Engineers studying how to cool everything from solar collectors to high-power electronics may soon have a sharper computational tool, thanks to a new numerical study of a thixotropic Jeffery nanofluid flowing past an inclined stretching cylinder. Published in Results in Chemistry, the research combines a state-dependent microstructural model of a non-Newtonian fluid with MXene (Ti3C2) nanoparticles, Newtonian heating boundary conditions, and a full second-law thermodynamic analysis of entropy generation. The work is entirely computational: it solves transformed boundary-layer equations with a parametric continuation method and reports how skin friction, heat transfer, and irreversibility respond to changes in key physical parameters.
The Jeffery fluid model is an advanced form of non-Newtonian viscoelastic fluid that captures both elastic and viscous behavior, including relaxation and retardation times that reflect fluid memory and delayed response to applied stress. Such models are relevant to industrial polymers and biological fluids, and to processes like extrusion, crude oil transport, and blood flow in microcirculation. What sets the new formulation apart, the authors argue, is that existing traditional studies treat the viscosity of non-Newtonian fluids as a material constant, whereas their model couples the constitutive rheological tensor with a scalar microstructural variable that undergoes advective transport, physical network restoration, and shear-induced breakdown of the matrix.
That microstructural variable, denoted lambda, ranges between zero and one, with one representing a fully structured state and zero a fully destroyed fluid network. The dynamic viscosity depends on this state through a thixotropic structural viscosity parameter, so the fluid effectively stiffens where its internal network is intact and softens where shear has broken it down. Two competing parameters govern the evolution: a structural recovery parameter and a structural breakdown parameter, which together determine how the network rebuilds over time and how quickly shear destroys it near the moving boundary.
The physical setup is a steady, two-dimensional, axisymmetric magnetohydrodynamic flow of a Ti3C2 nanofluid in a water-ethylene glycol base over an inclined stretching porous cylinder. The cylinder surface stretches linearly, a porous surface permits mass suction, and an external magnetic field is applied at an inclination angle. The energy equation includes viscous dissipation, thermal radiation, and Joule heating, and the thermal boundary condition is Newtonian heating, in which heat transfer along a closed surface is directly proportional to the real temperature of that surface. Under this condition the supply of heat is controlled by a convective process from an external hot fluid source, so the surface temperature depends on the flow dynamics rather than being fixed, a situation relevant to heat exchanger design, solar radiation absorbers, and petroleum engineering.
On the numerical side, the team used the parametric continuation method, which introduces an embedded parameter to trace the whole solution path and is preferred over Newton-Raphson iteration for its reliability and stability with stiff nonlinear systems. The semi-infinite domain was truncated at a similarity threshold of 6.0 and discretized with a uniform mesh of 200 elements. The absolute convergence tolerance was set at 10 to the minus 8, with maximum absolute residuals below the same bound, and the scheme converged within 4 to 6 iterations. Validation against previously published studies showed agreement to four decimal places in skin friction and Nusselt number values across a range of stretching parameters, which the authors cite as confirmation of the accuracy of the formulation and calculation process.
A grid independence study from 60 to 250 nodes showed the computed skin friction coefficient and local Nusselt number stabilizing completely by 200 nodes, where the residual error fell to 3.04 times 10 to the minus 9 and the microstructural uncertainty reached plus or minus 2.1 times 10 to the minus 6. Refining the grid beyond 200 nodes produced no numeric differences beyond six significant figures. Because CPU time rose only from 0.45 seconds to 2.58 seconds across the refinement range, the 200-node grid was chosen as the default for all subsequent parametric calculations, offering what the authors describe as an ideal balance between low uncertainty and computational performance.
A derivative-based sensitivity analysis quantified how each governing parameter affects the two engineering quantities of interest. The magnetic parameter and the thixotropic structural viscosity parameter were the most influential on surface drag, raising the skin friction coefficient by 18.5342 percent and 14.2854 percent respectively when doubled from their baseline settings, because the resistive Lorentz force and the unsheared structural dynamic viscosity are directly amplified. The structural breakdown parameter showed a strong negative sensitivity, decreasing surface shear stress by 11.6421 percent as continuous shear destruction of the internal microstructural network lowers the effective viscosity near the boundary. For heat transfer, the thermal radiation parameter was the dominant positive factor, with a doubling of the radiative flux parameter producing a 22.6461 percent increase in the local Nusselt number, while a larger Eckert number reduced the Nusselt number by 14.5387 percent, since greater internal frictional heating shrinks the temperature difference between the cylinder wall and the ambient stream.
The profile results follow consistent physical logic. Velocity declines as the magnetic factor grows because the Lorentz force acts as a magnetic damper opposing the flow, and it also drops with increasing nanoparticle concentration, since the added Ti3C2 raises the mixture’s dynamic viscosity and internal flow resistance. Velocity decreases with the porosity parameter, which acts as an inverse permeability factor, but rises with the velocity slip factor, the mixed convection factor, and the surface curvature factor, the last because greater curvature thins the boundary layer and raises velocities near the wall. On the thermal side, the temperature field increases with the Newtonian heating factor, the Eckert number, and the radiation parameter, and decreases with the Deborah number, which pushes the fluid toward solid-like elastic behavior, and with nanoparticle volume fraction, which enhances the mixture’s thermal conductivity and cooling performance.
The entropy generation analysis decomposes irreversibility into heat transfer, fluid friction, and magnetic or porous medium contributions, with the Bejan number indicating whether thermal gradients or frictional and Joule heating effects dominate. Increasing the thixotropic structural viscosity parameter raised entropy generation while lowering the Bejan number, because a stiffer boundary layer matrix produces greater fluid friction irreversibility relative to thermal transport. Raising the Brinkman number from 1.0 to 3.0 nearly tripled the surface entropy generation number, from 1.52718 to 3.15652, and drove the Bejan number down from 0.48195 to 0.23318, showing that fluid dynamic drag dominates at high Brinkman values. Increasing the dimensionless temperature difference parameter reversed the trend, cutting the entropy number to 1.61843 and lifting the Bejan number to 0.45521, while raising the radiation parameter pushed the Bejan number toward an upper value of 0.62143, signaling dominance of radiative heat transfer irreversibility.
Among the headline quantitative results, the surface skin friction decreased by up to 41.5343 percent and 85.1217 percent with variations of the Deborah number and the velocity slip factor respectively, and increased by up to 3.22616 percent with the porosity factor. The rate of energy transfer rose by up to 75.4138 percent when the Ti3C2 nanoparticle fraction changed from 0.01 to 0.04, and by 57.6461 percent when the Newtonian heating factor was enhanced from 1.0 to 4.0. Increasing the thixotropic parameter raised overall surface friction by 21.1492 percent while reducing surface heat transfer rates by 19.5214 percent. The authors also note the study’s limitations: the model assumes steady, two-dimensional flow on a rigid inclined cylinder, a homogeneous nanoparticle dispersion that ignores clustering or sedimentation, and a truncated similarity domain, with future work planned toward transient three-dimensional simulation, ternary nanofluids, and experimental validation for real-world thermal management applications.
Subject of Research: Numerical simulation of thixotropic Jeffery nanofluid flow with entropy generation analysis over an inclined stretching cylinder
Article Title: Irreversibility analysis of thixotropic Jeffery nanofluid flow subject to newtonian heating with parametric sensitivity analysis and grid convergence verification past an inclined cylinder
Article References: Hussain, S. M., Jubair, S., Ali, B., Rehman, M. U., Abdullayev, S., & Khalilov, S. (2026). Irreversibility analysis of thixotropic Jeffery nanofluid flow subject to newtonian heating with parametric sensitivity analysis and grid convergence verification past an inclined cylinder. Results in Chemistry, 32, Article 103971. https://doi.org/10.1016/j.rechem.2026.103971
Image Credits: AI Generated
DOI: 10.1016/j.rechem.2026.103971
Keywords: thixotropic fluid, Jeffery nanofluid, MXene, entropy generation, Bejan number, Newtonian heating, magnetohydrodynamics, skin friction, Nusselt number, parametric continuation method, grid independence, inclined cylinder
News Source: Audrey Campbell. (October 28, 2026). Thixotropic MXene Nanofluid Flow Over an Inclined Cylinder Yields New Irreversibility Map. Scienmag.
