∂ y 2 ,

(4) ∂ C ∂ t + u ∂ C ∂ x + v ∂ C ∂ y = D B ∂ 2 C ∂ y 2 − K 0 ( C − C ∞ ) .

Figure 1

Physical configuration of the problem.

Table 1

Thermophysical properties of hybrid nanofluid

Physical properties	Water	Al₂O₃ (alumina)	TiO₂ (titanium oxide)
C _p (J/kg K)	4179.0	765.0	686.20
ρ (kg/m³)	997.100	3970.00	4250.0
k (w/mK)	0.61300	40.00	8.95380
σ (Ω/m)	0.0500	1 × 10¹⁰	0.85 × 10¹⁰

Boundary conditions related to the problem are given as follows [38]:

(5) u = U w + L ∂ u ∂ y , v = v w , T = T w + k 1 ∂ T ∂ y , C = C w + k 2 ∂ C ∂ y at y = 0 ,

(6) u → 0 , T → T ∞ C → C ∞ as y → ∞ .

The similarity transformations convert partial differential equations (2)–(4) to ordinary differential equations (ODEs). Similitude transformations used have variables of the form [38]

(7) η = y a v f ( 1 − c t ) , v = x a v f ( 1 − c t ) f ( η ) , θ ( η ) = T − T ∞ T w − T ∞ , φ ( η ) = C − C ∞ C w − C ∞ , u = a x 1 − c t f ′ ( η ) , T = T ∞ + T 0 U w x v 1 − c t , C = C ∞ + C 0 U w x v 1 − c t , B ( t ) = B 0 1 − c t .

These similarity transformations used in Eqs. (2)–(4) result in

(8) A 1 A 2 1 + 1 γ f ‴ + f f ″ − f ′ 2 − δ f ′ + η 2 f ″ − M f ′ + λ 1 [ θ + N ] − D a f ′ = 0 ,

where N = β C β T ( C − C ∞ ) ( T − T ∞ ) .

(9) 1 + A 4 4 3 R d θ ″ − Pr A 3 A 4 δ 2 ( η θ ′ + 2 θ ′ ) + Pr ( N b φ ′ θ ′ ( η ) + N t θ ′ 2 ( η ) ) = 0 ,

(10) φ ″ − S c δ 2 ( η φ ′ + 2 φ ′ ) + R c S c φ = 0 .

The non-dimensional form of boundary conditions with the use of similarity transformations is given as follows:

(11) f ′ ( η ) = 1 + λ f ″ ( η ) , f ( η ) = V 0 , θ ′ ( η ) = 1 + ξ θ ′ ( η ) , φ ( η ) = 1 + β φ ′ ( η ) at η = 0 θ ( η ) → 0 , f ′ ( η ) → 0 , φ ( η ) → 0 as η → ∞ .

The parameters involved in nondimensional form of ODEs and boundary conditions in Eqs. (8–10) are as follows:

Pr = ν f α f is the Prandtl factor,

δ = c χ is the unsteady parameter,

λ = L a 2 v is the velocity slip parameter,

M = σ B 0 2 ρ a is the magnetic field parameter,

ξ = k 1 a 2 v is the temperature slip parameter,

N b = τ D B ( C w − C ∞ ) ν is the Brownian motion parameter,

N t = τ D T ( T w − T ∞ ) ν T ∞ is the thermophoresis parameter,

β = k 2 a 2 v is the concentration slip parameter,

Sc = v D B is the Schmidt factor,

K 0 = R C a is the chemical reaction,

R d = 16 σ ∗ T ∞ 3 3 k f k ∗ is the thermal radiation,

λ 1 = g [ β T ( T − T ∞ ) ( 1 − C t ) 2 ] a 2 x is mixed convection,

D a = ϑ k a ( 1 − C t ) is porous media.

The thermophysical properties of hybrid nanofluid in Eqs. (8)–(10) are given as follows: μ hnf μ f = 1 ( 1 − φ 1 ) 2.5 ( 1 − φ 2 ) 2.5 ,

ρ hnf = φ 2 ρ s 2 + [ φ 1 ρ s 1 + ( 1 − φ 1 ) ρ f ] ( 1 − φ 2 ) ,

( ρ C p ) hnf = φ 2 ( ρ C p ) s 2 + [ φ 1 ( ρ C p ) s 1 + ( 1 − φ 1 ) ( ρ C p ) f ] ( 1 − φ 2 )

k hnf = k nf k s 2 + 2 k nf − 2 φ 2 ( k nf − 2 k s 2 ) k s 2 + 2 k nf + 2 φ 2 ( k nf − 2 k s 2 ) ,

where k nf = k f k s 1 + 2 k nf − 2 φ 1 ( k nf − 2 k s 1 ) k s 1 + 2 k nf + 2 φ 1 ( k nf − 2 k s 1 ) ,

A 1 = 1 ( 1 − φ 1 ) 2.5 ( 1 − φ 2 ) 2.5 ,

A 2 = φ 2 ρ s 2 ρ f + φ ρ s 1 ρ f + ( 1 − φ 1 ) ( 1 − φ 2 ) ,

A 3 = φ 2 ( ρ C p ) s 2 ( ρ C p ) f + φ 1 ( ρ C p ) s 1 ( ρ C p ) f + ( 1 − φ 2 ) ( 1 − φ 2 ) ,

A 4 = k hnf k f .

Table 1 describes the thermophysical properties of the hybrid nanofluid used in this hybrid nanofluid.

3 Physical quantities of interests

In this article, the substantial physical quantities are skin friction (Cf_x) or drag force, Nusselt factor (Nu_x), and Sherwood factor (Sh_x), which are given as

(12) Cf x = τ w ρ f U w 2 , Nu x = − x q w k f ( T w − T ∞ ) , Sh x = x q m k f ( C w − C ∞ ) .

Here, τ w = μ hnf ∂ u ∂ y y = 0 , q w = − k hnf ∂ T ∂ y y = 0 , q m = − D B ∂ C ∂ y y = 0 .

The non-dimensionalized form of these physical quantities with the use of non-dimensional parameters is given as follows:

(13) Re x 1 / 2 Cf x = μ hnf μ f f ″ ( 0 ) , Re x − 1 / 2 Nu x = − k hnf k f + Rd θ w 3 θ ′ ( 0 ) , Re x − 1 / 2 Sh x = − k hnf k f φ ′ ( 0 ) ,

where Re = x U w v f is the Reynolds number.

4 Method of solution

MATLAB bvp4c is constructed on the basis of the Runge–Kutta technique in conjunction with the shooting method in order to solve Eqs. (8)–(11). We present functions that analyze differential equations according to the corresponding boundary residuals. We take into account the following: From the nonlinear differential Eqs. (8)–(11), it is possible to deduce further first-order differential equations.

The first task to carry out the computation is to convert the boundary value problem to an initial value problem. Let by using the following notations:

(14) f = y 1 , f ′ = y 2 , f ″ = y 3 , f ‴ = y ′ 3 , θ = y 4 , θ ′ = y 5 , θ ″ = y ′ 5 , φ = y 6 , φ ′ = y 7 , φ ″ = y ′ 7 .

By using the above variables, the system of first‐order ODEs is

(15) y ′ 1 = y 2 ,

(16) y ′ 2 = y 3 ,

(17) y ′ 3 = A 2 A 1 1 + 1 γ y 1 y 3 − ( y 2 ) 2 − δ y 2 + η 2 y 3 − M y 2 + λ [ y 4 + N ] − D a y 2 ,

(18) y ′ 4 = y 5 ,

(19) y ′ 5 = 1 1 + A 4 4 3 R d Pr A 3 A 4 δ 2 ( η y 5 + 2 y 5 ) − Pr [ Nb y 5 y 7 + Nt ( y 5 ) 2 ] ,

(20) y ′ 6 = y 7 ,

(21) y ′ 7 = Sc δ 2 ( η y 7 + 2 y 7 ) − R c Sc y 6 .

The boundary conditions are given as

(22) y 1 ( 0 ) − V 0 = 0 , y 1 ( 0 ) − λ y 3 ( 0 ) − 1 = 0 , y 5 ( 0 ) − ξ y 5 ( 0 ) − 1 = 0 , y 6 ( 0 ) − β y 7 ( 0 ) − 1 = 0 y 2 ( ∞ ) → 0 , y 4 ( ∞ ) → 0 , y 6 ( ∞ ) → 0 .

The boundary conditions in eq (22) are utilized via the use of a finite value of η _max as given

(23) f ′ ( η max ) → 0 , θ ′ ( η max ) → 0 , φ ′ ( η max ) → 0 .

The step is taken Δη = 0.001, and the convergent criterion is 10⁻⁶ for the desired accuracy.

5 Code validation

In order to verify the validity and accuracy of the present analysis, the results for the heat transfer –θ″(0) were compared with those reported by Santhi et al. [38].

6 Results and discussion

The graphical findings for velocity, temperature, and concentration profiles are evaluated and compared to different values of various physical factors that are involved in the equations that regulate the system. To facilitate numerical computation, we set the following values: λ = 0.5, δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Nb = 0.5, Nt = 0.5, Pr = 6.2, Rc = 0.1, Sc = 1.0, and D _a = 0.5. In the graphs, f′(η), θ(η), and ϕ(η) signify the flow fields (velocity, temperature, and concentration, respectively) (Table 2).

Table 2

Comparison of (−θ′(0)) with the results of Santhi et al. [38] for miscellaneous significances of Pr and λ ₁ = D _a = 0

Parameter (Pr)	Santhi et al. [38] (−θ′(0))	Present study (−θ′(0))
2.00	0.911341	0.916732
6.130	1.759676	1.752672
7.00	1.895397	1.882626
20.00	3.353915	3.357256

Table 3 shows that skin friction coefficient, Nusselt number, and Sherwood number decrease in the TiO₂/Al₂O₃–water-based hybrid nanoliquid as the quantities of M increase. However, the values of the Sherwood number increase as the values of Rd improve. This is despite the fact that the values of the nondimensional skin friction coefficient and the Nusselt number are decreasing. This table clearly shows that the values of Cf and Shx in hybrid nanoliquid increase in tandem with the intensification of the elements of (Rc). Table 3 also shows that the impact of (Rc) on the rates of heat transfer coefficient is recognized, and we determined that the values of Nux have decreased. If the values of (Pr) increase, the dimensionless values of (Cf) and (Nu_x) will also increase. On the contrary, the values of (Sh_x) will follow the opposite trend. The values of the skin friction coefficient decrease as the values of (λ) increase, while the values of the Nusselt number and the Sherwood number increase in a hybrid nanoliquid composed of TiO₂/Al₂O₃ and water.

Table 3

Values of skin friction coefficient f ″ ( 0 ) , Nusselt number ‒ θ ′ ( 0 ) , and Sherwood number ‒ φ ′ ( 0 ) for fixed values of λ = 0.5, ξ = 0.5, β = 0.5, M = 0.25, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0, D _a = 0.5, and t = 0.1

M	Rd	Da	Pr	Rc	f ″ ( 0 )	‒ θ ′ ( 0 )	‒ φ ′ ( 0 )
0.25	1.0	0.5	6.2	0.1	0.3456	0.2987	0.4673
0.3					0.3256	0.2653	0.4214
0.6					0.3089	0.2342	0.3986
0.9					0.2874	0.2098	0.3542
	0.25				0.2753	0.1982	0.3152
	0.5				0.2984	0.2142	0.3672
	0.72				0.3183	0.2651	0.3974
		0.5			0.3345	0.2546	0.2135
		1.0			0.3678	0.2789	0.2456
		1.5			0.3986	0.2987	0.2675
			8.0		0.1876	0.6548	0.5789
			11.0		0.1986	0.5437	0.5437
			12.0		0.2130	0.4573	0.5189
				0.3	0.1974	1.1678	0.6156
				0.6	0.2164	0.1531	0.5290
				1.0	0.2368	0.1341	0.4178

Diagrammatic representation of the velocity field f ′ ( η ) for a variety of magnetic values (M) is shown in Figure 2. Figure 2 depicts the pattern of a decrease in the velocity of the fluid as the quantity of M increases. An increase in the magnitude of the magnetic parameter results in the production of a resistive force that is often referred to as the Lorentz force. This Lorentz force serves to impede the velocity of the fluid, resulting in a decrease in the velocity of the fluid. As a result, the velocity distribution f ′ ( η ) decreases occurs whenever the quantity of M increases. The relationship between the porosity parameter Da and the momentum profile f ′ ( η ) of the hybrid nanofluid is shown in Figure 3. As can be seen from Figure 3, there is a clear relationship between the porosity parameter Da and the momentum profile f ′ ( η ) of the hybrid nanofluid.

$Figure 2 Interpretation of f ′ ( η ) f^{\prime} (\eta ) for M with λ = 0.5, ξ = 0.5, δ = 1.0, β = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D a = 0.5, Nb = 0.5, Nt = 0.5.$

Figure 2

Interpretation of f ′ ( η ) for M with λ = 0.5, ξ = 0.5, δ = 1.0, β = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D _a = 0.5, Nb = 0.5, Nt = 0.5.

$Figure 3 Interpretation of f ′ ( η ) f^{\prime} (\eta ) for Da with λ = 0.5, ξ = 0.5, δ = 1.0, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, and Sc = 1.0, Nb = 0.5, Nt = 0.5.$

Figure 3

Interpretation of f ′ ( η ) for Da with λ = 0.5, ξ = 0.5, δ = 1.0, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, and Sc = 1.0, Nb = 0.5, Nt = 0.5.

In Figures 4–6, the profiles of velocity f ′ ( η ) , temperature θ(η), and concentration ϕ(η) are examined for various values of an unsteadiness parameter δ. These profiles are analyzed for a variety of values. As can be apparent in Figures 4 and 5, an increase in the value of δ results in a decrease in the velocity field and temperature distribution, which are referred to as the momentum and thermally barrier layer. As seen in Figure 6, there is a noticeable increase in the concentration of nanoparticles when the value of δ is increased.

$Figure 4 Interpretation of f ′ ( η ) f^{\prime} (\eta ) for δ with λ = 0.5, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D a = 0.5, Nb = 0.5, Nt = 0.5.$

Figure 4

Interpretation of f ′ ( η ) for δ with λ = 0.5, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D _a = 0.5, Nb = 0.5, Nt = 0.5.

$Figure 5 Interpretation of θ ( η ) \theta (\eta ) for δ with λ = 0.5, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D a = 0.5, Nb = 0.5, Nt = 0.5.$

Figure 5

Interpretation of θ ( η ) for δ with λ = 0.5, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D _a = 0.5, Nb = 0.5, Nt = 0.5.

$Figure 6 Interpretation of φ ( η ) \varphi (\eta ) for δ with λ = 0.5, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D a = 0.5, Nb = 0.5, Nt = 0.5.$

Figure 6

Interpretation of φ ( η ) for δ with λ = 0.5, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D _a = 0.5, Nb = 0.5, Nt = 0.5.

Numerous values of mixed convective components are shown in Figure 7, which displays the velocity profile for those variables. Furthermore, it demonstrates that the is magnified as a result of the presence of buoyancy force. The fact that a higher buoyancy force creates a favorable pressure gradient, causing the flow to be assisted in the upward direction, and as a result, the velocity increases, is well established.

$Figure 7 Interpretation of f ′ ( η ) f^{\prime} (\eta ) for λ with δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D a = 0.5, Nb = 0.5, Nt = 0.5.$

Figure 7

Interpretation of f ′ ( η ) for λ with δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D _a = 0.5, Nb = 0.5, Nt = 0.5.

The influence of the Brownian motion parameter Nb on velocity and temperature profiles is shown in Figures 8 and 9. The velocity in the boundary layer decreases as Nb increases from 0.3 to 1.0, as seen in Figures 8 and 9, which show the impact of Nb on dimensionless temperature and velocity. The dimensionless temperature rises as Nb increases. This is because increasing the number of nanofluid particle collisions generates thermal energy, which raises the temperature of the nanofluid. The analysis of the thermophoresis parameter Nt reveals the influence of the thermophoresis phenomena. As illustrated in Figures 10 and 11, raising this parameter causes the temperature of the boundary layer to rise and the concentration of nanoparticles to rise. Increasing Nt from 0.3 to 1.0 also enhances the velocity and temperature of nanoparticles at the surface, as shown in Figures 10 and 11. The thermophoresis parameter can have a positive or negative value, with a negative sign suggesting a hot surface and a positive value indicating a cold surface.

$Figure 8 Interpretation of f ′ ( η ) f^{\prime} (\eta ) for Nb with δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D a = 0.5, Nt = 0.5.$

Figure 8

Interpretation of f ′ ( η ) for Nb with δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D _a = 0.5, Nt = 0.5.

$Figure 9 Interpretation of φ ( η ) \varphi (\eta ) for Nb with δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D a = 0.5, Nt = 0.5.$

Figure 9

Interpretation of φ ( η ) for Nb with δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D _a = 0.5, Nt = 0.5.

$Figure 10 Interpretation of f ′ ( η ) f^{\prime} (\eta ) for Nt with δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D a = 0.5, Nb = 0.5.$

Figure 10

Interpretation of f ′ ( η ) for Nt with δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D _a = 0.5, Nb = 0.5.

$Figure 11 Interpretation of φ ( η ) \varphi (\eta ) for Nt with δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D a = 0.5, Nb = 0.5.$

Figure 11

Interpretation of φ ( η ) for Nt with δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D _a = 0.5, Nb = 0.5.

Figures 12–14 illustrate the influence that the radiation quantity has on the velocity of the fluid, the temperature, and the distribution of the concentration. As can be seen from these photos, the acceleration and temperature of the liquid increase as a result of the radiation factor being magnified. From a physical standpoint, the radiation causes an increase in the surface heat flow, and as a result, a higher temperature in the boundary-layer area should be estimated. In terms of concentration, the opposite tendency has always been observed.

$Figure 12 Interpretation of f ′ ( η ) f^{\prime} (\eta ) for Rd with λ = 0.5, δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D a = 0.5, Nb = 0.5, Nt = 0.5.$

Figure 12

Interpretation of f ′ ( η ) for Rd with λ = 0.5, δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D _a = 0.5, Nb = 0.5, Nt = 0.5.

$Figure 13 Interpretation of θ ( η ) \theta (\eta ) for Rd with λ = 0.5, δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D a = 0.5, Nb = 0.5, Nt = 0.5.$

Figure 13

Interpretation of θ ( η ) for Rd with λ = 0.5, δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D _a = 0.5, Nb = 0.5, Nt = 0.5.

$Figure 14 Interpretation of φ ( η ) \varphi (\eta ) for Rd with λ = 0.5, δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D a = 0.5, Nb = 0.5, Nt = 0.5.$

Figure 14

Interpretation of φ ( η ) for Rd with λ = 0.5, δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Pr = 6.2, Rc = 0.1, Sc = 1.0 and D _a = 0.5, Nb = 0.5, Nt = 0.5.

Decrementing trend in temperature against Prandtl number (Pr) is displayed in Figure 15. It is because by enhancing (Pr), thermal diffusion of fluid decreases, due to which temperature distribution shows decreasing behavior. The effect of the chemical reaction parameter (Rc) on the concentration profile is illustrated in Figure 16. Generally, it is observed that, in most cases, the decreased concentration field is noticed for destructive chemical reactions (Figure 17). Schmidt number is an important physical quantity in the problem to consider while investigating the mass transfer. It is an analogue of the Prandtl number; similar to the effect of (Pr) on the temperature field, a decreasing influence of (Sc) on concentration distribution is revealed. This is because, with an increase in the Schmidt number (Sc), viscous diffusion rises, causing particles to move extensively and increasing convective potential.

$Figure 15 Interpretation of θ ( η ) \theta (\eta ) for Pr with λ = 0.5, δδ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Rc = 0.1, Sc = 1.0 and D a = 0.5, Nb = 0.5, Nt = 0.5.$

Figure 15

Interpretation of θ ( η ) for Pr with λ = 0.5, δδ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Rc = 0.1, Sc = 1.0 and D _a = 0.5, Nb = 0.5, Nt = 0.5.

$Figure 16 Interpretation of φ ( η ) \varphi (\eta ) for Rc with λ = 0.5, δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Sc = 1.0 and D a = 0.5, Nb = 0.5, Nt = 0.5.$

Figure 16

Interpretation of φ ( η ) for Rc with λ = 0.5, δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Sc = 1.0 and D _a = 0.5, Nb = 0.5, Nt = 0.5.

$Figure 17 Interpretation of φ ( η ) \varphi (\eta ) for Sc with λ = 0.5, δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, and D a = 0.5, Nb = 0.5, Nt = 0.5.$

Figure 17

Interpretation of φ ( η ) for Sc with λ = 0.5, δ = 1.0, ξ = 0.5, β = 0.5, M = 0.5, Rd = 1.0, Pr = 6.2, Rc = 0.1, and D _a = 0.5, Nb = 0.5, Nt = 0.5.

7 Conclusion

The influence of the pertinent factors on the flow, heat, and mass transport factors was shown using a graph, and a short discussion took place. In light of this, the following are some concluding observations about this topic. The findings were determined to be satisfactory when they were evaluated in light of the previous research that had been conducted.

As the magnetic field factor (M) enhances, the resulting fluid velocity has a greater tendency to decrease.
Additionally, as the Prandtl factor (Pr) enhances, the temperature profiles have a tendency to decrease.
The concentration profile in relation to the Sc and the chemical reaction (Rc) all exhibit a declining tendency.
The velocity and temperature increase with higher Brownian motion, while the velocity increases and temperature decreases with higher thermophoresis.
The velocity and temperature have decreased as the unsteady parameter has increased, but they have increased as the concentration has increased.
The resultant velocity has been shown to increase as the buoyancy force increases; however, the behavior of the resultant velocity has been shown to reverse when the Darcy parameter increases.

Funding information: The authors state no funding involved.
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: The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-01-18

Revised: 2024-04-22

Accepted: 2024-05-24

Published Online: 2024-06-14

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

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https://doi.org/10.1515/phys-2024-0043

Keywords for this article

Brownian motion; thermophoresis; thermal radiation; porous media; chemical reaction

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