Bernstein collocation method for Jeffery–Hamel nanofluid flow and heat transfer: a comparative study with the Chebyshev collocation method

1 Introduction

Nanofluids are a special kind of fluid formed by distributing nanometer-sized particles into a standard base fluid such as water, oil, or blood. The term “nanofluid” was initially coined by Choi (1995), who found that suspending tiny quantities of nanoparticles in a base fluid significantly enhances its thermal conductivity compared to that of the base fluid on its own. According to the application, several types of nanomaterials are introduced into the base fluid, including metallic particles such as copper and silver, oxide-based particles such as alumina and titanium oxide, and carbon-based particles such as graphene and carbon nanotubes. Due to their modified thermal conductivity, viscosity, and heat-transfer properties, a variety of nanofluids have been developed as a viable alternative to standard fluids in complex engineering and therapeutic systems. When small particles flow away from a source or toward a sink across converging and diverging channel configurations, the Jeffery–Hamel (JH) flow-based framework may adequately explain the flow characteristics (Ahmad and Ilyas, 2019; Namoos and Jasim, 2025). This physical structure leads to the phenomenon that is mathematically termed as Jeffery–Hamel flow, which was initially described by Jeffery (1915) and Hamel (1916). This flow depicts the uniform radial motion of a viscous incompressible fluid over a wedge-shaped channel, driven by two non-parallel walls that may converge or diverge depending on the flow pattern. The governing equations for this type of flow are naturally non-linear and are not calculated precisely in most cases. As a result, many scientists have used a variety of analytic and numerical techniques to obtain approximate solutions and examine the effects of various physical factors on flow behavior. Given its real-world relevance, this framework is widely used in engineering and biomedical research, including studies of MHD flows and blood flow modeling (Joneidi et al., 2010; Rahman et al., 2021).

Many studies on magnetohydrodynamic JH flow can be performed using a variety of analytical and computational methods. Magnetohydrodynamics (MHD) is the analysis of the properties of electrically conducting fluids under the influence of a magnetic field. An interaction involving magnetic fields and the fluid generates a Lorentz force that directly alters the fluid motion; its influence on Jeffery–Hamel flow has been extensively studied. Hamrelaine et al. (2019) analyzed the non-linear governing equations and found that the velocity profile improves significantly as the magnetic field strength increases. The role of an inclined magnetic field in thermal and mass transfer was examined by Onyango et al. (2020), who showed that these factors significantly modify the velocity and temperature distributions. The stability of the resultant solution was further analyzed by Ullah et al. (2024), who report that both the flow dynamics and the stable state are extremely sensitive to variations in the Reynolds and Hartmann numbers. The combined effect of a nanofluid and a magnetic field on Jeffery–Hamel flow has gained attention in recent years. The presence of an electromagnetic field alters the nanofluid’s thermal and physical properties, thereby affecting the velocity and temperature profiles within the channel. The suspended form of the nanoparticles further improves the fluid’s heat-tranfer properties and when integrated into a magnetic field, offering greater control over the flow pattern in converging and diverging channels (Sheikholeslami et al., 2012; Imani et al., 2012; Biswal and Chakraverty, 2022).

The impact of porous medium on Jeffery–Hamel flow was also studied. Darcy’s law, which states that the fluid velocity across a porous medium is directly proportional to the pressure gradient and inversely proportional to the fluid viscosity and medium’s stiffness, is the basic model for characterizing porous media phenomena. An analysis of flow behavior through converging/diverging channels embedded in a porous material was performed by Hemayat et al. (2025), revealing that the porous framework imposes resistance on flowing fluids and markedly changes the velocity profile. Ahmad and Farooq, (2023) examined the double-diffusive JH flow of a nanofluid through a permeability medium, thereby showing that the porous form creates a resistance to the nanofluid flow, while the nanoparticle dispersion additionally improves thermal conductivity as well. An accurate representation of fluid flow includes not only the velocity field but also the temperature and concentration domains. The thermal field is the dispersion of heat energy within a fluid due to temperature variations between a fluid and its boundaries. On the other hand, the concentration field describes the distribution of the dissolved substances or dispersed particles within a fluid, which is determined by concentration variations across regions. Each field plays crucial role in medication delivery, chemical interactions, and biological mechanisms. Extending JH flow to incorporate such impacts leads to a broader physically relevant model (Ullah et al., 2024a; Ullah et al., 2024b; Haq et al., 2025).

The combined effect of a magnetic field and porous media on nanofluid flow in the Jeffery–Hamel geometry was studied using a range of analytical and computational methods. Pourabdian et al. (2016) studied three types of nanoparticles, namely, copper, alumina, and titania, and discovered that the volume fraction showed a greater influence on heat transmission than the type of particles that was used. Li et al. (2018) utilized the Galerkin technique and found that the nanoparticle volume fraction and Reynolds number both impact the velocity and the temperature profile features, with the Nusselt number improving as both factors increased. Akinshilo et al. (2020) applied the homotopy perturbation approach to show that higher Reynolds numbers increase fluid viscosity, whereas the Darcy number regulates heat transmission inside the channel. Abdulridah and Jasim, (2023) used the perturbation iteration method to show that the Prandtl number (the ratio of momentum diffusivity to thermal diffusivity) has a major effect on temperature. Jan et al. (2025) used the NDSolve approach to analyze the combined effects of the magnetic field and heat radiation, finding that the Hartmann number and nanoparticle volume ratio significantly influenced the velocity and temperature fields, respectively.

In this regard, advanced mathematics introduces collocation-based methods for approximating solutions of differential equations using a variety of orthogonal polynomials, such as Chebyshev, Legendre, Fibonacci, and Bernstein, among others. In general, the Chebyshev collocation method is a well-known spectral mathematical approach that approximates solutions by extending them in terms of Chebyshev polynomials based on Gauss-Lobatto nodes, which are recognized for their superior convergence rate, high accuracy, and mathematical stability, transforming differential equations into a system of algebraic equations using a set of collocation points (Lovetskiy et al., 2024; Khidir and Alfaydi, 2025; El-Baghdady and El-Azab, 2016; Karunakar Chakraverty, 2019). In a similar framework, BCM relies on Bernstein polynomials, which create a full basis across a specific interval and are renowned for their neutrality, consistent convergence, and stable computation without overcoming ill-conditioning challenges, making them highly appropriate for approximate solutions for complicated non-linear problems (Coluccio et al., 2008; Bataineh et al., 2021; Shahni and Singh, 2021). Jafarian et al. (2013) demonstrated the efficiency of BCM in resolving standardized systems of integral equations; Hammad (2021) used it to normalize extended wave equations; Ali (2021) verified its spectral adequacy for functional differential equations. Bataineh et al. (2022) used BCM for the first time to solve the non-linear MHD Jeffery–Hamel vascular blood-flow problem, confirming its performance compared with that of the homotopy perturbation method (HPM).

To further advance this framework beyond numerical techniques, the current study uses response surface methodology (RSM) to statistically optimize and assess the heat transfer properties of MHD Jeffery–Hamel nanofluid blood flow in a stenosed arterial channel. Although the Bernstein collocation method provides precise numerical solutions to the governing flow equations, it does not inherently quantify how sensitive output responses, such as the Nusselt number and skin friction coefficient, are to changes in the input flow variables, nor does it identify which variable has the greatest impact on heat transfer. RSM bridges this gap by establishing a functional relationship between independent input factors and the corresponding output response variables through a statistically optimized set of experiments, thereby enhancing the numerical results derived from BCM. The central composite design (CCD) proposed by Box and Wilson forms the foundation of RSM and allows the prediction of linear, quadratic, and interaction impacts among the chosen parameters with only a small number of experimental runs. In the context of nanofluid flow problems, RSM has been widely utilized by many researchers to improve heat-transfer rates and skin friction in relation to key flow characteristics. Algehyne et al. (2023) studied entropy optimization methods and response surface methodology for the blood hybrid nanofluid movement through an integrated stenosis artery with magnetically embedded gold–tantalum particles, and the sensitivity analysis demonstrated that the Hartmann number shows positive sensitivity, while thermal emission and the nanoparticle volume fraction indicate negative sensitivity for all magnetized field values. Tang et al. (2023) utilized RSM-driven optimization methods combined with analysis of variance (ANOVA) to biomagnetic-based Carreau nanofluid blood flow in a stenosed artery. Ul Haq et al. (2025) conducted a sensitivity analysis of ternary nanofluid flow over a curved stretching surface using RSM and ANOVA, demonstrating that the Hartmann number is the most significant factor affecting both the skin friction and the heat-transfer rate.

Despite notable advancements in the analysis of nanofluid flow, certain key factors remain inadequately addressed in the published works. Although copper ( C u ) nanoparticles have been studied in Jeffery–Hamel flow (Chandra and Das, 2023), comparative analysis involving iron oxide ( F e 3 O 4 ) particles with blood as a base fluid in the same flow structure has not been carried out. Moreover, while F e 3 O 4 nanoparticles with blood as the base fluid have been extensively studied in various configurations (Bouazzi et al., 2025), their application to Jeffery–Hamel flow in the presence of MHD and porous medium effects has received limited attention. Additionally, the implementation of RSM for the statistical evaluation of controlling the flow parameters in Jeffery–Hamel blood flow problems has not been adequately addressed in the existing literature.

Motivated by this constraint, from a physiological perspective, human arteries exhibit a non-uniform curving geometry that plays a crucial role in determining the blood-flow dynamics. The Jeffery–Hamel flow structure is used to precisely capture such behavior as it closely reflects the converging/diverging characteristics of arterial blood flow. In this regard, the 2D non-linear Navier–Stokes equations governing nanofluid blood flow are reduced to a 1D non-linear ordinary differential equation via appropriate similarity transformations. To solve the equation thus obtained, the BCM is implemented, in which the solution is approximated as a linear combination of Bernstein polynomials. The unknown coefficient values are obtained by incorporating the Chebyshev collocation nodes into the governing equation, thereby collocating the residual at certain points over the domain. This method reduces the differential equation into a system of algebraic equations, which is subsequently solved effectively to obtain the solution.

The novelty of the present study is encapsulated in the following findings:

• The MHD Jeffery–Hamel flow was examined through a constricted channel by integrating distinct nanoparticles, namely, C u and F e 3 O 4, separately, so as to develop a broader and more extensive understanding of nanofluid blood flow behavior.
• The solutions of the BCM were compared with that of the CCM to validate the precision and effectiveness of the proposed approach.
• RSM was used to statistically assess and analyze the effects of the governing parameters on flow behavior

2 Description of the physical problem

Blood constantly circulates through arteries for supplying the body with the essential nutrients. Although blood exhibits non-Newtonian behavior due to red blood cell aggregation and deformation, these effects are particularly dominant at microcirculatory vessels or under low shear rate conditions. However, at the relatively high shear rates found in large arteries, the impact of non-Newtonian effects minimizes substantially, and the apparent viscosity reaches a nearly constant value, thus making the Newtonian assumption reasonable. To study this, blood is considered as a Newtonian fluid and the Jeffery–Hamel flow concepts is used, which help to evaluate the analysis of the variations in blood flow inside converging/diverging channels based on idealized large arterial walls. To ensure consistency and clarity, the proposed model is influenced by the following assumptions,

• The flow is assumed to be two-dimensional, steady, laminar, and incompressible, which is expressed in cylindrical polar coordinates systems ( r , θ ).
• The flow geometry is an arterial segment characterized by two intersecting non-parallel rigid walls enclosing an angle, forming either diverging or converging vascular channels.
• Blood is considered as a Newtonian base fluid that conducts electricity.
• F e 3 O 4 nanoparticles are dispersed in blood owing to their biological compatibility and magnetic behavior.
• C u nanoparticles have exceptional heat conductivity, which causes them to be dispersed uniformly in blood.
• To capture the eccentric effect of F e 3 O 4 and C u nanoparticles on the flow dynamics, a comprehensive study is indispensable.
• Furthermore, the body temperature tends to be higher than the wall temperature ( T > T w ).

3 Formulation of the problem

Based on the assumptions, showcasing that the flow of blood originates from a point source or sink at the intersection of two non-parallel walls with an angle 2 α, the flow geometry is depicted in Figure 1. The present flow structure is specified through cylindrical polar coordinates ( r , θ , z ) with the channel walls. It is defined as convergent when α < 0 and divergent when α > 0. Two-dimensional steady flow ensures that the flow field is dependent on only r and θ and that there is no fluid parameter change in the z-direction. Accordingly, the velocity component is given as V = ( u ( r , θ ) , 0,0 ). The flow is exposed to an external magnetic field B ( r ) = ( 0 , B 0 r , 0 ) acting normal to the radial direction. The induced magnetic field is ignored under the low magnetic Reynolds number assumption. The reduced form of the flow governing equation with respect to the aforementioned assumptions is as follows:

Continuity equation:

ρ n f r ∂ ∂ r r u r , θ = 0 . (1)

Momentum equation:

In r-direction,

u r , θ ∂ u r , θ ∂ r = − 1 ρ n f ∂ P ∂ r + ν n f ∂ 2 u r , θ ∂ r 2 + 1 r ∂ u r , θ ∂ r + 1 r 2 ∂ 2 u r , θ ∂ θ 2 − u r , θ r 2 − σ f B 0 2 ρ n f r 2 u r , θ − ν f ρ n f K u r , θ . (2)

In θ-direction,

1 ρ n f r ∂ P ∂ θ − 2 ν n f r 2 ∂ u r , θ ∂ θ = 0 . (3)

Energy equation:

u r , θ ∂ T r , θ ∂ r = k n f ρ C P n f ∂ 2 T r , θ ∂ r 2 + 1 r ∂ T r , θ ∂ r + 1 r 2 ∂ 2 T r , θ ∂ θ 2 . (4)

The appropriate boundary conditions for the present mathematical framework are outlined below:

At the channel centerline: u = U max, ∂ u ∂ θ = 0, and ∂ T ∂ θ = 0 at θ = 0.

At the wall of the channel: u = 0 , T = T w at θ = α.

Here, P is the fluid pressure, σ f denotes electrical conductivity of base fluid, B 0 represents the electromagnetic induction, T is the fluid temperature, T w denotes the upper wall temperature, ρ n f is the effective density, k n f represents the effective thermal conductivity, μ n f is the effective dynamic viscosity, ν n f is the effective kinematic viscosity of the nanofluid, and C p n f represents the specific heat of nanofluid. Here, ϕ 1 is the volume fraction, μ f is the viscosity of the base fluid, and ρ f and ρ s denote the density of the base fluid and nanoparticle, respectively.

ρ n f = 1 − ϕ 1 ρ f + ϕ 1 ρ s .

μ n f = μ f 1 − ϕ 1 2.5 .

ν n f = μ n f ρ n f .

ρ C P n f = 1 − ϕ 1 ρ C P f + ϕ 1 ρ C p s .

k n f k f = k s + 2 k f − 2 ϕ 1 k s − k f k s + 2 k f + ϕ 1 k s − k f . (5)

For solely radial flow, the velocity component can be calculated by assuming u θ = 0. On integrating Equation 1, it becomeswhere ω ( θ ) is the arbitrary function of θ.

r u r = ω θ , (6)

Introducing ξ = θ α, the dimensionless form of the velocity iswhere u a is the maximum velocity value.

ω ξ = ω θ u a , T T w = Θ ξ r 2 , ξ = θ α , (7)

With the help of Equations 6, 7, the following expressions are obtained:

∂ u ∂ r = − u a r 2 ω ξ ∂ 2 u ∂ r 2 = 2 u a r 3 ω ξ ∂ u ∂ θ = u a α r ω ′ ξ ∂ 2 u ∂ θ 2 = u a α 2 r ω ″ ξ . (8)

∂ T ∂ r = − 2 T w r 3 Θ ξ ∂ 2 T ∂ r 2 = 6 T w r 4 Θ ξ ∂ T ∂ θ = T w α r 2 Θ ′ ξ ∂ 2 T ∂ θ 2 = T w α 2 r 2 Θ ″ ξ . (9)

Integrating Equation 3 with respect to θ gives the pressure in the form:

P 2 ν n f ρ n f = u r , θ r .

The following expression is derived by differentiating P with respect to r and utilizing Equation 7:

1 ρ n f ∂ p ∂ r = − 4 ν n f u a r 3 ω ξ . (10)

Substituting Equations 5, 8, 10 into Equation 2 and the reduced form of ODE is given by,where B 1 = 2 α R e A 1 and B 2 = ( 4 − ( 1 − ϕ 1 ) 1.25 H a − k p 1 A 1 )

ω ‴ ξ + B 1 ω ξ ω ′ ξ + B 2 α 2 ω ′ ξ = 0 , (11)

On utilizing Equation 9 and substituting it into the energy Equation 4, the ODE can be obtained as:where

Θ ″ ξ + 4 α 2 Θ ξ + B 3 ω ξ Θ ξ = 0 , (12)

B 3 = 1 − ϕ 1 + ρ C p s ρ C p f ϕ 1 k s + 2 k f − 2 ϕ 1 k s − k f k s + 2 k f + ϕ 1 k s − k f 2 α 2 P r .

It is subjected to the following boundary conditions:

ω 0 = 1 , ω ′ 0 = 0 , ω 1 = 0 , Θ 1 = 1 , Θ ′ 0 = 0 . (13)

Here, R e is the Reynolds number, and it can be defined as: is the Hartmann number.is the porosity parameter.is the Prandtl number.

R e = α ω max ν f = U max r α ν f d i v e r g e n t − c h a n n e l , α > 0 , ω max > 0 c o n v e r g e n t − c h a n n e l , α < 0 , ω max < 0 .

H a = σ f B 0 2 ρ f ν f .

k p 1 = ν f K U max

P r = ω max ρ C p f k f

A 1 = 1 − ϕ 1 2.5 1 − ϕ 1 + ρ s ρ f ϕ 1 .

The skin friction coefficient ( C f ) and wall shear stress ( τ w ) are represented mathematically as follows: Substituting Equation 7 into Equations 14, 15 gives:

C f = τ w ρ f U max 2 . (14)

τ w = μ n f 1 r ∂ u r , θ ∂ θ . (15)

C f = 1 R e 1 − ϕ 1 2.5 ω ′ 1 . (16)

The mathematical formulas for the local Nusselt number ( N u ) and the rate of heat are provided below:

N u = r q w | θ = α k f T w , q w = − k n f ∇ T . (17)

Using Equation 7, the above Equation 17 can be simplified to:

N u = − 1 α k n f k f Θ ′ 1 . (18)

4 Methodology

The BCM is an efficient numerical approach that represents the unknown solution with a collection of Bernstein basis polynomials and solves the specified differential equation at several precisely selected collocation points within the domain.

This section discusses the Bernstein polynomials defined over the interval [0, 1] and provides a basic structure for creating approximation solutions using BCM, which are given as:

Definition: the Bernstein polynomial of degree m defined on the interval [0,1] is given as in Equation 19 (Bhatti and Bracken, 2007; Dascioglu and Isler, 2013)where the binomial coefficient is

B j , m ξ = m j ξ j 1 − ξ m − j , j = 0,1 , … . , m , (19)

m j = m ! j ! m − j ! .

Set B j , m as zero for any index j that is negative or exceeds m. The function of ω ( ξ ) and Θ ( ξ ), which defines an exact solution to Equations 11–13, is then calculated using the BCM technique. Here, C and D are constant matrices of size 1 × ( m + 1 ), with unknown values to be obtained, and Φ ( ξ ) is a matrix of size ( m + 1 ) × 1, where the coefficients are the Bernstein basis polynomials, with each of them defined as follows.

ω ξ ≈ ω m ξ .

= ∑ j = 0 m c j B j , m ξ .

= C Φ ξ . (20)

Θ ξ ≈ Θ m ξ .

= ∑ j = 0 m d j B j , m ξ .

= D Φ ξ . (21)

C = c 0 , c 1 , c 2 , … . , c m 1 × m + 1 , (22)

D = d 0 , d 1 , d 2 , … . , d m 1 × m + 1 , (23)

Φ ξ = B 0 , m ξ , B 1 , m ξ , … , B m , m ξ m + 1 × 1 . (24)

The nth order derivative of ω ( ξ ) can be represented by the Bernstein basis polynomial elements as follows:

ω n ξ ≈ ω m n ξ = C Φ n ξ , n = 1,2,3 , … . (25)

Θ n ξ ≈ Θ m n ξ = D Φ n ξ , n = 1,2,3 , … . (26)

The BCM scheme is utilized to obtain the solution of Equations 11, 12 subject to the boundary conditions, Equation 13. Applying Equations 20–26 on Equation 11, we obtain the following residual functions R e ( ξ ) from Equation 27 together with the boundary condition, Equation 28 representing the velocity profile, which is as follows:

R e ξ = C Φ ‴ ξ + B 1 C Φ ξ C Φ ′ ξ + B 2 α 2 C Φ ′ ξ . (27)

ω m 0 = C Φ 0 = 1 ,

ω m ′ 0 = C Φ ′ 0 = 0 , (28)

ω m 1 = C Φ 1 = 0 .

Likewise, the relevant residual function with respect to the temperature profile as seen in Equation 29 along with the boundary condition, Equation 30 is computed as follows:

R θ ξ = D Φ ″ ξ + 4 α 2 D Φ ξ + B 3 C Φ ξ D Φ ξ . (29)

Θ m 1 = D Φ 1 = 1 ,

Θ m ′ 0 = D Φ ′ 0 = 0 . (30)

Now, the collocation nodes ξ j are replaced into R e ( ξ ) and R θ ( ξ ).

Setting R e ( ξ ) = R θ ( ξ ) = 0, we obtain a system of algebraic equations at the collocation node ξ j, where collocation points ξ j are defined as follows in Equation 31:

ξ j = 1 2 + 1 2 c o s 2 j + 1 π 2 m − 1 j = 0,1 , … . , m − 2 . (31)

Finally, a total of ( m + 1 ) linear or non-linear algebraic equation are generated. Solving this set of equation offers the unknown coefficient of the vector C and D. After determining C and D, an approximation ω m ( ξ ) and Θ m ( ξ ) is completely attained.

5 Results and discussion

The main objective of this research is to use the BCM to construct a numerical approach for the JH blood-flow problem under a high magnetic field using a nanofluid model. This subject is extremely important in biomedical sciences, especially for defining blood flow between narrowing and enlarging blood arteries. The numerical results achieved by BCM are contrasted to those independently assessed by utilizing the CCM. This comparative study reveals the accuracy and utility of the proposed collocation technique.

The validity of the developed BCM is determined by formulating the absolute error compared to the BCM solution and the numerical findings, which is stated as follows:where ω n ( ξ ) is the numerical solution and ω A ( ξ ) is the approximated solution derived by the proposed technique.

E r r o r = ω n ξ − ω A ξ , (32)

On assuming blood as the base fluid suspended with different nanoparticles such as copper and iron oxide, the nanofluid flow is analyzed with a no-slip condition. The temperature equilibrium and thermo-physical properties of the base fluid and the nanoparticle that are obtained are computed and presented in Table 1.

TABLE 1. Thermo-physical properties of copper( C u ) and iron oxide(F e 3 O 4, commonly known as magnetite).

Figure 2a (i,ii) and Figure 3a (i,ii) show the significance of the proportion of the nanoparticle volume on the velocity distribution for R e = 100 , H a = 50, k p 1 = 50. In the diverging channel ( α > 0 ), the velocity declines with the increase in volume of the nanoparticle due to the increase in effective viscosity of the blood, which improves the flow resistance inside the artery. A moderate backflow is observed at lower values of ϕ 1 at high R e, indicating an inverse flow in the widening arterial region. In contrast, in the converging channel, the velocity increases because of increasing ϕ 1, establishing that the narrowing artery develops the blood flow and intensifies the upward trend of the nanofluid.

The influence of elevating the magnetic field ( H a ) on the velocity profile ω ( ξ ) is shown in Figure 2b (i,ii) and Figure 3b (i,ii) with respect to convergent and divergent channels, while R e, k p 1, and ϕ 1 are kept constant. The findings show that the rate of acceleration improves for a larger Hartmann number in both channel structures. Biologically, the applied magnetic field reacts with the electrically conductive blood and generates a Lorentz force, which alters the flow of blood inside the artery. The velocity achieves its peak at the central line of the artery and declines at the arterial walls due to wall resistivity. In the convergent channel, the flow is steady without backflow, but in the diverging channel, backflow emerges at a higher Hartmann number due to the larger electromagnetic field opposing the blood velocity in the wider artery area.

Figure 2c (i,ii) and Figure 3c (i,ii) reveal the significance of the Reynolds number on the velocity graph ω ( ξ ) for both the converging and diverging channels, while H a = 50, k p 1 = 50, and ϕ 1 = 0.01 remain constant. The results confirm that in the divergent channel ( α > 0 ), the velocity is reduced as the Reynolds number increases due to stronger inertial forces, which enhances the flow resistivity in the wider arterial region. Conversely, in the convergent channel, the velocity increases with the Reynolds number due to the arterial constriction that speeds up the blood flow, moving toward the center. Backflow is not observed in the converging channel because the contraction preserves forward motion, but backflow in the diverging channel occurs at higher Reynolds numbers due to the strength of inertial forces. The existence of a magnetic field can assist in controling blood flow by creating a Lorentz force that suppresses backflow within the artery.

Figure 2d (i,ii) and Figure 3d (i,ii) depict the variation of the velocity profile according to the permeability parameter k p 1 for both α < 0 and α > 0. The results reveal that permeability factor improves and the velocity profiles likewise evolve. Biologically, a higher level of permeability decreases the resistance generated by the porous medium found inside the artery, which permits blood flow more easily. Because the drag force declines with increasing porosity, the flow grows stronger, eventually leading to enhanced velocity profiles. The identical scenario emerges for both arterial channels, showing that the dominant effect of k p 1 is a lower level of hydraulic resistance in the blood vessel.

Figure 4a displays the effect of R e on the temperature profile in the converging channel. The thermal temperature increases with increasing R e for both C u and F e 3 O 4 nanoparticles. Actually, a higher R e generates greater inertial stresses, which intensify the blood flow inside the constricting artery. This improves convectional heat transfer and enhances the temperature within the channel. The existence of nanoparticles in a fluid increases its heat conductivity, causing high temperature in the arterial region.

Figure 4b reveals the influence of the inclination angle α on the temperature distribution in the converging channel. The findings indicate that the temperature declines as the angle is increased for both C u and F e 3 O 4 particles. Biologically, an increase in the convergent angle increases channel narrowing, which accelerates the fluid velocity toward the center. This acceleration decreases the thermal boundary layer thickness and restricts the residence time of the fluid, which increases the rate of convective heat transfer inside the channel, thereby reducing the degree of temperature in the arterial section.

Figure 4c displays the significance of the magnetic field ( H a ) on thermal profile, where the outcomes reveal that the temperature improves because of the increasing Hartmann number for both C u and F e 3 O 4 nanoparticles. From fluid flow mechanism, a higher magnetic field generates a Lorentz force caused by the interaction between the magnetic field and electrically conductive blood, which increases the resistance to the motion. This resistance transforms mechanical energy into thermal energy, consequently augmenting the temperature in the converging arteries.

The expression of Skin friction coefficient and Nusselt number are given by Equation 16 and Equation 18. Figure 5 shows the efficiency of the skin friction coefficient ( C f ) and Nusselt number ( N u ) based on different Reynolds numbers with several volume fraction factors. A gradual decrease in C f is observed as R e increases, while Nu continues to increase with R e, exhibiting divergent behavioral trends for these two factors.

Table 2 provides the comparative analysis of the flow velocity ( ω ( ξ ) ), defined by Equation 11, calculated by the BCM, CCM, and Adomian decomposition method (ADM) for α = − 5 ° , R e = 25 , H a = 250 , k p 1 = 50 , ϕ 1 = 0.05. Here, the three approaches are compared to show that the results of BCM are compatible not just with that of the CCM but also with the ADM, an entirely distinct analytical method that is not dependent on the collocation framework, thereby offering a stronger basis for validation. It is evident from Table 2 that the findings derived by all three methods are in high accordance with each other, with values equivalent to up to four decimal places throughout the entire domain, a result that confirms both the numerical accuracy and the uniformity of the proposed BCM.

TABLE 2. Comparative analysis of flow velocity across distinct methodologies: BCM, CCM, and ADM ( α = − 5 ° , R e = 25 , H a = 250 , k p 1 = 50 , ϕ 1 = 0.05 ).

Figure 6 depicts the thermal profile Θ ( ξ ) determined by the BCM, CCM, and ADM. It proves that the BCM and CCM curves correspond with each other across the entire domain, owing to their concordance of up to six decimal places, while the ADM curves display the same trend in close proximity, demonstrating excellent mutual consistency in the temperature distribution among all three methods. This graphical comparison shows that the proposed technique captures the temperature field just as well, supporting the current BCM’s dependability and uniformity.

In order to address the biomedical importance of this study, the velocity and temperature distribution of C u and F e 3 O 4 particles are analyzed quantitatively using the BCM, and the outcomes are presented in Table 3 for α = 5 ° , R e = 30 , H a = 300 , k p 1 = 25 , ϕ 1 = 0.02 , P r = 6.2. It is notable that F e 3 O 4 nanoparticles produce accelerated velocity compared to copper nanoparticles at all the interior grid points ξ, which can be physically clarified by the lower density of F e 3 O 4 that minimizes inertial resistance and allows for smoother blood flow through the stenosed arterial region. On the other hand, a reverse trend is seen in the temperature values, where C u nanoparticles exhibit significantly greater temperature values in comparison with F e 3 O 4 nanoparticles across the arterial cross-section, which is attributed to the substantially higher thermal conductivity of copper. From a biological perspective, the high velocity, intrinsic biocompatibility, low toxicity, and magnetic responsiveness of F e 3 O 4 particles are highly desirable for targeted delivery of drugs, magnetic hyperthermia therapy, and MRI-directed cardiac treatment in stenosed arteries. However, the enhanced thermal behavior of C u nanoparticles is clinically relevant for thermal ablation and hyperthermia-related treatment within the arterial region.

TABLE 3. BCM of ω ( ξ ) and Θ ( ξ ) for C u and F e 3 O 4 nanoparticles at α = 5 ° , R e = 30 , H a = 300 , k p 1 = 25 , ϕ 1 = 0.02 , P r = 6.2.

Table 4 analyzes the convergence results of the present numerical framework for calculating the value of ω ″ ( 0 ) based on F e 3 O 4 nanoparticles at prescribed parameter values. In order to examine the convergence of the optimum solution, the degree of approximation is gradually enhanced. The computational results imply that the calculated values exhibit variations for low levels of approximation, while asymptotically fixed values can be achieved for greater levels, which indicates that the method convergences to a peculiar uniform solution. This confirms the numerical reliability, stability, and dependability of the present mathematical framework.

TABLE 4. Convergence study of the BCM outcome for ω ′ ′ ( 0 ) of F e 3 O 4 nanoparticles at specified R e = 25, H a = 50, k p 1 = 25, and ϕ 1 = 0.01.

To validate the proposed numerical technique, Supplementary Table S1 evaluates the results of the JH blood flow presented in the published works derived through VIM and the numerical methods. The factors are specified as α = 5 ° , R e = 50, H a = 0, k p 1 = 0, and ϕ 1 = 0, and the analysis is executed based on the value of ω ″ ( ξ ) at specific intervals in the diverging channel. The findings provide a significant uniformity, and a high level of exactness is observed for m = 15, which supports the consistency of the proposed method.

To extend the validation, Table 5 shows the velocity profile ω ( ξ ), governed by Equation 11, for both C u and F e 3 O 4 nanoparticle instances at selected grid points for α = 5 ° , R e = 50 , H a = 500 , k p 1 = 50 , ϕ 1 = 0.02, along with the corresponding absolute errors computed using Equation 32 between the BCM and the numerical solutions. The purpose of determining these absolute errors is to numerically show the proximity of the two solutions and verify that the BCM outcomes are in good agreement with the numerical result. As evident from Table 5, the absolute errors for both the C u and F e 3 O 4 cases are quite small, ranging between O ( 1 0 − 6 ) and O ( 1 0 − 8 ) across the entire domain. This small magnitude of error, confined between the sixth and eighth decimal places, clearly demonstrates the high precision of the proposed BCM in obtaining the velocity profile for both the nanoparticle structures. Furthermore, the errors are uniformly small across the whole domain with no amplification, proving the computational stability and uniformity of the method under this case.

TABLE 5. Absolute error of blood flow against ξ by BCM (α = 5 °, R e = 50, H a = 500, k p 1 = 50, and ϕ 1 = 0.02).

BCM provides a numerical solution to the problem based on the degree of the Bernstein polynomial and the number of collocation points used for the analysis (here, the degree of polynomial (11) and the number of collocation points (11)). However, for better accuracy, increasing the degree of the Bernstein polynomials leads to congestion of the underlying algebraic matrices, and the system becomes hyper-sensitive to small rounding errors. Likewise, with a large number of collocation points, the method diverges entirely and undermines the reliability of the results. In order to address this issue, RSM is implemented as a statistical approximation model based on the numerical data produced by BCM. Instead of repeatedly solving the underlying equations for each parameter combination, the RSM relies on a predetermined set of BCM solutions to develop an explicit mathematical framework that describes a relationship between the controlling variables and the corresponding flow outcomes. The established response surface provides a simple computational representation for the numerical outcomes, allowing the flow properties to be predicted rapidly within the selected parameter range. Furthermore, RSM assists in identifying the parameter values that generate the expected response and enables the interaction of multiple variables to be examined efficiently. As a result, accurate predictions can be produced without repeatedly addressing the high-order BCM equation.

6 Response surface methodology

The contemporary statistical technique identifies the system’s optimal state and reveals the controlling factors on the skin friction coefficient. Despite both nanoparticles being analyzed in the study, RSM is performed for the C u particle as the F e 3 O 4 fluid suspension and the C u fluid suspension exhibits substantially similar traits on the flow distribution of fluid and temperature. The experimental model addresses three critical variables, namely, the nanoparticle volume fraction ( 0.01 ≤ ϕ 1 ≤ 0.07 ), the magnetic field parameter ( 100 ≤ H a ≤ 300 ), and the Reynolds number ( 25 ≤ R e ≤ 75 ).

The factors and their corresponding optimum values are listed in Table 6. RSM analyzes the statistical significance of these variables on skin friction and emphasizes the key factor among different parameters across various conditions. A 3 × 3 (level and factor) face-centered CCD–RSM structure indicating the correlation between the predictor and response variable is given below in Equation 33.

C f Reg = f 1 ϕ 1 + f 2 H a + f 3 R e + f 4 ϕ 1 ∗ H a + f 5 ϕ 1 ∗ R e + f 6 H a ∗ R e + f 7 ϕ 1 2 + f 8 H a 2 + f 9 R e 2 + f 10 . (33)

TABLE 6. The key parameters and their corresponding levels for skin friction analysis for RSM.

Here, f i, i = 1,2,3 … are known as the regression coefficients. A face-centered CCD is implemented to create a test plan with 20 sets of runs, and the combination of each collection is shown in Supplementary Table S2. Additionally, the response surface design includes six axial points, 10 factorial vertical points, and six center points.

6.1 Accuracy of the model

The reliability and precision of the established response surface model are evaluated by applying the analysis of variance (ANOVA) test, while the residual analysis justifies the model competency. These applied statistical techniques assess the validity of the method and verify the accuracy of the predictions with the calculated data. Computed ANOVA findings for this specific approach are listed in Supplementary Table S3, and it was found that the corresponding linear, squared, and interactions terms of selected variables have a major impact on the statistical response, which confirms the validity of the theoretical model. The vital role of predictors is easily assessed with the calculated p-value, and hence, we accept the hypothesis that the chosen variable has a significant effect on the response provided that the p-value is less than 0.05. Furthermore, the F-value navigates the SD of the predictors and indicates the statistically significant differences between the group means.

Adopting the F-test, if the test statistic is more than one and the p-value is below 0.05, the relevant terms are considered to be statistically noteworthy and incorporated in the statistical analysis. The ANOVA findings prove that the quadratic effects of the Hartmann number and the volume fraction, along with the interaction term H a × ϕ 1, are substantially less statistically significant ( p > 0.05 ). As a result, these factors have a minor effect on the skin-friction coefficient when compared to the significant model variables.

Supplementary Table S3 shows that the statistical predictors of the simplified model remain significant, and the reduced, precise, and un-coded regression model thus obtained in Equation 34:

C f Reg = − 0.14393 + 0.0038307 R e − 0.0000677 H a − 0.1823 ϕ 1 − 0.000027442 R e ∗ R e + 0.0000000007 H a ∗ H a − 0.245 ϕ 1 ∗ ϕ 1 + 0.0000006941 R e ∗ H a + 0.003433 R e ∗ ϕ 1 − 0.0000533 H a ∗ ϕ 1 . (34)

The computed R 2 score establishes the degree of variation in the response by the explanatory variable and reflects the level of convergence between the observed and estimated data. For this specific design, the evaluated positive correlation coefficient assures that the model is perfect. Furthermore, the values of S = 0.0006454, R 2 = 99.94 % adjusted, R 2 = 99.97 %, and predicted R 2 = 99.75 % are all close to unity, which confirms that the proposed model offers the most suitable fit and verifies that the chosen model is precise, reliable, and well-compatible with the present study.

6.2 Residual plot

The residual plots for C f are depicted in Figure 7, which help to validate the reliability and precision of the formulated response surface technique. The normal probability plot showcases that the obtained data points are denser around the regression fit, confirming that the residuals exhibit a normal distribution. Following that, the histogram illustrates an approximate axi-symmetric pattern with minor skewness, indicating that the data acquired are reliable.

The plot of residuals versus the fitting values together with the observational order represents no specific behavior, demonstrating that the estimated and observed results are consistent. It is further explained that the apex residual value equals to approximately 0.0011, which implies that the developed model is accurate and well-fitted for predictors.

6.3 Pareto chart

The Pareto chart in linear, square interaction, and quadratic form for the three parameters, namely, R e, H a, and ϕ 1, is displayed in Figure 8. It is shown that if the normalized effect is higher than 2.2 (the critical threshold value), the corresponding variable is statistically notable. The Pareto chart articulates the order of control by the pertinent parameter over the skin friction coefficient C f as: first the Reynolds number, followed by the Hartmann number, and, finally, the volume fraction. Based on the results, the alteration of R e has a supremacy over H a and ϕ on the variation of the response variable.

6.4 Interaction plot for c f

Figure 9 depicts the interaction plots for C f, which are used to analyze the pooled impact of the controlling parameters on the response variable and to find any major interaction between them, if present. The interaction plots for C f with respect to the predictor variables R e, H a, and ϕ 1 are analyzed. Visual representation of R e × H a versus the mean C f shows that the average C f grows gradually with the increment of R e by fixing H a and vice versa, dictating that the value of R e and H a interplays well. Though the divergence between the observed lines is small, C f increases with R e for all ϕ 1 indices, which can be noted in the R e × ϕ 1 plot. This supports that the relation between R e and ϕ 1 is substantial, while the impact of ϕ 1 is comparatively lower than that of R e. Based on the H a × ϕ 1 graph, it is apparent that the response variable shifts slightly for different values of H a and ϕ 1. It is observed that the interaction across H a and ϕ 1 is trivial, as the viewed traces are almost parallel. On the whole, the predicator R e has an extensive impact on C f, even as the Hartmann number has a modest effect, and the volume fraction has minimal impact.

6.5 Contour plot and surf plot

Figure 10 displays the pair of variations in the skin friction factor model corresponding to distinct combinations including Reynolds number ( R e ), Hartmann number ( H a ), and nanoparticle volume fraction( ϕ 1 ). In plots shown in Figures 10a,d, it is seen that the skin friction coefficient is amplified for the increment of both R e and H a ( ϕ 1 = 0.04 ), which means that the response is more reliant on R e. Both graphs in Figures 10b,e reveal that H a is retained with a fixed value 200, and C f increases mainly with R e, while the influence of ϕ 1 is comparatively small. Likewise, graphs in Figures 10c,f display that with a fixed value of R e = 50, the skin friction coefficient points a minimum variance in H a and ϕ 1 and confines the contour threshold value to a narrow range. Thus, the blended effect of H a and ϕ 1 is almost peripheral. Eventually, the Reynolds number exerts a paramount effect on the response compared to that of the other predictors such as the magnetic field ( H a ) and volume fraction ( ϕ 1 ), which have a nominal impact. The surface plot discloses the consistency of the involved parameters, ensuring slow modification in response, while the normal parameters have no interaction. Precisely, the surface plot agrees with both schematic representations, namely, the Pareto equation and interaction plot parameters.

In addition, ANOVA, residual plot, the Pareto chart, interaction plots, contour plots, and response surface plots confirm the accuracy and predictive ability of the developed RSM model. The strong agreement among these statistical evaluations indicates that the response surface methodology precisely reflects a relationship between the governing variables and the associated flow response within the examined domain. Furthermore, the presented model provides an effective framework for response predictions and optimization that eliminates the need for recurrent BCM computations. Overall, the relative relevance of the controlling parameters has the relationship R e > H a > ϕ 1. This tendency is consistent across all RSM analyses and supports the resilience of the proposed statistical technique.

7 Conclusion

This study analyzes the Jeffery–Hamel flow of nanofluid blood infused with C u and F e 3 O 4 nanoparticles in both converging and diverging channels using the BCM. The non-linear governing equations related to the flow problem are determined quantitatively, and the resultant solution reveals adequate consistency with those derived using the Chebyshev collocation technique. The convergence analysis precisely identifies that the proposed mathematical technique yields firmly stable solutions at high levels of approximation; however, the error estimation reflects very minor variances for both nanoparticles. In addition, RSM was applied to examine the effect of the controlling parameters on the skin friction coefficient. Despite C u and F e 3 O 4 nanoparticles being analyzed separately, F e 3 O 4 is identified to be more reliable and harmonious for blood-based flow processes owing to its higher biological compatibility than the C u nanoparticles.

The key findings of current study are listed as follows:

• Reversal flow occurs in the diverging channel for the increase of R e values.
• An increase in Ha inhibits the backflow. Eventually, with a suitable values of R e or having a wider channel symmetry, one can suppress the flow inversion through high-level magnetic intensity.
• The velocity profile can be increased by adjusting the R e value, as the velocity of both C u-blood and F e 3 O 4-blood nanofluids decreases in the diverging channel and increases in the converging channel at high R e.
• Despite the blood velocity being rapid, the targeted medications often encounter turbulence and high shear stress, which can damage organs before the completion of tasks. However, the concept of centering half angle slows the blood speed with less shear force, facilitating a notable decrease in the velocity profile, as measured for both C u and F e 3 O 4 nanoparticles with the increase of the angle.
• In the diverging channel, the nanoparticle volume fraction ϕ 1 and the fluid velocity are negatively correlated: as ϕ 1 increases, the nanofluid velocity decreases. Instead, it has opposite effects in a converging channel. This phenomenon is dynamic in biomedical applications, enabling the medication to be trapped at the desired disease site before the bloodstream washes it out.
• Under specific conditions, exceptional angles can control the variation of temperature distribution and maintain an almost constant temperature profile.

The RSM-based substantial results are:

• Reynolds number is the major influencing factor that affects the velocity distribution and skin friction coefficient.
• Hartmann number exerts a moderate effect on blood-flow dynamics.
• Nanoparticle volume fraction indicates comparatively less significant effects.
• Combined influence of H a and ϕ 1 is weaker than that of R e.

Furthermore, the RSM analysis confirms the BCM results, demonstrating that the Reynolds number has a greater effect on flow behavior than the Hartmann number and the volume fraction, thereby determining the consistency of both techniques.

Extending the current study by integrating hybrid nanofluids to improve the thermal and flow efficiency, pulsatile blood flow conditions to better simulate the heartbeat-driven flow behaviors, and non-Newtonian fluid modeling to precisely represent the complex rheological properties of blood would yield more biologically accurate hemodynamic findings in stenosed arteries.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fnano.2026.1872948/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Ritika Gupta, University of Nebraska Medical Center, United States

Reviewed by: Nitesh Priyadarshi, National Agri-Food Biotechnology Institute, India

Sadique Rehman, Kanazawa University, Japan