Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field

Shuguang Li; Nainaru Tarakaramu; Muhammad Ijaz Khan; Narsu Sivakumar; Panyam Venkata Satya Narayana; Sherzod Abdullaev; Nissren Tamam; Sayed M. Eldin

doi:10.1515/phys-2023-0131

Article Open Access

Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field

Shuguang Li , Nainaru Tarakaramu , Muhammad Ijaz Khan , Narsu Sivakumar , Panyam Venkata Satya Narayana , Sherzod Abdullaev , Nissren Tamam and Sayed M. Eldin

Published/Copyright: February 23, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

A mathematical model is envisaged that discusses the motion of 3D nanofluids (NFs) with anisotropic slip influence magnetic field past a stretching sheet. The heat transportation phenomenon is analysed by melting effect, heat generation, and chemical reaction. The main motivation of this study is to analyse the behaviour of liquid motion and heat transfer (HT) of NFs because this study has huge applications in boiling, solar energy, and micropower generation, which are used in the engineering process. The physical governing partial differential equation is transformed into a coupled non-linear system of ordinary differential equations using suitable appropriate transformations. The translated equations are calculated using Runge–Kutta–Fehlberg method via shooting procedure. The physical characteristics of various parameters on velocities, concentration, and thermal fields are explored in detail. The HT is high in NFs when compared to pure or regular liquids for ascending values of heat source parameter and slip factor. Also, the skin friction coefficients via coordinate axes and rate of Nusselt number were analysed.

Keywords: melting effect; nanofluid; MHD; heat source; chemical reaction; slip condition; thermal radiation

Nomenclature

( x 1 , y 1 ): Cartesian coordinates
u 1 , u 2 , and u 3: velocity components along x 1 , y 1 , z 1 -axis
C 1: volume fraction of nanoparticle
C f: skin friction coefficient
c p: specific heat
C ∞: uniform ambient concentration
D B: Brownian diffusion
D T: thermophoresis diffusion
f: dimensionless stream function
f ': dimensionless velocity
H: heat source parameter = Q 0 / a 1 ( ρc p )
k 1: thermal conductivity
Le: Lewis number = α 1 / D B
M: magnetic field parameter = σ 1 B 0 2 a 1 ρ f
Mt: melting parameter c f ( T ∞ − T m ) λ 1 + c s ( T m − T s )
N t: thermophoresis parameter = τ 1 D T α 1 T ∞ ( T w − T ∞ )
N b: Brownian motion coefficient = D B τ 1 ( C w − C ∞ ) α 1 υ 1
Pr: Prandtl number = υ 1 α 1
q r: radiative heat flux
Re x: Reynolds number
R d: radiation parameter = 16 σ 1 T ∞ 3 3 ( ρc p ) α 1 k 1
T 1: temperature of the fluid
T s: solid surface temperature
T m: temperature of the melting surface
T ∞: fluid temperature far away from the surface
T w: constant fluid temperature of the wall
U w: stretching velocity
U ∞: free stream velocity

Greek symbols

μ 1: dynamic viscosity
ϕ: dimensionless concentration
σ 1: Boltzmann’s constant
γ: chemical reaction parameter = k 0 / a 1
λ 1: slip factors = N i μ 1 a 1 / υ 1
υ 1: kinematic viscosity of the fluid
α 1: thermal diffusivity = k 1 / ( ρ c p )
τ 1: ratio of the nanoparticle to the fluid ( ρ c ) p / ( ρ c ) f
( ρ c ) f: heat capacity of the fluid
( ρ c ) p: heat capacity of the nanoparticle to the fluid
ρ: density

Subscripts

∞: condition at free stream

Abbreviation

HT: heat transfer
TR: thermal radiation
TD: thermal diffusivity
3D: three-dimensional
CR: chemical reaction
MT: melting transfer
BL: boundary layer
HGT: homogenous reaction
NFs: nanofluids
HTM: heat and mass transfer
MHD: magnetohydrodynamic
SS: stretching sheet
2D: two-dimensional
RFs: regular fluids
CLAM: China low activation martensitic
HR: heterogeneous reaction

1 Introduction

The melting transfer is more popular topic in upcoming researchers because it is closely related to a wide range of technologically significant processes. The impact of melting (fusion) is a physical process (phase conversion of a material from a solid to a fluid). It has various natural applications of heat or pressure (such as thermocouples, semiconducting materials, storage of latent heat, sanitization, optical material processing, permafrost melting, crystal growth, heat transfer (HT) and heat engines, metal casting, glass industry, solidifying magma, defrosting in frozen grounds, freezing of soil around the heat exchanger coils of a thermal energy storage, ground-based pump), laser manufacturing (drilling welding and selective sintering), etc. Epstein and Cho [1] examined the exact solutions for melting motion of HT via flat plate. On the other hand, Kamierczak et al. [2,3] studied melting of a vertical flat plate via porous medium with convection motion. Rahman et al. [4] presented the melting effect on magnetohydrodynamic (MHD) laminar and HT motion via the moving surface by the applied lie group method. Das [5] discussed the mathematical model of MHD motion of HT from electrically conducting liquid via parallel melting surface. Recently, Harish Babu et al. [6] explored the melting technology applied to magneto-NF motion via non-linear stretching surface. Venkateswarlu et al. [7] investigated numerical analysis of viscous dissipation and heat source on MHD motion via melting surface. Hayat et al. [8] explored the melting HT on 3D motion of NFs via impermeable SS. The melting HT and heat absorption characteristics in radiated stagnation point (SP) motion of Carreau liquid were created by Khan et al. [9]. The melting HT on MHD motion of Sisko liquid via nonlinear stretching velocity was examined by Hayat et al. [10]. Sheikholeslami and Rokni [11] presented the Buongiorno model that is applied to NF motion via stretching plate with magnetic field. Hayat et al. [12] explained the MHD SP motion of Jeffrey material via nonlinear SS. The microstructure of selective laser melting-built China low activation martensitic steel plates was analysed by Huang et al. [13]. Hajabdollahi et al. [14] investigated the close contact melting process generated by rotation. Fauzi et al. [15] examined the effect of melting effect on mixed convection boundary layer (BL) motion past a vertical surface via non-Darcian porous medium. Sheikholeslami et al. [16] analysed the impact of melting HT on NF motion with Lorentz forces.

The new-generation researchers are interested in doing the liquid motion when CR (combination of heterogeneous reaction [HR] and homogeneous reaction) is present. The physical condition values mentioned in HR and homogeneous reaction depend on weight, shape, distribution, architectural, size, appearance, colours, income, radioactivity, disease, temperature, and so on. Homogeneous reaction is very simple in comparison with HRs because homogeneous reaction depends upon only one nature reacting species. But the HR depends upon the two or more than two distinct nature reacting species. The heat generation rate is transpiring in liquid, whereas HR is supported on some catalyst surfaces only. Homogeneous catalyst transpires in gaseous state, whereas heterogeneous catalyst chances in solid state. These reactions’ major role applications of chemical reaction in industries (such as hydrometallurgical industry, atmospheric flows, fog disposition and dissipation, biochemical systems, combustion, catalysis, ceramics and polymer production, etc.). Recently, DelaRosa et al. [17] analysed the use of heterogeneous catalysts with sulfonic group in hydrolysis of cellulose. Muto et al. [18] developed the unsteady numerical simulation with a detailed CR mechanism. Hayat et al. [19] presented the third-grade NF motion via stretchable rotating disk. Sulochanaa et al. [20] found that the variable porosity parameter has tendency to enhance HMT rate. Naukkarinen and Sainio [21] developed the field of CR engineering using the virtual laboratory (VL) concept. Yang et al. [22] explored the 2D surface and semi-finite wedge instability of oblique detonation waves. Sambath et al. [23] considered the CR and thermal radiation (TR) effect on natural convective hydromagnetic motion of viscous liquid via vertical cone. Some of the CR model problems were analysed in previous studies [24–26]. The CR on unsteady MHD NFs motion via SS was studied by Tarakaramu and Satya Narayan [27]. Veera Krishna and Gangadhar Reddy [28] presented the unsteady MHD free convection in a BL motion via a porous moving vertical plate with CR. Bhatti et al. [29] examined the non-linear TR and CR effects on 3D MHD motion of viscous NFs with gyrotactic microorganisms via stretching porous cylinder. Recently, some important works on fluid flow are highlighted in previous studies [30–35].

The impact of heat source mechanism is known as a good regulatory mechanism of HT. This mechanism has more significant effect on NF motion owing to its engineering applications (metal waste, spent nuclear fuel, reactor safety analysis, radioactive materials, fire, and combustion). Jain and Bohra [36] presented the entropy generation on MHD liquid motion and HT via stretching cylinder with slip regime. Qayyum et al. [37] examined the impact of buoyancy force on MHD SP motion of tangent hyperbolic NFs. Hosseinzadeh et al. [38] discussed the non-uniform generation or absorption of HT in NFs motion via porous stretching sheet (SS). Ahmed and Elshehabey [39] explained that the buoyancy-driven HT enhances NFs motion with heat generation or absorption effect. Kanchana and Zhao [40] introduced the nonlinear stability analyses of Rayleigh–Benard convection with internal heat source in NFs. Recently, the generation or absorption influence on 3D NF motion models was developed [41–44]. Some of authors [45–48] developed numerical study of entropy generation on NFs via stretching surface. Some other studies regarding material applications, mathematical modelling and techniques, fluid flow, HT rate, and nanomaterial are addressed as follows: fluid flow behaviour examination [49–51], thermal mechanics [52–54], first hidden-charm pent quark with strangeness [55], and material characteristics [56–58].

The main intention and objective of this study to determine the nonlinear TR on 3D NF motion via SS with the anisotropic slip and melting effect. The major motivation of this work is to incorporate NFs to enhance the thermal conductivity and to make efficient HT. Also, the model is developed for the evaluation of the behaviour of TR parameter, melting parameter, Prandtl number, Brownian motion parameter, and thermophoresis parameter. The BL thickness via SS is increasingly being used in mechanical and physical processes, boiling, solar energy, maritime processes, aeronautical, and constructions. Flow of liquid and HT across SS is used in many technical processes, including polymer extrusion, food, and paper processing, fiberglass manufacturing, plastic film stretching, wire drawing, and continuous casting. The primary goal of the ongoing research is to learn about the properties of TR of 3D NF motion.

2 Mathematical formulation

Let us assume that the melting effect on 3D MHD motion of NFs via linear SS z 1 = 0 with heat source and CR. As shown in Figure 1, assume that the Cartesian coordinate system ( x 1 , y 1 , z 1 ) corresponding to velocity components ( u 1 , u 2 , u 3 ) is examined. The nanoparticles (NPs) are taken on the linear stretching surface. The surface velocity is assumed to be constant ( U 1 , V 1 , 0 ) . A constant magnetic field is applied in the direction of z 1 . The liquid motion on surface will be determined by the potential flow assumed as (ref. [59]):

Figure 1

Physical model of the problem.

u 1 = a 1 x 1 , u 2 = a 1 y 1 , u 3 = − 2 a 1 z 1 , p 1 = − ρ f a 1 2 2 ( x 1 2 + y 1 2 ) + p 0 ,

As per the aforementioned liquid motion consideration, we can formulate the continuity, momentum, conservative, and energy equations, which are as follows (ref. [59]):

(1) ∂ u 1 ∂ x 1 + ∂ u 2 ∂ y 1 + ∂ u 3 ∂ z 1 = 0 ,

(2) u 1 ∂ u 1 ∂ x 1 + u 2 ∂ u 1 ∂ y 1 + u 3 ∂ u 1 ∂ z 1 = − 1 ρ f ∂ p 1 ∂ x 1 + υ 1 ∂ 2 u 1 ∂ x 1 2 + ∂ 2 u 1 ∂ y 1 2 + ∂ 2 u 1 ∂ z 1 2 − σ 1 B 0 2 ρ f u 1 ,

(3) u 1 ∂ u 2 ∂ x 1 + u 2 ∂ u 2 ∂ y 1 + u 3 ∂ u 2 ∂ z 1 = − 1 ρ f ∂ p 1 ∂ y 1 + υ 1 ∂ 2 u 2 ∂ x 1 2 + ∂ 2 u 2 ∂ y 1 2 + ∂ 2 u 2 ∂ z 1 2 − σ 1 B 0 2 ρ f u 2 ,

(4) u 1 ∂ u 3 ∂ x 1 + u 2 ∂ u 3 ∂ y 1 + u 3 ∂ u 3 ∂ z 1 = − 1 ρ f ∂ p 1 ∂ z 1 + υ 1 ∂ 2 u 3 ∂ x 1 2 + ∂ 2 u 3 ∂ y 1 2 + ∂ 2 u 3 ∂ z 1 2 − σ 1 B 0 2 ρ f u 3 ,

(5) u 1 ∂ T 1 ∂ x 1 + u 2 ∂ T 1 ∂ y 1 + u 3 ∂ T 1 ∂ z 1 = α 1 ∂ 2 T 1 ∂ x 1 2 + ∂ 2 T 1 ∂ y 1 2 + ∂ 2 T 1 ∂ z 1 2 + τ 1 D B ∂ C 1 ∂ x 1 ∂ T 1 ∂ x 1 + ∂ C 1 ∂ y 1 ∂ T 1 ∂ y 1 + ∂ C 1 ∂ z 1 ∂ T 1 ∂ z 1 + τ 1 D T T ∞ ∂ T 1 ∂ x 1 2 + ∂ T 1 ∂ y 1 2 + ∂ T 1 ∂ z 1 2 − Q 0 ( T 1 − T ∞ ) ( ρ c p ) f − 1 ( ρ c p ) ∂ q r ∂ z 1 ,

(6) u 1 ∂ C 1 ∂ x 1 + u 2 ∂ C 1 ∂ y 1 + u 3 ∂ C 1 ∂ z 1 = D B ∂ 2 C 1 ∂ x 1 2 + ∂ 2 C 1 ∂ y 1 2 + ∂ 2 C 1 ∂ z 1 2 + D T T ∞ ∂ 2 T 1 ∂ x 1 2 + ∂ 2 T 1 ∂ y 1 2 + ∂ 2 T 1 ∂ z 1 2 − k 0 ( C 1 − C ∞ ) .

Subject to boundary conditions are the anisotropic slip-on moving surface, constant NP volume fraction, and constant surface temperature as follows:

(7) u 1 − U = N 1 μ 1 ∂ u 1 ∂ z 1 , u 2 − V = N 2 μ 1 ∂ u 2 ∂ z , u 3 = 0 T 1 = T w , C = C w , k 1 ∂ T 1 ∂ z 1 z 1 = 0 = ρ ( λ + c s ( T m − T s ) ) u 2 ( x , 0 ) , at z 1 → 0 u 1 = a 1 x 1 , u 2 = a 1 y 1 , u 3 = − 2 a 1 z 1 , T 1 = T ∞ , C 1 = C ∞ , as 1 z → ∞ .

According to (ref. [60]), the radiative heat flux q r is given by:

(8) q r = − 4 σ 1 3 k 1 ∂ T 1 4 ∂ y 1 .

The temperature differences within motion are as follows:

(9) T 1 4 = T ∞ 4 + 4 T ∞ 3 ( T 1 − T ∞ ) + 6 T ∞ 2 ( T 1 − T ∞ ) 3 + …

Now, by removing the higher-order terms beyond the first degree in ( T 1 − T ∞ ) , then

(10) T 1 4 = 4 T ∞ 3 T 1 − 3 T ∞ 4 .

Solving Eq. (8) using Eq. (10), it can be transformed as follows:

(11) ∂ q r ∂ z 1 = − 4 σ 1 T ∞ 3 3 k 1 ∂ 2 T 1 ∂ z 1 2 .

Therefore, using Eq. (11), then the energy Eq. (5) becomes

(12) u 1 ∂ T 1 ∂ x 1 + u 2 ∂ T 1 ∂ y 1 + u 3 ∂ T 1 ∂ z 1 = α 1 ∂ 2 T 1 ∂ x 1 2 + ∂ 2 T 1 ∂ y 1 2 + ∂ 2 T 1 ∂ z 1 2 + 16 σ 1 T ∞ 3 3 ( ρ c p ) k 1 ∂ q r ∂ z 1 ∂ 2 T 1 ∂ z 1 2 − Q 0 ( T 1 − T ∞ ) ( ρ c p ) + τ 1 D B ∂ C 1 ∂ x 1 ∂ T 1 ∂ x 1 + ∂ C 1 ∂ y 1 ∂ T 1 ∂ y 1 + ∂ C 1 ∂ z 1 ∂ T 1 ∂ z 1 + τ 1 D T T ∞ ∂ T 1 ∂ x 1 2 + ∂ T 1 ∂ y 1 2 + ∂ T 1 ∂ z 1 2 .

The suitable similarity variable is as follows:

(13) u 1 = a 1 x 1 f ' ( η ) , u 2 = a 1 y 1 g ′ ( η ) , u 3 = − a 1 υ 1 ( f ( η ) + g ( η ) ) , η = a 1 υ 1 z 1 , θ ( η ) = T 1 − T ∞ T f − T ∞ , ϕ ( η ) = C 1 − C ∞ C f − C ∞ .

Eqs. (2)–(4), (6), and (12) takes the following following form after implementing Eq. 13, we have

(14) f ‴ = M f ′ − f ″ ( f + g ) + ( f ′ ) 2 − 1 ,

(15) g ‴ = ( g ′ ) 2 + g ′ M − g ″ ( f + g ) − 1 ,

(16) θ ″ ( 1 + R d ) + Pr ( θ ′ ( f + g ) + N b θ ′ ϕ ′ + N t ( θ ′ ) 2 − H θ ) = 0 ,

(17) ϕ ″ = γ ϕ − Le Pr ( f + g ) ϕ ′ − N t N b θ ″ .

The converted boundary conditions are

(18) f ' ( 0 ) − λ 1 f ″ ( 0 ) = 0 , g ( 0 ) = 0 , g ′ ( 0 ) − λ 2 g ″ ( 0 ) = 0 θ ( 0 ) = 1 , Pr f = − H θ ′ , ϕ ′ ( 0 ) = 0 , as η → 0 f ′ ( ∞ ) = 1 , g ′ ( ∞ ) = 1 , θ ( ∞ ) = 0 , ϕ ( ∞ ) = 0 , at η → ∞ .

In some of physical important quantities in this work, via x 1 - and y 1 -directions of local Nusselt number Nu x 1 and Nu y 1 , local skin friction C f x 1 and C f y 1 . It is defined as:

(19) Nu x 1 = x 1 q u 3 k 1 ( T w − T ∞ ) , C f x 1 = τ u 3 x 1 ρ f U 2 , Nu y 1 = y 1 q u 3 k 1 ( T w − T ∞ ) , C f y 1 = τ u 3 y 1 ρ f V 2 ,

where τ u 3 x 1 = μ 1 ∂ u 1 ∂ z 1 z 1 = 0 , τ u 3 y 1 = μ 1 ∂ u 2 ∂ z 1 z 1 = 0 , q m = − k 1 ∂ T 1 ∂ z 1 z 1 = 0 .

Substituting Eqs. (13) into Eq. (19), we obtain

(20) Re x 1 − 1 / 2 Nu x 1 = − ( 1 + R d ) θ ′ ( 0 ) , Re x 1 − 1 / 2 C f x 1 = − f ″ ( 0 ) , Re y 1 − 1 / 2 C f y 1 = − g ″ ( 0 ) ,

where the local Reynolds numbers Re x 1 = a 1 x 1 2 / u 2 and Re y 1 = a 1 y 1 2 / u 2 are along x 1 - and y 1 -directions, respectively.

2.1 Numerical analysis

Using shooting technique to solve the nonlinear system of Eqs. (14)–(17). To obtain a numerical solution by selecting a step size Δ η = 0 .001 , we consider the momentum equation in the x and y directions, as well as the energy equations of third and second order, respectively. Momentum equation in the direction of x and y , energy equations are third and second order, respectively:

(21) f ‴ = M f ′ − f ″ ( f + g ) + ( f ′ ) 2 − 1 ,

(22) g ‴ = ( g ′ ) 2 + g ′ M − g ″ ( f + g ) − 1 ,

(23) θ ″ = ( − Pr / ( 1 + R d ) ) ( θ ′ ( f + g ) + N b θ ′ ϕ ′ + N t ( θ ′ ) 2 − H θ ) ,

(24) ϕ ″ = γ ϕ − Le Pr ( f + g ) ϕ ′ − N t N b θ ″ .

Considering dependent variables ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξ 5 , ξ 6 , ξ 7 , ξ 8 , ξ 9 , ξ 10 , ξ 12 , and ξ 13 and substituting in Eqs. (21)–(24), then we obtain the following format:

(25) f = ξ 1 ,

(26) f ′ = ξ 2 ′ = ξ 2 ,

(27) f ″ = ξ 3 ′ = ξ 3 ,

(28) f ‴ = ξ 4 ′ = ξ 4 = M ξ 2 − ξ 4 ( ξ 1 + ξ 5 ) + ( ξ 2 ) 2 − 1 ,

(29) g = ξ 5 ,

(30) g ′ = ξ 6 ′ = ξ 6 ,

(31) g ″ = ξ 7 ′ = ξ 7 ,

(32) g ‴ = ξ 8 ′ = ξ 8 = ( ξ 6 ) 2 − ( ξ 1 + ξ 5 ) ξ 7 − 1 ,

(33) θ = ξ 9 ′ = ξ 9 ,

(34) θ ′ = ξ 10 ′ = ξ 10 ,

(35) θ ″ = ξ 11 ′ = ξ 11 = ( − Pr / ( 1 + R d ) ) ( ξ 10 ( ξ 1 + ξ 5 ) + N b ξ 10 ξ 12 + N t ( ξ 10 ) 2 − H ξ 9 ) ,

(36) ϕ = z 11 ′ = z 11 ,

(37) ϕ ′ = z 12 ′ = z 12 ,

(38) ϕ ″ = ξ 13 = γ ξ 11 − Le Pr ( f + g ) ξ 12 − N t N b ( ( − Pr / ( 1 + R d ) ) ( ξ 10 ( ξ 1 + ξ 5 ) + N b ξ 10 ξ 12 + N t ( ξ 10 ) 2 − H ξ 9 ) ) .

Associated boundary layer as formated below

(39) f ′ ( 0 ) − λ 1 f ″ ( 0 ) = 0 , g ( 0 ) = 0 , g ′ ( 0 ) − λ 2 g ″ ( 0 ) = 0 θ ( 0 ) = 1 , Pr f = − H θ ′ , ϕ ′ ( 0 ) = 0 , as η → 0 f ′ ( ∞ ) = 1 , g ′ ( ∞ ) = 1 , θ ( ∞ ) = 0 , ϕ ( ∞ ) = 0 , at η → ∞ .

To solve system of Eqs. (21)–(25) ten initial conditions must be known, but ξ 4 , ξ 8 , ξ 11 , and ξ 13 are unknown, at ζ → ∞ , boundary condition of f ( ζ ) , g ( ζ ) , θ ( ζ ) , ϕ ( ζ ) are unknown, and these three unknown conditions are indicated by ξ 1 , ξ 5 , ξ 9 , and ξ 11 . In order to achieve an error of less than 10 − 10 , the parameters taken must be approximated to a finite value, denoted as ζ ∞ , using Newton’s scheme.

3 Results and discussion

The variation of M (magnetic field parameter) on f ′ ( η ) and g ′ ( η ) is presented in Figure 2. It is seen that the f ′ ( η ) and g ′ ( η ) declined with various statistical values of M . Furthermore, a variation of relative study reveals that Re x 1 / 2 C f x (skin friction coefficient via axial direction) decays with various distinct values of M t (Melting parameter) against λ 1 (slip parameter via x 1 -axis) for both cases M = 0 and M > 0 , as explored in Figure 3. It is noted that the hydrodynamic ( M = 0 ) is lower than the hydromagnetic case ( M > 0 ) along x 1 -direction. Physically, M is direct proportional to EC (electrical conductivity); due to this, the magnetic field and resistive force applied act more into opposite direction of liquid motion, and low EC, and then liquid speed goes to very slow motion in stretching surface.

$Figure 2 Impact of M M on f ′ ( η ) and g ′ ( η ) f^{\prime} (\eta )\hspace{.5em}\text{and}\hspace{.5em}g^{\prime} (\eta ) .$

Figure 2

Impact of M on f ′ ( η ) and g ′ ( η ) .

$Figure 3 Impact of M t Mt on Re x 1 / 2 C f x {\mathrm{Re}}_{x}^{1/2}{C}_{\text{f}x} .$

Figure 3

Impact of M t on Re x 1 / 2 C f x .

Figure 4 presents the effect of M t (melting parameter) on θ ( η ) (temperature profile). It is noted that the θ ( η ) dwindles with distinct statistical values of M t . Physically, the melting parameter is a combination of Stefan numbers for solid and liquid aspects. Also, it is proportional to specific heat at constant pressure. Due to this fact, the fluid particles yield low temperature.

$Figure 4 Impact of M t Mt on θ ( η ) \theta (\eta ) .$

Figure 4

Impact of M t on θ ( η ) .

The influence of R d (TR parameter) on θ ( η ) is shown in Figure 5a. It is noted that θ ( η ) improves with distinct statistical values of R d , while the reverse behaviour follows concentration as predicted in Figure 5b. Physically, the TR is inversely proportional to TD (thermal diffusivity); due to this, the low TD in liquid motion at stretching surface released high temperature and low concentration.

$Figure 5 (a) Impact of R d {R}_{\text{d}} on θ ( η ) \theta (\eta ) . (b) Impact of R d {R}_{\text{d}} on ϕ ( η ) \phi (\eta ) .$

Figure 5

(a) Impact of R d on θ ( η ) . (b) Impact of R d on ϕ ( η ) .

The variation of Pr (Prandtl number) on θ ( η ) is shown in Figure 6. As expected, thermal BL decays for different statistical values of Pr . Physically, Pr is inversely proportional to TD. Due to this, the TD released low temperature in NF motion at surface area.

$Figure 6 Impact of Pr \Pr on θ ( η ) \theta (\eta ) .$

Figure 6

Impact of Pr on θ ( η ) .

Figure 7 indicates the behaviour of N t (thermophoresis parameter) on θ ( η ) . It is seen that the thermal BL converges high in region 1 ≤ η ≤ 0.8 (approximate region) with ascending statistical values of N t , whereas the opposite behaviour displays N b (Brownian motion parameter), as shown in Figure 8. Physically, the Brownian motion and thermophoresis parameters are inversely proportional to kinematic viscosity. The kinematic viscosity released high temperature in NF motion at SS.

$Figure 7 Impact of N t {N}_{\text{t}} on θ ( η ) \theta (\eta ) .$

Figure 7

Impact of N t on θ ( η ) .

$Figure 8 Impact of N b {N}_{\text{b}} on θ ( η ) \theta (\eta ) .$

Figure 8

Impact of N b on θ ( η ) .

Figure 9 shows the effect of Le (Lewis number) on ϕ ( η ) (concentration profile). It is observed that the low concentration BL for ascending numerical values of Le . Physically, the Lewis number is ratio of TD to Brownian diffusivity. The low TD released low concentration in motion of NFs.

$Figure 9 Impact of Le \text{Le} on ϕ ( η ) \phi (\eta ) .$

Figure 9

Impact of Le on ϕ ( η ) .

The impact of λ 1 (slip parameter via axial direction direction) and λ 2 (slip parameter via transverse direction) on Re x 1 / 2 C f y is shown in Figure 10. It is clear that the skin friction enhances along the y 1 -direction for enlarged values of λ 1 . Physically, the slip factor is proportional to dynamic viscosity. NFs with low dynamic viscosity produce low coefficient of skin friction at SS.

$Figure 10 Impact of λ 1 {\lambda }_{1} on Re x 1 / 2 C f y {\mathrm{Re}}_{x}^{1/2}{C}_{\text{f}y} .$

Figure 10

Impact of λ 1 on Re x 1 / 2 C f y .

Figure 11 presents the behaviour of NFs and RFs (regular fluids) with various ascending values of H (heat source parameter) against λ 1 on the HT rate. It is observed that the Re x − 1 / 2 Nu x declined for distinct ascending statistical values of H . It is also stated that the high HT rate is reduced in case of NFs when compared to RF motion.

$Figure 11 Impact of H H on Re x − 1 / 2 Nu x {\mathrm{Re}}_{x}^{-1/2}{\text{Nu}}_{x} .$

Figure 11

Impact of H on Re x − 1 / 2 Nu x .

4 Conclusion

This study provides valuable insights into the behaviour of NFs in terms of liquid motion and HT. The findings can be used to optimize the design and performance of systems involving NFs, and to enhance the efficiency of processes such as boiling, solar energy utilization, and micro power generation. The results highlight the importance of considering factors such as slip influence, magnetic field, and chemical reactions in the analysis of NF behaviour. The findings also emphasize the potential of NFs to significantly enhance heat transfer compared to regular liquids. Further research can build upon these findings to explore additional parameters and conditions, and to develop more accurate and comprehensive models for NF behaviour. Overall, this research contributes to the understanding and advancement of NF technology, opening up new possibilities for its application in various engineering fields. The main results are mentioned as follows:

The f ' ( η ) and g ' ( η ) decrease for distinct enlarged statistical values of M . On the other hand, the Re x 1 / 2 C f x decays along the x 1 -direction with various ascending values of M t for the cases of M = 0 and M > 0 .
The Re x 1 / 2 C f y enhances along the y 1 -direction with distinct enlarged values of λ 1 against λ 2
The variation of Re x 1 / 2 Nu x decreases for the cases of NFs and RFs with distinct enlarged values of H .

Acknowledgments

The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project (Grant No. PNURSP2024R12), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

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.

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Received: 2023-07-01

Revised: 2023-08-26

Accepted: 2023-10-12

Published Online: 2024-02-23

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

Articles in the same Issue

https://doi.org/10.1515/phys-2023-0131

Keywords for this article

melting effect; nanofluid; MHD; heat source; chemical reaction; slip condition; thermal radiation

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