Abstract:
In 2006, the fourth author proposed a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle that can be either red or blue. New red and blue particles alternately get emitted from their respective bases and perform random walk. On encountering a particle of the opposite color they kill it and occupy its position. We prove that on the cylinder graph (the product of a path and a cycle) an interface spontaneously forms between red and blue and is maintained in a predictable position with high probability.

Abstract:
Particles added to a fluid interface can be used as a surface stabilizer in the food, oil and cosmetic industries. As an alternative to rigid particles, it is promising to consider highly deformable particles that can adapt their conformation at the interface. In this study we compute the shapes of soft elastic particles using molecular dynamics simulations of a cross-linked polymer gel, complemented by continuum calculations based on linear elasticity. It is shown that the particle shape is not only affected by the Young's modulus of the particle, but also strongly depends on whether the gel is partially or completely wetting the fluid interface. We find that the molecular simulations for the partially wetting case are very accurately described by the continuum theory. By contrast, when the gel is completely wetting the fluid interface the linear theory breaks down and we reveal that molecular details have a strong influence on the equilibrium shape.

Abstract:
It is shown that dynamics of the interface between ideal fluid and light viscous fluid is exactly integrable in the approximation of small surface slopes for two-dimensional flow. Stokes flow of viscous fluid provides a relation between normal velocity and pressure at interface. Surface elevation and velocity potential of ideal fluid are determined from two complex Burgers equations corresponding to analytical continuation of velocity potential at the interface into upper and lower complex half planes, respectively. The interface loses its smoothness if complex singularities (poles) reach the interface.

Abstract:
The `no-slip' is a fundamental assumption and generally-accepted boundary condition in rheology, tribology and fluid mechanics with strong experimental support. The violations of this condition, however, are widely recognized in many situations, especially in the flow of non-Newtonian fluids. Wall slip could lead to large errors and flow instabilities, such as sharkskin formation and spurt flow, and hence complicates the analysis of fluid systems and introduces serious practical difficulties. In this article, we discuss slip at fluid-solid interface in an attempt to highlight the main issues related to this diverse complex phenomenon and its implications.

Abstract:
In magnetic fluid dynamics appears the problem of reconstruction of free boundary between conducting fluids, e.g. in aluminum electrolysis cells. We have investigated how the interface between two fluids of different conductivity of a highly simplified model of an aluminum electrolysis cell could be reconstructed by means of external magnetic field measurements using simple genetic algorithm.

Abstract:
We present modeling of the incompressible viscous flows in the domain containing an unconfined fluid and a porous medium. For such setting a rigorous derivation of the Beavers-Joseph-Saffman interface condition was undertaken by J\"ager and Mikeli\'c [SIAM J. Appl. Math. \rm 60 (2000), p. 1111-1127] using the homogenization method. So far the interface law for the pressure was conceived and confirmed only numerically. In this article we justify rigorously the pressure jump condition using the corresponding boundary layer.

Abstract:
We extend the volume of fluid method for the computation of two-phase flow to a higher order accurate method in two dimensions. The interface reconstruction by the PLIC method is thereby replaced by a periodic interface reconstruction. The advection step is reformulated and extended to higher order in order to account for the present interface representation. This periodic interface reconstruction describes the interface in terms of higher order periodic B-splines. Numerical tests verify that the theoretical order of convergence is indeed exhibited by the present method.

Abstract:
We study the boundary conditions at a fluid-solid interface using molecular dynamics simulations covering a broad range of fluid-solid interactions and fluid densities, and both simple and chain-molecule fluids. The slip length is shown to be independent of the type of flow, but rather is related to the fluid organization near the solid, as governed by the fluid-solid molecular interactions.

Abstract:
Recently, it was shown that the extreme Kerr black hole is the only candidate for a (Kerr) black hole limit of stationary and axisymmetric, uniformly rotating perfect fluid bodies with a zero temperature equation of state. In this paper, necessary and sufficient conditions for reaching the black hole limit are presented.

Abstract:
Erosion occurring in equipment dealing with liquid-solid mixtures such as pipeline parts, slurry pumps, liquid-solid stirred reactors and slurry mixers in various industrial applications results in operational failure and economic costs. A slurry erosion tank test rig is designed and was built to investigate the erosion rates of materials and the influencing parameters such as flow velocity and turbulence, flow angle, solid particle concentration, particles size distribution, hardness and target material properties on the material loss and erosion profiles. In the present study, a computational fluid dynamics (CFD) tool is used to simulate the erosion rate of sample plates in the liquid-solid slurry mixture in a cylindrical tank. The predictions were made in a steady state and also transient manner, applying the flow at the room temperature and using water and sand as liquid and solid phases, respectively. The multiple reference frame method (MRF) is applied to simulate the flow behavior and liquid-solid interactions in the slurry tank test rig. The MRF method is used since it is less demanding than sliding mesh method (SM) and gives satisfactory results. The computational domain is divided into three regions: a rotational or MRF zone containing the mixer, a rotational zone (MRF) containing the erosion plates and a static zone (outer liquid zone). It is observed that changing the MRF zone diameter and height causes a very low impact on the results. The simulated results were obtained for two kinds of hard metals namely stainless steel and ST-50 under some various operating conditions and are found in good agreement with the experimental results.