This study investigates the unsteady flow and heat transfer of copper-alumina/water hybrid nanofluid over a permeable rotating disk with internal heat generation. The governing time-dependent partial differential equations are reduced to nonlinear ordinary differential equations using similarity transformations and solved numerically with the Matlab bvp4c boundary-value solver. Three similarity-solution branches are obtained over the investigated decelerating range. Suction alters the radial and tangential wall shear stresses and increases the local Nusselt number for the first and second branches, whereas the third branch exhibits an opposite response and a negative local Nusselt number. Increasing the heat generation parameter raises the temperature and thickens the thermal boundary layer. These branch-dependent responses extend the isothermal unsteady rotating-disk model by quantifying how internal heat generation modifies thermal transport in the triple-solution regime.
A comprehensive computational framework is established to examine the magnetohydrodynamic (MHD) boundary-layer characteristics of three viscoelastic non-Newtonian fluid models - Maxwell, Williamson, and Casson over an exponentially extending surface situated within a Darcy-Forchheimer porous medium. The formulation incorporates Joule heating, viscous dissipation, radiative heat flux, Brownian and thermophoretic diffusion, internal heat generation/absorption, and a uniform first-order chemical reaction. The governing coupled nonlinear partial differential equations are transformed into a high-order system of ordinary differential equations by similarity transformations and are solved numerically using the Lobatto IIIA-based bvp4c method. The influences of the governing physical parameters on the velocity, temperature, nanoparticle concentration, skin-friction coefficient, local Nusselt number, and local Sherwood number are analyzed and discussed in detail. The results reveal that increasing the magnetic parameter and porous medium resistance suppresses the fluid velocity, whereas thermal radiation, Joule heating, viscous dissipation, and heat generation significantly enhance the temperature distribution. Brownian motion increases the thermal boundary-layer thickness while reducing the nanoparticle concentration, whereas thermophoresis enhances both the temperature and concentration fields. Furthermore, the comparative analysis demonstrates distinct rheological responses of Maxwell, Williamson, and Casson fluids under identical physical conditions, providing valuable insights for the design and optimization of industrial processes involving non-Newtonian nanofluids, porous media, and magnetohydrodynamic transport phenomena.
Transverse jets in cross flows are found to be quite common in various engineering applications, such as film cooling in gas-turbine, dilution of combustion, fuel injection, ventilation, environmental discharge, and mixing applications. This review highlights the effect of the interaction among jet trajectory, coherent vortices, and scalar transport on heat/mass transfer performance in steady and pulsatile cross flows. Momentum flux ratio is one basic criterion for the interaction of the jet with the cross flow, while the flow dynamics is further affected by the geometrical configuration, boundary-layer features, density ratio, Reynolds number, and jet exit conditions. The shear-layer, horseshoe, and wake vortices affect the flow dynamics near the nozzle exit region, while the counter-rotating vortex pairs control the plume deformation and scalar transport in the far field. The pulsation adds some more factors such as frequency, amplitude, waveform, and phase. Velocity, concentration, vortex location, and phase-resolved responses are then used to compare diagnostic and computational approaches instead of just trajectories. For reliable comparisons, identical steady and pulsating flow conditions, definitions, uncertainties, and benchmark data are needed. Performance should be evaluated according to the application, since penetration useful for combustion or dilution may be inefficient for film cooling.
The magnetohydrodynamic (MHD) stagnation-point flow of a micropolar nanofluid over a linearly stretching sheet is investigated numerically to analyze the effects of a magnetic field. Although several numerical studies exist on MHD nanofluid and micropolar fluid flows, a detailed comparative assessment of advanced spectral techniques, particularly the spectral relaxation method (SRM) and the spectral local linearization method (SLLM), for this class of coupled nonlinear problems remains limited. Moreover, the combined effects of parameters like Brownian motion, thermophoresis, and micropolar forces on heat and mass transfer characteristics have not been thoroughly explored. The governing nonlinear partial differential equations are transformed into a system of coupled nonlinear ordinary differential equations using an appropriate similarity transformation. A comprehensive numerical analysis is conducted to examine the effects of key dimensionless parameters on the velocity, angular velocity, temperature, and concentration profiles, as well as on the Nusselt and Sherwood numbers. Results obtained using SRM and SLLM are presented graphically and in tabular form and compared with results from the existing literature. Excellent agreement is observed, validating the accuracy and reliability of both numerical methods. The results reveal that increasing the Brownian motion parameter (Nb) enhances fluid temperature due to improved thermal transport, whereas higher thermophoresis (Nt) and Prandtl number (Pr) reduce the temperature, highlighting their significant role in controlling heat and mass transfer in MHD micropolar nanofluid flows.
To support the optimized control for the modern drying system, this paper presents a physics-informed deep learning approach for simulating coupled heat and mass transfer during the drying of wood particles, a critical unit operation in modern biomass energy processing. Firstly, a physics-based reference model was developed using Fick’s second law with moisture-dependent effective diffusivity to capture internal evaporation kinetics. Afterward, to overcome the computational bottleneck of standard partial differential equation (PDE) solvers, a Physics-Informed Neural Operator (PINO) framework was evaluated against the physical benchmark. Despite significant temperature PDE losses due to strong nonlinearities in surface heat balances and coupled thermal dynamics compared to moisture transport, the PINO model exhibits excellent predictive agreement with the reference model across parameter space, with the deviation in drying time lower than 5%. It physically resolves both fundamental drying periods: the constant-rate drying period governed by the adiabatic saturation temperature and the falling-rate period following surface moisture depletion. The proposed PINO surrogate offers a fast, reliable tool for real-time monitoring and optimization of wood drying systems.
