We perform a quantitative assessment of different strategies to compute the contribution due to surface tension in incompressible two-phase flows using a conservative level set (CLS) method. More specifically, we compare classical approaches, such as the direct computation of the curvature from the level set or the Laplace-Beltrami operator, with an evolution equation for the mean curvature recently proposed in literature. We consider the test case of a static bubble, for which an exact solution for the pressure jump across the interface is available, and the test case of an oscillating bubble, showing pros and cons of the different approaches.
The Volume-of-Fluid (VoF) method for simulating incompressible two-phase flows is widespread in academic and commercial simulation software because of its many advantages: a high degree of volume conservation, applicability to unstructured domain discretization (relevant for engineering applications), straightforward parallel implementation with the domain-decomposition and message-passing approach (important for large-scale simulations), and intrinsic handling of strong deformations and topological changes of the fluid interface. However, stable and accurate handling of small-scale capillary …
In this paper we study the motion of an internal water wave and an internal wave in a porous medium. For these problems we establish that, if the free boundary and, in the case of the Euler equations, also the tangential velocity at the interface are sufficiently smooth, the depth cannot vanish in finite time. This results holds regardless of gravity and surface tension effects or, if applicable, the stratification in multiphase flows.
The linearized Navier-Stokes equations for a system of superposed immiscible compressible ideal fluids are analyzed. The results of the analysis reconcile the stabilizing and destabilizing effects of compressibility reported in the literature. It is shown that the growth rate $n$ obtained for an inviscid, compressible flow in an infinite domain is bounded by the growth rates obtained for the corresponding incompressible flows with uniform and exponentially varying density. As the equilibrium pressure at the interface $p_\infty$ increases (less compressible flow), $n$ increases towards the unif…
Viscous linear surface waves are studied at arbitrary wavelength, layer thickness, viscosity and surface tension. We find that in shallow enough fluids no surface waves can propagate. This layer thickness is determined for some fluids, water, glycerin and mercury. Even in any thicker fluid layers, propagation of very short and very long waves is forbidden. When wave propagation is possible, only a single propagating mode exists for a given horizontal wave number. In contrast, there are two types of non-propagating modes. One kind of them exists at all wavelength and material parameters, and th…
Statistical solutions, which are time-parameterized probability measures on spaces of square-integrable functions, have been established as a suitable framework for global solutions of incompressible Navier-Stokes equations (NSE). We compute numerical approximations of statistical solutions of NSE on two-dimensional domains with non-periodic boundary conditions and empirically investigate the convergence of these approximations and their observables. For the numerical solver, we use Monte Carlo sampling with an H(div)-FEM based deterministic solver. Our numerical experiments for high Reynolds …
In the present note we review some recent results for a class of singular perturbation problems for a Navier-Stokes-Korteweg system with Coriolis force. More precisely, we study the asymptotic behaviour of solutions when taking incompressible and fast rotation limits simultaneously, in a constant capillarity regime. Our main purpose here is to explain in detail the description of the phenomena we want to capture, and the mathematical derivation of the system of equations. Hence, a huge part of this work is devoted to physical considerations and mathematical modeling.
The viability of the Whitham equation as a nonlocal model for capillary-gravity waves at the surface of an inviscid incompressible fluid is under study. A nonlocal Hamiltonian system of model equations is derived using the Hamiltonian structure of the free surface water wave problem and the Dirichlet-Neumann operator. The system features gravitational and capillary effects, and when restricted to one-way propagation, the system reduces to the capillary Whitham equation. It is shown numerically that in various scaling regimes the Whitham equation gives a more accurate approximation of the free-…
A comprehensive scheme for the spatial discretisation of continuity equation, momentum advection and normal and shear stresses at the fluid interfaces is presented for numerically simulating the incompressible two phase flows based on the finite volume and volume of fluid methods. The scheme uses the mass conservation for the advection of the interface between the two fluids, physical considerations in determining the momentum flux at the surfaces of the finite volume so that it is consistent with that for the advection of the interfaces, and correctly accounts for the contributions to the nor…
We study immiscible two-phase flow of a compressible and an incompressible fluid inside a capillary tube of varying radius under steady-state conditions. The incompressible fluid is Newtonian and the compressible fluid is an inviscid ideal gas. The surface tension associated with the interfaces between the two fluids introduces capillary forces that vary along the tube due to the variation in the tube radius. The interplay between effects due to the capillary forces and the compressibility results in a set of properties that are different from incompressible two-phase flow. As the fluids move …
We consider the sloshing problem for an incompressible, inviscid, irrotational fluid in an open container, including effects due to surface tension on the free surface. We restrict ourselves to a constant contact angle and seek time-harmonic solutions of the linearized problem, which describes the time-evolution of the fluid due to a small initial disturbance of the surface at rest. As opposed to the zero surface tension case, where the problem reduces to a partial differential equation for the velocity potential, we obtain a coupled system for the velocity potential and the free surface displ…
The notion of tissue surface tension has provided a physical understanding of morphogenetic phenomena such as tissue spreading or cell sorting. The measurement of tissue surface tension so far relied on strong approximations on the geometric profile of a spherical droplet compressed between parallel plates. We solved the Laplace equation for this geometry and tested its solution on true liquids and embryonic tissue fragments as well as multicellular aggregates. The analytic solution provides the surface tension in terms of easily and accurately measurable geometric parameters. Experimental res…
The general pressure equation (GPE) is a new method proposed recently by Toutant (J. Comput. Phys., 374:822-842 (2018)) for incompressible flow simulation. It circumvents the Poisson equation for the pressure and performs better than the classical artificial compressibility method. Here it is generalized for two-phase incompressible viscous flows with variable density and viscosity. First, the pressure evolution equation is modified to account for the density variation. Second, customized discretizations are proposed to deal with the viscous stress terms with variable viscosity. Besides, addit…
The free surface lattice Boltzmann method (FSLBM) is a combination of the hydrodynamic lattice Boltzmann method (LBM) with a volume of fluid (VOF) interface apturing technique for the simulation of incompressible free surface flows. Capillary effects are modeled by extracting the curvature of the interface from the VOF indicator function and imposing a pressure jump at the free boundary. However, obtaining accurate curvature estimates from a VOF description can introduce significant errors. This article reports numerical results for three different surface tension models in standard test cases…
Stability of cylindrical interface between two ideal incompressible fluids, including the magnetic field, surface tension and gravitational field is studied in linear approximation. We found that helical waves arising both in plasma comet tails and on the vertical cylindrical water jet in the air are described by the same dispersion equation where the comet tail magnetic field plays the same stabilizing role as surface tension for water jet. Hence they represent the same phenomenon of Kelvin-Helmholtz instability. Thus helical waves in comet tails and astrophysical jets may be simulated in the…
From a columnar approximation of the Euler equations of an incompressible fluid with surface tension, we derive in the short-wave approximation a new integrable classical 1+1 dimensional field theory for the motion of the surface. Together with a Lorentz invariance,this system has the novel feature of solutions which become multiple valued in finite time.
We deploy linear stability analysis to find the threshold wavelength ($λ$) and surface tension ($γ$) of Rayleigh-Plateau type "peristaltic" instabilities in incompressible neo-Hookean solids in a range of cylindrical geometries with radius $R_0$. First we consider a solid cylinder, and recover the well-known, infinite wavelength instability for $γ\ge6 μR_0$, where $μ$ is the solid's shear modulus. Second, we consider a volume-conserving (e.g.\ fluid filled and sealed) cylindrical cavity through an infinite solid, and demonstrate infinite wavelength instability for $γ\ge 2 μR_0$. Third, we cons…
In 1994, Eggers and Dupont suggested the slender jet model, a one-dimensional model that describes the motion of a thin axisymmetric column of viscous, incompressible fluid with a free surface. In their model, the momentum equation was derived in a manner that was not completely self-consistent. Consequently, the viscosity term was described with less accuracy than the surface tension term. In this paper, we derive a novel slender jet momentum equation in completely self-consistent manner that allows us to describe both the viscosity and the surface tension forces with the same accuracy as the…
We study the influence of the surrounding gas in the dynamics of drop impact on a smooth surface. We use an axisymmetric 3D model for which both the gas and the liquid are incompressible; lubrication regime applies for the gas film dynamics and the liquid viscosity is neglected. In the absence of surface tension a finite time singularity whose properties are analysed is formed and the liquid touches the solid on a circle. When surface tension is taken into account, a thin jet emerges from the zone of impact, skating above a thin gas layer. The thickness of the air film underneath this jet is a…
