A Mathematica-based algorithm to trace stable/unstable solutions in transient free convection of non-Newtonian fluids Zafar Hayat Khan, Guilin Zhang, Alexander Trounev, Rashid Ahmad, Waqar Ahmed Khan Canadian Journal of Chemical Engineering, 2026 This paper presents a Mathematica‐based computational framework for tracing stable and unstable solutions in the transient free convection of power‐law fluids governed by Stokes' second problem. The method combines high‐order finite difference discretization with an event‐driven stability detector implemented through WhenEvent , enabling nonlinear stability assessment without relying on linearization or eigenvalue analysis. Stability maps constructed in the extended parameter space reveal that the critical Grashof number increases with the power‐law index, from for shear‐thinning fluids to for shear‐thickening fluids . Variations in the magnetic parameter and Eckert number shift the stability boundaries significantly, with magnetic damping stabilizing flows for and destabilizing flows for , while viscous dissipation produces strong thermal amplification, thereby narrowing the stable regime. Long‐time periodic solutions and phase‐plane portraits further demonstrate that oscillation intensity and waveform deformation increase with . Wall‐based heat‐transfer and shear‐stress quantities show a consistent hierarchical ordering , confirming the dominant role of shear‐dependent viscosity in organizing momentum–thermal coupling. The proposed algorithm provides a reproducible and efficient tool for exploring nonlinear stability in oscillatory non‐Newtonian convection.
Time-fractional analysis of free convection in a heated square cavity Zafar Hayat Khan, Meijiang Zhou, Waqar Ahmed Khan, Alexander Trounev, Zaitang Huang Journal of Computational Design and Engineering, 2026 This study presents a comprehensive numerical analysis of time-fractional free convection in a square cavity with a heated vertical wall, governed by the time-fractional Navier–Stokes equations incorporating Caputo derivatives to account for memory-dependent and non-local effects. A fourth-order finite difference scheme is employed for spatial discretization, while implicit time-integration schemes of orders $2 - \\alpha ,\\,\\,3 - \\alpha ,$ and $4 - \\alpha $ are used to approximate the fractional time derivatives, with $0 < \\alpha \\le 1$. After comparative evaluation, the $2 - \\alpha \\,\\,$ scheme is selected for its balance between accuracy and computational cost. The resulting system is solved using a linear solver implemented in Mathematica 14, utilizing the ‘Pardiso’ method, a parallel direct sparse solver, to ensure efficient and stable computation. Results show that the fractional-order parameter $\\alpha $ has a significant impact on the flow structure and heat transfer characteristics. Higher values of $\\alpha $ lead to classical behaviour with faster stabilization and organized convection cells, while lower values introduce pronounced memory effects, delayed transitions, and oscillatory dynamics. The proposed method is validated against classical benchmark solutions, demonstrating high accuracy and convergence. This work not only underscores the potential of fractional models in capturing transient convection behaviour but also establishes a benchmark framework for future studies on fractional-order heat transfer systems.
Entropy generation in time-fractional magnetohydrodynamic nanofluid Couette flow between permeable walls Zafar Hayat Khan, Oluwole Daniel Makinde, Waqar Ahmed Khan, Abdulaziz Alasiri, Alexander Trounev Physics of Fluids, 2025 This study investigates entropy generation in magnetohydrodynamic radiative nanofluid Couette flow through a permeable channel using the Caputo time-fractional derivative framework. The model incorporates thermal radiation, magnetic field effects, viscous and Joule heating, nanoparticle volume fraction, and wall suction/injection, with the upper permeable wall moving and subject to convective heat exchange with its surroundings. The fractional derivative captures memory-dependent transport processes, extending classical models of transient nanofluid flow and heat transfer. The governing fractional partial differential equations for velocity and temperature are solved numerically via a hybrid Euler wavelet collocation method for spatial discretization and a fractional finite difference scheme for temporal discretization, with Newton's method employed for nonlinear resolution. Key thermophysical quantities such as Bejan number, entropy generation rate, skin friction and Nusselt number are analyzed to assess irreversibilities and transport mechanisms. The parametric results reveal that increasing the fractional-order parameter and nanoparticles volume fraction enhances thermal performance but also augment entropy generation , while stronger magnetic fields and wall permeability significantly alter momentum transport and entropy distribution. These findings provide valuable guidelines for entropy management in nanofluid-based thermal–fluid systems with applications in energy conversion, microfluidics, and advanced cooling technologies.
Modeling liquid-vapor fronts in porous media using time-fractional derivatives: An innovative framework Zafar Hayat Khan, Meijiang Zhou, Alexander Trounev, Waqar Ahmed Khan Physics of Fluids, 2025 This study presents an innovative framework for modeling the liquid–vapor phase change interface in porous media by employing time-fractional derivatives. Traditional heat transfer models often rely on integer-order derivatives, assuming local and instantaneous diffusion processes, which fail to fully capture the memory effects and nonlocal dynamics inherent in many real-world phase transition processes. To address this limitation, we incorporate time-fractional derivatives into the energy balance for both the liquid and vapor phases and the dynamics of the phase-change front. Using the Caputo fractional derivative, we model the nonlocal temporal behavior, offering a more accurate and comprehensive representation of heat transfer and phase transition dynamics. The study focuses on the time-fractional dynamics of liquid–vapor front in porous media in a geothermal context, but the methodological approach is broadly applicable to systems exhibiting anomalous diffusion and memory effects, particularly those involving phase transitions. Numerical solutions are computed using a finite difference method with fourth-order differentiation matrices, ensuring high accuracy and stability. Simulations reveal that increasing the fractional order parameter α slows the phase-change front, indicating sub-diffusive behavior characteristic of porous structures. Governing parameters such as heat generation, heat absorption, density ratio, porosity, temperature contrast, and fractional order parameter are analyzed, demonstrating combined impact on heat transfer and liquid–vapor front dynamics. These findings provide critical insights for optimizing energy extraction and environmental engineering applications, offering a fresh perspective on phase transition modeling in complex systems.
Effect of variable thermal conductivity on heat transfer from a hollow sphere with heat generation using homotopy perturbation method 2008 Proceedings of the ASME Summer Heat Transfer Conference Ht 2008, 2009
Heat transfer from solids with variable thermal conductivity and uniform internal heat generation using homotopy perturbation method 2008 Proceedings of the ASME Summer Heat Transfer Conference Ht 2008, 2009
