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Flow optimization and irreversibility reduction in electroosmotic MHD Buongiorno nanofluid systems within complex wavy arteries for biological fluids

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Published/Copyright: January 8, 2026
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Open Physics
From the journal Open Physics Volume 24 Issue 1

Abstract

This study examines the optimization of flow and reduction of irreversibility in electroosmotic magnetohydrodynamic (MHD) Buongiorno nanofluid systems within complex wavy arteries for biological fluids. The research focuses on the coupled effects of electroosmotic forces and MHD on nanofluid behavior in biologically relevant geometries. By employing Buongiorno’s model for nanofluids, which accounts for both Brownian motion and thermophoresis, we analyze the transport phenomena within a non-Newtonian fluid environment typical of biological fluids. The complex wavy structure of arteries is modeled to reflect realistic physiological conditions, emphasizing the impact of arterial geometry on flow characteristics. Nanoparticles move randomly due to thermal energy and temperatures gradient, influencing flow efficacy and irreversibility. Complex peristaltic waves may form on infected vein walls. In electro-osmotic magnetohydrodynamics, the MHD flow field and irreversibility system are simulated mathematically. We studied the long-wavelength, low-Reynolds-number approximation. The nonlinear model of partial differential equations (PDEs) is approximated utilizing Adomian decomposition methodology (ADM). System irreversibility and entropy creation are also analyzed. Biophysical and thermal flow properties are displayed and explained. Increased nanofluid particle interfacial lengths boost Bejan counts. Developing Brownian and thermophoresis diffusion increases the Bejan numbers, which is essential. The Jeffery and Debye–Huckel parameters reduce the movement energy of particles, which decreases the temperature and entropy rate of the nanofluid, while the other factors increase the movement energy.

Keywords: Jeffery nanofluid; irreversibility process; electroosmosis force; induced magnetic field; peristaltic motion

Nomenclature

U, V

velocity fixed frame components (m/s)

B X , B Y

induced magnetic field components in fixed frame

u, v

non-dimensional components of velocity

B x , B y

induced magnetic field components in wave frame

P m

magnetic pressure k g / m s 2

Ψ

stream function

F

magnetic force function

X, Y

cartesian coordinate in the fixed frame (m)

J 1

ratio of relaxation to retardation time (−)

J 2

retardation time (s)

ρ f

fluid density k g / m 3

t

time (s)

T

temperature of fluid in K

T r , T l

temperature on right and left sides

ϕ

concentration k g / m 3

ϕ r , ϕ l

concentricity at the right and left walls, respectively

Θ

dimensionless temperature

χ

non-dimensional concentration

E x

potential of electrical

ζ

chemical reactive parameter

β

heat generating/absorbing variable

UHS

Helmholtz–Smoluchowski velocity

R e

Reynolds number

Pr

Prandtl number

S r

Soret number

S c

Schmidt number

β 1

mobility of the medium

β 2

slip velocity parameter

β 3

thermal slip parameter

β 4

concentration slip parameter

E c

Eckert number

B e

Bejan number

M

Hartmann number

N b

Brownian movement

N t

thermophoretic variable

G 1

thermal Grashof number

G 2

concentration Grashof number

R m

magnetic Reynolds number

E g

entropy production

MHD

magnetohydrodynamic

1 Introduction

One of the most important things that Gouy–Stodola theory confirms, depending on the 2nd law of thermodynamics, is that whenever usable energy is lost, this is closely linked to the increase in entropy production, as we note that the latter represents the most important measure for measuring the size of irreversibility phenomena within the thermodynamic system. It can also be said that entropy arises primarily based on a group of multiple factors, for example, thermal conductivity and internal friction (viscosity), which confirms to us that the dissipation of energy and its transformation into forms that cannot be recovered [1], 2]. Therefore, after reducing entropy, which is a basic goal in designing such engineering systems as an optimal design, this leads to reducing the rates of thermodynamic irreversibility and thus works to boost the efficiency of heat transference and improve the overall performance of the system [3], 4]. From this standpoint, this vision is based primarily on the principle that every real process must be accompanied by some loss of energy, and the reason for this is due to non-ideal phenomena. This is the reason why controlling entropy is a critical factor in creating and designing highly efficient systems. Consequently, performing an entropy production analysis is essential for various treatments in several manufacturing procedures, such as metallic twisting, coating of optical fiber, and tumor therapy [5]. The research performed an analytical assessment of entropy production because of heat transfer and the flow of nanoliquids among co-rotating containers with sides exhibiting a fixed heat fluxing. The study [6] noted that the incorporation of the viscous dissipation effect indicates a meaningful increment in the total entropy production of H2O–Al2O3 nanoliquid flowing inside a micro-channel. A distinctive method to enhance entropy formation is presented for the flowing of nonlinear Sisko nanostructure working a rotating stretchable disk, whereby the total irreversibility rates are analyzed across different flowing characteristics [7]. The impact of velocity slippery and entropic production on magneto-peristalsis in an incompressible liquid inside a divergence tube was derived from the examination [8]. Vaidya et al. [9] conducted a computer analysis to measure and examine the heterogeneity-homogeneity response and entropy production in a magneto-peristalsis flow of a Jeffery fluid, including the influences of slippage mechanisms. The study used Buongiorno’s concept to investigate the influence of nanoparticles insertion and distributions on the intrinsic irreversibility inside a micro-channel under varied parameters [10]. In 3-dimensions, Mburu and his colleagues [11] presented a computational solution to examine entropy reduction in the magnetohydrodynamic flowing of Oldroyd-B liquid, including relaxing-retarding viscous dissipative flux and coupled chemical reactions. Lately, Kumar et al. [12] examined entropic production in relation to inertial forces and streamline curvature.

One of the most important basic criteria in analyzing and conducting irreversibility distributions in different systems is the Bejan number, as the thermal Bejan number is one of the most important pivotal numbers that play a fundamental part in the development of understanding and evaluating the performance of heat transfer processes across different applications [13]. It is important that it can be represented in four basic areas, namely the study of sizing systems based on forced convection, as it helps in making improvements in the design of these systems based on compressed flow. It also helps in cooling electronic components by providing us with an accurate indicator of the efficiency of the heat dissipation process, in addition to preventing thermal damage. As for the evaluation consistent with the 2nd law of thermodynamics for contact melting and lubrication processes, its role is prominent in improving the performance of mechanical systems, and finally in the construction and design of heat exchangers, as it contributes to measuring entropic loss and improving the effectiveness of heat transfer. From the above, we note that this number creates a balance between many of the effects of thermal diffusion and the influencing dynamic factors, which makes it one of the tools Indispensable in making improvements to thermal systems and reducing energy loss. Using four fundamental flow configurations, Bejan [14] proposed the 2nd law of thermodynamics to clarify heat transfer by forced convective flowing. Bejan [15] examined approaches for heat exchanger design, augmentation of heat flow, and development of insulating schemes. Involving changes in the Bejan number, Samal et al. [16] computationally analyzed the immigration of hybridized non-Darcy magnetohydrodynamics nanomaterials within a porous micro-annulus using the control volume finite element technique and entropic development.

Nanofluids are essentially enhanced fluids because they contain very small nanoparticles suspended in different base fluids [17], 18]. One of the most important features of these fluids is their superior physical and thermal properties when compared to base fluids. This makes them always the ideal choice for improving the efficiency of heat transfer in many engineering and industrial applications [19], 20]. When we use nanoparticles of materials with high thermal conductivity, these fluids are heat conductors, meaning they work greatly to enhance their thermal conductivity capacity [21]. They also play an essential position in enhancing the productivity of thermal stability and heat dissipation [22]. Choi and Eastman [23] performed the preliminary analysis on nanoparticles, which significantly boosts the thermal conductance of the fundamental liquid. Ali et al. [24] explored the influences of heat radiative flux on the peristalsis blood circulation of Jeffrey fluids with double diffusions, in the existence of zirconia nanosolid molecules. Buongiorno [25] proposed an innovative mathematical system for analyzing heat transfer and flow in nanoliquids. Rafique et al. [26] managed the Buongiorno framework to explore the influences of thermophoretic and Brownian diffusions induced by nanomolecules. Vigneshwari et al. [27] managed Buongiorno’s nanofluid model to investigate the influence of heat generation porous materials and magnetized fields inside a complex, undulating-walled enclosure containing nanofluids. This study was performed in the existence of viscous dissipative flow via a porous medium. The research [28] examined the effects of facilitated and magnetic Reynolds numbers and curvature on the activities of a couple stress inside the Buongiorno nanofluid system in a resistant curved conduit. There are many studies and research papers that have dealt with the flow of nanofluids based primarily on the Bongiorno model, including, for example Refs. [29], 30], which can be used by reviewing and studying them in detail to determine the values of the different parameters that represent the optimal values for use in many new studies.

There is a significant connection between the chemistry of reactions and the physics of flows, which regulates changes in the features of the flow liquid. The dynamics of the chemical process clearly demonstrate this connection. Therefore, their potential uses in the fields of medicine, biological science, and manufacturing include important operations in the preparation of food, the production of polymers, and the extrusion of plastic. The research [31], 32] investigated the impact of chemical reactive processes on flows by using various theoretical structures and a wide range of techniques. Hina et al. [33] investigated the heat transmission characteristics of a nanoliquid inside a microchannel by using the Buongiorno concept. They took into consideration the impacts of magnetohydrodynamics, chemical processes, viscous dissipation, and slip boundary conditions in their investigation. Anjum et al. [34] demonstrated the chemical reaction that occurs among the magnetized micropolar nanofluid flow that occurs inside Buongiorno’s model while it is contained within an impermeable cylinder. With the use of Buongiorno’s model, Rasheed et al. [35] investigated three chemical processes to explore solutal and heat transmission in nanofluid movement.

Micropumps frequently use electroosmotic flow (EF), which operates on the principle of electroosmosis. It is regulated by exterior electrical fields at the contact among a solid and a fluid. This phenomenon leads to the formation of an electrical double layer at the contact. The charged molecule is enveloped by an ionic milieu, formed by the existence of side charges and their opposing ions. Consequently, it is extensively used for numerous productive purposes [36], 37], making it the most favored option. It may serve as an efficient micro-pump to generate power for diverse liquids in biological apparatus. Hafez and Abd-Alla [38] investigated the entropic generation in the electroosmotic flowing of a nanofluid with temperature-dependent viscosity during peristaltic motion. Numerous mathematical models of peristaltic motion concerning electroosmotic fluxes using micro-pumps are delineated in [39], 40]. Magnetohydrodynamics (MHD) is an efficient method used in medicine for identifying ailments such as lung cancer, and dehydration, and conducting MRI scans [41]. Abd-Alla et al. [42] worked on an examination of the influences of the produced magnetized forces on the peristalsis flowing of a liquid exhibiting micropolar characteristics and coupled stress inside a symmetrical channel. While the examination [43] derived an analytical solution for the flowing of a magnetic Jeffrey liquid in a porous pipe with walls that moreover expanding or contracting. Magesh et al. [44] explored the influences of the induced magnetized forces on the behaviour of Jeffrey fluid in a curved conduit throughout peristaltic. The examination [45] conducted research that accurately assessed the influence of a tri-hybridized nanoliquid on the ciliate arteries, including the effects of generated magnetized force and curvature without any assumptions.

One of the methods used in the mathematical analysis process is the Adomian decomposition methodology (ADM), which is considered one of the effective methods for solving complex differential equations, whether these equations are linear or nonlinear. In addition to solving integral equations and partial and ordinary differential equations, it is characterized by its ability to perform accurate approximate solutions [46]. It also enables it to perform a computational simulation for vital applications in the areas of sciences and medical engineering. In the eighties of this century, George Adomian [47] developed this method, and it was updated in the twentieth century. It is a methodology based on analyzing mathematical problems into simple components that can be solved iteratively. This method has gained wide fame for its accuracy and flexibility, which has made it an effective tool in many engineering, medical, and biological systems. This method has also been successfully used in conducting research and studies, proving its great efficiency in dealing with these complex problems, as it provides a comprehensive solution without the need for excessive simplification that would lose the model’s accuracy [48].

The mathematical framework that defines the flow of Jeffrey-induced magneto-Buongiorno nanofluid within a complex tapering peristaltic conduit is being investigated. We are considering the impacts of electro-osmotic forces and slippery constraints in our investigation. This study aims to approximate the combined effects of low Reynolds numbers and expanded wavelengths. Obtaining the approximate solution for the nonlinear system of PDEs is accomplished via the use of ADM. In addition to this, the evaluation is based on the irreversibility of the model as well as the formation of entropy. The parameters of the flow are displayed and analyzed for both bio-physical and thermal aspects. An increment in the interstitial distance between the particles that make up the nanoliquid results in an increase in the Bejan number. It was at this moment that several questions were raised:

  1. How do the molecular kinetic energy and consequently the temperature distribution in the nanofluid vary as functions of the Jeffery parameter and Debye–Hückel potential?

  2. What are the thermodynamic consequences of these parameters (Jeffery and Debye–Hückel) for the system rate of entropy production?

  3. How do Brownian and thermophoretic diffusions affect the Bejan number?

  4. Should the Hartmann and further dimensionless numbers such as Eckert, Prandtl, and Schmidt numbers dictate the behavior of the kinetic energy and the rates of entropic generation in some physical sense?

  5. Importance of the induced magnetic field on the thermal and hydrodynamic boundary layer in nanofluids under molecular slip constraints.

  6. What are the effects of slip conditions in the case of magnetic fields over nanofluid flow?

This will be addressed in the subsequent sections of the present investigation. This work focuses on biofluids, including blood, synovial fluid, and mucus, showing non-Newtonian behavior due to suspended cells (e.g., erythrocytes), proteins, and nanoparticles. Crucially, as part of the body’s transport systems, these fluids let magnetic and electric effects control their flow. The way blood flows and its thickness change, for example, reflects the Jeffery fluid model used here; the elastic and thick quality of mucus is like the Buongiorno nanofluid model when it flows in waves.

2 Mathematical formulations

Take into consideration an incompressible nanofluid flow in the complex peristaltic tapering conduit characterized by induced MHD non-Newtonian (Jeffery model) Buongiorno flow. Specified as follows is the formula that illustrates the complex tapering sinusoidal wave that flows along the tube sides in Figure 1 [49], [50], [51]:

(1) r X , t = S 0 + S 1 λ X − C t + S 2 ⁡ cos 2 π λ X − C t + S 3 ⁡ cos 4 π λ X − C t .

Figure 1: Geometric model of the flow.
Figure 1:

Geometric model of the flow.

As illustrated in Figure 1, the geometry of the problem is a symmetric wavy channel with complicated peristaltic walls. The equation (1), where S 0 represents the mean wall thickness, S 1 regulates the tapering effect, and S 2 and S 3 determine the amplitude of the main and secondary peristaltic waves and define the upper and lower walls. Combining electroosmotic forces, magnetic field, and peristaltic motion in the axial direction X induces the flow. In this context, C, λ, and t represent the wave speed, the wavelength, and the time on their own.

What follows is a description of the equations that control the system [43], 48]:

(2) ∇ ⋅ V ̲ = 0 , ∇ ⋅ B ̲ = 0 ,

(3) ρ f D V ̲ D t = ∇ ⋅ τ ̲ ̲ + 1 μ e B ̲ * ⋅ ∇ B ̲ * − 1 2 ∇ B * 2 + ρ p − ρ f ϕ − ϕ l g ̲ w + ρ f g ̲ α 1 − ϕ 0 × T − T l + ρ e E x ̲ ,

(4) ∂ B ̲ * ∂ t = ∇ × V ̲ × B ̲ * + 1 η ∇ 2 B ̲ * ,

(5) ρ c p f D T D t = k f ∇ 2 T + S i j e i j + σ f B 0 2 V 2 + τ ρ c p f D T T m ∇ T ⋅ ∇ T + τ ρ c p f D B ∇ ϕ ⋅ ∇ T + Q 0 ,

(6) D ϕ D t = D B ∇ 2 ϕ + D T T m ∇ 2 T − k 1 ϕ − ϕ m .

By use of the boundary constraints, which are specified as follows [26], 33], 45]:

(7) U + β ̄ 2 S X Y = 0 , T + β ̄ 3 ∂ T ∂ Y = T r , ϕ + β ̄ 4 ∂ ϕ ∂ Y = ϕ r , F ̄ = 0 , B X = 0 a t y = r X , t , U − β ̄ 2 S X Y = 0 , T − β ̄ 3 ∂ T ∂ Y = T l , ϕ − β ̄ 4 ∂ ϕ ∂ Y = ϕ l , B X = 0 a t y = − r X , t .

As well as the stress tensor for the Jeffery concept is characterized as the following [52]:

τ ̲ ̲ = − P I ̲ ̲ + S ̲ ̲ .

One expression that may be used to represent the extra stress tensor S ̲ ̲ is as below [45]:

S ̲ ̲ = μ f 1 + J 1 γ ̇ + J 2 γ ̈ , γ ̇ = ∇ V + ∇ V T .

The flowing field can be influenced by an external force known as the MHD force. Therefore, the normal magnetized force that affects the movement of the conduit is represented by the symbol B 0. The elements of the induced magnetized field are denoted by the equation B * B X X , Y , t , B Y X , Y , t + B 0 , 0 . An additional external force that exerts its influence on the wall of the tapering conduit is known as the electro-osmotic force (EOF). An examination of the EOF is carried out by applying a Poisson–Boltzmann distribution to the probability distribution of the electric potential [48].

(8) ∇ 2 Γ ̄ = − ρ e ε 0 ε , ρ e = n + − n − e z .

The variables ρ e , ɛ 0, ɛ, n +, n −, e, and z represent the net charge density, vacuum permittivity, dielectric constant, the number of cations density, the volume of anions density, electron charge, and charge balance, correspondingly. The electrical charge density, derived from Boltzmann’s distribution as:

ρ e = − 2 n 0 e z ⁡ sinh z e Γ ̄ K B T m ,

where n 0, K B , and T m are bulkiness concentration, Boltzmann constant, and average temperature, separately. After dimensionless and applying the Debye–Huckel linearization, the electrical potential distribution (Poisson–Boltzmann equation) be:

(9) ∇ 2 Γ = k 2 Γ .

To transfer our formulae from the fixed coordinates X , Y to a movable once x , y , the subsequent alterations should be utilized:

(10) Y = y , X = x + C t , U = u + C , V = v .

Dimensionless parameters are:

(11) x ̂ = x λ , y ̂ = y S 0 , v ̂ = v C , u ̂ = u C , P ̂ = S 0 2 P λ μ f C , δ = S 0 λ , Θ = T − T l T r − T l , χ = ϕ − ϕ l ϕ r − ϕ l , B X = B x B 0 , B Y = B y B 0 .

The streamline and magnetic force functions are defined as follows:

(12) u = ∂ Ψ ∂ y , v = − δ ∂ Ψ ∂ x , B x = ∂ F ∂ y , B y = − δ ∂ F ∂ x .

By employing Eqs. (9)–(12) into Eqs. (1)–(8), we have:

(13) Ψ x y − Ψ y x = 0 ,   F x y − F y x = 0 ,

(14) R e δ Ψ y Ψ x y − Ψ x Ψ y y = − ∂ P m ∂ x + 2 δ 2 1 + J 1 ∂ ∂ x Ψ x y + J 2 − Ψ x Ψ xyy + δ Ψ y Ψ xxy + 1 1 + J 1 ∂ ∂ y Ψ y y − δ 2 Ψ x x + J 2 δ Ψ y − δ 2 Ψ xxx + Ψ xyy − δΨ x − δ 2 Ψ x x y + Ψ y y y + S 2 R e δ F y F y x + − δ F x + 1 F y y + G 1 Θ + G 2 χ + β 1 k 2 Γ ,

(15) δ 3 R e Ψ x Ψ x y − Ψ y Ψ x x = − ∂ P m ∂ y + 1 1 + J 1 δ 2 ∂ ∂ x − δ 2 Ψ x x + Ψ y y + J 2 δ Ψ y Ψ x y y − δ 2 Ψ x x x − δ Ψ x Ψ y y y − δ 2 Ψ x x y − 2 δ 2 1 + J 1 ∂ ∂ y J 2 δ Ψ y Ψ x x y − δ Ψ x Ψ x y y + Ψ x y − δ 3 R e S 2 F y F x x + F x F x y ,

(16) 1 R m ∇ 2 F − δ Ψ y F x − Ψ x F y + Ψ y = E ,

(17) R e Pr δ Ψ y Θ x − Ψ x Θ y = ∇ 2 Θ + E c Pr 1 1 + J 1 4 Ψ x y δ 2 Ψ x y + J 2 δ 3 Ψ y Ψ y x x − Ψ x Ψ y y x + Ψ y y − δ 2 Ψ x x 2   + J 2 δ Ψ y y − δ 2 Ψ x x Ψ y Ψ y y x − δ 2 Ψ y Ψ x x x − Ψ x Ψ y y y + δ 2 Ψ x Ψ x x y + M 2 E c Pr Ψ y + 1 2 + N t Pr δ 2 Θ x 2 + Θ y 2 + N b Pr δ 2 χ x Θ x + χ y Θ y ,

(18) R e δ Ψ y χ x − δ Ψ x χ y = S r δ 2 Θ x x + Θ y y + 1 S c δ 2 χ x x + χ y y − ζ χ ,

(19) δ 2 Γ x x + Γ y y = k 2 Γ ,

(20) Ψ y + 1 + β 2 1 + λ 1 Ψ y y = 0 , Ψ = q 2 , θ + β 3 ∂ θ ∂ y = 0.5 , χ + β 4 ∂ χ ∂ y = 0.5 , ∂ F ∂ y = 0 , F = 0 a t y = r x , Ψ = − q 2 , Ψ y + 1 − β 2 1 + λ 1 Ψ y y = 0 , θ − β 4 ∂ θ ∂ y = − 0.5 , χ − β 4 ∂ χ ∂ y = − 0.5 a t y = − r x , ∂ F ∂ y = 0 a t y = 0 ,

where r x = 1 + l 1 x + l 2 ⁡ cos 2 π x + l 3 ⁡ cos 4 π x , y = r x represents an upper wall, y = − r x indicates a lower wall, and y = 0 implies the centerline.

Moreover, the nondimensional parameters that are specified as follows:

k 2 = 2 S 0 2 n 0 e 2 z 2 ε ε 0 k B T m , β 1 = − ε ε 0 ς C μ f , β = Q 0 S 0 2 k f T r − T l , N b = τ ρ f D B μ f ϕ r − ϕ l , Pr = μ c p f k f , E x = U H S C , N t = τ ρ f D T T m T r − T l , P m = p + B 0 2 δ R e 2 μ e ρ f C 2 B * 2 , E c = C 2 c p f T l − T u , M 2 = σ f B 0 2 S 0 2 μ f , S 2 = M 2 R e R m , R m   =   σ f CS 0 μ e ,   ζ = k 1 ρ f S 0 2 μ f , R e = ρ f c S 0 μ f , S c = μ f ρ f D B , G 1 = S 0 2 ρ f g α 1 − ϕ 0 T r − T l C μ f , G 2 = S 0 2 ρ p − ρ f ϕ r − ϕ l g w C μ f , S r = ρ f D T T r − T l μ f T m ϕ r − ϕ l , l 1 = S 1 S 0 , l 2 = S 2 S 0 , l 3 = S 3 S 0 .

We obtain the following results when we use the extended wavelength λ → ∞ and the modest Reynolds numbers R e → 0 approximation:

(21) 0 = − ∂ P m ∂ x + 1 1 + J 1 Ψ yyy + S 2 R e F y y + G 1 Θ + G 2 χ + β 1 k 2 Γ ,

(22) 0 = ∂ P m ∂ y ,

(23) F y y = R m E − Ψ y ,

(24) 0 = Θ y y + E c Pr 1 + J 1 Ψ y y 2 + M 2 E c Pr Ψ y + 1 2 + N t   Pr Θ y 2 + N b Pr χ y Θ y + β ,

(25) 0 = 1 S c χ y y + S r Θ y y − ζ χ ,

(26) Γ y y = k 2 Γ .

By taking the derivative concerning y:

(27) 0 = 1 1 + J 1 Ψ yyyy − M 2 Ψ y y + G 1 Θ y + G 2 χ y + β 1 k 2 Γ y .

Taking into account the boundary constraint Γ ± r x = 1 , the solution to the potential formula [21] may be expressed as follows:

(28) Γ x , y = cosh k y s e c h k r x .

3 Semi-analytical solutions by ADM

It is necessary to make use of ADM to assess the semi-analytical solutions for the non-linear model of the regulating equations (19)–(22) once equation (23) has been used. According to the decomposition series, the ADM for the functions Ψ x , y , Θ x , y , and χ x , y that are not yet recognized is as follows:

(29) Ψ x , y = ∑ n = 0 ∞ Ψ n x , y , Θ x , y = ∑ n = 0 ∞ Θ n x , y , χ x , y = ∑ n = 0 ∞ χ n x , y .

The functions that enter the boundary or initial conditions, non-homogeneity bounds, and integral constants are denoted by the symbols Ψ0, Θ0 and χ 0 respectively.

Equations of zero-order:

(30) Ψ 0 y y y y = − β 1 k 2 1 + J 1 Γ y , Θ 0 y y = − β + M 2 E c Pr , χ 0 y y = S c S r β + M 2 E c Pr .

We derive the zero-order formulas by utilizing the boundary restrictions that are provided, which are as follows:

(31) Ψ 0 x , y = α 0 ⁡ sinh k y + α 3 y 3 + α 1 y ,

(32) Θ 0 x , y = γ 2 y 2 + γ 1 y + γ 0 ,

(33) χ 0 x , y = λ 2 y 2 + λ 1 y + λ 0 .

There are many different shapes that the 1st-order solution takes:

(34) Ψ 1 x , y = α 16 sinh k y + α 15 y 5 + α 14 y 4 + α 13 y 3 + α 12 y 2 + α 11 y + α 10 ,

(35) Θ 1 x , y = γ 10 ( sinh k y ) 2 + γ 11 sinh k y 2 2 + γ 12 − 1 + cosh k y + γ 13 y 6 + y 2 γ 14 + γ 15 cosh k y + γ 16 y 3 + γ 17 y 4 + γ 18 + y γ 19 sinh k y + γ 110 ,

(36) χ 1 x , y = λ 10 − 1 + cosh k y + λ 11 ( sin h k y ) 2 + λ 12 1 + cosh 2 k y + λ 13 y 6 + y 2 λ 14 + λ 15 cosh k y + λ 16 y 3 + λ 17 y 4 + y λ 18 sinh k y + λ 19 + λ 110 ,

where

α 0 = − s e c h k r 1 + J 1 β 1 k ,

α 1 = 1 4 k r r 1 + J 1 + 3 β 2 k r 1 + J 1 3 q − 2 r − 1 + 1 + J 1 β 1 + 6 q β 2 + 2 y 1 + J 1 β 1 3 r 1 + J 1 + 6 − k 2 r 2 β 2 tanh k r ,

α 3 = 1 4 k r 2 r 1 + J 1 + 3 β 2 1 + J 1 − k q − 2 1 + J 1 β 1 tanh k r   + 2 k r − 1 + β 1 1 + J 1 + k β 2 tanh k r ,

γ 0 = 1 2 E c M 2 Pr + β r r + 2 β 3 , γ 1 = 1 2 r + β 3 , γ 2 = 1 2 − E c M 2 Pr − β ,

λ 0 = − 1 2 S c S r E c M 2 Pr + β r r + 2 β 4 ,

λ 1 = 1 2 r + β 4 , λ 2 = 1 2 S c S r E c M 2 Pr + β ,

α 10 = r 1 + J 1 − 12 β 2 − r 3 1 + J 1 2 G 1 γ 1 + G 2 λ 1 + r 3 β 2 2 G 1 γ 1 + G 2 λ 1 24 1 + J 1 2 − β 2 2 ,

α 11 = 1 60 k 2 r 30 k c o s h k r r − 3 sinh k r k 2 + M 2 + M 2 J 1 α 0 + 1 1 + J 1 2 − β 2 2 k 2 45 q 1 + J 1 2 − β 2 2 + 30 r − 1 + J 1 2 − 1 + 2 α 1 + 2 α 1 β 2 2   + r 5 1 + J 1 1 + J 1 2 − β 2 2 3 M 2 α 3 − G 1 γ 2 − G 2 λ 2 ,

α 12 = 1 + J 1 6 β 2 + r 3 1 + J 1 2 G 1 γ 1 + G 2 λ 1 − r 3 β 2 2 G 1 γ 1 + G 2 λ 1 12 r 1 + J 1 2 − β 2 2 ,

α 13 = 1 60 r 3 − 15 q + 30 ⁡ sinh k r k 2 + M 2 + M 2 J 1 α 0 k 2 − 60 r 3 α 3 + 30 r − cosh k r k 2 + M 2 + M 2 J 1 α 0 k − 1 + J 1 2 1 + J 1 2 − β 2 2   − 2 r 5 1 + J 1 3 M 2 α 3 − G 1 γ 2 − G 2 λ 2 ,

α 14 = − 1 24 1 + J 1 G 1 γ 1 + G 2 λ 1 , α 15 = 1 60 1 + J 1 3 M 2 α 3 − G 1 γ 2 − G 2 λ 2 , α 16 = M 2 1 + J 1 α 0 k 2 ,

γ 10 = − E c Pr k 2 + M 2 + M 2 J 1 α 0 2 4 1 + J 1 , γ 11 = 48 E c Pr α 0 α 3 k + k J 1 , γ 12 = − 2 E c M 2 Pr α 0 k 2 1 + α 1 + 18 α 3 k 3 , γ 13 = − 3 10 E c M 2 Pr α 3 2 ,

γ 14 = − 1 4 1 + J 1 Pr E c k 2 − k 2 + M 2 + M 2 J 1 α 0 2   + 2 1 + J 1 E c M 2 α 1 2 + α 1 + γ 1 N t γ 1 + N b λ 1 ,

γ 15 = Pr − 24 E c M 2 1 + J 1 α 0 α 3 4 k 1 + J 1 , γ 16 = − 1 3 Pr N b γ 2 λ 1 + γ 1 2 N t γ 2 + N b λ 2 ,

γ 17 = − Pr 3 E c M 2 1 + J 1 1 + α 1 α 3 + 18 E c α 3 2 + 2 1 + J 1 γ 2 N t γ 2 + N b λ 2 6 1 + J 1 ,

γ 18 = 1 120 k 3 1 + J 1 15 E c k 3 Pr α 0 2 2 k 2 r 2 − k 2 + M 2 + M 2 J 1 + 4 k 2 r − k 2 + M 2 + M 2 J 1 β 3 + k 2 + M 2 + M 2 J 1   × − 1 + cosh 2 k r + 2 k s i n h 2 k r β 3 + 240 E c Pr α 0 k 2 M 2 − 1 + cosh k r − k 2 M 2 J 1 − k 2 M 2 α 1 − k 2 M 2 J 1 α 1 + 12 k 2 α 3 − 18 M 2 α 3 − 18 M 2 J 1 α 3 + k s i n h k r 6 r k 2 − 2 M 2 − 2 M 2 J 1 α 3 + k 2 M 2 1 + J 1 1 + α 1 + 3 k 2 M 2 r 2 1 + J 1   + 2 − k 2 + M 2 + M 2 J 1 α 3 β 3 + cosh k r k 2 M 2 α 1 + M 2 J 1 k 2 + k 2 α 1   + 3 α 3 6 + k 2 r r − 2 β 3 + 3 α 3 − 4 k 2 + 6 M 2   + k 2 r M 2 r + 2 k − M k + M β 3 + 4 k 3 9 E c M 2 Pr r 6 1 + J 1 α 3 2 + 15 r 2 1 + J 1 × − 2 γ 2 + Pr E c M 2 α 1 2 + α 1 + γ 1 N t γ 1 + N b γ 1 + 30 r 1 + J 1 β 3 − 2 γ 2 + Pr E c M 2 α 1 2 + α 1   + γ 1 N t γ 1 + N b λ 1 + 5 r 4 P R 3 E c M 2 1 + J 1 γ 2 N t γ 2 + N b λ 2 + 20 r 3 Pr β 3 3 E c M 2 1 + J 1 1 + α 1 α 3 + 18 E c α 3 2   + 2 1 + J 1 γ 2 N t γ 2 + N b λ 2 ,

γ 19 = 12 E c Pr − k 2 + 2 M 2 + 2 M 2 J 1 α 0 α 3 k 2 1 + J 1 ,

γ 110 = 1 6 r + β 3 3 − 6 r γ 1 − 6 β 3 γ 1 + 2 r 3 Pr N b γ 2 λ 1 + γ 1 2 N t γ 2 + N b λ 2 + 6 r 2 Pr β 3 N b γ 2 λ 1 + γ 1 2 N t γ 2 + N b λ 2

λ 10 = 2 E c Pr S c S r α 0 k 2 M 2 1 + J 1 1 + α 1 + 6 − 2 k 2 + 3 M 2 + 3 M 2 J 1 α 3 k 3 1 + J 1 ,

λ 11 = E c k 2 + M 2 Pr S c S r α 0 2 4 1 + J 1 , λ 12 = − E c M 2 Pr S c S r J 1 α 0 2 8 1 + J 1 , λ 13 = 3 10 E c M 2 Pr S c S r α 3 2 ,

λ 14 = 1 4 1 + J 1 S c E c k 2 Pr S r − k 2 + M 2 + M 2 J 1 α 0 2   + 2 1 + J 1 ζ λ 0 + Pr S r E c M 2 α 1 2 + α 1 + γ 1 N t γ 1 + N b λ 1 ,

λ 15 = 6 E c M 2 Pr S c S r α 0 α 3 k , λ 16 = 1 6 S c ζ + 2 N b Pr S r γ 2 λ 1 + 2 Pr S r γ 1 2 N t γ 2 + N b λ 2 ,

λ 17 = 1 12 1 + J 1 S c 6 E c M 2 Pr S r 1 + J 1 1 + α 1 α 3   + 36 E c Pr S r α 3 2 + 1 + J 1 ζ λ 2 + 4 Pr S r γ 2 N t γ 2 + N b λ 2 ,

λ 18 = 12 E c Pr S c S r k 2 − 2 M 2 − 2 M 2 J 1 α 0 α 3 k 2 1 + J 1 ,

λ 19 = − 1 6 r + β 4 − 3 + 6 r λ 1 + 6 β 4 λ 1 + S c r 3 ζ + 2 N b Pr S r γ 2 λ 1   + 2 Pr S r γ 1 2 N t γ 2 + N b λ 2 + 3 S c r 2 β 4 ζ + 2 N b Pr S r γ 2 λ 1 + 2 Pr S r γ 1 2 N t γ 2 + N b λ 2 ) ) ,

λ 110 = − 1 120 k 3 1 + J 1 15 E c k 3 Pr S c S r α 0 2 × 2 k 2 r 2 − k 2 + M 2 + M 2 J 1 + 4 k 2 r × − k 2 + M 2 + M 2 J 1 β 4 + k 2 + M 2 + M 2 J 1   × − 1 + cosh 2 k r + 2 k s i n h 2 k r β 4 . + 240 E c Pr S c S r α 0 k 2 M 2 − 1 + cosh k r − k 2 M 2 J 1 − k 2 M 2 α 1 − k 2 M 2 J 1 α 1 + 12 k 2 α 3 − 18 M 2 α 3 − 18 M 2 J 1 α 3 + k s i n h k r × 6 r k 2 − 2 M 2 − 2 M 2 J 1 α 3 + k 2 M 2 1 + J 1 1 + α 1 + 3 k 2 M 2 r 2 1 + J 1   + 2 − k 2 + M 2 + M 2 J 1 α 3 β 4 + cosh k r × k 2 M 2 α 1 + M 2 J 1 k 2 + k 2 α 1 + 3 α 3 6 + k 2 r r − 2 β 4 + 3 α 3 − 4 k 2 + 6 M 2 + k 2 r M 2 r + 2 k − M k + M β 4 + 2 k 3 18 E c M 2 Pr S c S r r 6 1 + J 1 α 3 2 + 108 E c M 2 Pr S c S r r 5   × 1 + J 1 α 3 2 β 4 + 60 1 + J 1 λ 0 + 30 r 2 1 + J 1 × S c ζ λ 0 + Pr S r E c M 2 α 1 2 + α 1   + γ 1 N t γ 1 + N b λ 1 + 2 λ 2 + 60 r 1 + J 1 β 4 × S c ζ λ 0 + Pr S r E c M 2 α 1 2 + α 1 + γ 1 N t γ 1 + N b λ 1 + 2 λ 2 + 5 S c r 4 6 E c M 2 Pr S r 1 + J 1 1 + α 1 α 3   + 36 E c Pr S r α 3 2 + 1 + J 1 ζ λ 2 + 4 Pr S r γ 2 N t γ 2 + N b λ 2 + 20 S c r 3 β 4 6 E c M 2 Pr S r 1 + J 1 1 + α 1 α 3 + 36 E c Pr S r α 3 2 + 1 + J 1 ζ λ 2 + 4 Pr S r γ 2 N t γ 2 + N b λ 2 .

The following is a structure that represents the final semi-analytic solution to the non-linear model:

Ψ x , y = Ψ 0 x , y + Ψ 1 x , y + ⋯ ,

Θ x , y = Θ 0 x , y + Θ 1 x , y + ⋯ ,

χ x , y = χ 0 x , y + χ 1 x , y + ⋯ .

From Eq. (23) and boundary condition, the induced magnetic force takes the form:

F x , y = 1 60 k r R m ( 60 cosh k r r α 0 + α 16 + 30 k y 2 − r 2 s i n h k r α 0 + α 16 + r k y − r 2 − 15 y + r 2 α 3 − 10 2 y + r α 12   − 15 y + r 2 α 13 − 6 2 y 3 + r 4 y 2 + 3 r 2 y + r α 14 − 10 k y 6 − 3 y 2 r 4 + 2 r 6 α 15 − 60 ⁡ cosh k y α 0 + α 16 .

4 Entropy generation

The generation of thermal and technical elements is negatively impacted by irreversible losses, which in turn leads to a decline in the efficacy of the system and an increment in the amount of entropy found in the system. It is of the highest significance to determine the elements that increase the efficiency of the motion mechanism and lower the amount of entropy that is produced. By providing a quantitative measure of the amount of entropy that is produced, the 2nd law of thermodynamics makes it easier to compute irreversibilities with more ease. A number of phenomena, including viscous dissipation, convective heat transfer, and heat transfer across narrow temperature fluctuations, are responsible for the occurrence of irreversibility. For purposes that need a significant amount of energy, such as the cooling of electronic systems, solar power collectors, and geothermal energy systems, the formation of entropy is absolutely necessary. The primary focus of the investigation is on the investigation of fluid dynamics in conjunction with heat transmission via the application of various boundary restrictions. The mathematical estimate is utilized to ascertain the rate at which entropy is produced as a consequence of the irreversible influence of heat and viscosity [8], 9]:

E g = k f T r − T l 2 ▿ T ⋅ ▿ T + 1 T r − T l S i j e i j + 1 T r − T l σ f B 0 2 u 2 + D B T r − T l ▿ ϕ ⋅ ▿ T + D B T r − T l ▿ ϕ ⋅ ▿ ϕ .

We obtain after employing the nondimensional variables and the long wavelength approximation the following form:

N s = ∂ Θ ∂ y 2 + E c P r 1 1 + J 1 Ψ y y 2 + M 2 E c P r Ψ y + 1 2 + L ∂ χ ∂ y ∂ Θ ∂ y + L ∂ χ ∂ y 2 ,

where L = D B ϕ r − ϕ l k f is the diffusion parameter and N s = s 0 2 k f E g .

The Bejan number, often known as B e , is a mathematical term that measures the ratio of irreversible heat transmission to the total formation of entropy. As a result, it defines the ratio of entropy generation [53]:

B e = ∂ Θ ∂ y 2 ∂ Θ ∂ y 2 + E c P r 1 1 + J 1 Ψ y y 2 + M 2 E c P r Ψ y + 1 2 + L ∂ χ ∂ y ∂ Θ ∂ y + L ∂ χ ∂ y 2 ,

5 Results and discussion

This section will interpret physically the numerical outcomes for the axial velocity u, temperature outlines Θ, concentration outlines χ, entropic production E g , Bejan number B e , the horizontal element of the induced magnetized force B x and magnetic force F in the Jeffery model of Buongiorno nanoliquid with induced magnetic field and electroosmotic force flow in complicated peristaltic tapering conduit with velocity, thermal, and concentration slip boundary conditions. Mathematica programming is used to calculate computational outcomes. Unless otherwise specified, all dimensionless parameters are selected within realistic physiological and microfluidic ranges to ensure practical relevance to blood flow and nanofluid transport applications Then we advance to explain and evaluate these results and present a physical interpretation of these results. Figure 2(a–d) explain the influences of the Jeffrey parameter J 1, thermal and concentration Grashof G 1 and G 2, Soret number S r , and Schmidt S c numbers, electroosmosis force β 1 and velocity slip parameter β 2 respectively, on the axial velocity u. Figure 2(a) shows how the particle dynamics (Jeffery parameter J 1) and buoyancy forces (thermal and concentration Grashof numbers G 1, G 2) impact on the axial velocity u. It can be viewed that increasing in J 1 that take values of 0–3 lead to more complexity and nonlinearity in flow behavior due to the presence of suspended particles, the axial speed of the Newtonian fluid is less than that of the case of the Jeffery model. We can also notice that in the case of buoyancy forces (G 1 = G 2 = 2), the axial speed u enhances by advancing the mid, and the inverse behaviour happens by flowing far from the midway of the channel. Figure 2(b) inspects how buoyancy forces reshape the velocity distribution across the channel due to the changes in the thermal and concentricity Grashof numbers G 1, G 2. It is remarked that the thermal Grashof number G 1 reflects buoyancy forces due to temperature differences in the fluid whereas G 1 increases, the buoyant force drives stronger fluid motion, hence the peak velocities increase. Also, a comparison between dashed and thick lines shows the effect of solutal buoyancy G 2 where dashed lines (G 2 = 3) show more pronounced velocity profiles, indicating the combined effect of thermal and concentration buoyancy enhancing the flow. Thick lines (G 2 = 0) are less intense since only thermal buoyancy is active. Effect of Soret S r and Schmidt S c Numbers presented in Figure 2(c). It is clear from this chart that as the thermal diffusion effects S r (how heat drives mass diffusion) increase, the temperature gradient causes more significant mass flux that tends to increase in velocity profile near the channel walls while reducing it in the central region. The explanation for this may be ascribed to the opposite relationship that exists between the Soret number and the concentration variation ϕ r − ϕ l , which is responsible for producing the impact that is seen. Also, Schmidt number S c is the ratio of momentum diffusivity (viscosity) to mass diffusivity: lower S c  = 0.4 implies higher mass diffusivity, which enhances mixing and shifts the flow profile while higher S c  = 0.8 limits diffusion, resulting in steeper gradients in velocity. Figure 2(d) portrays the adjustments in the axial rapidity with the mobility of the medium β 1 and slip velocity β 2, we illustrate that the slip velocity parameter enhances the velocity profile at all channels whole except the region − r x , − 0.4 that it has a reverse effect. We also notice that the case of the presence of electroosmotic force (the mobility of the medium β 1 = 100) enhances the flow significantly near the walls and leads to non-symmetric velocity profiles, while the reverse appears at β 1 = 0.

Figure 2: Variations in horizontal velocity u with (a) J 1, (b) G 1, G 2, (c) S r , S c and (d) β 1, β 2.
Figure 2:

Variations in horizontal velocity u with (a) J 1, (b) G 1, G 2, (c) S r , S c and (d) β 1, β 2.

Figure 3(a–d) display the differences that happen in Jeffery fluid temperature Θ corresponding to the changes in each of the Jeffery variable J 1 with the thermal slip parameter β 3, thermophoresis parameter N t with Brownian motion N b , electroosmotic force β 1 with Hartman number M and Eckert number E c with concentration slip parameter β 4. Figure 3(a) shows that as J 1 increases, the Jeffrey fluid temperature decreases due to increased cooling caused by flow in Jeffrey fluids, so the temperature of Newtonian fluid is more than that in the Jeffrey model case. Also, we show that the case of thermal slip (β 3 = 0.2) increases the temperatures near the sides by allowing additional heat to transmission through the boundary layer compared to the case of no thermal slippery (β 3 = 0). Both thermophoresis N t and Brownian motion N b enhance temperature distribution due to enhanced nanoparticle movement and intensified microscopic thermal fluctuations and dispersion as exhibited in Figure 3(b). Figure 3(c) indicates that as magnetic parameter M increases tends to increase in temperature due to the intensification of the Lorentz force, which suppresses motion and enhances internal energy dissipation, raising the fluid temperature. Also, this figure shows that the electroosmosis force reduces the fluid temperature. The ratio of kinetic energy to thermal energy is indicated by Eckert number (E c ), so by rising E c results in a significant rise in temperature due to viscous dissipation as seen in Figure 3(d). Furthermore, the concentration slip parameter (β 4 = 4) reduce the thermal profile, especially near the channel walls.

Figure 3: Variations in temperature Θ with (a) J 1, β 3, (b) N t , N b , (c) M, β 1 and (d) E c , β 4.
Figure 3:

Variations in temperature Θ with (a) J 1, β 3, (b) N t , N b , (c) M, β 1 and (d) E c , β 4.

Figure 4(a–f) display the changes in concentration profile with various parameters. Figure 4(a) shows that Jeffrey parameter J 1 increases, the concentration profile slightly increases, particularly near the center. This is due to the enhanced velocity in Jeffery fluids at the center, which enhances nanoparticle migration. Also, the case β 4 = 0.6 show lower concentrations compared to (β 4 = 0.2), highlighting the role of concentration slip in facilitating wall-particle interaction. Figure 4(b) appears to reverse the effect between concentration and temperature profiles with thermophoresis N t and Brownian motion N b parameters. Effects of Eckert number E c and electroosmotic parameter β 1 is showed in Figure 4(c): Elevated E c results in more dissipation in viscosity, which influences concentration indirectly through enhanced thermal gradients. Electroosmotic parameter (β 1 = 0.4) enhances particle movement due to electric field interaction which produces more concentrations compared to the no-electroosmotic force (β 1 = 0). Figure 4(d) explores the impact of thermal and concentration slippery parameters β 2 and β 3, correspondingly on concentration profile. Increasing β 2 allows more freely fluid layers near the wall, which indirectly alters particle transport and enhances nanoparticle concentration distribution. Thermal slip (β 3 = 1) reduces thermal boundary layer thickness, resulting in increasing nanoparticle transport compared to the no thermal slip (β 3 = 0). Effect of Soret S r and Schmidt S c Numbers on concentration profile presented in Figure 4(e). These parameters reduce the concentration owing to the opposite relationship between the Soret number and the concentricity difference. Figure 4(f) illustrates the impact of the chemical reactive parameter ζ and the flow rate q on the nanofluid concentration χ. The figure shows that an increase in ζ significantly enhances the concentration on the negative side of the conduit axis, whilst this influence diminishes toward the positive side. Additionally, it is reflected that an increment in flow rate reduces the concentration on both sides of the channel.

Figure 4: Variations in concentration χ with (a) J 1, β 4, (b) N b , N t , (c) E c , β 1, (d) β 2, β 3 and (e) S r , S c , and (f) ζ, q.
Figure 4:

Variations in concentration χ with (a) J 1, β 4, (b) N b , N t , (c) E c , β 1, (d) β 2, β 3 and (e) S r , S c , and (f) ζ, q.

Entropy production E g is a determination of irregularity or disequilibrium in thermal and physical processes and expresses the extent of the loss of usable energy in each system. Figure 5(a–d) illustrate how the entropy generation rate E g responds to variations in key physical parameters. Figure 5(a) shows the effect of the Jeffery parameter J 1 and slip velocity β 1, while Figure 5(b) highlights the influence of Brownian motion N b and thermal slip β 3. In Figure 5(c), the impacts of the Debye–Hückel factor k with concentration slip β 4. Finally, Figure 5(d) displays the combined influences of the thermophoresis variable N t and Hartmann number M on entropy production. Figure 5(a) illustrates that the entropy generation decreases significantly at the wall with higher J 1 due to enhanced resistance in Jeffery fluid velocity near the wall. Entropic production for the Newtonian situation is less than that of the situation of non-Newtonian fluid. Also, slip velocity reduces the entropy generation compared to the case of no-slip velocity β 2, showing how wall slip boosts velocity and thus reduces irreversibility. Figure 5(b) establishes that the Brownian motion N b improves the temperature distribution, thus directing a boost in entropic production, indicating that the system loses more useful energy. The presence of thermal slip enhances entropy generation near the center of the channel and prevents further energy loss at the walls, allowing greater heat flow near the boundaries. Figure 5(c) shows the effect of the Debye–Huckel coefficient k and concentration slip β 4. As the Debye–Huckel coefficient k increases, entropy generation often increases near the walls. This is due to the enhanced electrostatic interactions and internal resistance in this region, which increases the irreversibility of the system. The opposite occurs near the middle of the channel due to the weak or almost non-existent effect of electrostatic interactions in the middle. Concentration slip also reduces entropy generation, demonstrating that wall-particle interaction via slip significantly affects dissipative behavior. Higher magnetic field strength M increases flow resistance, leading to increased entropy generation. Higher thermosphere effects N t also push the nanoparticles toward cooler regions, enhancing the thermal gradient and increasing the generated entropy, as shown in Figure 5(d).

Figure 5: Variations in entropy generation E g with (a) J 1, β 2, (b) N b , β 3, (c) k, β 4 and (d) M, N t .
Figure 5:

Variations in entropy generation E g with (a) J 1, β 2, (b) N b , β 3, (c) k, β 4 and (d) M, N t .

Bejan number B e is the ratio of the thermal conduction force to the frictional (viscous) force within a system and is used to estimate the rates of entropic production caused by heat transference compared to viscous friction. Figure 6(a) confirms that B e generally declines in the zone 0 , r x with increasing J 1, indicating that viscous dissipation becomes more dominant overheat transfer. Slip velocity reduces B e , showing that near-wall friction contributes more to entropy generation under slip conditions. High values of N b lead to thermal fluctuations that disrupt the energy transfer balance, reducing B e , the effect of which is most pronounced in the upper half of the channel. The presence of thermal slip significantly enhances this effect, especially in the boundary regions, as evidenced by the shift of the region of minimum Bejan number closer to channel walls as shown in Figure 6(b). Figure 6(c) displays that with increasing k, the Bejan number decreases, especially at the sides and near the middle of the channel, while it increases in the intermediate regions of the upper and lower halves of the channel, indicating that B e is greatly influenced by the coefficient k. It is also evident that concentration slip leads to a stronger shift from thermal entropy to frictional entropy, which enhances the Bejan number. Increasing magnetic fields M impedes fluid movement and promotes local heating, which leads to a surge in B e with a decline in frictional entropy. Thermophoresis N t also enhances this effect, especially at higher values, by increasing the flux of nanoparticles and changing the energy dissipation areas as shown in Figure 6(d).

Figure 6: Variations in Began number Be with (a) J 1, β 2, (b) N b , β 3, (c) k, β 4 and (d) M, N t .
Figure 6:

Variations in Began number Be with (a) J 1, β 2, (b) N b , β 3, (c) k, β 4 and (d) M, N t .

The horizontal element of the induced magnetized force B x is the part of the field that arises from the interaction between the fluid motion and the external magnetic field and is in the direction of flow. It plays an important role in influencing fluid thermodynamics, stability, and electromagnetic forces within MHD (magnetohydrodynamics) systems. Figure 7(a–d) study the effects of physical parameters on B x . Figure 7(a) refers to the horizontal magnetic field B x decreases with increasing J 1, especially in the presence of buoyancy forces G 1 = G 2 = 0.3 . This decrease is due to increased fluid resistance and the suppression of convection currents that contribute to magnetic induction. In the absence of buoyancy, higher values of B x result, indicating the role of convention in damping the magnetic field. Figure 7(b) shows that as M increases, a more complex waveform appears in B x due to increased magnetic damping and induced field interaction. The slip velocity also leads to larger amplitudes compared to the case of no-slip condition as the lower wall friction allows for stronger magneto-flux coupling. Figure 7(c) discusses the effect of magnetic Reynolds number, where a higher R m enhances the induced magnetized force caused by the increase in magnetized convection compared to diffusion, and a higher value leads to an increase in the curvature of the field lines and a decrease in the lower middle of the channel. Expanding G 1 enhances the effects of convection, affecting B x through the magnetic disturbance generated by the flow. Increasing G 2 indicates a weaker induced magnetic field in the lower middle of the channel, indicating deeper penetration of magnetic diffusion due to convection generated by the concentration as displayed in Figure 7(d). Figure 8(a–d) refer to the magnetic force F as a decreasing function with J 1 and M and increasing with R m and G 1. Also, these figures indicate that there is an obvious effect for buoyancy forces, slip velocity, and Debye–Huckel coefficient only at the lower wall of the channel.

Figure 7: Variations in horizontal component of induced magnetic field B x with (a) J 1, (b) M, β 2, (c) k, R m and (d) G 1, G 2.
Figure 7:

Variations in horizontal component of induced magnetic field B x with (a) J 1, (b) M, β 2, (c) k, R m and (d) G 1, G 2.

Figure 8: Variation in magnetic force F $\mathcal{F}$ with (a) J 1, (b) M, β 2, (c) R m , k and (d) G 1, G 2.
Figure 8:

Variation in magnetic force F with (a) J 1, (b) M, β 2, (c) R m , k and (d) G 1, G 2.

Figures 9–16 illustrate the influences of thermo-physical factors on trapping behaviour. Trapping in fluid mechanics implies the confinement of fluid molecules within closed streamlines or specific regions of the flow, preventing them from escaping. This phenomenon can result from various influences such as hydro-dynamical forces, pressure gradients, or connections with boundaries and surrounding molecules. Trapping is widely applied in cell manipulation, targeted drug delivery, and the investigation of microscale biological processes. As the ratio of relaxing to retarding time parameter J 1 upsurges, the size of the bolus increases and creates new closed contours, and some boluses vanish in the upper part of the 1st wave of the complicated peristalsis conduit. Meanwhile, the bolus volume increases, and new complete boluses form in other zones, as illustrated in Figure 9. Figure 10 illustrates that the streamlined patterns reveal how the increased wall slip velocity (β 2) modifies the overall flow behavior. The bolus size decreases and some of it disappears in the upper half, while the opposite occurs in the lower half of the channel. Thermal and concentration slips show the inverse effect of slip velocity on streamlines as seen in Figures 11 and 12. Figure 13 demonstrates the effects of the Grashof thermal number G 1 on streamlines. We observe that increasing G 1 results in an upsurge in the size of the bolus (confined vortices), as well as the formation of additional complete bolus at two wave crests. At higher values of G 1, buoyancy forces become dominant, enhancing natural convection. This results in more complicated flowing patterns like eddies, vortices, and possibly turbulent flows, which contribute to improved mixing within the fluid caused by stronger convective streams. Therefore, entrainment mechanisms must consider these dynamic and potentially chaotic conditions associated with an increased G 1. Figure 14 shows that the concentration of Grashof number G 2 has the opposite effect on streamlines compared to the thermal Grashof number. The cause of the tapered variable l 1 on the streamlines is exhibited in Figure 15. When l 1 = 0, the channel has a uniform width, and the flow exhibits symmetric and stable vortices. At l 1 ≠ 0, the taper introduces geometric asymmetry, resulting in vortex elongation and horizontal shifting, especially in the upper recirculation regions. This is a direct result of the wall curvature change, which modifies the pressure gradient distribution. The main distinction between the pure peristalsis waves and the complicated peristalsis waves can be observed in Figure 16. We demonstrate that one crest of the peristalsis wave denoted as l 2, is equal to zero, but the complicated peristalsis wave, denoted as l 2, has two crests that are not equal to zero. Increasing the complex amplitude l 2 also results in a boost in the size and quantity of boluses, as well as an elevation in the peak of the wave. Thus, increasing l 2 leads to greater fluid entrapment and recirculation, typical in physiological transport systems with complex contractions.

Figure 9: Variations in streamlined Ψ x , y ${\Psi}\left(x,y\right)$ with Jeffery parameter J 1.
Figure 9:

Variations in streamlined Ψ x , y with Jeffery parameter J 1.

Figure 10: Variations in streamlined Ψ x , y ${\Psi}\left(x,y\right)$ with slip velocity β 2.
Figure 10:

Variations in streamlined Ψ x , y with slip velocity β 2.

Figure 11: Variations in streamlined Ψ x , y ${\Psi}\left(x,y\right)$ with slip temperature β 3.
Figure 11:

Variations in streamlined Ψ x , y with slip temperature β 3.

Figure 12: Variations in streamlined Ψ x , y ${\Psi}\left(x,y\right)$ with slip concentration β 4.
Figure 12:

Variations in streamlined Ψ x , y with slip concentration β 4.

Figure 13: Variations in streamlined Ψ x , y ${\Psi}\left(x,y\right)$ with thermal Grashof numbers G 1.
Figure 13:

Variations in streamlined Ψ x , y with thermal Grashof numbers G 1.

Figure 14: Variations in streamlined Ψ x , y ${\Psi}\left(x,y\right)$ with concentration Grashof numbers G 2.
Figure 14:

Variations in streamlined Ψ x , y with concentration Grashof numbers G 2.

Figure 15: Variations in streamlined Ψ x , y ${\Psi}\left(x,y\right)$ with tapering parameter l 1.
Figure 15:

Variations in streamlined Ψ x , y with tapering parameter l 1.

Figure 16: Variations in streamlined Ψ x , y ${\Psi}\left(x,y\right)$ with complicated peristalsis parameter l 2.
Figure 16:

Variations in streamlined Ψ x , y with complicated peristalsis parameter l 2.

6 Validation

Figure 17 validate the current work to previous study as a limitation of present work. Figure 17(a) illustrates the comparison between present method (ADM) and previous analytical solution of “peristaltic transport of a Jeffrey fluid under the effect of magnetic field in symmetric channel by Kothandapani and Srinivas [54] by taking G 1 = G 2 = β 1 = β 2 = β 3 = β 4 = l 1 = l 3 = 0. It is observed that the results of present method are significantly close to analytical solution in Ref. [54]. If we take G 1 = G 2 = β 2 = β 3 = β 4 = l 1 = l 3 = 0 we get the same results to clear fluid of electroosmotically controlled by MHD peristalsis flowing in a uniformly channel by Maraj et al. [55] (Figure 17(b)).

Figure 17: Comparison results of axial velocity with (a) Ref. [54] and (b) Ref. [55].
Figure 17:

Comparison results of axial velocity with (a) Ref. [54] and (b) Ref. [55].

7 Concluding remarks

This research provided an advanced simulation of electroosmotic Jeffery–Buongiorno nanofluid flow within complex arterial channels. The novelty of the present work lies in the combined analysis of electroosmotic, magnetohydrodynamic, and slip effects in Jeffery–Buongiorno nanofluid flow through complex wavy arteries. This study reveals new interactions between electro-magnetic forces and thermodynamic irreversibility in biofluid systems. The inclusion of slip parameters, nonlinear wall shapes, and transport forces allowed for accurate characterization of thermofluidic behavior. The streamlined visualizations, entropy analyses, and induced magnetic effects confirmed the profound role of geometry and transport phenomena in shaping the system dynamics. For the purpose of providing approximate solutions for the non-linear model of PDEs, the ADM is applied. The following is a summary of the more significant discoveries that were discovered in the current investigation:

  1. Expanding the Jeffery variable causes nonlinear flow due to suspended particles and lower Newtonian fluid axial velocity than the Jeffery model. Buoyancy forces enhance the axial speed at the channel center and decline its distance.

  2. Increasing thermal and solutal buoyancy forces enhance fluid velocity, with combined effects producing stronger flow than thermal buoyancy alone.

  3. A greater Soret number enhances velocity along the walls while diminishing it in the center, due to improved thermal diffusion. A lower Schmidt number improves mixing by increasing how easily mass spreads out, while a higher Schmidt number reduces diffusion, leading to sharper changes in velocity.

  4. The slip velocity β 2 generally increases flow, except in a specific region. High medium mobility (β 1 = 100) due to electroosmotic forces enhancing near-wall velocity and causing flow asymmetry, unlike the symmetric profile at β 1 = 0.

  5. Jeffrey fluids have reduced temperatures compared to Newtonian fluids when the Jeffery parameter grows. Thermal slip improves heat transfer at the wall. Thermophoresis and Brownian motion enhance temperature distribution. A higher magnetic field raises fluid temperature by the Lorentz force, whereas electroosmosis diminishes it. Viscous dissipation associated with elevated Eckert numbers increases temperature, whereas concentration slip β 4 decreases it near the walls.

  6. An augmentation of the Jeffrey parameter, thermal slip, and electroosmotic parameter enhances concentration by promoting particle mobility. Conversely, increased concentration slip, Soret number, Schmidt number, and flow rate diminish concentration. Thermophoresis and Brownian motion demonstrate contrasting influences on concentration and temperature distributions. A greater chemical reaction rate enhances concentration on the negative side of the channel, whereas this impact declines towards the positive side.

  7. Entropy generation increases with factors such as the Brownian motion parameter, Debye–Hückel parameter, Hartmann number, and thermophoresis parameter due to enhanced thermal gradients and flow resistance, while it decreases with parameters like Jeffery parameter, slip velocity, and concentration slip which promote smoother flow and reduce wall friction.

  8. The Bejan numbers often reduce with an increment in the Jeffrey parameter and slip velocity, indicating dominant frictional influences. Increased N b and thermal slip further diminish B e , particularly along channel walls. An upsurge in the Debye–Hückel factor diminishes B e at the center and boundaries while increasing it in intermediate regions. The concentration slip increases frictional entropy, increasing B e . Enhanced magnetic fields and thermophoresis augment B e by increasing localized heating and nanoparticle flow, especially in the upper channel.

  9. The horizontal elements of the induced magnetic field are affected by fluid motion and external magnetic field. Increasing the Jeffery parameter reduces the horizontal magnetic field due to higher fluid resistance and suppressed convection. Higher magnetic parameters cause complex horizontal magnetic field patterns due to stronger damping and induced field interaction. Slip velocity enhances amplitude compared to no-slip cases, and an increased magnetic Reynolds number increases magnetic convection, enhancing the induced field. Increasing G 1 enhances convection affecting B x , while increasing G 2 weakens B x at the channel center due to deeper magnetic diffusion by concentration-driven convection.

  10. Magnetic force F decreases with J 1 and M and increases with R m and G 1. Furthermore, buoyancy forces, slip velocity, and the Debye–Huckel coefficient have significant effects alone at the lower wall of the channel.

  11. Increasing J 1 expands bolus size, generates new closed shapes, and results in the disappearance of certain boluses or the formation of others in different locations.

  12. Streamline patterns demonstrate how wall slip velocity affects flow behavior. The top part of the channel loses bolus size, while the lower half gains it.

  13. The slip velocity inversely affects streamlines for thermal and concentration slips.

  14. An upsurge in the Grashof thermal number is caused by a larger bolus (restricted vortices) and the appearance of additional full boluses at two wave crests.

  15. The concentration Grashof number exhibits an inverse influence on streamlines compared to the thermal Grashof number.

  16. The channel is uniformly wide and possesses symmetric and stable vortices when l 1 = 0. The taper at l 1 ≠ 0 causes vortex elongation and horizontal shifting, particularly in upper recirculation zones, due to geometric asymmetry. By comparing pure and complex peristaltic waves, demonstrating a single crest for l 2 = 0 and two crests for l 2 ≠ 0. Enhanced complex amplitude l 2 enhances the size and quantity of wave crests and bulges.

Future research should look into temporary interactions between electro-osmosis and MHD in pulsing blood flows, testing with blood vessel models that can change their flexibility based on individual patients and using machine learning to better adjust magnetic fields for effectively delivering nano-drugs. Especially for cardiovascular treatments and cancer therapy, these developments would link theoretical models with practical applications in precision medicine.

Corresponding authors: Mohamed R. Eid, Center for Scientific Research and Entrepreneurship, Northern Border University, Arar 73213, Saudi Arabia, E-mail: ; and Essam M. Elsaid, Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha, 61922, Saudi Arabia, E-mail: 

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number NBU-FFR-2025-3021-21. The authors are thankful to the Deanship of Graduate Studies and Scientific Research at University of Bisha for supporting this work through the Fast-Track Research Support Program.

  1. Funding information: The Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number NBU-FFR-2025-3021-21. The Deanship of Graduate Studies and Scientific Research at University of Bisha for supporting this work through the Fast-Track Research Support Program.

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

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

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

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Received: 2024-08-07
Accepted: 2025-12-17
Published Online: 2026-01-08

© 2026 the author(s), published by De Gruyter, Berlin/Boston

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

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https://doi.org/10.1515/phys-2025-0255
Keywords for this article
Jeffery nanofluid; irreversibility process; electroosmosis force; induced magnetic field; peristaltic motion
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BY 4.0

Articles in the same Issue

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