Investigating the behaviour of electro-magneto-hydrodynamic Carreau nanofluid flow with slip effects over a stretching cylinder

Muhammad Afzal; Faiza Zahid; Badar E Alam; Mohammed M. M. Jaradat; Imran Siddique; Bagh Ali; Binjian Ma

doi:10.1515/ntrev-2025-0166

Article Open Access

Investigating the behaviour of electro-magneto-hydrodynamic Carreau nanofluid flow with slip effects over a stretching cylinder

Muhammad Afzal , Faiza Zahid , Badar E Alam , Mohammed M. M. Jaradat , Imran Siddique , Bagh Ali and Binjian Ma

Published/Copyright: May 14, 2025

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From the journal Nanotechnology Reviews Volume 14 Issue 1

Abstract

This study primary focus to analyses the slip flow of electro-magneto-hydrodynamic (EMHD) Carreau nanofluid across a stretching cylinder in the context of Arrhenius activation energy, chemical reactions, and variable thermal conductivity within a porous medium. To observe the effects of Brownian motion and thermophoresis, which are critical in nanofluid dynamics, Buongiorno’s model is employed. Also, the effects of changing EMHD forces on fluid flow dynamics are investigated. Due to the strong Lorentz force that is produced when electric and magnetic fields interact, it is crucial to take into account their combined effects in a variety of industrial applications. By introducing nonsimilarity variables, the partial differential equations are transformed into a system of coupled ordinary differential equations (ODEs). The bvp4c solver, a MATLAB built-in solver, is implemented to compute the solution to the resultant set of ODEs. Graphs demonstrate how various parameters influence the profiles of velocity, microorganisms, concentration, and temperature. It is observed that thermophoresis significantly influences the thermal and concentration boundary layer regions. Skin friction increases as the curvature parameter value is enhanced. The local Nusselt number rises as the Prandtl number value increases. The local Sherwood number increases with the constant thermophoresis and the Schmidt number increases. As the curvature parameter, bioconvection Schmidt number rises and the value of local motile density increases.

Keywords: Carreau nanofluid; EMHD; variable thermal conductivity; permeable medium; chemical reaction; slip boundary condition; stretching cylinder; bvp4c

Nomenclature

b: chemotaxis constant (m)
C: nanoparticles volume fraction (–)
C ∞: ambient nanoparticle volume fraction (–)
C p: specific heat ( J kg − 1 K − 1 )
D B: Brownian diffusion coefficient ( m 2 s − 1 )
D m: microorganism diffusion coefficient ( m 2 s − 1 )
k r: rate of chemical reaction ( mol m − 1 s − 1 )
m: fitted rate constant (–)
M: magnetic parameter
n: power law index (–)
N ∞: ambient nanoparticle volume fraction (–)
Nu x: Nusselt number
Pr: Prandtl number
q r: radiative heat flux ( W s − 2 )
T: nondimensional temperature (K)
T w: temperature at bottom disk (K)
T ∞: ambient temperature (K)
W c: swimming speed of micro-organism ( m s − 1 )
( u , v ): velocity components
ν: kinematic viscosity ( m − 2 s − 1 )
ρ: density of fluid ( kg m − 3 )
κ: thermal conductivity of fluid ( W m − 1 K − 1 )
σ: fluid electrical conductivity ( kg − 1 m 3 A 2 )
α: thermal diffusivity ( m 2 s − 1 )
η: dimensionless distance (–)
κ 1: permeability of the porous medium ( m 2 )

1 Introduction

Recent research over the past few decades highlights the growing interest in the study of non-Newtonian fluids within the field of fluid dynamics. Non-Newtonian fluid flows find extensive applications in industrial and manufacturing processes [1], including the creation of insulating materials, extrusion, hot rolling, metal spinning, and metal extrusion. Moreover, integrating heat transfer with stretching flow concepts is crucial for enhancing the efficiency of these applications [2,3]. Shear rate and shear stress have a nonlinear connection in non-Newtonian fluids. From the modelling point of view, the Carreau model is fundamental since it illustrates power-law behaviour and is frequently employed as a core predictive method. The Carreau model is a four-parameter one first put forth by Carreau [4] in 1972 and was developed in a series of many different research works. Gomathi and De investigated the effect of ion slip on the dynamics of Casson-Williamson flow and reported that an increase in viscous forces, combined with yield stress, leads to a reduction in fluid velocity [5]. In the study by Raju and Sandeep [6], the researchers conducted a study on magnetohydrodynamic (MHD) Carreau fluid using cross-diffusion over a wedge. They observed that with an increase in wedge angle parameter, the velocity field of fluid also increases. Using different flow configuration, many other researchers focus on dynamics of various non-Newtonian fluid subject to different physical conditions [7–11]. Raju and Sandeep [12] conducted a comprehensive investigation of the effects of an uncertain heat source or sink and nonlinear thermal radiation in the presence of both homogeneous and heterogeneous phenomena. They extensively studied and analysed the implications of these factors on the heat transfer characteristics and behaviour of the system. The Buongiorno model was used by Khan et al. [13] to predict the behaviour of a time-dependent Carreau fluid along a wedge. They discovered that increasing the Weissenberg number raises temperature of the fluid.

A fluid and nanoparticles are combined to form a nanofluid. In nanofluids, nanoparticles are frequently formed of metals, oxides, carbides, and carbon nanotubes. Compared to their respective base fluids, nanofluids have very distinct thermophysical characteristics. Numerous industries [14], including cooling of transformer oil, cooling of electronics, residential refrigerators and freezers, solar water heating, and engine cooling, use nanofluids [15–17]. In an experiment conducted few years ago, Choi and Eastman [18] and Choi et al. [19] introduced nanoparticles to a base fluid and looked at the thermophysical properties of the nanofluids in comparison to the base fluids. He found that the efficiency of nanofluids is improved, especially in terms of their thermal conductivity, which is much higher than that of the base fluid. Numerous scholars have engaged in theoretical and experimental works in this area since this ground-breaking experiment [9,20,21]. Microbes residing in a fluid are responsible for the occurrence of bioconvection, a phenomenon that plays a role in stabilising nanoparticles and increasing the fluid’s susceptibility to thermal and mass transport. This aspect has been studied by Kuznetsov [22]. Fields such as biomedical engineering, biological technology, and environmental systems rely on the suspension of nanoparticles and bioconvection, as highlighted by Chan et al. [23].

Static charges provide an electric field, whereas the changing motion of electric charges produces a magnetic field. MHD is the study of magnetic properties and dynamics of electrically conducting fluids. In the disciplines of engineering, transportation, and medical sciences, MHD has a broad spectrum of applications. For instance, a strong electric field could be used to control or halt the development of a brain tumour. The research conducted by Sparrow and Cess [24] presented a mathematical model that elucidates the behaviour of free convective flow in the presence of a magnetic field. The study focused on examining the specific characteristics of MHD flow under forced and free convective conditions between two parallel plates. The study conducted by Mazumder et al. [25] revealed that Hall currents have a substantial impact on heat transfer characteristics and flow. The research conducted by Raptis [26] focused on analysing the dynamics of unsteady convective flow through a porous medium. In their numerical investigation, Rudraiah et al. [27] examined the natural convective fluid flow within a rectangular enclosure. They observed that the impact of a magnetic field on the rate of heat transfer is more pronounced in a section with a small Grashof number. Certainly, the study carried out by AboEldahab [28] focused on examining the heat transfer characteristics of free convection MHD flow as it passes a stretching sheet. The analysis took into consideration the impact of radiation and buoyancy effects on flow and heat transfer phenomena in this context. Awais et al. [29] made notable observations regarding the dynamics of MHD Sisko fluid flowing towards a stretching cylinder. They discovered that velocity and temperature are affected differently depending on the magnetic field’s strength. As a result of the heating effect caused by the magnetic field, they specifically observed a drop in velocity and a rise in temperature. The movement of the electrically conducting fluids is influenced by an external magnetic field [30]. A Lorentz force results from this interaction.

The electric field [31] generates the strong Lorentz force needed for flow control in industrial applications, and Sajid and Hayat [32] examined the impact of heat radiation on boundary layer flow caused by an exponentially stretched sheet. Shehzad et al. [33] examined the 3D flow properties of an Oldroyd-B fluid with a heat source, a heat sink, and varying thermal conductivity. In the studies conducted by Waqas et al. [34] and Dogonchi and Ganji [35] the characteristics of heat transfer mechanisms are examined with respect to variable thermal conductivity and heat source/sink. Akbar et al. [36] observed and investigated the effects of heat transfer on a vertical stretching plate. The study considered the influence of variable thermal conductivity and ohmic heating on heat transmission outcomes. The least amount of energy necessary for reactants to go through a chemical reaction or physical transport is called activation energy. Due to its many uses in compound innovation, geothermal artificial lakes, the recovery of thermal lubricant, and the simmering of atomic reactors, Arrhenius energy with mass transport phenomena and chemical reactions has been extensively studied. It might be challenging to calculate the activation energy using the Arrhenius equation when the temperature varies greatly with the rate constant. Since conversions can have a variety of impacts on reactants, it is imperative to be efficient with the reaction and minimize energy waste. In their experimental study [37], they focused on the convection flow of heat transmission through the combination of chemical reactions and thermal radiation. By analysing the obtained experimental data, they successfully developed a fouling resistance-based model for heat transfer. Khan et al. [38] examined how the cone and plate in a porous medium promoted the formation of chemically reactive species and mixed convection at the MHD Williamson nanofluid.

Based on the aforementioned discussion, the primary aim of this study is to investigate the effects of changing electro-magneto-hydrodynamic (EMHD) forces on fluid flow dynamics in the slip flow of an EMHD bioconvective Carreau nanofluid over a stretching cylinder within a porous medium. In addition, the study incorporates factors such as chemical reactions, Arrhenius activation energy, and variable thermal conductivity. Microorganisms are responsible for the occurrence of bioconvection, a phenomenon that plays a role in stabilising nanoparticles and increasing the fluid’s susceptibility to thermal and mass transport. The existing literature indicates that the current constructed problem has not yet been thoroughly investigated. The outcomes of this research hold significant relevance and practical implications across multiple industries. Specifically, fields such as metallurgical procedures, blow moulding, glass fibres, and extrusion processes can benefit from the findings and insights derived from this study. A wise scope of nanofluid and non-Newtonian fluids motivated us to investigate present elaborated problem. The objective of this report is to find the answer of these questions:

What do Arrhenius activation energy, chemical reactions, and variable thermal conductivity within a porous medium have to influence dynamics of Carreau nanofluid?
What is the variation in quantities of engineering interest against involved physical involved parameters, to optimise the thermal management?
What is the role of nanoparticles thermophoresis inside the Carreau nanofluid?

The content of this article is organised as follows: Section 1 presents a detailed review of the relevant literature and the research gap. Section 2 and 3 outline the mathematical modelling and governing equations and quantities of engineering interest. Section 4 explains the numerical method employed to solve the equations. Section 5 discusses the results obtained and provides a comprehensive analysis of the findings. Finally, Section 6 concludes the study, summarising the key outcomes and suggesting potential areas for future research.

2 Mathematical formulation

By incorporating thermal radiation, Arrhenius activation energy, variable thermal conductivity, and slip boundary conditions and zero mass boundary conditions, we investigate a two-dimensional electromagnetic hydrodynamics (EMHD) system where a Carreau bio-nanofluid slip flows steadily over a stretching cylinder. In our analysis, we examine the incompressible flow of a viscous fluid while taking into account the simultaneous presence of an applied magnetic field B and an electric field E . The surface is located at r = 0 , variable magnetic field B ( x ) = B * x l , and variable electric field E ( x ) = E * x l has been applied orthogonally to the fluid flow. It is considered that the stretching velocity in the x -direction is u w = a x L . The magnetic Reynolds number is of such a low magnitude that it does not generate a significant magnetic field. In Figure 1, the cylindrical axis is aligned along the x -axis, while the radial direction is represented by the r -axis.

Figure 1

Problem geometry.

The expression for the stress tensor of Cauchy in the case of a Carreau fluid is expressed by:

(1) τ ¨ = μ ˆ ́ A ˆ ́ ,

(2) μ ˆ ́ − μ ˆ ́ ∞ μ ˆ ́ 0 − μ ˆ ́ ∞ = [ 1 + ( Γ γ ˘ ) 2 ] n − 1 2 .

The parameter μ ˆ ́ ∞ corresponds to the viscosity at an infinite shear rate. The term μ ˆ ́ 0 represents the viscosity at a zero shear rate. The symbol A ˆ ́ refers to the first kind Rivlin-Erickson tensor. Γ represents a material time constant. The variable μ ˆ ́ denotes the shear-rate viscosity [39,40]:

(3) A ˆ ́ = ∇ V + ( ∇ V ) T ,

(4) γ ˘ = 1 2 ∑ i ∑ j γ ˘ i j γ ˘ i j = 1 2 ∏ = 1 2 t r ( A ˆ ́ 2 ) .

The second invariant strain tensor is ∏ . For simplification of research, we assumed η ˆ ˘ ∞ = 0 and Γ γ ˘ < 1 and can be defined thus by using binomial expansion equation (2) as follows:

(5) μ ˆ ́ = μ ˆ ́ ∘ 1 + n − 1 2 ( Γ γ ˘ ) 2 .

Equation (1) can be rewritten as follows:

(6) τ ¨ = μ ˆ ́ ∘ 1 + n − 1 2 ( Γ γ ˘ ) 2 A ˆ ́ .

For a Carreau nanofluid, the variable thermal conductivity can be described as follows:

κ ( T ) = κ ∞ ( 1 + ε θ ) .

The equations governing are given as as follows [41–44]:

(7) ∂ ( r u ) ∂ x + ∂ ( r v ) ∂ r = 0 ,

(8) u ∂ u ∂ x + v ∂ u ∂ r = ν ∂ 2 u ∂ r 2 + ν 1 r ∂ u ∂ r + ν 3 Γ 2 ( n − 1 ) 2 × ∂ u ∂ r 2 ∂ 2 u ∂ r 2 + ν Γ 2 ( n − 1 ) 2 r ∂ u ∂ r 3 − σ ρ ( B * 2 ( x ) u − E * ( x ) B * ( x ) ) − ν κ 1 u ,

(9) u ∂ T ∂ x + v ∂ T ∂ r = 1 ρ C p 1 r ∂ ∂ r κ ( T ) r ∂ T ∂ r + τ D B ∂ C ∂ r ∂ T ∂ r + D T T ∞ ∂ T ∂ r 2 − 1 ρ C p r ∂ r q r ∂ r + Q 1 ( T − T ∞ ) ρ C p ,

(10) u ∂ C ∂ x + v ∂ C ∂ r = D B ∂ 2 C ∂ r 2 + D B r ∂ C ∂ r + D T T ∞ r ∂ T ∂ r + D T T ∞ ∂ 2 T ∂ r 2 − κ r ( C − C ∞ ) T T ∞ m exp − E a κ b T ∞ ,

(11) u ∂ N ∂ x + v ∂ N ∂ r + b W C ( C w − C ∞ ) ∂ ∂ r N ∂ C ∂ r = D m r ∂ ∂ r r ∂ N ∂ r ,

where u and v are velocity components, ν is the kinematic viscosity, n is the power law index, g is the gravitational field, κ 1 is the porosity, σ is the electrical conductivity, ρ is the fluid density, T ∞ is the ambient temperature, T is the temperature, T w is the temperature at wall, α is the thermal conductivity, C p is the specific heat, q r is the thermal radiation, Q 1 is the heat source, C is the concentration, D T is the thermophoretic diffusion, C w is a concentration at the wall, C ∞ is the ambient concentration, N is the concentration of microorganisms, N w is the concentration of microorganisms at the wall, N ∞ is the ambient concentration of microorganisms, κ r is the chemical reaction rate, m is the fitted rate constant, D B is the Brownian diffusion, D m is the microorganism diffusivity, W C is the swimming speed of maximum cell, and b is the chemotaxis constant. The thermal radiation is approximated by the Rosseland model as follows [45]:

(12) ∂ q r ∂ r = − 16 σ * T ∞ 3 3 κ * ∂ 2 T ∂ r 2 ,

where κ * is the mean absorption coefficient and σ * is Stefan-Boltzmann constant. The boundary conditions as follows [46,47]:

(13) u = u w + u slip = u 0 x l + l 0 ∂ u ∂ r 1 + n − 1 2 Γ 2 ∂ u ∂ r 2 , T = T w + l 2 ∂ T ∂ r , v = 0 , D B ∂ C ∂ r + D ∞ T ∞ ∂ T ∂ r = 0 , N = N w at r = R ,

(14) u → 0 , T → T ∞ , C → C ∞ , N → N ∞ , as r → ∞ .

The similarity transformations are defined as follows [44,46]:

(15) ξ = u 0 l ν r 2 − R 2 2 R , Ψ ( ξ ) = ν u 0 l x R f ( ξ ) , u = 1 r ∂ Ψ ∂ r , v = − 1 r ∂ Ψ ∂ x , θ ( ξ ) = T − T ∞ T w − T ∞ , χ ( ξ ) = N − N ∞ N w − N ∞ , ϕ ( ξ ) = C − C ∞ C w − C ∞ .

In view of equation (15), the governing system of equations (7) and (14) is expressed as follows:

(16) ( 1 + 2 ζ ξ ) f ‴ + 2 ζ f ″ + 3 ( n − 1 ) 2 We 2 ( 1 + 2 ζ ξ ) × ( f ″ ) 2 ( ( 1 + 2 ζ ξ ) f ‴ ( ξ ) + ζ f ″ ( ξ ) ) + ( n − 1 ) 2 We 2 ( 1 + 2 ζ ξ ) ζ ( f ″ ( ξ ) ) 3 − ( f ′ ( ξ ) ) 2 + f ( ξ ) f ″ ( ξ ) − M 1 2 ( f ′ ( ξ ) − F ) − K p f ′ ( ξ ) = 0 ,

(17) ( 1 + 2 ζ ξ ) ( ( 1 + ε θ ) θ ″ + ε θ ′ 2 ) + ( 1 + ε θ ) 2 ζ θ ′ + ( 1 + 2 ζ ξ ) P r ( Nt ( θ ′ ) 2 + Nb θ ′ ϕ ′ ) + P r ( f θ ′ + Q θ ) + 4 3 Rd ( ( 1 + 2 ζ ξ ) θ ″ + ζ θ ′ ) = 0 ,

(18) ( 1 + 2 ζ ξ ) ϕ ″ + 2 ζ ϕ ′ + Sc f ϕ ′ + Nt Nb ( 2 ζ θ ′ + ( 1 + 2 ζ ξ ) θ ″ ) − Sc γ r ( 1 + δ θ ) m ϕ exp − E 2 ( 1 + δ θ ) = 0 ,

(19) ( 1 + 2 ζ ξ ) χ ″ + 2 ζ χ ′ + Sb f χ ′ − Pe ( ( 1 + 2 ζ ξ ) ( ϕ ′ χ ′ + ( χ + δ 1 ) ϕ ″ ) + ( χ + δ 1 ) ζ ϕ ′ ) = 0 ,

with boundary conditions:

(20) f ( 0 ) = 0 , f ′ ( 0 ) = 1 + h 1 f ″ ( 0 ) 1 + n − 1 2 We 2 ( f ″ ( 0 ) ) 2 , f ′ ( ∞ ) = 0 , θ ( 0 ) = 1 + h 2 θ ′ ( 0 ) , θ ( ∞ ) = 0 , Nb ϕ ′ ( 0 ) + Nt θ ′ ( 0 ) = 0 , ϕ ( ∞ ) = 0 , χ ( 0 ) = 1 , χ ( ∞ ) = 0 ,

where ζ = 1 R ν l u 0 is the curvature parameter, M 1 2 = σ B * 2 l ρ u 0 is a magnetic parameter, E = E * B * u w 2 refers to the electric parameter, Q = l Q 1 ρ u 0 C p depicts the heat generation parameter, K p = ν l κ 1 u 0 is the porosity parameter, We 2 = Γ 2 x 2 u 0 3 ν l 3 is the Weissenberg number, E 2 = E a κ b T ∞ represents the activation parameter, Rd = 4 σ * T ∞ 3 κ * κ depicts the radiation parameter, Nb = τ D B ( C w − C ∞ ) ν depicts the Brownian parameter, Nt = τ D T ( T w − T ∞ ) ν T ∞ depicts dimensionless thermophoresis constant, γ r = κ r l u 0 is the dimensionless parameter of chemical reaction, P r = α ν depicts the Prandtl number, Sc = ν D B depicts the Schmidt number, δ 1 = N w N w − N ∞ depicts the microorganism difference dimensionless parameter, E 1 = − E a κ b T ∞ , Sb = ν D m depicts the dimensionless bioconvection Schmidt number, Pe = b W c D m denotes the dimensionless Peclet number, Bi = h f k ν l a depicts the Biot number, h 1 = l 0 u 0 l ν is the slip velocity parameter, h 2 = l 1 u 0 l ν is the slip temperature parameter, and δ = T w − T ∞ T ∞ depicts the temperature difference.

3 Physical characteristics

3.1 Skin friction coefficient

Frictional drag on a surface is defined by the dimensionless factor C f . It is described as follows:

(21) C f = τ w ¨ 2 ρ u w 2 .

The term for shear stress is:

(22) τ w ¨ = μ ˆ ́ ∂ u ∂ r + Γ 2 ( n − 1 ) 2 ∂ u ∂ r 3 r = R ,

By adding the aforementioned equation to equation (21), we obtain

(23) C f ( Re x ) − 1 2 2 = f ″ ( 0 ) + ( n − 1 ) 2 We 2 ( f ″ ( 0 ) ) 3 ,

where Re x stands for the local Reynolds number, which is Re x = u w x ν .

3.2 Local Nusselt number

To examine the heat transmission rate at the wall, Nu x is defined as follows:

(24) Nu x = x q w ˆ ̀ κ ( T w − T ∞ ) .

At the wall, Fourier’s law is defined as follows:

(25) q w ˆ ̀ = − κ ∂ T ∂ r r = R + ( q r ) r = R .

By adding the aforementioned equation to equation (24), we obtain

(26) Nu x ( Re x ) − 1 2 = − 1 + 4 3 Rd θ ′ ( 0 ) .

3.3 Local Sherwood number

The local Sherwood number Sh x is determined as follows:

(27) Sh x = x j w ˆ D B ( C w − C ∞ ) .

By applying Fick’s law, the mass transfer can be represented as follows:

(28) j w ˆ = − D B ∂ C ∂ r r = R .

By adding the aforementioned equation to equation (28), we obtain

(29) Sh x ( Re x ) − 1 2 = − ϕ ′ ( 0 ) .

3.4 Local density of motile microorganisms

The local density of motile microorganisms is given as follows:

(30) Nn x = x i w ˆ D m N W − N ∞ .

(31) i w ˆ = − D m ∂ N ∂ r r = R .

By adding the aforementioned equation to equation (30), we obtain

(32) Nn x ( Re x ) − 1 2 = − χ ′ ( 0 ) .

4 Numerical approach

In the realm of engineering and scientific simulations, complex and higher-order differential equations find frequent application. Various mathematical approaches are available in the current literature for implementation in computational simulations. Achieving high precision and efficiency, the dimensionless observations (16)–(19) and boundary conditions (20) can be effectively addressed using the bvp4c solver provided by MATLAB’s computational tool. The utilisation of this technique is reliable, very smooth, and simple. For this present elaborated model of Carreau nanofluid across a stretching cylinder in the context of Arrhenius activation energy, chemical reactions, and variable thermal conductivity within a porous medium, the B V P 4 ′ c choice is motivated by its an excellent performance in accuracy and stability to solve ordinary differential equations (ODEs) [48]. To begin the process, the coupled nonlinear ODEs are initially transformed into first-order representations to facilitate the utilisation of the given equation:

f = ϒ 1 , f ′ = ϒ 1 ′ = ϒ 2 , f ″ = ϒ 2 ′ = ϒ 3 , θ = ϒ 4 , θ ′ = ϒ 4 ′ = ϒ 5 , ϕ = ϒ 6 , ϕ ′ = ϒ 7 , χ = ϒ 8 , χ ′ = ϒ 8 ′ = ϒ 9 ,

(33) ϒ 3 ′ = 1 ( 1 + 2 ζ ξ ) + 3 ( n − 1 ) 2 ( 1 + 2 ζ ξ ) 2 We 2 ( ϒ 3 ) 2 × − 2 ζ ϒ 3 − ( n − 1 ) 2 We 2 ζ ( 1 + 2 ζ ξ ) ( ϒ 3 ) 3 − 3 ( n − 1 ) 2 We 2 ζ ( 1 + 2 ζ ξ ) ( ϒ 3 ) 3 + M 1 2 ( ϒ 2 + G ) + K p ϒ 2 + ( ϒ 2 ) 2 ,

(34) ϒ 5 ′ = 1 ( 1 + 2 ζ ξ ) ( 1 + ε ϒ 4 ) + 4 3 ( 1 + 2 ζ ξ ) Rd × ( − 2 ζ ( 1 + ε ϒ 4 ) ϒ 5 − ( 1 + 2 ζ ξ ) ε − P r ( 1 + 2 ζ ξ ) ( Nt ( ϒ 5 ) 2 + Nb ϒ 5 ϒ 7 ) − P r ( Q ϒ 4 + ϒ 1 ϒ 5 ) − 4 3 ζ Rd ϒ 5 ,

(35) ϒ 7 ′ = 1 ( 1 + 2 ζ ξ ) − 2 ζ ϒ 7 − 2 ζ Nt Nb ϒ 5 − ( 1 + 2 ζ ξ ) Nt Nb ϒ 5 ′ − Sc ϒ 1 ϒ 7 + γ r Sc ( 1 + δ ϒ 4 ) m ϒ 6 e x p − E 2 ( 1 + δ ϒ 4 ) ,

(36) ϒ 9 ′ = 1 ( 1 + 2 ζ ξ ) ( − 2 ζ ϒ 9 − Sb ϒ 1 ϒ 9 + Pe ( ( 1 + 2 ζ ξ ) × ( ϒ 7 ϒ 9 + ( δ 1 + ϒ 8 ) ϒ 7 ′ ) + ( δ 1 + ϒ 8 ) ζ ϒ 7 ) ) ,

(37) ϒ 1 ( 0 ) = 0 , ϒ 2 ( 0 ) = 1 + h 1 ϒ 3 ( 0 ) 1 + n − 1 2 We 2 ( ϒ 3 ( 0 ) ) 2 , ϒ 2 ( ∞ ) = 0 ,

(38) ϒ 4 ( 0 ) = 1 + u 1 ϒ 5 , ϒ 4 ( ∞ ) = 0 , Nb ϒ 7 ( 0 ) + Nt ϒ 5 ( 0 ) = 0 , ϒ 6 ( ∞ ) = 0 , ϒ 8 ( ∞ ) = 0 , ϒ 8 ( 0 ) = 1 .

The detailed about build-in scheme, initial guess, and convergence criteria are well explained by Ali et al. [49]. To observe the validity of present MATLAB code, we validate our current outcomes against already published numerical value for specific case. For ζ curvature distinct strength, it is reported that our code outcomes match with aforementioned studies. The present obtained outcomes are tested with the already existing literature [50,51] against the limited case and are found to be an excellent agreement, as indicated in Table 1. Therefore, we are confident that present findings are reliable and correct.

Table 1

When analysing the variation of C f ( Re ) − 1 2 2 with different values of ζ , the following parameter values are considered: We = G = M 1 = K p = h 1 = 0 and n = 1

ζ	Rangi et al. [50]	Hashim et al. [51]	Current result
0	− 1.0000	− 1.0000	− 1.0000
0.25	− 1.094378	− 1.094373	− 1.090845
0.5	− 1.188715	− 1.188727	− 1.188672
0.75	− 1.281833	− 1.281819	− 1.281823

5 Results and discussion

This section examines the influence of various key physical parameters involved in the problem, including the magnetic parameter ( M ) , heat generation ( Q ) , porosity ( K p ) , Weissenberg number (We), activation energy ( E 2 ) , radiation ( Rd ) , Brownian motion ( Nb ) , chemical reaction ( γ r ) , thermophoresis ( Nt ) , Prandtl number ( P r ) , Schmidt number ( Sc ) , microorganism difference dimensionless parameter ( δ 1 ) , dimensionless bioconvection Schmidt number ( E 1 ) , dimensionless Peclet number ( Pe ) , Biot number (Bi), slip velocity ( h 1 ) , slip temperature ( h 2 ) , and temperature difference ( δ ) , on fluid speed, temeprature, and concentration functions are analysed in detail. Table 2 illustrates how the C f rises with ζ , permeability parameter, and has a negligible effect when n , M 1 , We , and E are increased and diminishes as the slip parameter surges. Table 3 demonstrates that Nu x increases with Pr, while it decreases with an increase in Rd , Q , h 2 , ζ , and Nt parameters and has a negligible effect on Nu x when the value of Brownian motion factor surges. Table 4 demonstrates that the Sh x boosted with the Nt , Sc increased while it diminishing with an increment in ζ , Nb and negligible effect on Sh x when increased γ r , m , E 2 , δ . In Table 5, the value of Nn x surges as ζ and Sb increase and then diminishes as the Pe , and δ 1 increase.

Table 2

Taking into account the impacts of various factors on the coefficient of skin friction when ε = 0.1 , Rd = 0.8 , Sc = E 2 = 0.2 , δ = 0.2 , γ r = 0.8 , m = 1.3 , Nb = Q = 0.1 , Sb = 0.5 , Pe = δ 1 = 0.2 , h 2 = Nt = 0.1 , Pr = 7

ζ	M 1	We	n	E 2	K p	h 1	− C f ( Re ) − 1 2 2
0.2	0.1	0.1	0.5	0	0.1	0.1	0.976474
0.5	—	—	—	—	—	—	1.06186
0.7	—	—	—	—	—	—	1.11586
0.2	0.3	—	—	—	—	—	1.00852
—	0.5	—	—	—	—	—	1.0682
—	0.7	—	—	—	—	—	1.14919
—	0.1	0.1	—	—	—	—	0.976474
—	—	0.3	—	—	—	—	0.971673
—	—	0.5	—	—	—	—	0.961282
—	—	0.1	0.5	—	—	—	0.976474
—	—	—	0.7	—	—	—	0.976709
—	—	—	0.9	—	—	—	0.976942
—	—	—	0.5	0	—	—	0.976474
—	—	—	—	0.1	—	—	0.974657
—	—	—	—	0.2	—	—	0.97287
—	—	—	—	0	0.1	—	0.976474
—	—	—	—	—	0.3	—	1.05377
—	—	—	—	—	0.5	—	1.12318
—	—	—	—	—	0.1	0.1	0.976474
—	—	—	—	—	—	0.3	0.776998
—	—	—	—	—	—	0.5	0.650122

Table 3

Analysing the influence of multiple factors on the Nu x when We = M 1 = h 2 = K p = 0.1 , n = 0.5 , E = 0.1 , h 1 = 0.1 , E 2 = Sc = 0.2 , δ = 0.2 , γ r = 0.8 , Sb = 0.5 , δ 1 = Pe = 0.2 , m = 1.3

ζ	Nt	Nb	ε	Rd	Pr	Q	h 2	− θ ( 0 )
0.2	0.2	0.1	0.1	0.8	7	0.1	0.1	0.869277
0.4	—	—	—	—	—	—	—	0.84314
0.6	—	—	—	—	—	—	—	0.81526
0.2	0.2	—	—	—	—	—	—	0.869277
—	0.5	—	—	—	—	—	—	0.844893
—	0.7	—	—	—	—	—	—	0.828069
—	0.2	0.3	—	—	—	—	—	0.869277
—	—	0.5	—	—	—	—	—	0.869277
—	—	0.7	—	—	—	—	—	0.869277
—	—	0.1	0.1	—	—	—	—	0.869277
—	—	—	0.3	—	—	—	—	0.823928
—	—	—	0.5	—	—	—	—	0.783784
—	—	—	0.1	0.8	—	—	—	0.869277
—	—	—	—	1	—	—	—	0.800687
—	—	—	—	1.2	—	—	—	0.740961
—	—	—	—	0.8	5.5	—	—	0.742537
—	—	—	—	—	6.5	—	—	0.829494
—	—	—	—	—	6.8	—	—	0.853618
—	—	—	—	—	7	0.1	—	0.869277
—	—	—	—	—	—	0.3	—	0.393361
—	—	—	—	—	—	0.4	—	0.30945
—	—	—	—	—	—	0.1	0.1	0.869277
—	—	—	—	—	—	—	0.3	0.745021
—	—	—	—	—	—	—	0.5	0.577464

Table 4

Assessing the implications of various factors influencing the local Sherwood index when We = M 1 = Kp = 0.1 , n = 0.5 , E = 0.1 , h 1 = 0.1 , ε = 0.1 , h 2 = 0.1 , Pr = 7 , Q = 0.2 , Sb = δ 1 = Pe = 0.2 , Rd = 0.8

ζ	Nb	Nt	Sc	γ r	m	δ	E 2	− ϕ ( 0 )
0.2	0.5	0.4	0.2	0.8	1.3	0.2	0.2	0.677332
0.4	—	—	—	—	—	—	—	0.65419
0.6	—	—	—	—	—	—	—	0.629095
0.2	0.5	—	—	—	—	—	—	0.677332
—	0.7	—	—	—	—	—	—	0.483808
—	0.9	—	—	—	—	—	—	0.376296
—	0.5	0.4	—	—	—	—	—	0.677332
—	—	0.6	—	—	—	—	—	0.991908
—	—	0.8	—	—	—	—	—	1.28935
—	—	0.4	0.2	—	—	—	—	0.677332
—	—	—	0.4	—	—	—	—	0.991908
—	—	—	0.6	—	—	—	—	1.28935
—	—	—	0.2	0.8	—	—	—	0.677332
—	—	—	—	1	—	—	—	0.673225
—	—	—	—	1.2	—	—	—	0.669307
—	—	—	—	0.8	1.3	—	—	0.677332
—	—	—	—	—	1.5	—	—	0.668738
—	—	—	—	—	1.7	—	—	0.668159
—	—	—	—	—	1.3	0.2	—	0.677332
—	—	—	—	—	—	0.4	—	0.669307
—	—	—	—	—	—	0.6	—	0.669307
—	—	—	—	—	—	0.2	0.2	0.677332
—	—	—	—	—	—	—	0.4	0.673121
—	—	—	—	—	—	—	0.6	0.676453

Table 5

Taking into account the influence of various factors on motile microorganisms when We = M 1 = K p = 0.1 , n = 0.5 , E = 0.1 , h 1 = 0.1 , h 2 = 0.1 , ε = 0.1 , Nt = 0.1 , Q = 0.1 , Rd = 0.8 , Sc = E 2 = 0.2 , δ = 0.2 , γ r = 0.8 , m = 1.3 , Nb = 0.1 , P r = 7

ζ	Sb	Pe	δ 1	− χ ( 0 )
0.2	0.9	0.2	0.2	0.390263
0.4	—	—	—	0.448539
0.6	—	—	—	0.521424
0.2	0.9	—	—	0.390263
—	1.2	—	—	0.65271
—	1.4	—	—	0.731399
—	0.9	0.2	—	0.390263
—	—	0.4	—	0.506205
—	—	0.6	—	0.491207
—	—	0.2	0.2	0.438326
—	—	—	0.4	0.38609
—	—	—	0.7	0.307736

Figure 2 exhibits the consequence ζ has on the velocity distribution. The ζ exhibits an adverse relationship with R . The radius of the cylinder declines with an enhancement in ζ . A contact area between the fluid and the boundary layer diminishes. As a result, the resistive force exerted by the fluid decreases, leading to a boost in velocity distribution. Figure 3 shows temperature profile is influenced by the ζ . ζ leads to an increase in the temperature distribution. Physically, for greater values of the curvature parameter, the cylinder’s radius decreases, leading to a reduction in the surface area near the solid boundary. It is important to highlight that heat transfer occurs in two modes: conduction at the surface and convection within the fluid domain. As the surface area of the cylinder diminishes, a slight decrease in the temperature profile is observed near the cylinder’s surface. This occurs because a smaller amount of heat energy is conducted from the surface to the fluid. Simultaneously, the thermal boundary layer thickness increases due to enhanced heat transfer within the fluid, driven by the improved convection process around the cylinder. Figure 4 demonstrates concentration distribution is influenced by ζ . Concentration distribution dwindles when ζ has surged.

$Figure 2 Velocity appearance against ζ \zeta .$

Figure 2

Velocity appearance against ζ .

$Figure 3 Temperature appearance against ζ \zeta .$

Figure 3

Temperature appearance against ζ .

$Figure 4 Concentration appearance against ζ \zeta .$

Figure 4

Concentration appearance against ζ .

Figure 5 illustrates that increasing the values of Nt leads to temperature curves enhanced. Figure 6 shows that an enhanced Nt value results in a surge in the concentration curve. Actually, the growing power of Nt , a larger number of nanoparticles migrate from the hot surface to the cold surface. This movement ultimately leads to an increase in both the temperature and concentration distributions. Figure 7 demonstrates that the velocity profile is primarily influenced by M 1 . The Lorentz force arises from a change in M 1 , leading to resistive forces that cause the fluid flow to reverse direction and reduce velocity. The velocity distribution diminishes as we boost h 1 , as observed in Figure 8. It is noted that when slip occurs, the velocity near the stretching surface does not match the wall’s stretching velocity. Under slip conditions, the momentum imparted by the stretching surface to the fluid is only partially transferred. This results in a reduction in fluid velocity, thereby causing a deceleration in the boundary layer flow. Figure 9 demonstrates that the K p influences the velocity appearance. Distribution of velocity is diminished with enhanced K p . The resistance to fluid flow, which arises due to permeable medium, is responsible for reduced velocity appearance. Figure 10 illustrates the impact of the Pr parameter on the temperature profile. Increasing Pr caused a reduction in the thermal boundary layer and a decrease in the temperature distribution.

$Figure 5 Temperature appearance against Nt {\rm{Nt}} .$

Figure 5

Temperature appearance against Nt .

$Figure 6 Concentration appearance against Nt {\rm{Nt}} .$

Figure 6

Concentration appearance against Nt .

$Figure 7 Velocity appearance against M 1 {M}_{1} .$

Figure 7

Velocity appearance against M 1 .

$Figure 8 Velocity appearance against h 1 {h}_{1} .$

Figure 8

Velocity appearance against h 1 .

$Figure 9 Velocity appearance against K p {K}_{p} .$

Figure 9

Velocity appearance against K p .

Figure 10

Temperature appearance against Pr.

Figure 11 demonstrates the influence of the radiation parameter. Fluid is heated by radiation, consequently resulting in an enhanced temperature curve. Physically, a more heat transfer is occur due to greater radiation values so temperature maximise. The temperature distribution diminishes as we boost h 2 parameter, as illustrated in Figure 12. The impact of γ r , Nb , and Sc on the concentration distribution is depicted in Figures 13, 14, and 15, respectively. The graphs demonstrate that surging γ r , Sc , and Nb values resulted in a dwindle in the concentration distribution. Physically, a rise in Nb leads to an increase in the random motion of nanoparticles. As the interaction among particles intensifies, kinetic energy is converted into thermal energy, resulting in a reduction in the nanoparticle concentration within the fluid. As the Sc strength increases, the weaker Brownian diffusion coefficient leads to a reduction in boundary layer thickness and a decline in ϕ ( ξ ) . This is attributed to the influence of the increasing Brownian diffusion coefficient with higher Sc . Figure 16 indicates how the Sb influences the microorganism profile, with a boost in Sb leading to a reduction in the motile microorganism profile.

$Figure 11 Temperature appearance against Rd {\rm{Rd}} .$

Figure 11

Temperature appearance against Rd .

$Figure 12 Temperature appearance against h 2 {h}_{2} .$

Figure 12

Temperature appearance against h 2 .

$Figure 13 Concentration appearance against γ r {\gamma }_{r} .$

Figure 13

Concentration appearance against γ r .

$Figure 14 Concentration appearance against Sc {\rm{Sc}} .$

Figure 14

Concentration appearance against Sc .

$Figure 15 Concentration appearance against Nb {\rm{Nb}} .$

Figure 15

Concentration appearance against Nb .

$Figure 16 Microorganism appearance against Sb {\rm{Sb}} .$

Figure 16

Microorganism appearance against Sb .

6 Conclusion

This investigation primary focus to analyses the slip flow of EMHD Carreau nanofluid across a stretching cylinder in the context of Arrhenius activation energy, chemical reactions, and variable thermal conductivity within a porous medium. The study explicitly considers the consequences of chemical reactions, the presence of a porous medium, and changes in thermal conductivity. Buongiorno’s model is employed, and also, the effects of changing EMHD forces on fluid flow dynamics are investigated. By introducing nonsimilarity variables, the partial differential equations are transformed into a system of coupled ODEs. The bvp4c solver, a MATLAB built-in solver, is implemented to compute the solution to the resultant set of ODEs. The outcomes of present report would be beneficial in the fluids, which are designed for optimisation of heat transfer in various industrial zone and processing material. The present study main finding points are shown below:

The velocity profile dwindles as the porosity ( K p ) and magnetic ( M 1 ) values increase and rises as the curvature strength ( ζ ) value rises.
The temperature profile rises with the value of curvature ( ζ ) , and radiation ( Rd ) rises and falls as the values of Pr, u 1 , and Nb rise. The enhancing power of radiation, thermophoresis, and heat source are cause to Nn x decreases, but reverse trend is reported for Pr.
As the ζ , γ r , Sc , and Nb values rise, the concentration profile dwindles and increases as do the Nt and E parameter values.
As Nb and ζ increase the Sh x falls, but when the Nt value rises, it increases. As Sb and ζ values increased, the result in motile microorganisms rise.

For future study, it can be extended further to optimise the host fluid temperature of different models like Casson, Ellis, Maxwell, and micropolar. In addition, these models can be expanded to incorporate hybrid nanoparticles for enhanced performance.

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. Conceptualization; Muhammad Afzal, Imran Siddique, Mohammed M. M. Jaradat and Bagh Ali. Methodology; Imran Siddique and Badar E Alam, Funding acquisition; Mohammed M. M. Jaradat. Investigation; Bagh Ali. Validation; Binjian Ma. Data curation; Badar E Alam and Faiza Zahid. Writing – original draft preparation; Faiza Zahid and Bagh Ali. Software; Bagh Ali and Mohammed M. M. Jaradat. Writing – review and editing; Muhammad Afzal, Imran Siddique and Bagh Ali. Supervision; Binjian Ma.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-05-21

Revised: 2024-11-27

Accepted: 2025-04-08

Published Online: 2025-05-14

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

Articles in the same Issue

https://doi.org/10.1515/ntrev-2025-0166

Keywords for this article

Carreau nanofluid; EMHD; variable thermal conductivity; permeable medium; chemical reaction; slip boundary condition; stretching cylinder; bvp4c

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