Yuri Feldman

Volume of fluids (VOF) methods are attractive for two-phase flow simulations because they transport a conservative phase indicator without explicitly tracking the interface. Their accuracy, however, depends critically on the velocity field used for this transport. In high-density-ratio flows, sharp jumps in material properties and localized surface-tension forces make the incompressible pressure–velocity formulation stiff, and splitting errors or discrete divergence errors can appear directly as phase-volume drift, interface smearing, or spurious pressure oscillations. Thus, an essential difficulty of robust VOF simulation lies not only in the advection of the phase indicator, but in the efficient implicit solution of the coupled pressure–velocity system that governs the flow.We present a monolithic Vanka–VOF framework for incompressible twophase flows characterized by large density and viscosity variations. The method solves the momentum and continuity equations in a fully coupled pressure–velocity system using a global Big-Box Vanka relaxation strategy. This coupling enforces the discrete incompressibility constraint to machine precision and removes the pressure–velocity splitting errors typical of projection and segregated formulations. The resulting divergence-free velocity field is then used to advance the phase indicator by a conservative algebraic VOF method with bounded, high-resolution cubic upwind interpolation. The formulation is verified against two- and three-dimensional advection, Rayleigh–Taylor, shear-deformation, and rising-bubble benchmarks at density and viscosity ratios up to 103, while its applicability to strongly contrasted flows with topological change is demonstrated through a large three-dimensional bubble-coalescence simulation.

The current study introduces a novel fully coupled monolithic solver for the direct forcing immersed boundary method (IBM) in incompressible flows. The solver simultaneously integrates pressure, velocity, nonlinear convection terms, and Lagrangian forces into a unified framework, leveraging a modified big-box Vanka smoother extended with additional Lagrange multipliers arising from the IBM formulation. Central to the approach is the use of a Schur complement decomposition, which reduces the operator size by two-thirds while preserving both stability and accuracy. The solver’s monolithic structure eliminates splitting errors and artificial pressure boundary conditions, common drawbacks of segregated methods. Additionally, the developed methodology enables high CFL numbers (up to 0.5), making it particularly effective for moving boundary simulations. Verification studies cover a broad set of benchmark problems, including both stationary and moving immersed bodies across a wide range of Reynolds numbers. These tests confirm that the solver achieves computational times comparable to existing semi-implicit methods while enhancing accuracy and stability. By addressing key challenges in high-fidelity incompressible flow simulations, the proposed method offers a robust and broadly applicable monolithic solver.

The interaction between thin sheets and fluid droplets has recently attracted attention due to its frequent occurrence in natural systems and extensive use in technological applications. Motivated by these applications, this paper focuses on the interaction resulting from a drop that rests at the center of a circular superhydrophobic thin sheet. We investigate how the size and surface tension of the drop, together with the elastic and geometric parameters of the sheet, determine the overall deformation of the system. To this end, we formulate an analytical model based on energetic considerations that can predict the mutual interaction between the drop and the sheet. We show that the deformation of the sheet outside the contact region is almost unaffected by the exact shape of the drop, and thus mimics the problem of a rigid indentation. However, inside the contact region, there is a strong coupling between the drop's shape and the sheet's deformation. We derive analytical predictions for the size of the contact region and show that its shape comprises a constant curvature. These findings are in good quantitative agreement with the numerical minimization of the energy.

Snap-through is a buckling instability that allows slender objects, including those in plant and biological systems, to generate rapid motion that would be impossible if they were to use their internal forces exclusively. In microfluidic devices, such as micromechanical switches and pumps, this phenomenon has practical applications for manipulating fluids at small scales. The onset of this elastic instability often drives the surrounding fluid into motion – a process known as snap-induced flow. To analyse the complex dynamics resulting from the interaction between a sheet and a fluid, we develop a prototypical model of a thin sheet that is compressed between the two sides of a closed channel filled with an inviscid fluid. At first, the sheet bends towards the upstream direction and the system is at rest. However, once the pressure difference in the channel exceeds a critical value, the sheet snaps to the opposite side and drives the fluid dynamics. We formulate an analytical model that combines the elasticity of thin sheets with the hydrodynamics of inviscid fluids to explore how external pressure differences, material properties and geometric factors influence the system's behaviour. To analyse the early stages of the evolution, we perform a linear stability analysis and obtain the growth rate and the critical pressure difference for the onset of the instability. A weakly nonlinear analysis suggests that the system can exhibit a pressure spike in the vicinity of the inverted configuration.