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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.

This work presents an analytical method for determining vertical dynamic impedance and displacement response factor of a rigid cylindrical foundation embedded in unsaturated poroelastic soils. The foundation is assumed to be perfectly boned to its surrounding soil and its overlying half-space of unsaturated poroelastic soil, subjected to harmonic vertical loadings. The soil surrounding the circumference of the cylinder is modeled as a number of infinitely thin horizontal soil layers. Based on the Biot-type soil constitutive model, the equations governing the interaction of unsaturated soils with the cylindrical foundation are derived. Solutions are obtained by solving ordinary differential equations transformed from partial differential governing equations using the Hankel transform. The proposed solutions are verified against existing solutions of benchmark elastodynamic problems for embedded cylindrical foundations in dry and saturated soils. Using the derived solutions, several influencing parameters defining the stiffness and mass of the foundation system are examined to investigate the dynamics of the foundation interacting with it adjacent soils. It is concluded that the dynamic displacement response factor is sensitive to soil saturation. It is believed that the proposed solution should be beneficial to dynamic design with cylindrical foundations embedded in unsaturated soils.