Wakes and the dynamic interactions of multiple wakes have been a focal point of numerous research endeavours. Traditionally, wake interaction studies have focused on wakes produced by similar bodies. In contrast, the present study positions a non-shedding porous disc adjacent to periodically shedding solid discs of varying diameters and dimensional shedding frequencies. Using hot-wire measurements, we explore the intriguing interaction between these wakes. Remarkably, our findings reveal that the wake of the non-shedding disc acquires oscillations from the wake of the shedding disc, irrespective of their distinct frequencies. We demonstrate high receptivity of the porous disc’s wake and connect our findings to real-life applications.

The inertial migration of neutrally buoyant spherical particles in viscoelastic fluids flowing through square channels is experimentally and numerically studied. In the experiments, using dilute aqueous solutions of polymers with various concentrations that have nearly constant viscosities, we measured the distribution of suspended particles in downstream cross-sections for the Reynolds number ($\textit{Re}$) up to 100 and the elasticity number ($El$) up to 0.07. There are several focusing patterns of the particles, such as four-point focusing near the centre of the channel faces on the midlines for low $\textit{El}$ and/or high $\textit{Re}$, four-point focusing on the diagonals for medium $\textit{El}$, single-point focusing at the channel centre for relatively high $\textit{El}$ and low $\textit{Re}$, and five-point focusing near the four corners and the channel centre for high $\textit{El}$ and very low $\textit{Re}$. Among these focusing patterns, various types of particle distributions suggesting the presence of a new equilibrium position located between the midline and the diagonal, and multistable states of different equilibrium positions were observed. In general, as $\textit{El}$ increases from 0 at a constant $\textit{Re}$, the particle focusing positions shift from the midline to the diagonal in the azimuthal direction first, and then inward in the radial direction to the channel centre. These focusing patterns and their transitions were numerically well reproduced based on a FENE-P model with measured values of viscosity and relaxation time. Using the numerical results, the experimentally observed focusing patterns of particles are elucidated in terms of the fluid elasticity-induced lift and the wall-induced elastic lift.

A partial differential equation governing the evolution of the joint probability distribution of multicomponent flow observations, drawn randomly from one or more control volumes, is derived and applied to examples involving irreversible mixing. Unlike local probability density methods, this work adopts an integral perspective by regarding a control volume as a sample space with an associated probability distribution. A natural and general definition for the boundary of such control volumes comes from the magnitude of the gradient of the sample space distribution, which can accommodate Eulerian or Lagrangian frames of reference as particular cases. The formulation exposes contributions made by uncertain or stochastic boundary fluxes and internal cross-gradient mixing in the equation governing the observables’ joint probability distribution. Advection and diffusion over a control volume’s boundary result in source and drift terms, respectively, whereas internal mixing, in general, corresponds to the sign-indefinite diffusion of probability density. Several typical circumstances for which the corresponding diffusion coefficient is negative semidefinite are identified and discussed in detail. The framework is a natural setting for examining available potential energy, the incorporation of uncertainty into bulk models, and establishing a link with the Feynman–Kac formula and Kolmogorov equations that are used to analyse stochastic processes.

This study is devoted to the analysis of capillary oscillations of a gas bubble in a liquid with an insoluble surfactant adsorbed on the surface. The influence of the Gibbs elasticity, the viscosities of the liquid and gas, as well as the shear and dilatational surface viscosities, on the damping of free oscillations is examined. Dependences of the frequency shift and the damping rate on the parameters of the problem are determined. In the limit of small viscosities and neglecting the surfactant surface diffusion, a simplified dispersion relation is obtained, which includes finite parameters of surface viscosities and Gibbs elasticity. From this relation, conditions are identified under which the damping of capillary oscillations can occur with a small frequency. Numerical solutions of the full dispersion relation demonstrate that a non-oscillatory regime is impossible for the considered configuration. An additional mode associated with Gibbs elasticity is discovered, characterized as a rule by low natural frequency and damping rate. Approximate relations for the complex natural frequency of bubble oscillations in a low-viscosity liquid in the presence of a surfactant are derived, including an estimate of the contribution of the gas inside the bubble to viscous dissipation. An original Lagrangian–Eulerian method is proposed and used to perform direct numerical simulations based on the full nonlinear Navier–Stokes equations and natural boundary conditions at the interface, accounting for shear and dilatational viscosities. The numerical data on the damping process confirm the results of the linear theory.