This study investigates the dynamics of two oscillating rigid spheres moving through an infinite porous medium saturated with Stokes fluid flow, addressing the problem of how fluid properties, permeability, frequency, and slip length influence the system. The objective is to model the interactions between the spheres, which differ in size and velocity as they move along the axis connecting their centers while applying slip boundary conditions to their surfaces. We derive the governing field equations using a semi-analytical method and solve the resulting system of equations numerically through a collocation technique. Our novel quantitative results include insights into the drag force coefficients for both in-phase and out-of-phase oscillations of each hydrophobic sphere, considering parameters such as diameter ratio, permeability, frequency, velocity ratios, slip lengths, and the distances between the spheres. Notably, when the spheres are sufficiently far apart, the normalized drag force coefficients behave as if each sphere is moving independently. Additionally, we present streamlines that illustrate the interactions between the spheres across a range of parameters, highlighting the novelty of our findings. A purely viscous medium and no-slip conditions are used to validate the numerical approach and results.

This study investigates the intricate dynamics of synchronized and desynchronized oscillations of rigid spheres with varying diameters immersed in magnetorheological ferrofluids. It delineates two distinct motion types: synchronized oscillation, where spheres resonate in unison due to magnetic forces, and desynchronized oscillation, characterized by out-of-phase movements stemming from disparate frequencies and hydrodynamic interactions. Employing Stokes flow theory, the research formulates a spherical solution and applies boundary conditions through a sophisticated collocation technique. Noteworthy findings indicate that drag force coefficients rapidly converge to definitive parameters, including the ratio of sphere diameters, the Hartmann number, frequency parameters, and slippage conditions, signifying independent motion at sufficient inter-sphere separations. Furthermore, the study rigorously examines the influence of variations in fluid properties and magnetic fields on sphere interactions, while also evaluating the precision of the numerical methodologies employed. The implications of this research are profound, offering significant insights for applications necessitating meticulous control over particle dynamics.

Planar Kolmogorov flow (a pure sinusoidal velocity field restricted to one direction) has various applications in geophysics, astrophysics, and stability theory. This flow can generate large-scale instabilities, which are similar to the jet formation found in many models of fluid flows with a random forcing (known as zonostrophic instability). An important extension of the Kolmogorov flow is the inclusion of a background vorticity gradient at an angle to the flow (known as the β-effect). In this article, we consider the Kolmogorov flow influenced by a β-effect and coupled with a horizontal magnetic field, finding initial suppression of instability in the weak field branch and appearance of a range of instabilities and jet-like structures in the strong field branch. For weak fields, there is suppression of the pure hydrodynamic instability, but another instability branch appears as the field strength is increased. Alfvén waves interact with Rossby waves to generate magnetohydrodynamic Rossby waves, which have no hydrodynamic analogue. For high magnetic field strengths, these waves return to being Alfvén waves. Numerical results were found to compare well with results obtained using matrix eigenvalue perturbation theory. Although the flow we study is laminar, our results are relevant for a better understanding of instabilities for a wide class of turbulent magnetohydrodynamic flows.