Discontinuous Galerkin finite element method applied to the coupled unsteady Stokes/Cahn-Hilliard equations

Two-phase flows driven by the interfacial dynamics are studied by tracking implicitly interfaces in the framework of the Cahn-Hilliard theory. The fluid dynamics is described by the Stokes equations with an additional source term in the momentum equation taking into account the capillary forces. A discontinuous Galerkin finite element method is used to solve the coupled Stokes/Cahn-Hilliard equations. The Cahn-Hilliard equation is treated as a system of two coupled equations corresponding to the advection-diffusion equation for the phase field and a nonlinear elliptic equation for the chemical potential. First, the variational formulation of the Cahn-Hilliard equation is presented. A numerical test is achieved showing the optimal order in error bounds. Second, the variational formulation in discontinuous Galerkin finite element approach of the Stokes equations is recalled, in which the same space of approximation is used for the velocity and the pressure with an adequate stabilization technique. The rates of convergence in space and time are evaluated leading to an optimal order in error bounds in space and a second order in time with a backward differentiation formula at the second order. Numerical tests devoted to two-phase flows are provided on ellipsoidal droplet retraction, on the capillary rising of a liquid in a tube, and on the wetting drop over a horizontal solid wall.     

Transfer of a microfluid to a stochastic fibre network

The transfer of a microscopic fluid droplet from a flat surface to a deformable stochastic fibre network is investigated. Fibre networks are generated with different levels of surface roughness, and a two-dimensional, two-phase fluid-structure model is used to simulate the fluid transfer. In simulations, the Navier-Stokes equations and the Cahn-Hilliard phase-field equations are coupled to explicitly include contact line dynamics and free surface dynamics. The compressing fibre network is modelled as moving immersed boundaries. The simulations show that the amount of transferred fluid is approximately proportional to the contact area between the fluid and the fibre network. However, areas where the fluid bridges and never actually makes contact with the substrate must be subtracted.     

A sharp condition for existence of an inertial manifold

It is shown that a perturbation argument that guarantees persistence of inertial (invariant and exponentially attracting) manifolds for linear perturbations of linear evolution equations applies also when the perturbation is nonlinear. This gives a simple but sharp condition for existence of inertial manifolds for semi-linear parabolic as well as for some nonlinear hyperbolic equations. Fourier transform of the explicitly given equation for the tracking solution together with the Plancherel's theorem for Banach valued functions are used.     

An invariant-manifold analysis of electrophoretic traveling waves

A new, geometric proof of a theorem of Fife, Palusinski, and Su on electrophoretic traveling waves is presented. The proof is based upon the perturbation theory for invariant manifolds due to Fenichel. The results proved here reproduce the existence, uniqueness, and asymptotic approximation theorem proved by Fifeet al. The proof given here is substantially simpler, and in addition, it provides additional insight into the geometric structure of the phase space of the traveling wave equations for this system.     

Non-linear normal modes,invariance, and modal dynamics approximations of non-linear systems

Non-linear systems are here tackled in a manner directly inherited from linear ones, that is, by using proper normal modes of motion. These are defined in terms of invariant manifolds in the system's phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach to which the theory developed for discrete systems can be applied-are simultaneously applied to the same study case-an Euler-Bernoulli beam constrained by a non-linear spring-and compared as regards accuracy and reliability. Numerical simulations of pure non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the non-linear normal modes are demonstrated, and it is also found that, for a pure non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differential equation.     

A phase field model for failure in interconnect lines due to coupled diffusion mechanisms

We describe a diffuse interface, or phase field model for simulating electromigration and stress-induced void evolution and growth in interconnect lines. Microstructural evolution is tracked by defining an order parameter, which takes on distinct uniform values within solid material and voids, and varying rapidly from one to the other over narrow interfacial layers associated with the void surfaces. The order parameter is governed by a form of the Cahn-Hilliard equation. An asymptotic analysis demonstrates that the zero contour of order parameter tracks the motion of a void evolving by coupled surface and lattice diffusion, driven by stress, electron wind and vacancy concentration gradients. Efficient finite element schemes are described to solve the modified Cahn-Hilliard equation, as well as the equations associated with the accompanying mechanical, electrical and bulk diffusion problems. The accuracy and convergence of the numerical scheme is investigated by comparing results to known analytical solutions. The method is applied to solve various problems involving void growth and evolution in representative interconnect geometries.     

Two-phase flow analysis based on a phase-field model using orthogonal basis bubble function finite element method

In this study, a finite element method based on a phase-field model for gas–liquid two-phase flow is proposed. MINI element based on a bubble function element stabilisation method is employed for the incompressible Navier–Stokes equations. The Cahn–Hilliard equation is employed to estimate the interface of gas and liquid. The orthogonal basis bubble function element is used to solve the Cahn–Hilliard equation. In particular, a detailed explanation for solving the Cahn–Hilliard equation based on a finite element method is given.     

The invariant manifold of beam deformations

The subcentre invariant manifold of elasticity in a thin rod may be used to give a rigorous and appealing approach to deriving one-dimensional beam theories. Here I investigate the analytically simple case of the deformations of a perfectly uniform circular rod. Many, traditionally separate, conventional approximations are derived from within this one approach. Furthermore, I show that beam theories are convergent, at least for the circular rod, and obtain an accurate estimate of the limit of their validity. The approximate evolution equations derived by this invariant manifold approach are complete with appropriate initial conditions, forcing and, in at least one case, boundary conditions.     

Sharp-crack limit of a phase-field model for brittle fracture

A matched asymptotic analysis is used to establish the correspondence between an appropriately scaled version of the governing equations of a phase-field model for fracture and the equations of the two-dimensional sharp-crack theory of Gurtin and Podio-Guidugli (1996) that arise on assuming that the bulk constitutive behavior is nonlinearly elastic, requiring that surface energy provides the only factor limiting crack propagation, and assuming that the fracture kinetics are isotropic. Consistent with the prominence of the configurational momentum balance at the crack tip in the latter theory, the approach capitalizes on the configurational momentum balance that arises naturally in the context of the phase-field model. The model developed and utilized here incorporates irreversibility of the phase-field evolution. This is achieved by introducing a suitable constraint and by carefully heeding the influence of that constraint on the kinetics underlying microstructural changes associated with fracture. The analysis is predicated on the assumption that the phase-field variable takes values in the closed interval between zero and unity.     

On a Model for Soft Landing with State-Dependent Delay

Automatic soft landing is modeled by a differential equation with state-dependent delay. It is shown that in the model soft landing occurs for an open set of initial data, which is determined by means of a smooth invariant manifold.      

Spectral barriers and inertial manifolds for dissipative partial differential equations

In recent years, the theory of inertial manifolds for dissipative partial differential equations has emerged as an active area of research. An inertial manifold is an invariant manifold that is finite dimensional, Lipschitz, and attracts exponentially all trajectories. In this paper, we introduce the notion of a spectral barrier for a nonlinear dissipative partial differential equation. Using this notion, we present a proof of existence of inertial manifolds that requires easily verifiable conditions, namely, the existence of large enough spectral barriers.     

Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities

The paper addresses the occurrence of possible restrictions on the flows defined by scalar retarded functional differential equations (FDEs), locally around certain simple singularities, compared with the possible flows of ordinary differential equations (ODEs) with the same singularities. It is found that for the Hopf and the Bogdanov-Takens singularities, there are no restrictions on the local flows defined by scalar FDEs, even when the nonlinearities depend on just one delayed value of the solutions. On the other hand, for the singularity associated with a zero and a conjugated pair of pure imaginary numbers as simple eigenvalues, it is shown that there occur restrictions on the flows defined by scalar FDEs with nonlinearities involving just one delay, as well as two delays satisfying a certain resonance condition. These restrictions are of geometric significance, since they amount to the impossibility of observing the homoclinic orbits that occur in arbitrarily small neighborhoods of the singularity for ODEs. Versal unfolfings for the considered singularities by FDEs and the possible restrictions on the associated flows are also studied.     

Invariant and Partially Invariant Solutions of the Green-Naghdi Equations

All invariant and partially invariant solutions of the Green-Naghdi equations are obtained that describe the second approximation of shallow water theory. It is proved that all nontrivial invariant solutions belong to one of the following types: Galilean-invariant, stationary, and self-similar solutions. The Galilean-invariant solutions are described by the solutions of the second Painleve equation, the stationary solutions by elliptic functions, and the self-similar solutions by the solutions of the system of ordinary differential equations of the fourth order. It is shown that all partially invariant solutions reduce to invariant solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 26–35, November–December, 2005.     

Lie Symmetry Analysis of Differential Equations in Finance

Lie group theory is applied to differential equations occurring as mathematical models in financial problems. We begin with the complete symmetry analysis of the one-dimensional Black–Scholes model and show that this equation is included in Sophus Lie's classification of linear second-order partial differential equations with two independent variables. Consequently, the Black–Scholes transformation of this model into the heat transfer equation follows directly from Lie's equivalence transformation formulas. Then we carry out the classification of the two-dimensional Jacobs–Jones model equations according to their symmetry groups. The classification provides a theoretical background for constructing exact (invariant) solutions, examples of which are presented.     

Optimal estimates for the uncoupling of differential equations

Our aim in this note is to give optimal conditions on the spectral gap for the existence of an uncoupling of a differential equation of the form = Cz + H(=) into a system ofuncoupled equations of the form (x, y) = (Ax, By) + (F(x, (x)), G((y),y)), whereC=A×B is a bounded linear operator on a Banach spaceZ=X×Y satisfying a spectral gap condition, andH=(F,G) is a Lipschitz function withH(0) = 0. We also give optimal conditions for the regularity of the manifoldsgraph andgraph , and optimal conditions for the regularity of the leaves of two foliations of the phase space associated to the uncoupling. Sharp estimates for the Lipschitz constant of and and for the Hölder exponent of the uncoupling homeomorphism and its inverse are also given.     

Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations

In this paper, we study the long-time behavior of a class of nonlinear dissipative partial differential equations. By means of the Lyapunov-Perron method, we show that the equation has an inertial manifold, provided that certain gap condition in the spectrum of the linear part of the equation is satisfied. We verify that the constructed inertial manifold has the property of exponential tracking (i.e., stability with asymptotic phase, or asymptotic completeness), which makes it a faithful representative to the relevant long-time dynamics of the equation. The second feature of this paper is the introduction of a modified Galerkin approximation for analyzing the original PDE. In an illustrative example (which we believe to be typical), we show that this modified Galerkin approximation yields a smaller error than the standard Galerkin approximation.     

Regularity of Invariant Manifolds for Nonuniformly Hyperbolic Dynamics

For any sufficiently small perturbation of a nonuniform exponential dichotomy, we show that there exist invariant stable manifolds as regular as the dynamics. We also consider the general case of a nonautonomous dynamics defined by the composition of a sequence of maps. The proof is based on a geometric argument that avoids any lengthy computations involving the higher order derivatives. In addition, we describe how the invariant manifolds vary with the dynamics.      

Center manifold and stability for skew-product flows

We study the existence and smoothness of global center, center-stable, and center-unstable manifolds for skew-product flows. Smooth invariant foliations to the center stable and center unstable manifolds are also discussed.     

Reaction-diffusion problems in cell tissues

In this paper we consider reaction diffusion problems describing a reaction occurring in a planar domain (regarded as cell tissue). The diffusivity is assumed to be large except in the neighborhood of curves (regarded as membranes), around which it is assumed to be small. The subregions determined by the membranes and by the boundary of the domain (tissue wall) are regarded as cells. We assume that the tissue wall is a barrier through which no substance can pass. We prove that the dynamics is described by a system of ordinary differential equations that can be explicitly exhibited through the parameters defining the reaction diffusion problem, the length of the membranes and the area of the cells. The tools employed are a detailed analysis of an eigenvalue problem and the invariant manifold theory.