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An evolution of a spherical region, subjected to uniform buoyancy force, is investigated. Incompressibility and axial symmetry are assumed, together with a buoyancy discontinuity at the boundary. The boundary turns into a vortex sheet and the system evolves into a ring. Contrary to the case of mechanically generated rings, buoyancy-driven rings are unstable. This is due to the generation of negative vorticity at the bottom. Furthermore, a sequence of Kelvin-Helmholtz instabilities arises along the buoyancy anomaly boundary. This sequence transfers the energy toward large scales with $\kappa^{-3}$ distribution. The vortex blob method has been used to simulate the system numerically. An optimization algorithm, used previously in two dimensions, has been extended to the axisymmetric case. It reduces computational complexity from $N^2$ to $N \log N$, where N is the number of nodes. Additionally, a new algorithm has been developed as a remedy for the exponential growth of the number of nodes required. It exploits a tendency of the vortex sheet to form many parallel stripes, by merging them together., Comment: 66 pages, 38 figures; affiliations corrected

This paper presents the first results of the two-phase flow simulation obtained using recently introduced physical, mathematical and numerical model of the intermittency region between two-phases (Wac{\l}awczyk 2017, 2021). The statistical interpretation of the intermittency region evolution equations allows to account for the non-equilibrium effects in the domain separating two phases. The source of non-equilibrium are spatial variations in the ratio of work done by volume and interfacial forces governing its width. As the statistical description of the two-phase flow differs from the deterministic two-phase flow models known in the literature, in the present work we focus discussion of the results on these differences. To this goal, the rising two dimensional gas bubble is studied; differences between equilibrium and non-equilibrium solutions are investigated. It is argued the statistical description of the intermittency region has potential to account for physical phenomena not considered previously in the computer simulations of two-phase flow., Comment: This is the manuscript of the paper recently submitted to Acta Mechanica Sinica

This paper concerns modeling of the evolution of intermittency region between two weakly miscible phases due to temporal and spatial variations of its characteristic length scale. First, the need of a more general description allowing for the evolution of intermittency region is rationalized. Afterwards, results of the previous work (Wac{\l}awczyk T., 2017, On a relation between the volume of fluid, level-set and phase field interface models, Int. J. Multiphas. Flow, Vol. 97) are discussed in context of the sharp interface models known in the literature and insight into droplet coalescence mechanism recently recognized in the molecular dynamics studies (Perumanath S., Borg M.K., Chubynsky M.V., Sprittles J.E., Reese J.M., 2019, Droplet coalescence is initiated by thermal motion, Phys. Rev. Lett., Vol. 122). Finally, the physical and numerical models extending applicability of the equilibrium solution to the case when intermittency region could also be in the non-equilibrium state is introduced and verified in several test cases., Comment: The paper accepted for publication to International Journal of Multiphase Flow

This paper discusses a relation between the re-initialization equation of the level-set functions derived by Wac{\l}awczyk [J.Comp.Phys., 299, (2015)] and the condition for the phase equilibrium provided by the stationary solution to the modified Allen-Cahn equation [Acta Metall., 27, (1979)]. As a consequence, the statistical model of the non-flat interface in the state of phase equilibrium is postulated. This new physical model of the non-flat interface is introduced based on the statistical picture of the sharp interface disturbed by the field of stochastic forces, it yields the relation between the sharp and diffusive interface models. Furthermore, the new techniques required for the accurate solution of the model equations are proposed. First it is shown, the constrained interpolation improves re-initialization of the level-set functions as it avoids oscillatory numerical errors typical for the second-order accurate interpolation schemes. Next, the new semi-analytical, second order accurate Lagrangian scheme is put forward to integrate the advection equation in time avoiding interface curvature oscillations introduced by the second-order accurate flux limiters. These techniques provide means to obtain complete, second-order convergence during advection and re-initialization of the interface in the state of phase equilibrium., Comment: This is the preprint of the paper accepted for publication in the International Journal of Multiphase Flow

The aim of this paper is to verify the new numerical implementation of a binary fluid, heat conduction dominated solidification model. First, we extend a semi-analytical solution to the heat diffusion equation, next, the range of its applicability is investigated. It was found that the linearization introduced to the heat diffusion equation negatively affects the ability to predict solidus and liquidus lines positions whenever the magnitude of latent heat of fusion exceeds a certain value. Next, a binary fluid solidification model is coupled with a flow solver, and is used in a numerical study of Al-4.1%Cu alloy solidification in a two-dimensional rectangular cavity. An accurate coupling between the solidification model and the flow solver is crucial for the correct forecast of solidification front positions and macrosegregation patterns., Comment: the paper is in the review in Chemical and Process Engineering

In this paper, a new re-initialization method for the conservative level-set function is put forward. First, it has been shown that the re-initialization and advection equations of the conservative level-set function are mathematically equivalent to the re-initialization and advection equations of the localized signed distance function. Next, a new discretization for the spatial derivatives of the conservative level-set function has been proposed. This new discretization is consistent with the re-initialization procedure and it guarantees a second-order convergence rate of the interface curvature on gradually refined grids. The new re-initialization method does not introduce artificial deformations to stationary and non-stationary interfaces, even when the number of re-initialization steps is large., Comment: the paper is currently under review after the first revision in the Journal of Computational Physics (72 pages)

In the present paper the classical point symmetry analysis is extended from partial differential to functional differential equations with functional derivatives. In order to perform the group analysis and deal with the functional derivatives we extend the quantities such as infinitesimal transformations, prolongations and invariant solutions. For the sake of example the procedure is applied to the continuum limit of the heat equation. The method can further lead to significant applications in statistical physics and fluid dynamics., Comment: 24 pages