This study aims to investigate the effect of a constant magnetic field on heat transfer, flow of fluid, and entropy generation of mixed convection in a lid-driven porous medium enclosure filled with nanofluids (TiO₂-water). Uniform constant heat fluxes are partially applied to the bottom wall of the enclosure, while the remaining parts of the bottom wall are considered to be adiabatic. The vertical walls are maintained at a constant cold temperature and move with a fixed velocity. A sinusoidal wall is assumed to be fixed and kept adiabatic at the top enclosure. Three scenarios are considered corresponding to different directions of the moving isothermal vertical wall (±1). The influence of pertinent parameters on the heat transfer, flow of fluid, and entropy generation in an enclosure are deliberated. The parameters are the Richardson number (${\tilde{R}}_i$ = 1, 10, and 100), the Hartmann number (0 ≤ $\tilde{H}$a ≤ 75 with a 25 step), and the solid volume fraction of nanoparticles (0 ≤ $\tilde{\Phi }$ ≤ 0.15 with a 0.05 step). The Grashof and Darcy numbers are assumed to be constant at 10⁴ and 10⁻³, respectively. The finite element method, utilizing the variational formulation/weak form, is applied to discretize the main governor equations. Triangular elements have been employed within the studied envelope, with the elements adapting as needed. The results showed that the streamfunction and fluid temperature decreased as the solid volume fraction increased. The local $\tilde{\mathrm{N}}\mathrm{u}$ number increased by more than 50% at low values of $\tilde{\mathsf{\Phi }}$ (up to 0.1). This percentage decreases between 25% and 40% when $\tilde{\mathsf{\Phi }}$ is in the range of 0.1 to 0.15. As $\tilde{\mathrm{H}}$a increases from 0 to 75, these percentages increase at low values of the value of $\tilde{\mathrm{R}}\mathrm{i}=1$ and 10. These variations are primarily dependent on the value of the Richardson number.

Complex fluids display flow heterogeneities and a variety of nonlinearities under large amplitude oscillatory shear (LAOS). LAOS has developed as a means of characterizing nonlinear fluid rheology. In this study, the spatial velocity distribution during LAOS is measured by rheo-nuclear magnetic resonance (Rheo-NMR) velocimetry. Rheo-NMR is frequently used to measure responses of complex fluids to steady shear deformations by imaging local velocity across a fluid gap noninvasively. Rheo-NMR analysis of oscillatory flow has been more limited due to hardware limitations. Recently developed Rheo-NMR equipment makes LAOS Rheo-NMR possible. LAOS Rheo-NMR provides measurement of spatial velocity profiles as a function of oscillatory period, providing time dependent spatially resolved local velocity time series, which show distinct features dependent on material response.

In this study, a two-dimensional numerical method is employed to discretize the governing equations for heat transfer and mixed convection inside a square cavity with a sinusoidal bottom wall. The numerical method is based on the variational formulation/finite element with triangular nonsymmetrical elements. The vertical walls of the cavity are kept at a constant cold temperature, while the constant temperature heat source is applied to the sinusoidal bottom wall. The top wall and the rotating cylinder inside the cavity are considered insulated. The dimensionless angular velocities of an insulated rotating solid cylinder that have been considered are 0, 3000, and 6000 clockwise/counterclockwise. The working fluid is Al2O3 nanofluids with porous media at varying ratios. This analysis examined various ratios of nanofluids/porous media (nanoporous) volumes to the total enclosure volume, ranging from 0% to 100%, with a 25% increment. The effect of nanoporous media volume on the main pertinent parameters (velocity vector, heatlines, isotherms, dimensionless entropy, Bejan number, streamlines, and Nusselt numbers) is investigated at various solid cylinders’ rotational angular velocities. A constant Rayleigh number (104), Darcy number (10-3), and nanopartical solid volume fraction (0.02) are used. The results of the present work have been validated against several published works, to test the accuracy of the present analysis. In the present analysis, the data is plotted for different main parameters. The findings of this work showed that the average Nusselt number values on the hot wall increase as angular velocity increases, regardless of the rotational direction. Additionally, increasing the ratio of porous media/enclosure volume above 75% does not significantly impact the Nusselt number.