A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows

This paper reports a new meshless Integrated Radial Basis Function Network (IRBFN) approach to the numerical simulation of interfacial flows in which the two-way interaction between a moving interface and the ambient viscous flow is fully investigated. When an interface between two immiscible fluids moves, not only its position and shape but also the flow variables (i.e. velocity field and pressure) change due to the presence of surface tension along the moving interface. The velocity field of the ambient flow, on the other hand, causes the interface to move and deform as a result of momentum transport between the two immiscible fluids on both sides of the interface. Numerical investigations of such a two-way interaction is reported in this paper where the level set method is used in combination with high-order projection schemes in the meshless framework of the IRBFN method. Numerical investigations on the meshless projection schemes are performed with typical benchmark incompressible viscous flow problems for verification purposes. The approach is then demonstrated with the numerical simulation of two bubbles moving, stretching and merging in an incompressible ambient fluid under the action of buoyancy force.     

Global existence of two phase nonhomogeneous viscous incompbessible fluid flow

We shall discuss the temporarily global solution for the two phase free boundary problem. Both fluids are regarded as immiscible, nonhomogeneous, viscous and incompressible and subject to surface tention on the interface. The global solution is obtained near the equilibrium state under the sufficiently small initial data and external forces.     

Surface tension effects on interaction between two fluids near a wall

Interaction between two fluids near a fixed solid surface ismodelled, with surface tension acting as an important influenceon the assumed planar motion. The two fluids are immiscible,incompressible and have small density and viscosity ratios;the heavier more viscous body of fluid is approaching the solidsurface and the other fluid is lying as a thin layer in between.In the so-called supercritical range where, for both fluids,inviscid forces dominate over viscous ones, a pair of pressure–shaperelations is found which leads to a nonlinear integro-differentialequation for the unknown interface shape. Analysis, computationand comparisons are applied to the equation. Travelling-statesolutions are found of periodic and non-periodic form, includinginteresting cases which exhibit parabolic growth of the layerthickness in the far field.     

弹性固体和充满两种互不相溶粘性流体的多孔固体之间界面的反射和折射及其衰减波

在充满两种互不相溶粘性流体的多孔固体中,研究弹性波的传播.用3个数性的势函数描述3个纵波的传播,用1个矢性的势函数单独描述横波的传播.根据这些势函数,在不同的组合相中,定义出质点的位移.可以看出,可能存在3个纵波和1个横波.在一个弹性固体半空间与一个充满两种互不相溶粘性流体的多孔固体半空间之间,研究其界面上入射纵波和横波所引起的反射和折射现象.由于孔隙流体中有粘性,折射到多孔介质中的波,朝垂直界面方向偏离.将入射波引起的反射波和折射波的波幅比,作为非奇异的线性代数方程组计算.进一步通过这些波幅比,计算出各个被离散波在入射波能量中所占的份额.通过一个特殊的数值模型,计算出波幅比和能量比系数随入射角的变化.超过SV波的临界入射角,反射波P将不再出现.越过界面的能量守恒原理得到了验证.绘出了图形并对不同孔隙饱和度以及频率的变化,讨论它们对能量分配的影响.     

Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous,incompressible fluids

We introduce a new sharp interface model for the flow of two immiscible, viscous, incompressible fluids. In contrast to classical models for two-phase flows we prescribe an evolution law for the interfaces that takes diffusional effects into account. This leads to a coupled system of Navier–Stokes and Mullins–Sekerka type parts that coincides with the asymptotic limit of a diffuse interface model. We prove the long-time existence of weak solutions, which is an open problem for the classical two-phase model. We show that the phase interfaces have in almost all points a generalized mean curvature.     

Kinetic Equation for Soliton Gas: Integrable Reductions

We study the dynamic behaviour of two viscous fluid films confined between two concentric cylinders rotating at a small relative velocity. It is assumed that the fluids are immiscible and that the volume of the outer fluid film is large compared to the volume of the inner one. Moreover, while the outer fluid is considered to have constant viscosity, the rheological behaviour of the inner thin film is determined by a strain-dependent power-law. Starting from a Navier–Stokes system, we formally derive evolution equations for the interface separating the two fluids. Two competing effects drive the dynamics of the interface, namely the surface tension and the shear stresses induced by the rotation of the cylinders. When the two effects are comparable, the solutions behave, for large times, as in the Newtonian regime. We also study the regime in which the surface tension effects dominate the stresses induced by the rotation of the cylinders. In this case, we prove local existence of positive weak solutions both for shear-thinning and shear-thickening fluids. In the latter case, we show that interfaces which are initially close to a circle converge to a circle in finite time and keep that shape for later times.

     

The Muskat problem and related topics

In the paper, we consider the evolution of the free boundary separating two immiscible viscous fluids with different constant densities in an absolutely rigid solid body and in an elastic skeleton. The motion of the liquids is described by the Stokes equations driven by the input pressure and the force of gravity. For flows in a bounded domain, we prove the existence and uniqueness of classical solutions and emphasize the study of the properties of the moving boundary separating the two fluids.     

A cut finite element method for a Stokes interface problem

We present a finite element method for the Stokes equations involving two immiscible incompressible fluids with different viscosities and with surface tension. The interface separating the two fluids does not need to align with the mesh. We propose a Nitsche formulation which allows for discontinuities along the interface with optimal a priori error estimates. A stabilization procedure is included which ensures that the method produces a well conditioned stiffness matrix independent of the location of the interface.     

On the Rayleigh-Taylor Instability for Two Uniform Viscous Incompressible Flows

The authors study the Rayleigh-Taylor instability for two incompressible immis- cible fluids with or without surface tension, evolving with a free interface in the presence of a uniform gravitational field in Eulerian coordinates. To deal with the free surface, instead of using the transformation to Lagrangian coordinates, the perturbed equations in Eule- rian coordinates are transformed to an integral form and the two-fluid flow is formulated as a single-fluid flow in a fixed domain, thus offering an alternative approach to deal with the jump conditions at the free interface. First, the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one with a free interface separating the two fluids, both fluids being in （unstable） equilibrium is analyzed. By a general method of studying a family of modes, the smooth （when restricted to each fluid domain） solutions to the linearized problem that grow exponentially fast in time in Sobolev spaces are constructed, thus leading to a global instability result for the linearized problem. Then, by using these pathological solutions, the global instability for the corresponding nonlinear problem in an appropriate sense is demonstrated.     

Flow past a sphere straddling the interface of a two-phase system

An explicit solution is found for flow past a sphere with perfect slip in a two phase flow of two immiscible viscous fluids. The interface remains flat and an expression is found for the drag on the sphere.

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Résumé Une solution de l'équation de Stokes est présentée pour une sphère entièrement immergée, la surface de séparation de deux fluides non mélangeables se situant á la hauteur du centre de cette sphère. Pour la surface de cette dernière la condition d'immobilité est remplacée par une condition de tension de cisaillement nulle.

     

The Immersed Interface Method for Simulating Two-Fluid Flows

We develop the immersed interface method (IIM) to simulate a two-fluid flow of two immiscible fluids with different density and viscosity. Due to the surface tension and the discontinuous fluid properties, the two-fluid flow has nonsmooth velocity and discontinuous pressure across the moving sharp interface separating the two fluids. The IIM computes the flow on a fixed Cartesian grid by incorporating into numerical schemes the necessary jump conditions induced by the interface. We present how to compute these necessary jump conditions from the analytical principal jump conditions derived in [Xu, DCDS, Supplement 2009, pp. 838-845]. We test our method on some canonical two-fluid flows. The results demonstrate that the method can handle large density and viscosity ratios, is second-order accurate in the infinity norm, and conserves mass inside a closed interface.     

On interfacial transition conditions in two phase gravity flow

A layer of ice and sediment is modelled as a mixture of two nonlinear, very viscous, constant density fluids interacting mechanically via Darcy- and Pick-type forces. An inclined layer of this mixture overlain by a layer of ice modelled as a viscous fluid is considered with boundary conditions of no-slip or viscous sliding at the base and no stress at the free surface. The interface is treated as a singular surface across which the jump conditions of mass and momentum for the constituents are assumed to hold. Furthermore, because the components are viscous fluids, a kinematic condition for the continuity of the tangential velocity is formulated. The momentum jump conditions involve surface production terms requiring additional surfacial constitutive relations.We show that the posed physical problem admits a mathematical solution only in the case that the interface momentum production is non-zero.Dedicated to Hans Roethlisberger on the occasion of his seventieth birthday.     

Smooth interface in a two-component Stokes flow

The problem considered is that of evolution of the free boundary Γ(t) separating two immiscible viscous fluids with different constant densities and viscosities. The motion is described by the Stokes equations driven by the gravity force. We prove the existence of classical solutions for small timet and establish that the free boundary Γ(t)∈C ^l+2 (l>0 is an arbitrary non-integer number)     

On Nonlinear Rayleigh-Taylor Instabilities

Some of the mathematical properties of the interface between two incompressible inviscid and immiscible fluids with different densities under the influence of a constant gravity field 9 are investigated. The purpose of this paper is to prove that linearly unstable modes for Rayleigh-Taylor instabilities give birth to nonlinear instabilities for the full nonlinear system. The main ingredient is a general instability theorem in an analytic framework which enables us to go from linear to nonlinear instabilities.     

Strong solutions for a 1D viscous bilayer shallow water model

In this paper, we consider a viscous bilayer shallow water model in one space dimension that represents two superposed immiscible fluids. For this model, we prove the existence of strong solutions in a periodic domain. The initial heights are required to be bounded above and below away from zero and we get the same bounds for every time. Our analysis is based on the construction of approximate systems which satisfy the BD entropy and on the method developed by A. Mellet and A. Vasseur to obtain the existence of global strong solutions for the one dimensional Navier–Stokes equations.     

Retrieving the Bingham model from a bi-viscous model: Some explanatory remarks

In this note we prove some analytical results on the Bingham model. In particular we show how to derive some constitutive and kinematical properties through a limit procedure in which the visco-plastic model is retrieved from a linear bi-viscous model. We also prove that, assuming a no-slip condition at the interface separating the two viscous fluids, no source of entropy can be present on such interface.     

Admixture stratification in the stagnation region of two streams

A self-similar 2D steady-state flow of two immisible viscous fluids with inertial particles is considered. It is assumed that two viscous streams, one of which contains the particles, collide forming a flat interface with a stagnation point. The general case is discussed, when the fluids have different viscosities and densities and the streams are directed at arbitrary angles. The far field corresponds to inviscid vortex flow near an “oblique” stagnation point. The limiting case of viscous dusty flow near a rigid plane is studied in detail within one-way and two-way coupling approximations. Thin zones of particle accumulation are detected. Threshold parameters corresponding to the change of flow regime are found. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities

We consider a diffuse interface model describing flow and phase separation of a binary isothermal mixture of (partially) immiscible viscous incompressible Newtonian fluids having different densities. The model is the nonlocal version of the one derived by Abels, Garcke and Grün and consists in a inhomogeneous Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. This model was already analyzed in a paper by the same author, for the case of singular potential and non-degenerate mobility. Here, we address the physically more relevant situation of degenerate mobility and we prove existence of global weak solutions satisfying an energy inequality. The proof relies on a regularization technique based on a careful approximation of the singular potential. Existence and regularity of the pressure field is also discussed. Moreover, in two dimensions and for slightly more regular solutions, we establish the validity of the energy identity. We point out that in none of the existing contributions dealing with the original (local) Abels, Garcke Grün model, an energy identity in two dimensions is derived (only existence of weak solutions has been proven so far).     

Dusty effects on pulsating viscous flow of two immiscible fluids between two parallel plates

The flow of two immiscible incompressible dusty viscous fluids between two parallel plates generated by a pulsating pressure gradient is investigated. Velocity fields for the fluid-particle system along with the expressions for the skin friction drag at the plates are obtained and studied graphically. It is found that there is an immediate response to pressure fluctuations in the first stream at low frequency range 0<σ≤4 being maximum at σ=4. On the contrary, the second stream is more responsive to fluctuations at relatively higher frequencies. The maximum response in this case is shifted to σ=16.     

One time‐step finite element discretization of the equation of motion of two‐fluid flows

We discretize in space the equations obtained at each time step when discretizing in time a Navier‐Stokes system modelling the two‐dimensional flow in a horizontal pipe of two immiscible fluids with comparable densities, but very different viscosities. At each time step the system reduces to a generalized Stokes problem with nonstandard conditions at the boundary and at the interface between the two fluids. We discretize this system with the “mini‐element” and establish error estimates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005