Some remarks on the flux-free finite element method for immiscible two-fluid flows

We give some theoretical considerations on the the flux-free finite element method for the generalized Stokes interface problem arising from the immiscible two-fluid flow problems. In the flux-free finite element method, the flux constraint is posed as another Lagrange multiplier to keep the zero-flux on the interface. As a result, the mass of each fluid is expected to be preserved at every time step. We first study the effect of discontinuous coefficients (viscosity and density) on the error of the standard finite element approximations very carefully. Then, the analysis is extended to the flux-free finite element method.     

An Acceleration Technique for the Augmented IIM for 3D Elliptic Interface Problems

A new fast algorithm based on the augmented immersed interface method and a fast Poisson solver is proposed to solve three dimensional elliptic interface problems with a piecewise constant but discontinuous coefficient. In the new approach, an augmented variable along the interface, often the jump in the normal derivative along the interface is introduced so that a fast Poisson solver can be utilized. Thus, the solution of the Poisson equation depends on the augmented variable which should be chosen such that the original flux jump condition is satisfied. The discretization of the flux jump condition is done by a weighted least squares interpolation using the solution at the grid points, the jump conditions, and the governing PDEs in a neighborhood of control points on the interface. The interpolation scheme is the key to the success of the augmented IIM particularly. In this paper, the key new idea is to select interpolation points along the normal direction in line with the flux jump condition. Numerical experiments show that the method maintains second order accuracy of the solution and can reduce the CPU time by 20-50%. The number of the GMRES iterations is independent of the mesh size.     

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.     

The IIM in polar coordinates and its application to electro capacitance tomography problems

In a numerical study of the electric field inside an Electro Capacitance Tomography (ECT) device and other applications, often a Poisson equation with a discontinuous coefficient needs to be solved in polar coordinates. This paper is devoted to the Immersed Interface Method (IIM) in polar coordinates and the application to the solution of the electric potential inside an ECT device. The numerical algorithm is based on a finite difference discretization on a uniform polar coordinates grid. The finite difference scheme is modified at grid points near and on the interface across which the coefficient is discontinuous so that the natural jump conditions are satisfied. The algorithm and analysis here is one step forward in applying the IIM for 3D problems in axisymmetric situations or in the spherical coordinates. Numerical examples against exact solutions and the application to ECT problems are also presented.     

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.     

Interface conditions for Stokes equations with a discontinuous viscosity and surface sources

The interface conditions, or jump conditions, for the pressure and the velocity of the solution to the incompressible Stokes equations with a discontinuous viscosity and a singular source along an interface are derived in this work. While parts of the results agree with those in the literature, some of the results are new. These theoretical results are useful for developing accurate numerical methods for the interface problem.     

Viscous linear instability of a sheared liquid-gas interface - Effect of a velocity deficit near the interface

The initial state of liquid atomization by a fast gas stream is considered by viscous linear spatial stability analysis for parallel two-fluid flow. The unbounded basic velocity profile is characterized by different asymptotic velocities and a velocity deficit near the interface. We examine the influence of the velocity deficit on the spatial growth rates of two competing modes originating from the Kelvin-Helmholtz and viscosity contrast mechanisms. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An interface-fitted adaptive mesh method for elliptic problems and its application in free interface problems with surface tension

This work presents a novel two-dimensional interface-fitted adaptive mesh method to solve elliptic problems of jump conditions across the interface, and its application in free interface problems with surface tension. The interface-fitted mesh is achieved by two operations: (i) the projection of mesh nodes onto the interface and (ii) the insertion of mesh nodes right on the interface. The interface-fitting technique is combined with an existing adaptive mesh approach which uses addition/subtraction and displacement of mesh nodes. We develop a simple piecewise linear finite element method built on this interface-fitted mesh and prove its almost optimal convergence for elliptic problems with jump conditions across the interface. Applications to two free interface problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented. In these applications, the interface surface tension serves as the jump condition or the Dirichlet boundary condition of the pressure, and the pressure is solved with the interface-fitted finite element method developed in this work. In this study, a level-set function is used to capture the evolution of the interface and provide the interface location for the interface fitting.     

Coupling of compressible and incompressible flow regions using the multiple pressure variables approach

In many cases, multiphase flows are simulated on the basis of the incompressible Navier–Stokes equations. This assumption is valid as long as the density changes in the gas phase can be neglected. Yet, for certain technical applications such as fuel injection, this is no longer the case, and at least the gaseous phase has to be treated as a compressible fluid. In this paper, we consider the coupling of a compressible flow region to an incompressible one based on a splitting of the pressure into a thermodynamic and a hydrodynamic part. The compressible Euler equations are then connected to the Mach number zero limit equations in the other region. These limit equations can be solved analytically in one space dimension that allows to couple them to the solution of a half‐Riemann problem on the compressible side with the help of velocity and pressure jump conditions across the interface. At the interface location, the flux terms for the compressible flow solver are provided by the coupling algorithms. The coupling is demonstrated in a one‐dimensional framework by use of a discontinuous Galerkin scheme for compressible two‐phase flow with a sharp interface tracking via a ghost‐fluid type method. The coupling schemes are applied to two generic test cases. The computational results are compared with those obtained with the fully compressible two‐phase flow solver, where the Mach number zero limit is approached by a weakly compressible fluid. For all cases, we obtain a very good agreement between the coupling approaches and the fully compressible solver. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonlinear instability of Hele-Shaw flows with smooth viscous profiles

We rigorously derive nonlinear instability of Hele-Shaw flows moving with a constant velocity in the presence of smooth viscosity profiles where the viscosity upstream is lower than the viscosity downstream. This is a single-layer problem without any material interface. The instability of the basic flow is driven by a viscosity gradient as opposed to conventional interfacial Saffman-Taylor instability where the instability is driven by a viscosity jump across the interface. Existing analytical techniques are used in this paper to establish nonlinear instability.     

A hybrid coupling interface method for elliptic complex interface problems

We propose a coupling interface method (CIM) under Cartesian grid for solving elliptic complex interface problems in arbitrary d dimensions, where the coefficients, the source terms, and the solutions may be discontinuous or singular across the interfaces. It consists of a first-order version (CIM1) and a second-order version (CIM2). In one dimension, this finite difference method at a grid point adjacent to the interface is derived based on piecewise linear (CIM1) or quadratic (CIM2) approximation of the solution and two jump conditions. The method is extended to high dimensions through a dimensionby-dimension approach. To connect information from each dimension, a coupled equation for the principal derivatives is derived through the jump conditions in each coordinate direction. For CIM2, one-side interpolation for cross derivatives is need. This coupling approach reduces number of grid point in the finite difference stencil. The hybrid method uses CIM1 or CIM2 adaptly for complex interface. Numerical tests demonstrate that CIM1 and CIM2 are respectively first order and second order in the maximal norm with less error as compared with other methods. In addition, the hybrid CIM can solve complex interface problems in two and three dimensions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A partially penalty immersed interface finite element method for anisotropic elliptic interface problems

In this article, we develop a partially penalty immersed interface finite element (PIFE) method for a kind of anisotropy diffusion models governed by the elliptic interface problems with discontinuous tensor‐coefficients. This method is based on linear immersed interface finite elements (IIFE) and applies the discontinuous Galerkin formulation around the interface. We add two penalty terms to the general IIFE formulation along the sides intersected with the interface. The flux jump condition is weakly enforced on the smooth interface. By proving that the piecewise linear function on an interface element is uniquely determined by its values at the three vertices under some conditions, we construct the finite element spaces. Therefore, a PIFE procedure is proposed, which is based on the symmetric, nonsymmetric or incomplete interior penalty discontinuous Galerkin formulation. Then we prove the consistency and the solvability of the procedure. Theoretical analysis and numerical experiments show that the PIFE solution possesses optimal‐order error estimates in the energy norm and norm.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1984–2028, 2014     

Global solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity and discontinuous initial data

In this paper, we study the evolutions of the interfaces between the gas and the vacuum for viscous one-dimensional isentropic gas motions. We prove the global existence and uniqueness for discontinuous solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity coefficient. Precisely, the viscosity coefficient μ is proportional to ρ^θ with 0<θ<1. Specifically, we require that the initial density be piecewise smooth with arbitrarily large jump discontinuities, bounded above and below away from zero, in the interior of gas. We show that the discontinuities in the density persist for all time, and give a decay result for the density as t→+∞.     

The Mather problem for lower semicontinuous Lagrangians

In this paper we develop the Aubry-Mather theory for Lagrangians in which the potential energy can be discontinuous. Namely we assume that the Lagrangian is lower semicontinuous in the state variable, piecewise smooth with a (smooth) discontinuity surface, as well as coercive and convex in the velocity. We establish existence of Mather measures, various approximation results, partial regularity of viscosity solutions away from the singularity, invariance by the Euler–Lagrange flow away from the singular set, and further jump conditions that correspond to conservation of energy and tangential momentum across the discontinuity.     

Convergence estimates for an optimized Schwarz method for PDEs with discontinuous coefficients

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. When the subdomains are overlapping or nonoverlapping, these methods employ the optimal value of parameter(s) in the boundary condition along the artificial interface to accelerate its convergence. In the literature, the analysis of optimized Schwarz methods rely mostly on Fourier analysis and so the domains are restricted to be regular (rectangular). As in earlier papers, the interface operator can be expressed in terms of Poincaré–Steklov operators. This enables the derivation of an upper bound for the spectral radius of the interface operator on essentially arbitrary geometry. The problem of interest here is a PDE with a discontinuous coefficient across the artificial interface. We derive convergence estimates when the mesh size h along the interface is small and the jump in the coefficient may be large. We consider two different types of Robin transmission conditions in the Schwarz iteration: the first one leads to the best estimate when h is small, whereas for the second type, we derive a convergence estimate inversely proportional to the jump in the coefficient. This latter result improves upon the rate of popular domain decomposition methods such as the Neumann–Neumann method or FETI-DP methods, which was shown to be independent of the jump. In memory of Gene Golub.     

Nonlinear inertial effects on the instability of a single long wave of finite amplitude at the interface of two viscous fluids

The evolution of a single long wave of finite amplitude at the interface of two immiscible fluids of different viscosities and densities, between two horizontal plates is solved, using a boundary layer flow approximation for the equation of motion in each fluid layer. It is found that when the nonlinear inertial effects are taken into account in a moderate manner, at least in the frame of the boundary layer approximation, the initial unperturbed flow with smooth interface is stable to a single wave perturbation at the interface, even in the presence of adverse density and viscosity stratifications. However, when the nonlinear effects are increased in a specific way, and the magnitudes of the parameters involved are kept within the order of magnitude established for the present theory, an unstable flow configuration can be obtained.     

A Well-Conditioned,Nonconforming Nitsche's Extended Finite Element Method for Elliptic Interface Problems

In this paper, we introduce a nonconforming Nitsche's extended finite element method (NXFEM) for elliptic interface problems on unfitted triangulation elements. The solution on each side of the interface is separately expanded in the standard nonconforming piecewise linear polynomials with the edge averages as degrees of freedom. The jump conditions on the interface and the discontinuities on the cut edges (the segment of edges cut by the interface) are weakly enforced by the Nitsche's approach. In the method, the harmonic weighted fluxes are used and the extra stabilization terms on the interface edges and cut edges are added to guarantee the stability and the well conditioning. We prove that the convergence order of the errors in energy and $L^2$ norms are optimal. Moreover, the errors are independent of the position of the interface relative to the mesh and the ratio of the discontinuous coefficients. Furthermore, we prove that the condition number of the system matrix is independent of the interface position. Numerical examples are given to confirm the theoretical results.     

Solvability of an inverse problem for discontinuous Sturm–Liouville operators

In this paper, we consider the Sturm–Liouville equation with the jump conditions inside the interval (0,π). The inverse problem is studied, which consists in recovering operator coefficients from two spectra, corresponding to different boundary conditions. We prove the uniqueness theorem and provide necessary and sufficient conditions for solvability of the inverse problem. We also obtain the oscillation theorem for the eigenfunctions of the considered discontinuous boundary value problem.     

Unsteady draining of a fluid from a circular tank

Three-dimensional draining flow of a two-fluid system from a circular tank is considered. The two fluids are inviscid and incompressible, and are separated by a sharp interface. There is a circular hole positioned centrally in the bottom of the tank, so that the flow is axially symmetric. The mean position of the interface moves downwards as time progresses, and eventually a portion of the interface is withdrawn into the drain. For narrow drain holes of small radius, the interface above the centre of the drain is pulled down towards the hole. However, for drains of larger radius the portion of the interface above the drain edge is drawn down first, rather than the central section. Non-linear results are obtained with a novel spectral technique, and are also compared against the predictions of linearized theory. Unstable Rayleigh–Taylor type flows, in which the upper fluid is heavier than the lower one, are also discussed.     

一个两流体系统中mKdV孤立波的迎撞*

本文从文[2]的基本方程出发,采用约化摄动方法和PLK方法,讨论了三阶非线性和色散效应相平衡的修正的KdV(mKdV)孤立波迎撞问题.这些波在流体密度比等于流体深度比平方的两流体系统界面上传播.我们求得了二阶摄动解,发现在不考虑非均匀相移的情况下,碰撞后孤立波保持原有的形状,这与Fornberg和whitham[6]的追撞数值分析结果一致,但当考虑波的非均匀相移后,碰撞后波形将变化.