Researchers have built a detailed numerical model of an unsteady, non-Newtonian nanofluid flowing over a curved, stretching surface, and their calculations show how a magnetic dipole, Stefan blowing, wall slip, thermal radiation, second-order chemical reactions, and bioconvection jointly shape the flow. The study, published in Discover Nano, was motivated in part by thermal management challenges in fusion reactor walls, where intense heat and particle fluxes must be dissipated while preserving structural integrity. Rather than treating these effects one at a time, the authors combined them into a single framework for a second-grade nanofluid, a viscoelastic fluid model that captures elastic memory effects that ordinary Newtonian models cannot.
The team, led by Muhammad Ramzan with co-authors including Saba Arshad, Nazia Shahmir, Yazeed Alkhrijah, Norah S. Barakat, Abdulkafi Mohammed Saeed and Wei Sin Koh, formulated governing equations for momentum, temperature, nanoparticle concentration, and the density of motile microorganisms. Using similarity transformations, they converted the partial differential equations into ordinary differential equations and solved them with the MATLAB bvp4c collocation method, which offers fourth-order accuracy, a mesh size of 0.01, and a convergence tolerance of 10 to the power of minus 6. A grid independence test showed that a grid of 15 by 15 points was sufficient, with the Nusselt number becoming insensitive to further mesh refinement.
A central feature of the model is the magnetic dipole. Unlike the uniform magnetic field used in classical magnetohydrodynamic models, a dipole produces a spatially varying field, generating localized magnetic body forces whose magnitude depends on the field gradient. The researchers represented magnetization as a linear function of temperature through a ferromagnetic index, coupling the magnetic field to the thermal field. In their results, increasing the ferrohydrodynamic interaction parameter reduced the velocity profile, which the authors attribute to a strengthened Lorentz-like retarding force that thickens the momentum boundary layer and suppresses fluid motion. At the same time, the surface friction coefficient increased abruptly as the ferrohydrodynamic interaction factor rose, because the magnetic body force accelerates fluid in regions of strong field and steepens the velocity gradient at the wall.
Stefan blowing, a wall-injection effect arising from species diffusion at the surface, had the opposite influence on the flow. The velocity profile showed an inflated trend as the Stefan blowing parameter increased, since greater mass injection from the surface into the boundary layer alters the momentum balance and raises the near-wall velocity. The second-grade fluid parameter also increased the velocity distribution, reflecting how greater elasticity reduces resistance to deformation under shear. By contrast, increasing the velocity slip factor lowered the velocity, because stronger wall slip weakens momentum transfer between the moving surface and the fluid.
The thermal analysis incorporated the Rosseland diffusion approximation for radiative heat transfer, a linearized formulation appropriate for optically thick media and modest temperature differences. Raising the radiation parameter enhanced the temperature field, because more radiative energy enters the fluid as the mean absorptivity index falls, thickening the thermal boundary layer. However, the study notes a subtlety: although the fluid temperature rises, the wall temperature gradient falls, so heat transfer efficiency decreases even as the thermal field grows. The unsteadiness parameter likewise increased the temperature distribution while reducing the Nusselt number, meaning less effective heat removal at the wall.
Quantitatively, the tabulated results show that increasing the unsteadiness parameter reduces the skin friction coefficient by 0.6083 percent and the Nusselt number by 0.8203 percent, a slight decline in heat transfer efficiency. When the curvature parameter is increased, meaning it becomes less negative, the skin friction coefficient rises by 2.6580 percent, while the Sherwood number and the motile microorganism density fall by 3.8926 percent and 2.6424 percent respectively. The surface drag increase with curvature reflects stronger velocity gradients near the wall, as streamline curvature intensifies momentum transport compared with a flat surface. The second-grade fluid parameter, by contrast, reduced the skin friction coefficient, with a decline of 13.40 percent observed as that parameter moved from 0.1 to 0.4.
Mass transfer followed its own pattern. The concentration distribution declined as the second-order reaction parameter rose, consistent with reactant consumption in bimolecular reactions of the kind common in polymerization, catalytic reactors, and wastewater treatment. The unsteadiness parameter strengthened the concentration field, which the authors link to enhanced particle transport under fluctuating flow. The impact of curvature and Brownian motion on the species transfer rate revealed a contradictory pattern in the reported results, and the Sherwood number increased with the Schmidt number, the chemical reaction parameter, and the Brownian motion parameter, attributed to steeper concentration gradients and improved diffusion mechanisms at the wall.
Bioconvection entered the model through gyrotactic microorganisms, denser-than-water swimmers that accumulate near the upper fluid layers and create instability-driven convection currents. These organisms respond to viscous and gravitational torques, and their collective swimming promotes fluid mixing and a more uniform nanoparticle suspension, which can reduce sedimentation. In the simulations, the microorganism concentration profile decreased with the bioconvection Schmidt number, because reduced microbial diffusivity limits dispersion, but increased with the bioconvective Peclet number and the dimensionless bioconvective factor, as stronger advective transport carries the organisms farther through the medium. The motile microorganism number itself rose with the bioconvection Schmidt number and fell with the bioconvective Peclet number, the bioconvective factor, and the curvature parameter.
To verify the model, the authors compared their skin friction results with a published study in a limiting case, finding discrepancies between 0.00375 percent and 0.06150 percent, which they describe as negligible confirmation of the model’s validity. Additional validation of the Nusselt and Sherwood numbers was performed in the limit of infinite curvature against earlier published benchmarks. Streamline and isotherm contour plots complemented the tabulated data, with closely spaced isotherms indicating stronger temperature gradients and heat transfer near the curved surface.
The authors are explicit about the limits of the work. The model assumes uniformly dispersed nanoparticles without agglomeration, constant thermophysical properties, and a two-dimensional flow configuration, and it rests entirely on numerical simulation without experimental validation. Similarity transformations apply only under the stated flow assumptions and boundary conditions, and solution accuracy depends on the computational settings and convergence tolerance. Within those constraints, the researchers suggest the findings can guide the design of microfluidic devices, biomedical transport systems, electronic cooling, solar energy systems, and curved-surface thermal systems where controlled heat transfer, mass transport, and bioconvective behavior matter, and they propose extending the model to triple-diffusive transport, inclined magnetic fields, additional non-Newtonian fluid classes, and eventually experimental confirmation.
Subject of Research: Numerical modeling of magnetic dipole, Stefan blowing, radiation and bioconvection effects on unsteady second-grade nanofluid flow over a curved stretching surface
Article Title: Magnetic dipole effects on unsteady nanofluid flow over a curved surface
Article References: Ramzan, M., Arshad, S., Shahmir, N., Alkhrijah, Y., Barakat, N. S., Saeed, A. M., & Koh, W. S. (2026). Magnetic dipole effects on unsteady nanofluid flow over a curved surface. Discover Nano, 21(1), Article 563. https://doi.org/10.1186/s11671-026-05063-x
Image Credits: AI Generated
DOI: 10.1186/s11671-026-05063-x
Keywords: nanofluid, magnetic dipole, bioconvection, Stefan blowing, thermal radiation, second-grade fluid, curved surface, heat transfer, mass transfer, bvp4c, ferrohydrodynamics, microfluidics
News Source: Denise Maddox. (October 27, 2026). Magnetic Dipoles Reshape Heat and Mass Transfer in Nanofluid Flow Over Curved Surfaces. Scienmag.
