Semi-analytical model maps heat transport in Maxwell nanofluid over stretching sheet

Engineers have long chased a stubborn problem: the working fluids that carry heat through industrial equipment are simply not very good at carrying heat. Conventional fluids have weak thermal conductivity, which caps the performance of heat exchangers, electronic cooling systems and countless other devices. A newly published study in Discover Mechanical Engineering tackles this limitation head-on by modeling how a Maxwell nanofluid—a viscoelastic base fluid seeded with nanoparticles—transports heat as it flows over a stretching sheet, using a modeling framework the authors describe as an innovative semi-analytical approach.

The research, led by Shahryar Hajizadeh, Zahra Poolaei Moziraji, Payam Jalili and Bahram Jalili, investigates the role of Maxwell nanofluids in enhancing thermal transport in fluid flow systems. The core premise is straightforward: replacing conventional working fluids with nanoparticle suspensions improves heat transfer capability and overall system efficiency, because various types of nanoparticles offer high thermal exchange efficiency. What distinguishes this study is the combination of effects it packs into a single model—Maxwell viscoelasticity, activation energy, mixed convection, magnetic interaction, Brownian diffusion and thermophoresis—all treated within a semi-analytical framework that the authors say offers reduced computational cost and rapid convergence compared with conventional numerical schemes.

The physical setup is a steady, laminar, two-dimensional Maxwell nanofluid flow over a stretching sheet, evaluated on a two-plane configuration under the effects of mixed convection and activation energy. A uniform transverse magnetic field is applied at right angles to the flow planes to balance the boundary layer with a constant heat source, and induced magnetic fields are neglected because of the low magnetic influence. The sheet’s stretching velocity and the external flow velocity both scale linearly with distance along the surface. Thermophysical properties are taken as constant except for nanoparticle-induced diffusion effects, and the boundary conditions prescribe the stretching velocity, surface temperature and nanoparticle concentration at the wall, with ambient values reached far from the surface.

Mathematically, the team integrated the equations of momentum, continuity, concentration and energy movement into the flow model as partial differential equations. By exploiting functional similarities—similarity transformations that collapse the spatial dependence of the flow into a single dimensionless variable—the constructed model was transmuted into a system of nonlinear ordinary differential equations governing the velocity, temperature and concentration fields. The dimensionless groups that emerge from this reduction include the Hartman number, which measures the strength of magnetic damping; the Prandtl number; the buoyancy and free convection parameters; the Brownian motion and thermophoresis factors; the Schmidt number; the Maxwell fluid parameter, which captures the fluid’s elastic relaxation; and the velocity ratio between the stretching surface and the ambient stream.

Solving these coupled nonlinear equations is the hard part, and the authors attacked it with two complementary tools. The first is Akbari-Ganji’s Method, or AGM, a semi-analytical technique that approximates the solution of each differential equation with a finite polynomial series whose constant coefficients are fixed by the boundary conditions. In this study, three test functions carrying fifteen constant coefficients were constrained by seven boundary-condition equations, with eight additional equations generated from the governing equations and their derivatives, producing a closed system of fifteen equations in fifteen unknowns. The second tool is the Finite Element Method, implemented in an adaptive framework that automatically refines the mesh based on residual error. The computations typically used 1,200 to 1,800 elements for different cases, with higher mesh density near the wall to resolve steep gradients, quadratic interpolation functions for the velocity, temperature and concentration fields, and an adaptive Newton iteration with a convergence tolerance of ten to the power of minus six.

The AGM’s appeal, the authors explain, is that it yields closed-form polynomial approximations, enabling rapid parametric studies at substantially lower computational cost than repeated finite element simulations. The resulting polynomial solutions for the dimensionless velocity, temperature and concentration profiles were then compared against the finite element results, and the close match between the two methods’ concentration profiles was taken as validation of the reaction-diffusion model and confirmation that the governing equations capture nanoparticle transport accurately.

The results trace how each parameter reshapes the flow. Increasing the stretching parameter reduces temperature near the wall, because stretching intensifies the tangential velocity and enhances convective removal of heat from the surface—a mechanical cooling effect in which the surface motion acts like a conveyor belt, pulling thermal energy away and thinning the thermal boundary layer. The magnetic field works in the opposite direction: the Lorentz force it introduces resists fluid motion, damping convective heat transport and causing the temperature to rise, effectively converting kinetic energy into internal energy and trapping heat within the boundary layer. Buoyancy forces enhance upward thermal motion, strengthening convective transport and reducing the near-wall temperature while thickening the thermal layer farther from the surface.

Nanoparticle-specific mechanisms leave their own distinct fingerprints. Brownian agitation increases the random movement of nanoparticles and enhances thermal diffusion, raising the temperature across the boundary layer and thickening the thermal layer, as the particles act as micro-scale energy carriers that boost the fluid’s effective thermal conductivity. Thermophoretic forces drive nanoparticles from hotter to cooler regions, transporting thermal energy outward, steepening the temperature gradient and expanding the thermal boundary layer—an effect that grows stronger at higher nanoparticle concentration. Internal heat generation injects thermal energy uniformly, producing a temperature rise and significant thickening of the thermal boundary layer that counteracts convective cooling. Increasing the Maxwell parameter, by contrast, lowers the temperature, because the elastic stresses of the viscoelastic fluid store mechanical energy and reduce viscous dissipation, limiting the conversion of mechanical work into heat and yielding a cooler boundary layer.

The concentration field reflects a tug-of-war between diffusion, chemical reaction and activation energy-controlled kinetics. Higher activation energy suppresses reaction activity, allowing more nanoparticles to remain unreacted and raising concentration throughout the boundary layer as the system shifts toward a diffusion-dominated regime governed more by Brownian mixing than chemical consumption. Increasing the reaction rate produces the opposite effect: a sharp drop in concentration near the wall, where reaction intensity is highest, and a thinner concentration boundary layer because reaction outpaces diffusion. On the velocity side, higher stretching accelerates fluid near the wall and thins the momentum boundary layer, buoyancy increases streamwise velocity by reducing the effective resistance to flow, and higher Maxwell relaxation parameters promote greater velocity as elastic recoil assists the flow—while simultaneously suppressing temperature. Increasing the Hartman number suppresses velocity and thickens the momentum boundary layer, because the Lorentz force acts as a momentum sink that dissipates kinetic energy.

At the wall itself, both the skin friction coefficient and the Nusselt number rise as the buoyancy parameter and Hartman number increase. Buoyancy-driven acceleration steepens the near-wall velocity gradient and raises shear stress, while magnetic damping steepens the wall velocity gradient in the same direction. On the thermal side, the magnetic field suppresses fluid motion in a way that increases the temperature gradient at the wall, strengthening heat transfer, and buoyancy accelerates the removal of heated fluid from the surface, boosting convective heat transfer. The study’s conclusions also note that as Brownian motion and chemical reaction parameters increase, nanoparticle concentration profiles decrease, while activation energy and thermophoresis raise them, and that Brownian motion and thermophoresis increase thermal diffusion, leading to higher temperatures and thicker thermal boundary layers.

Confidence in the numbers rests on external validation as well as internal cross-checking. The authors compared their temperature, velocity and concentration profiles, together with the Nusselt number and skin friction, against the analysis of Ahmad and colleagues, reporting minimal computational discrepancy across the five comparisons, with the numerical data tabulated alongside the graphical comparisons. The authors suggest the findings provide valuable insights for optimizing heat transfer in systems involving Maxwell nanofluids, such as thermal exchangers, cooling of electronic components, polymer processing and energy systems where precise control of heat and mass transfer is essential, and note relevance to cooling of stretching sheets in polymer extrusion and magnetically assisted heat transfer systems. The scope is deliberately bounded: the study is limited to two-dimensional steady flow and uniform magnetic fields, and the authors flag extensions to three-dimensional geometries, transient behavior, variable properties and hybrid nanoparticle suspensions as future work. Within those limits, the study offers a computationally cheap route to exploring a crowded parameter space—one where elasticity, magnetism, buoyancy and nanoparticle transport all pull the temperature field in different directions at once.

Subject of Research: Semi-analytical and numerical modeling of Maxwell nanofluid heat and mass transport over a stretching sheet

Article Title: Thermal transport behavior of Maxwell nanofluid using an innovative modeling framework on a stretched sheet

Article References: Hajizadeh, S., Poolaei Moziraji, Z., Jalili, P., & Jalili, B. (2026). Thermal transport behavior of Maxwell nanofluid using an innovative modeling framework on a stretched sheet. Discover Mechanical Engineering, 5(1), Article 209. https://doi.org/10.1007/s44245-026-00373-x

Image Credits: AI Generated

DOI: 10.1007/s44245-026-00373-x

Keywords: Maxwell nanofluid, stretching sheet, heat transfer, Akbari-Ganji method, finite element method, activation energy, mixed convection, Hartman number, thermophoresis, Brownian motion, skin friction, Nusselt number