This article surveys the formulation and discretization of inertial terms for incompressible two-phase flows with large viscosity and density ratios. The covered topics include progress in the discretization of the momentum equation, with a particular focus on pressure-velocity coupling, together with an algebraic momentum-preserving scheme. The implementation and exploration of approximate momentum conservation based on the integration of auxiliary density equations on a staggered mesh are also discussed. The new algebraic scheme is applied to challenging interfacial flows, such as droplet impacts at high density and viscosity ratios, demonstrating accuracy and stability.

We present a robust, accurate, and computationally efficient numerical strategy for modeling species transfer across immiscible fluid interfaces on Cartesian meshes. The method provides an efficient alternative to the classical one-fluid approach while retaining its algorithmic simplicity and limited computational overhead. The proposed framework relies on a two-phase formulation in cut cells, combining dedicated phase reconstructions with a simple and robust mass-flux approximation to accurately enforce interfacial jump conditions.The method is first assessed against analytical solutions for pure diffusion in both static and moving configurations. Compared with the standard one-fluid model, we demonstrate that appropriate phase reconstructions for the advection and the temporal scheme significantly improve the accuracy of the concentration field, including mean mass fluxes and interfacial concentrations. The approach is then validated on fully coupled three-dimensional simulations of mass transfer on rising bubbles. The computed Sherwood numbers are compared with established numerical results from the literature. While the Sherwood number is shown to be an unreliable indicator of concentration-field accuracy, the proposed schemes provide accurate and robust mass transfer predictions over a wide range of grid resolutions. Overall, the method achieves a substantial gain in accuracy for interfacial mass transfer at a computational cost that is comparable to the classical one-fluid formulations.

We present a space-time extension of a conservative Cartesian cut-cell finite-volume method for two-phase diffusion problems with prescribed interface motion. The formulation follows a two-fluid approach: one scalar field is solved in each phase with discontinuous material properties, coupled by sharp interface conditions enforcing flux continuity and jump laws. To handle moving boundaries on a fixed Cartesian grid, the discrete balance is written over phase-restricted space-time control volumes, whose geometric moments (swept volumes and apertures) are used as weights in the finite-volume operators. This construction naturally accounts for the creation and destruction of cut cells (fresh/dead-cell events) and yields strict discrete conservation. The resulting scheme retains the algebraic structure of the static cut-cell formulation while incorporating motion through local geometric weights and interface coupling operators. A series of verification and validation tests in two and three dimensions demonstrate super-linear accuracy in space, robust behavior under repeated topology changes and conservation across strong coefficient jumps and moving interfaces. The proposed space-time cut-cell framework provides a conservative building block for multiphase transport in evolving geometries and a foundation for future free-boundary extensions such as Stefan-type phase change.

We present a Cartesian cut-cell finite-volume method for sharp-interface two-phase diffusion problems in static geometries. The formulation follows a two-fluid approach: independent diffusion equations are discretized in each phase on a fixed staggered Cartesian grid, while the phases are coupled through embedded interface conditions enforcing continuity of normal flux and a general jump law. Cut cells are treated by integrating the governing equations over phase-restricted control volumes and faces, yielding discrete divergence and gradient operators that are locally conservative within each phase. Interface coupling is achieved by introducing a small set of interfacial unknowns per cut cell on the embedded boundary; the resulting algebraic system involves only bulk and interfacial averages. A key feature of the method is the use of a reduced set of geometric information based solely on low-order moments (trimmed volumes, apertures and interface measures/centroids), allowing robust implementation without constructing explicitly cut-cell polytopes. The method supports steady (Poisson) and unsteady (diffusion) regimes and incorporates Dirichlet, Neumann, Robin boundary conditions and general jumps. We validate the scheme on one-, two- and three-dimensional mono- and diphasic benchmarks, including curved embedded boundaries, Robin conditions and strong property/jump contrasts. The results demonstrate the expected convergence behavior, sharp enforcement of interfacial laws and excellent conservation properties. Extensions to moving interfaces and Stefan-type free-boundary problems are natural perspectives of this framework.

In this work, three classes of numerical methods are investigated to evaluate the mean curvature, the unit normal vector and the surface tension on a front tracking interface encountered in the simulation of multiphase flows with separated phases. The Laplace-Beltrami Operator discretization, the Integral Formulation of the surface tension and the Surface Reconstruction technique are well-known methods used in the literature, but whose accuracy, robustness and convergence properties are seldom studied. In a first step, different variants of these methods are presented and compared against each other on a static analytical surface to measure their sensitivity to the size and regularity of the mesh. Then, to assess the influence of the surface advection scheme and the remeshing procedures, two original and time dependent analytical surfaces have been developed, leading to a ligament formation or the birth of a drop/bubble on a flat surface/liquid film. Comparisons to such dynamical surfaces are especially useful and significant, since they highlight the sensitivity of numerical methods for the interface property calculation to the errors produced by the Lagrangian transport of the front-tracking surface. Finally, the accuracy of the different approximations is correlated to their computation time, so that the user may choose the most appropriate method according to the desired accuracy and cost.

We show that, using the Crouzeix-Raviart scheme, a cheap algebraic transformation, applied to the coupled velocity-pressure linear systems issued from the transient or steady Stokes or Navier-Stokes problems, leads to a linear system only involving as many auxiliary variables as the velocity components. This linear system, which is symmetric positive definite in the case of the transient Stokes problem and symmetric invertible in the case of the steady Stokes problem, with the same stencil as that of the velocity matrix, provides the exact solution of the initial coupled linear system. Numerical results show the increase of performance when applying direct or iterative solvers to the resolution of these linear systems.

In micro-channel gas flows, velocity slip and temperature jump must be applied at the fluid/solid interface to take the thermodynamic non-equilibrium into account in the vicinity of the wall. In this work, simulations at molecular scale have been carried out and shown a jump in the conduction heat flux, in agreement with the boundary condition proposed by Maslen (1958) [1] in continuum mechanics.