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This work is motivated by the importance of vertical vibration in nanofluids suspension, because of its wide applications in heat exchangers from one fluid to another, controlling the mixing in microvolumes, pharmaceutical industry, and engineering. The current study analyzes the effects of high-frequency vertical vibration on a horizontal Casson nanofluid layer heated from below saturated between rigid-rigid, rigid-free, and free-free boundary conditions. Field equations are derived by using time-average method, which describes the vibrational thermo-convection. These field equations are solved by utilizing linear stability analysis and Galerkin technique to obtain the thermal stability curve. By using this curve, the impacts of vertical vibration and physical governing factors on temperature profile are discussed. High-frequency vertical vibration is observed to have a stabilizing effect on Casson nanofluid suspension, thereby reducing thermo-convective heat transfer from one fluid to another. Stability threshold of vibrated suspension is increased approximately as $5.748\%$ and $81.937\%$ when $R_v\to 50$ and $R_v\to 100$, respectively. The significance of Brownian motion and thermophoretic force on convective heat transfer with vertical vibration is discussed. The resisting nature of Casson nanofluid on vibrational convective heat transfer is also tabulated.

A computational investigation is furnished to explore the responses of a Darcy–Forchheimer EMHD Williamson flow of a Sodium Alginate $({\mathrm{\text{C}}}_6{\mathrm{\text{H}}}_9{\mathrm{\text{NaO}}}_7)$-based Ag-Al₂O₃ hybrid nanofluid passing over a vertically exponentially stretching cylinder emerged through a porous region. The prime focus of this research is to encompass the inclusion of nonlinear variations in heat distribution, Newtonian boundary heating (NBH) effects, and the influence of thermo-convection alongside suction effects. Key parameters, including thermal buoyancy, Darcy porous medium effects, heat source/sink effects, Biot number, variable thermal index, and thermal convection factor, are comprehensively analyzed as these combining factors can play a crucial role in optimizing the efficacy of several systems such as heat exchanger, material processing and geothermal system that involve the concept of thermo-transportation mechanism. The physical flow dynamics are modeled, employing suitable similarity transformations, and subsequently translated into a dimensionless form. The ensuing collection of modified nonlinear ordinary differential equations is solved by employing the Bvp4c solver bundled into the MATLAB program. Several parameters are scrutinized through graphical presentations to elucidate their impacts on the velocity curve, temperature curve, skin friction coefficient, and Nusselt index. It is worth mentioning that the heat distribution profile significantly escalated for the rising values of several factors such as electric field parameter, varying thermal index, Biot number and shape factor, but the reverse is the pattern with suction and thermo-convection effect. Also, the thermal transportation rate at the proximity of the vertical cylindrical wall appears to exhibit an increment of about an average of 47% in SA-based Williamson hybrid nanofluid compared to Williamson fluid for thermo-convective effect, NBH, and thermal buoyancy. Furthermore, the proximate shear stress rate appears to be amplified by approximately 39% in Williamson hybrid nanofluid when contrasted with Williamson fluid for electric field parameter and thermo-convection effect alongside the raised Darcy–Forchheimer factor.