Long-Time Accuracy for Approximate Slow Manifolds in a Finite-Dimensional Model of Balance

We study the slow singular limit for planar anharmonic oscillatory motion of a charged particle under the influence of a perpendicular magnetic field when the mass of the particle goes to zero. This model has been used by the authors as a toy model for exploring variational high-order approximations to the slow dynamics in rotating fluids. In this paper, we address the long time validity of the slow limit equations in the simplest nontrivial case. We show that the first-order reduced model remains O(ε) accurate over a long 1/ε timescale. The proof is elementary, but involves subtle estimates on the nonautonomous linearized dynamics.     

Nonlinear stability of an axial electric field: Effect of interfacial charge relaxation

We introduce a nonlinear perturbation technique to third order, to study the stability between two cylindrical inviscid fluids, subjected to an axial electric field. The study takes into account the relaxation of electrical charges at the interface between the two fluids. At first order, a linear dispersion relation is obtained. Analytical and numerical results for the overstability and incipient instability conditions are given. For perfect dielectric fluids, the electric field has a stabilizing influence, while for leaky dielectric fluids, the electric field can have either a stabilizing or a destabilizing influence depending on the conductivity and permittivity ratios of the two fluids. At higher order, a nonlinear dispersion relation (nonlinear Ginzburg–Landau equation) is derived, describing the evolution of wave packets of the problem. For leaky dielectric fluids near the marginal state, a nonlinear diffusion equation (nonlinear incipient instability) is obtained. For perfect dielectric fluids, two cubic nonlinear Schrödinger equations are obtained. One of these equations to determine a nonlinear cutoff electric field separating stable and unstable disturbance, whereas the other is used to analyze the stability of the system. It is found that the nonlinear stability criterion depending on the ratio of permittivity, Such effects can only be explained successfully in the nonlinear sense, as the linear analysis unsuccessful to inform about them.     

A coupling between a 1D compressible-incompressible limit and the 1D p-system in the non smooth case

We consider two compressible immiscible fluids in one space dimension and in the isentropic approximation. The first fluid is surrounded and in contact with the second one. As the sound speed of the first fluid diverges to infinity, we prove the rigorous convergence for the compressible to incompressible limit of the coupled dynamic of the two fluids.     

Hill regions of charged three-body systems

For charged three-body systems, we discuss the configurations and orientations that are admissible for given values of the conserved total energy and angular momentum. The admissible configurations and orientations are discussed on a configuration space that is reduced by the translational, rotational and dilation symmetries of charged three-body systems. We consider the examples of the charged three-body systems given by the helium atom (two electrons and a nucleus) and the compound of two electrons and one positron. For comparison, the well known example of the Newtonian gravitational three-body system is discussed for the scheme presented in this paper first.     

Optimal control for two-dimensional stochastic second grade fluids

This article deals with a stochastic control problem for certain fluids of non-Newtonian type. More precisely, the state equation is given by the two-dimensional stochastic second grade fluids perturbed by a multiplicative white noise. The control acts through an external stochastic force and we search for a control that minimizes a cost functional. We show that the Gâteaux derivative of the control to state map is a stochastic process being the unique solution of the stochastic linearized state equation. The well-posedness of the corresponding stochastic backward adjoint equation is also established, allowing to derive the first order optimality condition.     

Martingale Solution to Equations for Differential Type Fluids of Grade Two Driven by Random Force of Lévy Type

In this article we study a system of nonlinear non-parabolic stochastic evolution equations driven by Lévy noise type. This system describes the motion of second grade fluids driven by random force. Global existence of a martingale solution is proved under general conditions on the noise. Since the coefficient of the noise does not satisfy a Lipschitz property, we could not prove any pathwise uniqueness result. We note that this is the first work dealing with a stochastic model for non-Newtonian fluids excited by external forces of Lévy noise type.     

Existence of global solutions to multidimensional equations for Bingham fluids

We consider equations describing the multidimensional motion of compressible viscous (non-Newtonian) Bingham-type fluids, i.e., fluids with multivalued function relating the stresses to the tensor of strain rates. We prove the global existence theorem in time and in the initial data for the first initial boundary-value problem corresponding to flows in a bounded domain in the class of “weak” generalized solutions. In this case, we admit an anisotropic relation between the stress and strain rate tensors and study admissible relations of this kind in detail.     

The Theoretical Aspects of the Orr-Sommerfeld Methods: The Case of Two Viscoelastic Fluids

In this article, conceived for physicists and mathematicians, we describe various Orr-Sommerfeld Equations (OSEs) and stress their differences, both in modeling, justification and in the results. These equations are derived from the Poiseuille flow of two viscoelastic or Newtonian fluids. The literature proposes a link between computation and experiment which is modeled by two different equations. We reinvestigate it and stress a hidden assumption. Then, we study extensively the long wave asymptotic stability of the flow of two viscoelastic fluids and exhibit a formula for characterization of loss of stability in a new case. Some waves are found through an OSE and cannot be found through the other. We give their growth rate implicitely for some of them. Last, we prove a theorem that says whether such a wave could be unstable or not.     

Two-dimensional computational simulation of eccentric annular cementing displacements

We consider a two-dimensional Hele–Shaw type model fordisplacement flows occurring in the primary cementing of anoil well. The fluids are visco-plastic and may get stuck inthe annulus if a critical pressure gradient is not exceeded.The model consists of solving a nonlinear elliptic variationalinequality equation for the stream function, coupled to an equationfor interface advection, or alternatively a concentration equationfor the mass fraction of each fluid. The key difficulty is toaccurately compute yielded and unyielded zones of the wellborefluids, which we accomplish by use of an augmented Lagrangianmethod to solve the stream function equation. We validate theaccuracy of our method against analytical solutions for stablesteady-state displacements. We study the convergence of theinterface to the steady state, showing that the apparent meta-stabilityis illusory. We then explore the effects of increasing eccentricity,showing that although the interface may remain stable it becomesunsteady. Initially fully mobile flows are found, but as theeccentricity increases further the narrow side fluids fail tomove in the far field. The narrow side interface can progressslowly through the static fluids by a burrowing motion, butfor still larger eccentricities even the interface becomes staticand a narrow-side mud channel forms.     

The formulation of linear theory of a reflected shock in cylindrical geometry

In this paper we formulate the linear theory for compressible fluids in cylindrical geometry with small perturbation at the material interface. We derive the first order equations in the smooth regions, boundary conditions at the shock fronts and the contact interface by linearizing the Euler equations and Rankine-Hugoniot conditions. The small amplitude solution formulated in this paper will be important for calibration of results from full numerical simulation of compressible fluids in cylindrical geometry.     

High-order nonlinear boundary value problems admitting multiple exact solutions with application to the fluid flow over a sheet

We frame a hierarchy of nonlinear boundary value problems which are shown to admit exponentially decaying exact solutions. We are able to convert the question of the existence and uniqueness of a particular solution to this nonlinear boundary value problem into a question of whether a certain polynomial has positive real roots. Furthermore, if such a polynomial has at least two distinct positive roots, then the nonlinear boundary value problem will have multiple solutions. In certain special cases, these boundary value problems arise in the self-similar solutions for the flow of certain fluids over stretching or shrinking sheets; examples given include the flow of first and second grade fluids over such surfaces.     

电场驱动直射流的动量方程

给出了电场驱动直射流的一维动量守恒方程．该方程是用应力分量表示的,适用于任何流体本构关系,只要流体是不可压缩的．结果显示,为了使方程封闭,需要沿轴向和径向两个方向的本构关系．然而,当附加应力张量的迹为0时,只需要沿轴向的一个本构关系就足够了．还发现,射流的第二主应力差的零阶近似总为0．与其他类型的动量方程做了比较．     

An inventory-transportation system with stochastic demand

We study a logistic system in which a supplier has to deliver a set of products to a set of retailers to face a stochastic demand over a given time horizon. The transportation from the supplier to each retailer can be performed either directly, by expensive and fast vehicles, or through an intermediate depot, by less expensive but slower vehicles. At most one time period is required in the former case, while two time periods are needed in the latter case. A variable transportation cost is charged in the former case, while a fixed transportation cost per journey is charged in the latter case. An inventory cost is charged at the intermediate depot. The problem is to determine, for each time period and for each product, the quantity to send from the supplier to the depot, from the depot to each retailer and from the supplier to each retailer, in order to minimize the total expected cost. We first show that the classical benchmark policy, in which the demand of each product at each retailer is set equal to the average demand, can give a solution which is infinitely worse with respect to the optimal solution. Then, we propose two classes of policies to solve this problem. The first class, referred to as Horizon Policies, is composed of policies which require the solution of the overall problem over the time horizon. The second class, referred to as Reoptimization Policies, is composed of a myopic policy and several rolling-horizon policies in which the problem is reoptimized at each time period, once the demand of the time period is revealed. We evaluate the performance of each policy dynamically, by using Monte Carlo Simulation.     

Lattice Boltzmann modeling of wall-bounded ternary fluid flows

Starting from the phase-field perspective, we first formulate a novel wetting boundary condition to describe the interactions among ternary fluids and a solid and then we propose a boundary scheme for its implementation in the framework of the lattice Boltzmann (LB) method. This scheme for three-phase fluids can preserve the reduction consistency property of the diphasic case such that it can give physically relevant results. Combining this wetting boundary scheme and the LB ternary fluid model based on multicomponent phase-field theory, we simulated several ternary fluid flow problems involving a solid substrate, including the spreading of binary drops on a substrate, the spreading of a compound drop on a substrate, the capillary intrusion of ternary fluids, and the shear of a compound liquid drop on a substrate. The numerical results are found to be good agreement with the analytical solutions and some available results. Finally, as an application, we use the LB model coupled with the present wetting boundary scheme to numerically investigate the impact of a compound drop on a solid circular cylinder. It is found that the dynamics of a compound drop can be remarkably influenced by the wettability of the solid surface and the dimensionless Weber number.     

On the global existence of weak solutions for the Cucker-Smale-Navier-Stokes system with shear thickening

We study the large-time dynamics of Cucker-Smale (C-S) flocking particles interacting with non-Newtonian incompressible fluids. Dynamics of particles and fluids were modeled using the kinetic Cucker-Smale equation for particles and non-Newtonian Navier-Stokes system for fluids, respectively and these two systems are coupled via the drag force, which is the main flocking (alignment) mechanism between particles and fluids. We present a global existence theory for weak solutions to the coupled Cucker-Smale-Navier-Stokes system with shear thickening. We also use a Lyapunov functional approach to show that sufficiently regular solutions approach flocking states exponentially fast in time.     

Electrodynamics with charged strings

We show that in a four-dimensional space–time a complex scalar field can be associated with a one-dimensionally extended object, called a charged string. The string is said to be charged because the complex scalar field describing it interacts with an electromagnetic field. A charged string is characterized by an extension of the symmetry group of the charge space to a group of stretch rotations. We propose relativistically invariant and gauge-invariant equations describing the interaction of a complex scalar field with an electromagnetic field, and each solution of them corresponds to a charged string. We achieve this by introducing the notion of a charged string index, which, as verified, takes only integer values. We establish equations from which it follows that charged strings fit naturally into the framework of the Maxwell–Dirac electrodynamics.     

Barotropic instability of shear flows

We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian partial differential equations. The last one is to study the singular and nonresonant neutral modes using Sturm-Liouville theory and hypergeometric functions.     

On the Two‐Dimensional Muskat Problem with Monotone Large Initial Data

We consider the evolution of two incompressible, immiscible fluids with different densities in porous media, known as the Muskat problem [21], which in two dimensions is analogous to the Hele‐Shaw cell [24]. We establish, for a class of large and monotone initial data, the global existence of weak solutions. The proof is based on a local well‐posedness result for the initial data with certain specific asymptotics at spatial infinity and a new maximum principle for the first derivative of the graph function.© 2016 Wiley Periodicals, Inc.     

Initial value problems for creeping flow of Maxwell fluids

We consider the flow of nonlinear Maxwell fluids in the unsteady quasistatic case, where the effect of inertia is neglected. We study the well-posedness of the resulting PDE initial-boundary value problem locally in time. This well-posedness depends on the unique solvability of an elliptic boundary value problem. We first present results for the 3D case with sufficiently small initial data and for a simple shear flow problem with arbitrary initial data; after that we extend our results to some 3D flow problems with large initial data.We solve our problem using an iteration between linear subproblems. The limit of the iteration provides the solution of our original problem.     

An Extrapolated Second Order Backward Difference Time-Stepping Scheme for the Magnetohydrodynamics System

In this article, we introduce a fully implicit, linearly extrapolated second-order backward difference time-stepping scheme for solving a time dependent non-homogeneous magnetohydrodynamic system for electrically conducting fluids. The extrapolated time-stepping scheme is used for time discretization and the mixed finite element method is used for spatial discretization. We first prove unconditional energetic stability without introducing an undesirable exponential Gronwall constant. Complete error analysis is provided without assuming any stability condition or restrictions on the time-step size. Numerical experiments are presented to confirm the theoretical convergence results and efficiency of the scheme.