The physical problem the team tackled is deceptively simple to state but notoriously hard to solve. When a viscous, electrically conducting fluid flows radially between two non-parallel walls meeting at a half-angle, the two-dimensional Navier–Stokes equations reduce — through similarity transformations — to a single non-linear ordinary differential equation for velocity, coupled to a second equation for temperature. The flow is called convergent when the walls narrow toward a sink and divergent when they widen from a source. Because the resulting equations are non-linear, no exact closed-form solution exists in most cases, forcing researchers to rely on approximate analytical and numerical schemes. Keerthana and Padma enriched the classical model by adding a transverse magnetic field, a porous medium characterized through Darcy’s law, and nanoparticle volume fraction effects, evaluated through well-established thermo-physical mixture models for effective density, viscosity, and thermal conductivity.

The choice of nanoparticles is where the study acquires its biomedical edge. The researchers dispersed two distinct particles separately into the blood base fluid: copper (Cu), prized for its exceptional thermal conductivity of roughly 400 watts per meter-kelvin, and magnetite (Fe3O4), the iron oxide celebrated for its magnetic responsiveness and biological compatibility. Blood itself — treated as a Newtonian fluid, a reasonable assumption at the high shear rates found in large arteries where non-Newtonian red-cell effects fade — has a thermal conductivity of only 0.492 watts per meter-kelvin and a density of 1,063 kilograms per cubic meter. Copper’s density of 8,933 kilograms per cubic meter makes Cu-blood suspensions heavier and more inertially resistant, while magnetite’s lower density of 5,200 kilograms per cubic meter permits smoother flow through constricted arterial regions.

The Bernstein collocation method works by expanding the unknown solution as a linear combination of Bernstein basis polynomials of degree m, defined on the interval [0,1] through binomial coefficients. The nth derivative of the approximation can itself be expressed in the same Bernstein basis, which allows the residual of the governing differential equation to be evaluated at precisely chosen collocation points — in this case, Chebyshev–Gauss–Lobatto nodes. Setting the residual to zero at these nodes converts the non-linear differential equation into a system of algebraic equations whose unknowns are the polynomial coefficients. Solving that system yields the velocity and temperature profiles directly. Bernstein polynomials are renowned for their neutrality, consistent convergence, and stable computation without succumbing to the ill-conditioning that plagues many high-degree polynomial bases.

To benchmark the new approach, the authors compared BCM results against the Chebyshev collocation method (CCM), a spectral technique that expands solutions in Chebyshev polynomials on Gauss–Lobatto nodes and is celebrated for its superior convergence rate and mathematical stability. They also cross-validated against the Adomian decomposition method (ADM), an entirely independent analytical framework that does not depend on collocation at all. The comparison revealed striking agreement: velocity values matched to four decimal places across the entire domain among all three methods, while temperature profiles from BCM and CCM corresponded to six decimal places, with ADM curves tracing the same trend in close proximity. Absolute errors between BCM and independent numerical solutions ranged between the sixth and eighth decimal places, uniformly small across the whole domain with no amplification — a signature of computational stability.

The convergence analysis added further weight to the reliability claim. By gradually increasing the degree of approximation from 9 to 21, the computed skin friction coefficient values shifted noticeably at low degrees but settled into asymptotically fixed values at higher levels — converging to −2.5606136 for the diverging channel and −1.4256114 for the converging channel for magnetite suspensions. The team also reproduced published results derived through the variational iteration method under matching parameter conditions, finding significant uniformity at degree 15. These checks confirm that the proposed framework yields firmly stable solutions at high approximation levels, a prerequisite for any method intended to serve as a dependable numerical engine for hemodynamic modeling.

The physics the solved equations revealed is equally compelling. In the diverging channel, increasing the nanoparticle volume fraction lowers velocity because the effective viscosity of blood rises, improving flow resistance inside the artery — and moderate backflow emerges at lower volume fractions under high Reynolds numbers, an inverse flow in the widening arterial region. In the converging channel, the trend reverses: velocity climbs with volume fraction because arterial narrowing accelerates the nanofluid. Raising the Hartmann number — the magnetic field parameter — improves the rate of acceleration in both channel structures, since the applied field reacts with electrically conductive blood to generate a Lorentz force that alters flow. Crucially, the magnetic field can suppress backflow in the diverging channel, offering a control lever for flow inversion. Higher porosity of the embedded porous medium reduces hydraulic drag and strengthens flow in both geometries.

Temperature behavior followed its own logic. In the converging channel, temperature increased with Reynolds number for both Cu and Fe3O4 suspensions, because stronger inertial stresses intensify convective heat transfer. Temperature declined as the convergent half-angle grew, since faster fluid spends less time near the walls, thinning the thermal boundary layer. Increasing the Hartmann number raised temperature, as the Lorentz force converts mechanical energy into thermal energy through resistance to motion. Quantitatively, magnetite suspensions produced faster velocity than copper at all interior grid points — attributable to lower density minimizing inertial resistance — while copper suspensions exhibited significantly higher temperatures, thanks to copper’s substantially greater thermal conductivity. From a clinical standpoint, Fe3O4’s high velocity, biocompatibility, low toxicity, and magnetic responsiveness suit targeted drug delivery, magnetic hyperthermia, and MRI-directed cardiac treatment, while copper’s enhanced thermal behavior is relevant for thermal ablation within the arterial region.

Perhaps the most forward-looking element of the study is its marriage of numerical computation with statistics. Because pushing the Bernstein polynomial degree too high congests the underlying algebraic matrices and makes the system hypersensitive to rounding errors, the authors implemented response surface methodology (RSM) built on a face-centered central composite design with 20 experimental runs — six axial points, ten factorial points, and six center points. The RSM model related the skin friction coefficient to three predictors: nanoparticle volume fraction, Hartmann number, and Reynolds number. Analysis of variance confirmed the model’s validity, with R-squared, adjusted R-squared, and predicted R-squared values all close to unity. Residual plots showed normally distributed errors with a maximum residual of roughly 0.0011, and the Pareto chart ranked the predictors in order of control: Reynolds number first, Hartmann number second, volume fraction last, with the normalized effect threshold set at 2.2.

The statistical layer did more than summarize — it cross-confirmed the numerical findings. Contour and surface plots showed the skin friction coefficient amplifying with both Reynolds and Hartmann numbers, with the response far more reliant on Reynolds number, while the combined influence of Hartmann number and volume fraction remained almost peripheral. Interaction plots revealed a substantial relationship between Reynolds and Hartmann numbers, whereas Hartmann–volume-fraction interactions were trivial, their traces nearly parallel. The convergence between RSM and BCM analyses — both identifying Reynolds number as the paramount factor — demonstrates the consistency of the two techniques and offers a computational shortcut: once the response surface is established, flow properties can be predicted rapidly within the parameter range without repeatedly solving high-order collocation equations. For a field racing toward personalized magnetic therapies in narrowed arteries, that combination of a stable polynomial solver and a statistically validated predictive surface may prove to be the framework that turns century-old channel-flow mathematics into practical clinical engineering.

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

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

Article References: Keerthana, N., & Padma, R. (2026). Bernstein collocation method for Jeffery–Hamel nanofluid flow and heat transfer: a comparative study with the Chebyshev collocation method. Frontiers in Nanotechnology, 8, Article 1872948. https://doi.org/10.3389/fnano.2026.1872948

Image Credits: AI Generated

DOI: 10.3389/fnano.2026.1872948

Keywords: Bernstein, collocation, method, Jeffery, Hamel, nanofluid, flow, heat, transfer, comparative, Chebyshev, scientific research