Numerical and neural-network analysis of Casson hybrid nanofluid flow and heat transfer in a sinusoidally stenosed artery

Abstract

We investigate unsteady flow and heat transfer in a blood-based Casson hybrid nanofluid containing Al₂O₃ and Fe₃O₄ nanoparticles as it passes through a rigid artery with a mild sinusoidal stenosis. The formulation includes a transverse magnetic field, Hall-current effects, viscous dissipation, and Joule heating. Under the long-wavelength approximation, the governing equations reduce to local radial initial–boundary-value problems. These problems are mapped onto a fixed domain and solved with a forward-time central-space (FTCS) scheme that uses a separate stencil at the centreline. Particular attention is given to the effects of the Casson, magnetic, Hall, Reynolds, Prandtl, and Eckert parameters, as well as the two nanoparticle volume fractions, on the velocity, temperature, wall shear stress, and local Nusselt number. We interpret the observed trends through the balance among viscous resistance, Lorentz damping, radial diffusion, internal heating, and the effective properties of the suspension. Parameter-specific Levenberg–Marquardt backpropagation neural networks are also trained to interpolate the FTCS data. In most of the sampled cases, the network follows the numerical profiles closely. However, a small number of negative centreline-temperature predictions make clear that a low overall error does not always imply physically acceptable behaviour at every point. The neural-network results should therefore be viewed as interpolation within the sampled ranges, while the physical conclusions remain tied to the assumptions of the idealized single-phase arterial model.

Clinical trial registration: Not applicable.

Introduction

Cardiovascular disease remains a major global health burden, and this has sustained interest in the mechanics of blood flow through diseased arteries. Blood transports oxygen, nutrients, hormones, and metabolic products throughout the vascular system, but it does not always behave like a simple Newtonian liquid. Its rheology is influenced by haematocrit, red-cell aggregation and deformability, plasma properties, and the local shear rate. These effects become particularly important in vessels of moderate size and in regions where the shear rate is low. A stenosis further complicates the flow by narrowing the lumen, accelerating the fluid locally, altering the pressure drop, and changing the stresses acting on the arterial wall. Mathematical models offer a controlled way to examine how geometry, rheology, and heat transfer interact in this setting [1–5] (Table 1).

Table 1.

Major components of blood and their principal physiological functions

Blood component	Principal function	Approximate volume (%)
Plasma	Transport medium for solutes, proteins, and metabolic products	50–60
Red blood cells	Oxygen and carbon-dioxide transport	40–50
White blood cells	Immune defence	<1
Platelets	Haemostasis and clot formation	<1
Plasma proteins	Osmotic balance, transport, and coagulation	included in plasma
Other dissolved constituents	Electrolytes, nutrients, hormones, and waste products	included in plasma

A substantial body of earlier work examined pulsatile motion, wall shear stress, and transport in stenosed arteries using both Newtonian and non-Newtonian descriptions [6–12]. Together, these studies showed that the predicted velocity, pressure, and wall stress depend strongly on the shape and severity of the narrowing, on vessel tapering, and on the chosen constitutive law [13–15]. To solve the resulting equations, researchers have used finite-difference, finite-volume, and finite-element methods in both idealized and anatomically inspired vascular geometries [16–23].

More recently, arterial-flow models have been extended to include magnetic forcing, heat transfer, suspended nanoparticles, and data-driven surrogates. Mohanavel et al. [24], for example, examined heat and mass transfer in a Casson hybrid nanofluid containing Cu and TiO₂ nanoparticles in a constricted artery. Zheng et al. [25] used the Levenberg–Marquardt algorithm for a magnetized Casson-flow problem with thermal and species-transport effects. Chand and Tripathi [26] combined finite-difference calculations with an ANN–PSO model for a porous stenosed artery, while Kumar et al. [27] studied magnetized hybrid nanofluid flow through a cosine-shaped stenosis. Neural-network and reduced-order approaches have also been applied to other non-Newtonian and haemodynamic transport problems [28–32].

Mahabaleshwar et al. [33] investigated ternary nanofluid film flow over a past stretching or shrinking surface with variable velocity. Sachhine et al. [34]investigated the stagnation point flow of a non-Newtonian tetra-Bingham nanofluid induced by a stretching plate and a circular cylinder, with particular emphasis on the rheological and thermodynamic effects and the influence of the viscosity ratio. [35] investigated the thermodynamic behaviour, considering viscous dissipation, Joule heating, Brownian motion, and thermophoresis by analysing activation energy, porous media, and coupled heat and mass transfer effects within an entropy-generation framework. [36] investigated chemically Prandtl-Eyring nanofluid flow through a Darcy-Forchheimer porous medium, incorporating the effects of Arrhenius activation energy to analyse the associated transport characteristics. [37] investigated Hiemenz stagnation-point flow of a ternary nanofluid over a permeable stretching or shrinking surface, incorporating magnetic-field effects, variable thermal conductivity, and thermal radiation in the heat-transfer analysis. Himanshu Upreti et al. [38] analysed heat transfer in Casson nanofluid stagnation-point flow over a stretching sheet, incorporating an induced magnetic field, the Cattaneo-Christov model, and Darcy-Forchheimer porous medium effects. [39] examined von Karman and Bodewadt rotating flows incorporating Hall current, magnetic field, thermal radiation, Darcy-Forchheimer porous effects, and slip conditions, with regression models used to assess their influence on skin-friction and Nusselt numbers. Authors studied a mathematical model for hybrid Casson nanofluid flow around a horizontal cylinder with a lower stagnation point, incorporating magnetic-field and free-convection effects in [40]. Bertwal et al. [41] investigated three-dimensional MHD tangent-hyperbolic fluid flow over a rotating disk with heat and mass transfer, employing a multilayer perceptron to predict skin-friction coefficient, Nusselt number, and Sherwood number. Singla et al. [42] examined MHD ternary hybrid nanofluid flow between rotating, stretchable disks in a porous medium, incorporating thermal radiation and velocity-slip effects using Al₂O₃, CuO, and Cu nanoparticles suspended in water. [43] studied water-based Ag − MoS₂ − TiO₂ and MgO − Cu − Al₂O₃ trihybrid nanofluids, examining the effects of nanoparticle shape on flow and thermal characteristics for potential hyperthermic cancer therapy applications. [44] examined the micropolar penta-hybrid nanofluid flow in a porous parallel-plate channel under magnetic and nonlinear inertial effects, showing that a stronger field suppresses velocity while enhancing microrotation. [45] examined the superior thermal performance of Au − CuO − Fe₃O₄-multiwalled carbon nanotubes nanofluid, whereas Ag − SiO₂ − TiO₂ − Al₂O₃ provided higher velocity and skin-friction profiles; an ANN with Bayesian regularisation was used to predict the skin-friction coefficient and Nusselt number.

The individual ingredients of this problem have been studied before, but their combined influence has received less attention. Here, Casson rheology is coupled with an Al₂O₃–Fe₃O₄ hybrid suspension, Hall-modified magnetic damping, viscous dissipation, and Joule heating in a sinusoidally stenosed artery. We first derive a reduced long-wavelength model and solve it with an explicit FTCS method. The numerical solutions are then used to train parameter-specific neural surrogates. Our aims are threefold: to determine how the main dimensionless parameters affect velocity, temperature, wall shear stress, and Nusselt number; to handle the cylindrical centreline carefully in the finite-difference scheme; and to examine how well an LM-BPNN can interpolate the FTCS solutions over the parameter ranges considered. The model is deliberately idealized and should not be interpreted as a patient-specific description or as a cellular-scale model of nanoparticle transport.

The novelty of the present study lies in the combined treatment of the Casson rheology and the Al₂O₃–Fe₃O₄ hybrid nanofluid in a sinusoidally stenosed artery and Hall–current effects, while accounting for viscous dissipation and Joule heating. The reduced model is solved using an explicit FTCS scheme with a dedicated centreline treatment, and the resulting numerical profiles are further represented using parameter–specific Levenberg–Marquardt backpropagation neural networks. This framework enables examination of the coupled effects of rheological, electromagnetic, thermal, and nanoparticle parameters on velocity, temperature, wall shear stress, and Nusselt number within a single formulation.

The principal parameters reflecting the distinctive aspects of the present model are the Hall parameter m, magnetic parameter M, Casson parameter β, and nanoparticle volume fractions ϕ₁ and ϕ₂, whose combined effects on the flow and thermal characteristics are examined together with Re, Pr, and Ec.

The selection of Al₂O₃ − Fe₃O₄ as the hybrid suspension is motivated by the complementary physical characteristics of the two nanoparticles. Al₂O₃ is chosen for its strong thermal-transport capability, whereas Fe₃O₄ provides magnetic responsiveness and modifies the suspension’s electrical and thermal behavior. This combination is therefore appropriate for the present magneto-thermal arterial model, which simultaneously considers magnetic damping, Joule heating, and heat transfer. However, this theoretical formulation does not establish the clinical superiority of this combination over alternative nanoparticles. Its biomedical suitability also depends on particle size, surface functionalization, aggregation, biocompatibility, toxicity, and allowable concentration, which require experimental assessment.

In the present theoretical analysis, relatively low nanoparticle volume fractions are considered, namely 0.1–1% for Fe₃O₄ and 0.1–3% for Al₂O₃. These ranges are selected to examine the influence of dilute nanoparticle loading on the effective flow and thermal properties without implying a clinically validated dose.

Mathematical model and assumptions

Consider an incompressible, axisymmetric, laminar Casson hybrid nanofluid flowing through a rigid artery with a mild, smooth sinusoidal stenosis. Cylindrical coordinates are adopted, where and denote the axial and radial velocity components, respectively. Blood is taken as the base fluid, while Al₂O₃ and Fe₃O₄ nanoparticles form the hybrid suspension. The particles and the base fluid are assumed to remain in local thermal equilibrium, with no relative slip between the phases. Because the magnetic Reynolds number is small, the induced magnetic field is neglected. The arterial wall is treated as rigid and impermeable and is maintained at a prescribed temperature.

The Casson constitutive relation is written as

1

where is the strain-rate tensor, Π = e_ije_ij is its second invariant, P_y is the yield stress, and Π_c is the critical invariant. Introducing the Casson parameter

the high-shear form used in the reduced model becomes

2

The Newtonian limit is recovered as β → ∞.

The dimensional arterial radius is prescribed by the sinusoidal stenosis profile

3

where R₀ is the healthy-artery radius, λ^* is the maximum stenosis height, l₀ is the stenosis length, and locates its upstream edge. Accordingly, we refer to the narrowing throughout the paper as a sinusoidal stenosis.

The pulsatile pressure gradient is represented by

4

Under axisymmetry, conservation of mass is

5

For a transverse imposed magnetic field and Hall parameter m, the electromagnetic contribution follows from the reduced Hall–MHD Ohm law

With a uniform transverse field, negligible imposed electric field, and negligible induced magnetic field, elimination of the transverse current gives an axial Lorentz force proportional to . The associated Joule-heating contribution is non-negative and proportional to . Under these assumptions, the dimensional momentum equations are [46]

6

7

where . The energy equation, including viscous dissipation and non-negative Joule heating, is

8

where Φ is the standard axisymmetric viscous-dissipation function. Radiation, Brownian motion, thermophoresis, nanoparticle concentration, and bioconvection are not part of the present formulation; including them would require additional transport equations and constitutive assumptions.

The initial and boundary conditions are [46]

9

The following nondimensional variables are introduced:

10

The governing dimensionless groups are

11

with , e = p₁/p₀, and c₁ = ωR₀/u₀. Effective-property ratios are defined by

12

The mixture density and heat capacity are evaluated sequentially as

13

14

The hybrid density is evaluated sequentially, with ϕ₁ and ϕ₂ denoting the volume fractions of the first and second nanoparticles, respectively. Thus, each nanoparticle contributes positively to the mixture density, while the remaining fraction represents the preceding mixture stage. The viscosity is modelled by . Thermal and electrical conductivities are evaluated by applying the Maxwell relation sequentially to the first and second nanoparticle species. For a generic transport property χ (where χ = k or σ),

15

Here, s1 and s2 denote Al₂O₃ and Fe₃O₄, respectively. Equation (15) is an effective-medium approximation and does not include particle aggregation, non-spherical shape effects, or interfacial resistance.

Long-wavelength reduction and radial mapping

For the present long, mildly stenosed artery, the aspect ratio . From (5), the radial velocity scales as u = O(εw), where w is the axial velocity. Thus, radial motion is asymptotically smaller than the axial motion and does not contribute at leading order to the local axial problem. Under this long-wavelength approximation, axial diffusion is O(ε²), while radial velocity is O(ε) relative to the axial velocity. The leading-order equations therefore reduce to a local radial problem at each prescribed axial station. Introducing the fixed-domain coordinate

16

produces

The quantitative basis of the long-wavelength reduction follows from the slenderness parameter . For ε ≪ 1, the radial velocity is O(ε) relative to the axial velocity, while axial diffusion enters at O(ε²). Hence, the neglected contributions are asymptotically smaller than the leading-order radial transport terms, providing the quantitative basis for the reduced formulation. A direct comparison with the unreduced two-dimensional model is beyond the scope of the present study. At leading order, the dimensionless axial momentum and energy equations are

17

18

The transformed conditions are

19

The discontinuity at (x, t) = (1, 0) represents sudden wall heating and is imposed explicitly in the numerical initialization. Here, τ_w is defined as the positive magnitude of the wall shear stress. Since w_x(1, t) < 0 for the no-slip velocity profile, the negative sign is introduced so that τ_w > 0. The corresponding signed shear stress is obtained without this sign reversal. The positive wall-shear magnitude and local Nusselt number are

20

Finally,

21

Numerical procedure

We solve Equations (17)–(19) at a sequence of prescribed axial locations z_n. For each location, the local radius R_n = R(z_n) is fixed during the time integration. Thus, every calculation is one-dimensional in the mapped radial coordinate and time; the index n labels the axial station rather than a second finite-difference direction.

Let

where Δx = 1/(K − 1) and . For k = 2, …, K − 1,

The FTCS velocity update is

22

The temperature update is

23

At x = 0, l’Hopital’s rule and symmetry give

Accordingly, the centreline updates are

The discrete initial and boundary conditions are

24

Stability, consistency, and verification

Define

and

A conservative positivity-preserving restriction that accounts for the special centreline stencil is

The inequalities are evaluated at the minimum stenotic radius and at the most restrictive combinations of the material and flow parameters. The interior FTCS discretization is first-order accurate in time and second-order accurate in space, with global truncation error O(Δt + Δx²).

Stability is checked using the largest values of D_w, D_θ, and S_M encountered over the parameter range. Sensitivity to the grid and time step is judged from the centreline velocity and temperature together with the wall shear stress and Nusselt number. When three systematically refined calculations are available, the observed order is estimated from

A residual diagnostic is defined explicitly as

but residual decay is not used as a substitute for grid independence or validation against an analytical/benchmark limiting case (Figs. 1, 2).

Fig. 1.

Residual histories based on successive-time-level differences. a Velocity residual history. b Temperature residual history

Fig. 2.

Neural-network definition used for the FTCS-generated database. a Input–output structure of the surrogate model. b Two-hidden-layer LM-BPNN architecture

Artificial neural network methodology

The neural network serves as a compact interpolator for the FTCS data. It is not used to validate the governing equations, and no claim is made beyond the parameter ranges represented in the numerical database.

Dataset construction and leakage prevention

The numerical experiments in Sect. 6 use a separate surrogate for each varied parameter. In each case, the mapped coordinate x and one parameter α ∈ {β, m, M, Re, Ec, Pr} form the network input,

whereas the remaining quantities are fixed at the baseline values stated in Sect. 5. Velocity and temperature are learned by separate output networks because the two fields have different scales and boundary behaviour. For parameters that do not enter one of the governing fields directly, only the physically relevant output is modelled.

The data are divided into training, validation, and test subsets in an 80/10/10 ratio using complete parameter cases rather than neighbouring points from the same radial profile. In other words, when a parameter value is held out for testing, every radial sample associated with that value remains in the test set. Inputs and targets are normalized using statistics calculated from the training data only, and the same scaling is then applied to the validation and test sets. This case-based split helps prevent leakage between strongly correlated neighbouring samples.

Architecture and training protocol

Each parameter-specific surrogate is a feed-forward multilayer perceptron with two hidden layers containing 15 and 10 neurons. Hyperbolic-tangent activation functions are used in the hidden layers, while the output layer is linear. Training is performed with the Levenberg–Marquardt algorithm by minimizing the mean-squared error. The procedure stops when one of the following conditions is met: 1000 epochs are reached, the validation patience is exhausted, the gradient falls below 10⁻¹², or the prescribed performance goal is achieved. For each model, the initialization, damping history, validation record, final epoch, and stopping reason are retained.

Because the network is an interpolator rather than a stand-alone numerical solver, its performance is assessed mainly on held-out parameter cases and on its pointwise physical behaviour, not simply on how well it fits the training data. Random initialization can influence the optimized weights; consequently, reproducible implementation requires reporting the random seed or summary statistics over repeated runs.

Physics-consistent assessment

An unconstrained network does not automatically satisfy the boundary conditions. Therefore, independent-test boundary errors are evaluated as

Any negative dimensionless temperature, non-zero wall velocity, or excessive centreline-gradient error is reported directly. A high regression coefficient can coexist with such local defects, so it is not taken as sufficient evidence of physical admissibility.

Performance measures

For N independent samples, the following measures are used:

25

26

27

28

The mean absolute error is also useful for diagnosing pointwise deviations. Percentage errors are excluded where the FTCS reference is zero or nearly zero. In the reported assessment, MSE, RMSE, and rounded R² values are complemented by direct profile and tabulated comparisons so that isolated local violations are not concealed by aggregate statistics.

For future deployment, computational efficiency should distinguish the offline training cost from online inference. A meaningful speed-up is defined by

Since no hardware-normalized timing study is available, we do not claim a quantitative computational advantage for the ANN.

Results and discussion

We now discuss how the main parameters change the velocity, temperature, wall shear stress, and local Nusselt number. Unless otherwise stated, the reference values are ϕ₁ = ϕ₂ = 0.025, Re = 2.0, e = 0.5, A₀ = 1.41, Pr = 14, R = 0.825, c₁ = 1.0, M = 0.5, Ec = 0.1, β = 0.5, and m = 0.1. Here, R = 0.825 represents the local dimensionless radius at the selected axial station. Tables 2 and 3 compare the radial velocity and temperature values for blood, a single-particle nanofluid, and the hybrid nanofluid. The axial velocity decreases from its centreline maximum to zero at the wall, in accordance with the no-slip condition. The dimensionless temperature increases from the cooler core toward the prescribed wall value θ = 1, which is consistent with heating from the arterial wall.

Table 2.

Tabulated radial variation of axial velocity for the base fluid, Al₂O₃/blood, and Al₂O₃–Fe₃O₄/blood hybrid nanofluid

	Velocity FTCS			Velocity LM-BPNN
	Blood	Nano	Hybrid	Blood LM-BPNN	Nano LM-BPNN	Hybrid LM-BPNN
0.0	0.6383	0.5993	0.5627	0.6383	0.5997	0.5748
0.1	0.6327	0.5941	0.5579	0.6322	0.5939	0.5575
0.2	0.6138	0.5764	0.5412	0.6161	0.5764	0.5420
0.3	0.5820	0.5465	0.5132	0.5818	0.5463	0.5119
0.4	0.5374	0.5046	0.4738	0.5378	0.5046	0.4732
0.5	0.4800	0.4507	0.4232	0.4748	0.4507	0.4233
0.6	0.4097	0.3847	0.3612	0.4065	0.3847	0.3547
0.7	0.3266	0.3066	0.2879	0.3306	0.3066	0.297
0.8	0.2306	0.2165	0.2033	0.2228	0.2165	0.1943
0.9	0.1218	0.1143	0.1073	0.1350	0.1143	0.1085
1.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

Table 3.

Tabulated radial variation of axial temperature for the base fluid, Al₂O₃/blood, and Al₂O₃–Fe₃O₄/blood hybrid nanofluid

	Temperature FTCS			Temperature LM-BPNN
	Blood	Nano	Hybrid	Blood LM-BPNN	Nano LM-BPNN	Hybrid LM-BPNN
0.0	0.000663	0.000663	0.000663	-0.023691	0.0024	0.000679
0.1	0.00467	0.004771	0.005618	0.004742	0.004638	0.005527
0.2	0.019662	0.020105	0.023784	0.018429	0.018824	0.02372
0.3	0.049408	0.050397	0.058527	0.052464	0.053494	0.059436
0.4	0.100286	0.101936	0.11541	0.099000	0.101036	0.105822
0.5	0.179715	0.181963	0.200404	0.176107	0.181802	0.197852
0.6	0.293696	0.296242	0.317615	0.305658	0.30052	0.322666
0.7	0.443511	0.445851	0.46664	0.436709	0.438806	0.463174
0.8	0.622664	0.624278	0.640423	0.624598	0.638054	0.646363
0.9	0.815686	0.816332	0.824742	0.846258	0.811687	0.810823
1.0	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000

Figure 3 presents surfaces assembled from the local radial solutions evaluated at successive axial stations. They should therefore be interpreted as a parametrized representation of the long-wavelength model rather than as the solution of a fully coupled two-dimensional axial–radial system. The velocity increases in the constricted region because the local radius decreases, while the no-slip condition enforces zero velocity at the wall. The temperature surface rises toward the wall and varies more sharply near the stenosis, where the mapped radial gradients are amplified by the factor R⁻²(z) in the energy equation.

Fig. 3.

Parametrized surfaces assembled from local radial solutions at successive axial stations: (a) axial velocity and (b) dimensionless temperature

As shown in Fig. 4a, the velocity increases with the Casson parameter β for all three fluids. This trend is consistent with the reduction of the factor (1 + 1/β) as the response approaches the Newtonian limit, thereby decreasing the effective Casson resistance. At any fixed β, the nanoparticle-containing fluids exhibit lower velocities because the effective viscosity and density modify the momentum balance. Figure 4b shows a rise in temperature with β for the selected parameter set. This response results from the combined change in the velocity gradients, viscous dissipation, and thermal transport coefficients and should not be attributed to a single mechanism alone.

Fig. 4.

Velocity (a) and temperature (b) distribution for blood, Fe₃O₄/blood, and Al₂O₃–Fe₃O₄ hybrid nanofluid for different values of Casson parameter β

Figure 5 shows the influence of the Hall parameter m. Because the magnetic damping and Joule-heating coefficients are divided by (1 + m²), increasing m weakens the direct magnetic resistance. Consequently, the velocity increases in Figure 5a. The temperature response in Figure 5b reflects two competing effects: reduced direct Joule heating and the accompanying change in the velocity field. For the present parameter values, the net result is an increase in temperature. The hybrid suspension remains warmer than the other fluids because all effective thermophysical properties change simultaneously.

Fig. 5.

Velocity (a) and temperature (b) distribution for blood, Fe₃O₄/blood, and Al₂O₃–Fe₃O₄ hybrid nanofluid for different values of Hall current parameter m

Figure 6 shows the effect of the magnetic parameter M. The velocity decreases as M increases because the Lorentz-force term opposes the motion of the electrically conducting fluid. The reduction is stronger for the nanoparticle suspensions owing to their modified electrical conductivity and viscosity. In contrast, Figure 6(b) shows an increase in temperature. This behaviour is consistent with magnetic dissipation and Joule heating, together with the altered velocity gradients and effective thermal properties.

Fig. 6.

Velocity (a) and temperature (b) distribution for blood, Fe₃O₄/blood, and Al₂O₃–Fe₃O₄ hybrid nanofluid for different values of magnetic parameter M

Figure 7 illustrates the influence of Re. For the selected transient state, the computed velocity increases with Re, whereas the temperature decreases. The thermal decrease is associated with the reduction of the diffusion and dissipation coefficients that appear with 1/Re in Equation (18). At a fixed Re, the nanoparticle suspensions produce lower velocities and higher temperatures than blood because their effective viscosity, heat capacity, thermal conductivity, density, and electrical conductivity differ from those of the base fluid.

Fig. 7.

Velocity (a) and temperature (b) distribution for blood, Fe₃O₄/blood, and Al₂O₃–Fe₃O₄ hybrid nanofluid for different values of Reynolds number Re

Figure 8 compares the effects of Pr and Ec. Increasing Pr reduces the coefficient of radial thermal diffusion in Equation (18), and the computed dimensionless temperature decreases for this wall-heating condition. Increasing Ec strengthens the viscous-dissipation and Joule-heating terms, producing the temperature rise shown in Figure 8b. The hybrid nanofluid exhibits the largest temperature values over the plotted range because of the combined changes in its effective transport properties.

Fig. 8.

Temperature distribution for blood, Fe₃O₄/blood, and Al₂O₃–Fe₃O₄ hybrid nanofluid for different values of (a) Prandtl number Pr and (b) Eckert number Ec

Figure 9 a shows how the wall-shear magnitude changes with nanoparticle volume fraction. Although the effective viscosity increases with particle loading, the wall velocity gradient decreases sufficiently in these simulations to yield a lower net wall shear stress. Figure 9b shows that wall shear stress increases with β, which is associated with the steeper near-wall velocity gradient as the fluid approaches the Newtonian limit. Because τ_w depends on both the prefactor (1 + 1/β)μ_hnf and w_x(1, t), the observed trend is a combined constitutive and kinematic effect.

Fig. 9.

Variation of wall shear stress with (a) different nanoparticle volume fractions of Fe₃O₄ and Fe₃O₄ − Al₂O₃ nanoparticles and (b) the Casson parameter for blood, nanofluid, and hybrid nanofluid

Figure 10 a shows the effect of nanoparticle volume fraction on the local Nusselt number. Because Nu is defined by the wall temperature gradient, the plotted response reflects the combined changes in thermal conductivity, volumetric heat capacity, velocity gradients, and internal heat generation rather than conductivity alone. Figure 10b shows the dependence on the Casson parameter. This trend results from the way β modifies the velocity gradient and the viscous-dissipation contribution to the thermal field. The panels therefore represent two distinct parameter sweeps with all other quantities held fixed. Because the plotted quantities originate from the final reduced model, these trends apply only to the assumed mild stenosis, rigid wall, prescribed wall temperature, and homogeneous single-phase suspension.

Fig. 10.

Variation of the local Nusselt number with (a) nanoparticle volume fraction and (b) Casson parameter for blood, nanofluid, and hybrid nanofluid

Figure 11 highlights the electromagnetic effects on wall shear stress. In Fig. 11a, increasing m weakens the direct magnetic damping through and changes the near-wall velocity gradient; for the cases shown, the resulting wall-shear magnitude decreases. Figure 11b shows that increasing M also lowers the wall-shear magnitude because the stronger Lorentz force suppresses axial motion. The differences among blood, nanofluid, and hybrid nanofluid reflect the simultaneous changes in viscosity, density, and electrical conductivity (Table 4, Figure 12).

Fig. 11.

Effect of (a) Hall current parameter and (b) magnetic parameter on wall shear stress for blood, nanofluid, and hybrid nanofluid

Table 4.

Percentage variation of the centreline velocity and temperature over the investigated parameter ranges

Parameter	Range	Blood:Δw	Nano:Δw	Hybrid:Δw	Blood:Δθ	Nano:Δθ	Hybrid:Δθ
β	0.5–1.5	+71.59%	+71.72%	+71.79%	+8.88%	+6.76%	+6.02%
m	0.5–2.5	+57.38%	+ 54.69%	47.17%	+11.03%	+10.31%	+7.60%
M	0.3–0.7	-5.48%	-5.23%	-4.53%	-2.85%	-2.61%	-1.79%
Re	2–10	+6.38%	+6.44%	+6.91%	-97.52%	-97.45%	-96.49%
Pr	5–20	–	–	–	-99.47%	-99.52%	-99.75%
Ec	0.1–1.0	–	–	–	16.96%	+16.05%	+11.86%

Fig. 12.

Percentage variation of centreline velocity and temperature over the investigated parameter ranges for blood, Al₂O₃/blood nanofluid, and Al₂O₃ − Fe₃O₄/blood hybrid nanofluid

ANN surrogate performance assessment

This section compares the LM-BPNN predictions with the velocity and temperature profiles obtained from the FTCS calculations. The LM-BPNN is trained entirely on the FTCS-generated database and therefore serves as a surrogate interpolator rather than an independent physical solver. Its agreement with FTCS data shows that the network can reproduce the numerical solution within the sampled parameter domain. Still, it cannot independently validate the governing equations or the FTCS discretization. Independent validation would require comparison with an alternative numerical formulation, analytical benchmark, CFD calculation, or experimental data. Figures 13, 14, 15, 16 present parameter-specific velocity surrogates for variations in β, m, M, and Re, whereas Figures 17, 18, 19, 20, 21, 22 present parameter-specific temperature surrogates for β, m, M, Re, Ec, and Pr.

Fig. 13.

Velocity profiles obtained by FTCS and LM-BPNN for different values of Casson parameter β. a Velocity profiles: FTCS and LM-BPNN for varying β. b Velocity-error profiles: FTCS and LM-BPNN for varying β. c Velocity function fit of the LM-BPNN for β. d Training performance of the LM-BPNN for β. e Training state of the LM-BPNN for β. f Error histogram of the LM-BPNN for β. g Regression analysis of the LM-BPNN for β

Fig. 14.

Velocity profiles obtained by FTCS and LM-BPNN for different values of Hall current parameter m. a Velocity profiles: FTCS and LM-BPNN for varying m. b Velocity-error profiles: FTCS and LM-BPNN for varying m. c Velocity function fit of the LM-BPNN for m. d Training performance of the LM-BPNN for m. e Training state of the LM-BPNN for m. f Error histogram of the LM-BPNN for m. f Regression analysis of the LM-BPNN for m

Fig. 15.

Velocity profiles obtained by FTCS and LM-BPNN for different values of magnetic parameter M. a Velocity profiles: FTCS and LM-BPNN results for varying M. b Velocity-error profiles: FTCS and LM-BPNN results for varying M. c Velocity function fit of the LM-BPNN for M. d Training performance of the LM-BPNN for M. e Training state of the LM-BPNN for M. f Error histogram of the LM-BPNN for M. g Regression analysis of the LM-BPNN for M

Fig. 16.

Velocity profiles obtained by FTCS and LM-BPNN for different values of Reynolds number Re. a Velocity profiles: FTCS and LM-BPNN for varying Re. b Velocity-error profiles: FTCS and LM-BPNN for varying Re. c Velocity function fit of the LM-BPNN for Re. d Training performance of the LM-BPNN for Re. e Training state of the LM-BPNN for Re. f Error histogram of the LM-BPNN for Re. g Regression analysis of the LM-BPNN for Re

Fig. 17.

Temperature profiles obtained by FTCS and LM-BPNN for different values of the Casson parameter β. a Temperature profiles: FTCS and LM-BPNN for varying β. b Temperature-error profiles: FTCS and LM-BPNN for varying β. c Temperature function fit of the LM-BPNN for β. d Training performance of the LM-BPNN for β. e Training state of the LM-BPNN for β. f Error histogram of the LM-BPNN for β. g Regression analysis of the LM-BPNN for β

Fig. 18.

Temperature profiles obtained by FTCS and LM-BPNN for different values of Hall current parameter m. a Temperature profiles: FTCS and LM-BPNN for varying m. b Temperature-error profiles: FTCS and LM-BPNN for varying m. c Temperature function fit of the LM-BPNN for m. d Training performance of the LM-BPNN for m. e Training state of the LM-BPNN for m. f Error histogram of the LM-BPNN for m. g Regression analysis of the LM-BPNN for m

Fig. 19.

Temperature profiles obtained by FTCS and LM-BPNN for different values of magnetic parameter M. a Temperature profiles: FTCS and LM-BPNN for varying M. b Temperature-error profiles: FTCS and LM-BPNN for varying M. c Temperature function fit of the LM-BPNN for M. d Training performance of the LM-BPNN for M. e Training state of the LM-BPNN for M. f Error histogram of the LM-BPNN for M. g Regression analysis of the LM-BPNN for M

Fig. 20.

Temperature profiles obtained by FTCS and LM-BPNN for different values of Reynolds number Re. a Temperature profiles: FTCS and LM-BPNN for varying Re. b Temperature-error profiles: FTCS and LM-BPNN for varying Re. c Temperature function fit of the LM-BPNN for Re. d Training performance of the LM-BPNN for Re. e Training state of the LM-BPNN for Re. f Error histogram of the LM-BPNN for Re. g Regression analysis of the LM-BPNN for Re

Fig. 21.

Temperature profiles obtained by FTCS and LM-BPNN for different values of Eckert number Ec. a Temperature profiles: FTCS and LM-BPNN for varying Ec. b Temperature-error profiles: FTCS and LM-BPNN for varying Ec. c Temperature function fit of the LM-BPNN for Ec. d Training performance of the LM-BPNN for Ec. e Training state of the LM-BPNN for Ec. f Error histogram of the LM-BPNN for Ec. g Regression analysis of the LM-BPNN for Ec

Fig. 22.

Temperature profiles obtained by FTCS and LM-BPNN for different values of Prandtl number Pr. a Temperature profiles: FTCS and LM-BPNN for varying Pr. b Temperature-error profiles: FTCS and LM-BPNN for varying Pr. c Temperature function fit of the LM-BPNN for Pr. d Training performance of the LM-BPNN for Pr. e Training state of the LM-BPNN for Pr. f Error histogram of the LM-BPNN for Pr. g Regression analysis of the LM-BPNN for Pr

For each group of plots, the profile panel compares the FTCS targets with the network predictions, the error panel displays the pointwise discrepancy, and the remaining panels summarize the function fit, training performance, optimization state, error histogram, and regression. The close overlap of most plotted curves indicates accurate interpolation over much of the sampled domain. Nevertheless, the tabulated values reveal that the largest discrepancies occur near locations where the target temperature is small; for example, the negative centreline prediction reported for blood in Table 3 is physically inadmissible and must be considered when interpreting global regression measures. Consequently, the tabulated RMSE and R² values are interpreted together with the visible pointwise discrepancies and boundary behaviour; the aggregate statistics do not remove the need for explicit physical-admissibility checks. Tables 2 and 3 provide representative radial comparisons, while Table 5 summarizes the reported optimization statistics. The very small training errors indicate that the smooth numerical profiles can be fitted accurately, but they do not by themselves establish generalization to parameter combinations outside the training set. Accordingly, all conclusions in this section are restricted to the sampled one-parameter domains represented in the corresponding FTCS datasets (Tables 6, 7, 8).

Table 5.

Results showing training, validation, and testing mean squared errors with other performance metrics

Cases	Mean Squared Error			Performance	Gradient	μ	Epochs
	Training	Validation	Testing
Velocity
β	7.07e-12	1.52e-11	1.68e-11	6.82e-11	7.56e-09	1e-09	1000
m	7.31e-12	2.43e-11	1.63e-11	4.49e-10	9.53e-08	1e-09	1000
M	8.57e-12	1.07e-11	8.66e-11	7.74e-11	7.35e-08	1e-08	1000
Re	4.47e-12	1.85e-10	2.10e-11	3.84e-10	1.92e-08	1e-09	1000
Temperature
β	6.06e-12	1.38e-11	1.33e-11	1.49e-10	5.67e-08	1e-10	1000
m	6.82e-12	9.08e-12	1.55e-11	1.08e-10	2.08e-08	1e-09	1000
M	1.00e-11	3.41e-11	5.37e-10	3.39e-10	3.39e-08	1e-08	1000
Re	1.20e-11	1.46e-11	1.70e-11	1.25e-10	5.43e-07	1e-09	1000
Ec	5.68e-12	1.63e-11	9.83e-12	1.59e-10	5.82e-08	1e-09	1000
Pr	4.26e-12	1.04e-11	4.24e-11	3.70e-11	1.71e-08	1e-09	1000

Table 6.

Performance evaluation of LM-BPNN predictions for velocity and temperature profiles in terms of MSE, RMSE, and R² for different governing parameters

Cases	MSE	RMSE	R2 (rounded)
Velocity
β	1.305043e-11	3.612538e-06	1.000 (rounded)
m	1.607322e-10	1.267802e-05	1.000 (rounded)
M	7.743155e-12	2.782652e-06	1.000 (rounded)
Re	7.080653e-12	2.660950e-06	1.000 (rounded)
Temperature
β	1.279462e-11	3.576957e-06	1.000 (rounded)
m	1.028973e-11	3.207760e-06	1.000 (rounded)
M	1.970576e-11	4.439117e-06	1.000 (rounded)
Re	1.339500e-11	3.659918e-06	1.000 (rounded)
Ec	8.062147e-12	2.839392e-06	1.000 (rounded)
Pr	1.459480e-11	3.820315e-06	1.000 (rounded)

Table 7.

Indicative comparison of the axial-velocity profile of the Al₂O₃–Fe₃O₄/blood hybrid nanofluid with previously published profiles

	Velocity
x	[23]	[47]	[46]	Present study	LM-BPNN
0.0	0.5881	0.5890	0.5085	0.5772	0.5717
0.1	0.5829	0.5828	0.5109	0.5722	0.5721
0.2	0.5725	0.5727	0.5103	0.5551	0.5563
0.3	0.5518	0.5539	0.4890	0.5264	0.5232
0.4	0.5215	0.5259	0.4481	0.4860	0.4873
0.5	0.4802	0.4865	0.4185	0.4341	0.4341
0.6	0.4257	0.4334	0.3820	0.3705	0.3668
0.7	0.3549	0.3608	0.2868	0.2953	0.2998
0.8	0.2634	0.2689	0.2273	0.2085	0.2061
0.9	0.1473	0.1481	0.0836	0.1101	0.1098
1.0	0.0000	0.0000	0.0000	0.0000	0.0000

Differences in modelling assumptions and parameter definitions should be considered when interpreting the comparison

Table 8.

Indicative comparison of the axial-temperature profile of the Al₂O₃–Fe₃O₄/blood hybrid nanofluid with previously published profiles

	Temperature
x	[48]	[46]	Present study	LM-BPNN
0.0	0.093989	0.0006628	0.000663	-0.027982
0.1	0.101855	0.000314	0.005444	0.005445
0.2	0.126176	0.001481	0.023033	0.023028
0.3	0.168902	0.003339	0.056884	0.061671
0.4	0.232505	0.015540	0.112719	0.113080
0.5	0.318937	0.032469	0.196773	0.169734
0.6	0.42428	0.065463	0.313474	0.313753
0.7	0.558456	0.224699	0.462688	0.461984
0.8	0.703227	0.373257	0.637426	0.637676
0.9	0.853959	0.792288	0.823235	0.821858
1.0	1.000000	1.000000	1.000000	1.000000

Differences in modelling assumptions and parameter definitions should be considered when interpreting the comparison

A useful computational check would be to compare the present FTCS results with a fully resolved axisymmetric CFD simulation using the same sinusoidal stenosis geometry, material properties, magnetic-field conditions, and boundary conditions. Such a comparison could be performed using the full governing equations without the long-wavelength reduction and could examine quantities such as the axial velocity. Temperature, wall shear stress, and pressure drop. Agreement between the two approaches would provide an additional check on the reduced formulation and the numerical implementation, particularly in the stenotic region where the radial gradients become stronger.

Experimental validation could be performed using a laboratory flow loop containing a stenosed artery-mimicking channel or compliant vascular phantom. Measurements of velocity and pressure drop could be compared with the predicted flow field, while controlled wall heating and suitable temperature measurements could provide a comparison for the thermal predictions. For the hybrid nanofluid cases, independently measured thermophysical properties would also be useful for assessing the effective property assumptions adopted in the present model. These comparisons are beyond the scope of the current theoretical study, but they would provide an important basis for assessing the model under more realistic conditions.

The present conclusions are therefore limited by the assumptions of a rigid axisymmetric artery, mild stenosis, a homogeneous single-phase suspension, prescribed wall temperature, and the absence of particle migration, deposition, wall compliance, patient-specific geometry, and experimental validation. Future investigations should address these limitations, include a documented rigid-body and time-step convergence study together with an analytical or independently computed benchmark, and examine the present predictions against full CFD and experimental measurements before broader predictive use.

Conclusion

The principal conclusions of the present study are summarized as follows:

1. Stenosis and rheological effects: The reduction in the arterial radius within the sinusoidal stenosis strengthens the radial gradients and produces the most pronounced changes in velocity, temperature, wall shear stress, and heat transfer. Increasing the Casson parameter β from 0.5 to 1.5 increases the centreline velocity by approximately 71.6–71.8% across the three fluid cases, reflecting the reduction in effective Casson resistance.

2. Magnetic and Hall effects: Increasing the magnetic parameter M from 0.3 to 0.7 decreases the centreline velocity by approximately 4.5–5.5% because of stronger Lorentz-force damping. In contrast, increasing the Hall parameter m from 0.2 to 2.5 increases the centreline velocity by approximately 47.2–57.5% owing to the weakening of the direct magnetic contribution through . These trends are consistent with the electromagnetic terms in the reduced model.

3. Thermal and inertial parameters: Over the investigated range, Re = 2 − 10 produces a net centreline velocity increase of approximately 6.4–6.9%, while the centreline temperature decreases by approximately 96.5–97.5%. Increasing Pr from 5 to 30 decreases the centreline temperature by approximately 99.5%, whereas increasing Ec from 0.1 to 1.0 increases it by approximately 11.9–17.0%. These changes arise from the respective effects of momentum transport, thermal diffusion, viscous dissipation, and Joule heating.

4. Hybrid-nanofluid effects: Introducing nanoparticles modifies the velocity and thermal fields through simultaneous changes in effective viscosity, density, heat capacity, thermal conductivity, and electrical conductivity. The resulting wall shear stress and Nusselt number responses therefore reflect the combined effects of rheology, velocity gradients, thermal transport, and internal heat generation rather than a single material property.

5. Numerical and neural-network findings: The explicit FTCS formulation, including the dedicated centreline treatment and Hall-modified magnetic and heating terms, provides the numerical database used by the parameter-specific LM-BPNNs. The networks reproduce most FTCS profiles satisfactorily within the sampled parameter ranges; however, the negative centreline-temperature prediction reported in the ANN comparison confirms that aggregate measures such as MSE and R² alone are insufficient to establish pointwise physical admissibility. The conclusions therefore remain restricted to the assumptions of the present idealized single-phase arterial model.

Abbreviations

CVDs

Cardiovascular diseases

NS

Navier–Stokes

FDM

Finite-difference method

FVM

Finite-volume method

FEM

Finite-element method

FTCS

Forward-time central-space

ANN

Artificial neural network

LM-BPNN

Levenberg–Marquardt backpropagation neural network

MSE

Mean-squared error

RMSE

Root-mean-squared error

MAE

Mean absolute error

R²

Coefficient of determination

CFD

Computational fluid dynamics

Nomenclature

▓

Dimensional radial and axial velocities (m s⁻¹)

u, w

Dimensionless radial and axial velocities (–)

Dimensional radial and axial coordinates(m)

r, z

Dimensionless radial and axial coordinates (–)

x

Mapped radial coordinate, x = r/R(z) (–)

Dimensional and dimensionless time (s, –)

Fluid temperature (K)

θ

Dimensionless temperature (–)

R₀

Healthy-artery radius (m)

R(z)

Dimensionless local arterial radius (–)

λ

Dimensionless stenosis height (–)

β

Casson parameter (–)

m

Hall parameter (–)

M

Magnetic parameter (–)

Re

Reynolds number (–)

Pr

Prandtl number (–)

Ec

Eckert number (–)

ϕ₁, ϕ₂

Nanoparticle volume fractions (–)

A_ρ

Effective density ratio (–)

B_μ

Effective viscosity ratio (–)

C_σ

Effective electrical-conductivity ratio (–)

Effective volumetric heat-capacity ratio (–)

E_k

Effective thermal-conductivity ratio (–)

Π, Π_c

Strain-rate invariant and its critical value (s⁻²)

P_y

Casson yield-stress parameter (Pa)

ε

Slenderness ratio, R₀/L (–)

ρ

Density (kg m⁻³)

μ

Dynamic viscosity (Pa s)

k

Thermal conductivity (W m⁻¹ K⁻¹)

C_p

Specific heat capacity (J kg⁻¹ K⁻¹)

σ

Electrical conductivity (S m⁻¹)

p

Pressure (Pa)

τ_w

Wall shear stress (Pa)

Nu

Local Nusselt number (–)

q_w

Wall heat flux (W m⁻²)

Author contributions

Muhammad Zeshan and Kashif Ali: Methodology, Conceptualization, Writing--Original Draft, Software. Shahzad Ahmad and Huma Gull: Writing--Review \& Editing, Formal Analysis. Muhammad Jawad: Investigation, Resources, Validation. Qasem M. Al-Mdallal and Muhammad Zeshan: Literature Review, Visualization, Resources. All authors have read and approved the final manuscript.

Data Availability

Data will be made available by the corresponding author upon reasonable request.

Declarations

Ethical approval

Not applicable; this is a theoretical computational study and uses no human participants, biological specimens, or patient-specific data.

Consent to publish

Not applicable.

Consent to participate

Not applicable.

Competing interests

The authors declare no competing interests.