Simulation of flow and electrokinetic transport in a micro‐analysis‐system

The aim of this work is the investigation of the flow and electrokinetic transport in a micro‐analysis‐system. A mathematical model for electroosmotic and electrophoretic phenomena is introduced in order to perform two‐dimensional, time‐dependent Finite Element simulations for an existing device. The model includes the feature of the flow field, the mass transfer, the external applied electric field and the involved chemistry. The results of the simulation are validated against experimental data and show good agreement. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spatio‐temporal Pattern Formation in Chemical Reactions

In our paper numerical simulations of chemical pattern in ionic reaction‐diffusion‐migration system assuming a “self‐consistent” electric field are presented. Chemical waves as well as stationary concentration pattern arise due to an interplay of an autocatalytic chemical reaction with transport processes. Concentration gradient inside the chemical pattern lead to electric diffusion‐potential which in turn affect the patterns. Thus, the model equations take the general form of the Fokker‐Planck equation.     

Eulerian Simulation of Gas‐Solid Particle Flow by BDIM

The numerical scheme based on the boundary domain integral method (BDIM) for the numerical simulation of twophase two‐component flows is presented. A program is being developed to model the hydrodynamics of fluidized bed systems by using the Eulerian approach in terms of velocity‐vorticity variables formulation. With the vorticity vector both phases motion computation scheme is partitioned into its kinematic and kinetic aspect. Influence of the drag coefficient on the two‐phase two‐component flow field is studied on the two‐phase gas‐solid particles vertical channel flow.     

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first‐order system in a stress‐velocity formulation for the Stokes problem and a volumetric flux‐hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well‐known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015     

Well‐posedness of Compressible Euler Equations in a Physical Vacuum

An important problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of a vacuum. In particular, physical vacuum, in which the boundary moves with a nontrivial finite normal acceleration, naturally arises in the study of the motion of gaseous stars or shallow water. Despite its importance, there are only a few mathematical results available near a vacuum. The main difficulty lies in the fact that the physical systems become degenerate along the vacuum boundary. In this paper, we establish the local‐in‐time well‐posedness of three‐dimensional compressible Euler equations for polytropic gases with a physical vacuum by considering the problem as a free boundary problem. © 2015 Wiley Periodicals, Inc.     

Numerical and asymptotic study of non‐axisymmetric magnetohydrodynamic boundary layer stagnation‐point flows

Both numerical and asymptotic analyses are performed to study the similarity solutions of three‐dimensional boundary‐layer viscous stagnation point flow in the presence of a uniform magnetic field. The three‐dimensional boundary‐layer is analyzed in a non‐axisymmetric stagnation point flow, in which the flow is developed because of influence of both applied magnetic field and external mainstream flow. Two approaches for the governing equations are employed: the Keller‐box numerical simulations solving full nonlinear coupled system and a corresponding linearized system that is obtained under a far‐field behavior and in the limit of large shear‐to‐strain‐rate parameter (λ). From these two approaches, the flow phenomena reveals a rich structure of new family of solutions for various values of the magnetic number and λ. The various results for the wall stresses and the displacement thicknesses are presented along with some velocity profiles in both directions. The analysis discovered that the flow separation occurs in the secondary flow direction in the absence of magnetic field, and the flow separation disappears when the applied magnetic field is increased. The flow field is divided into a near‐field (due to viscous forces) and far‐field (due to mainstream flows), and the velocity profiles form because of an interaction between two regions. The magnetic field plays an important role in reducing the thickness of the boundary‐layer. A physical explanation for all observed phenomena is discussed. Copyright © 2017 John Wiley & Sons, Ltd.     

Riemann‐Hilbert Problems for the Shapes Formed by Bodies Dissolving,Melting, and Eroding in Fluid Flows

The classical Stefan problem involves the motion of boundaries during phase transition, but this process can be greatly complicated by the presence of a fluid flow. Here we consider a body undergoing material loss due to either dissolution (from molecular diffusion), melting (from thermodynamic phase change), or erosion (from fluid‐mechanical stresses) in a fast‐flowing fluid. In each case, the task of finding the shape formed by the shrinking body can be posed as a singular Riemann‐Hilbert problem. A class of exact solutions captures the rounded surfaces formed during dissolution/melting, as well as the angular features formed during erosion, thus unifying these different physical processes under a common framework. This study, which merges boundary‐layer theory, separated‐flow theory, and Riemann‐Hilbert analysis, represents a rare instance of an exactly solvable model for high‐speed fluid flows with free boundaries.© 2017 Wiley Periodicals, Inc.     

Non‐Gaussian invariant measures for the Majda model of decaying turbulent transport

The problem of turbulent transport of a scalar field by a random velocity field is considered. The scalar field amplitude exhibits rare but very large fluctuations whose typical signature are fatter than Gaussian tails for the probability distribution of the scalar. The existence of such large fluctuations is related to clustering phenomena of the Lagrangian paths within the flow. This suggests an approach to turn the large deviation problem for the scalar field into a small deviation, or small ball, problem for some appropriately defined process measuring the spreading with time of the Lagrangian paths. Here, such a methodology is applied to a model proposed by Majda consisting of a white‐in‐time linear shear flow and some generalizations of it where the velocity field has finite, or even infinite, correlation time. The non‐Gaussian invariant measure for the (reduced) scalar field is derived and, in particular, it is shown that the one‐point distribution of the scalar has stretched exponential tails, with a stretching exponent depending of the parameters in the model. Different universality classes for the scalar behavior are identified which, all other parameters being kept fixed, display a one‐to‐one correspondence with a exponent measuring time persistence effects in the velocity field. © 2001 John Wiley & Sons, Inc.     

Macro‐hybrid mixed variational two‐phase flow through fractured porous media

Macro‐hybrid mixed variational models of two‐phase flow, through fractured porous media, are analyzed at the mesoscopic and macroscopic levels. The mesoscopic models are treated in terms of nonoverlapping domain decompositions, in such a manner that the porous rock matrix system and the fracture network interact across rock–rock, rock–fracture, and fracture–fracture interfaces, with flux transmission conditions dualized. Alternatively, the models are scaled to a macroscopic level via an asymptotic process, where the width of the fractures tends to zero, and the fracture network turns out to be an interface system of one less spatial dimension, with variable high permeability. The two‐phase flow is characterized by a fractional flow dual mixed variational model. Augmented two‐field and three‐field variational reformulations are presented for regularization, internal approximations, and macro‐hybrid mixed finite element implementation. Also abstract proximal‐point penalty‐duality algorithms are derived and analyzed for parallel computing. Copyright © 2016 John Wiley & Sons, Ltd.     

Stabilized boundary element methods for low‐frequency electromagnetic scattering

The electromagnetic scattering at a perfectly conducting object is usually initiated by an incoming electromagnetic field. It is well known that the classical boundary element implementations solving for the scattered electric field are not uniformly stable with respect to the frequency of the incoming signal. The subject of this article is to develop a stabilized boundary element formulation that does not suffer from the so‐called low‐frequency breakdown. The mathematical theory is verified by numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.     

An approach to understanding avalanche statistics in mean‐field driven threshold systems

Driven threshold models that produce complex histories of avalanches are used to simulate the dynamics of many complex interacting systems, such as earthquake generating faults and neural networks. A mean‐field model may be formulated in a way that makes avalanches Abelian, so the final size of the avalanche depends only on the initial conditions, not the algorithm. If the initial stress distribution is statistically stationary, the avalanche size distribution is generated by the first intersection of a random process with a curvilinear boundary. Solutions show that such mean‐field models are never truly critical, but always exhibit dissipation or finite‐size effects. © 2005 Wiley Periodicals, Inc. Complexity 10:68–72, 2005     

Asymptotic and numerical solutions of three‐dimensional boundary‐layer flow past a moving wedge

We consider a laminar boundary‐layer flow of a viscous and incompressible fluid past a moving wedge in which the wedge is moving either in the direction of the mainstream flow or opposite to it. The mainstream flows outside the boundary layer are approximated by a power of the distance from the leading boundary layer. The variable pressure gradient is imposed on the boundary layer so that the system admits similarity solutions. The model is described using 3‐dimensional boundary‐layer equations that contains 2 physical parameters: pressure gradient (β) and shear‐to‐strain‐rate ratio parameter (α). Two methods are used: a linear asymptotic analysis in the neighborhood of the edge of the boundary layer and the Keller‐box numerical method for the full nonlinear system. The results show that the flow field is divided into near‐field region (mainly dominated by viscous forces) and far‐field region (mainstream flows); the velocity profiles form through an interaction between 2 regions. Also, all simulations show that the subsequent dynamics involving overshoot and undershoot of the solutions for varying parameter characterizing 3‐dimensional flows. The pressure gradient (favorable) has a tendency of decreasing the boundary‐layer thickness in which the velocity profiles are benign. The wall shear stresses increase unboundedly for increasing α when the wedge is moving in the x‐direction, while the case is different when it is moving in the y‐direction. Further, both analysis show that 3‐dimensional boundary‐layer solutions exist in the range −1<α<∞. These are some interesting results linked to an important class of boundary‐layer flows.     

The Bernoulli Integral for a Certain Class of Non‐Stationary Viscous Vortical Flows of Incompressible Fluid

It has been shown in our paper [1] that there is a wide class of 3D motions of incompressible viscous fluid which can be described by one scalar function dabbed the quasi‐potential. This class of fluid flows is characterized by three‐component velocity field having two‐component vorticity field; both these fields can depend of all three spatial variables and time, in general. Governing equations for the quasi‐potential have been derived and simple illustrative example of 3D flow has been presented. Here, we derive the Bernoulli integral for that class of flows and compare it against the known Bernoulli integrals for the potential flows or 2D stationary vortical flows of inviscid fluid. We show that the Bernoulli integral for this class of fluid motion possesses unusual features: it is valid for the vortical nonstationary motions of a viscous incompressible fluid. We present a new very nontrivial analytical example of 3D flow with two‐component vorticity which hardly can be obtained by any of known methods. In the last section, we suggest a generalization of the developed concept which allows one to describe a certain class of 3D flows with the 3D vorticity.     

TOWARDS A STANDARD FOR THE INDIVIDUAL‐BASED MODELING OF PLANT POPULATIONS: SELF‐THINNING AND THE FIELD‐OF‐NEIGHBORHOOD APPROACH

ABSTRACT. In classical theoretical ecology there are numerous standard models which are simple, generally applicable, and have well‐known properties. These standard models are widely used as building blocks for all kinds of theoretical and applied models. In contrast, there is a total lack of standard individual‐based models (IBM's), even though they are badly needed if the advantages of the individual‐based approach are to be exploited more efficiently. We discuss the recently developed ‘field‐of‐neighborhood’ approach as a possible standard for modeling plant populations. In this approach, a plant is characterized by a circular zone of influence that grows with the plant, and a field of neighborhood that for each point within the zone of influence describes the strength of competition, i.e., growth reduction, on neighboring plants. Local competition is thus described phenomenologically. We show that a model of mangrove forest dynamics, KiWi, which is based on the FON approach, is capable of reproducing self‐thinning trajectories in an almost textbook‐like manner. In addition, we show that the entire biomass‐density trajectory (bdt) can be divided into four sections which are related to the skewness of the stem diameter distributions of the cohort. The skewness shows two zero crossings during the complete development of the population. These zero crossings indicate the beginning and the end of the self‐thinning process. A characteristic decay of the positive skewness accompanies the occurrence of a linear bdt section, the well‐known self‐thinning line. Although the slope of this line is not fixed, it is confined in two directions, with morphological constraints determining the lower limit and the strength of neighborhood competition exerted by the individuals marking the upper limit.     

Bayesian planning of step‐stress accelerated degradation tests under various optimality criteria

Step‐stress accelerated degradation testing (SSADT) has become a common approach to predicting lifetime for highly reliable products that are unlikely to fail in a reasonable time under use conditions or even elevated stress conditions. In literature, the planning of SSADT has been widely investigated for stochastic degradation processes, such as Wiener processes and gamma processes. In this paper, we model the optimal SSADT planning problem from a Bayesian perspective and optimize test plans by determining both stress levels and the allocation of inspections. Large‐sample approximation is used to derive the asymptotic Bayesian utility functions under 3 planning criteria. A revisited LED lamp example is presented to illustrate our method. The comparison with optimal plans from previous studies demonstrates the necessity of considering the stress levels and inspection allocations simultaneously.     

Cost‐efficient monitoring of continuous‐time stochastic processes based on discrete observations

Planning a cost‐efficient monitoring policy of stochastic processes arises from many industrial problems. We formulate a simple discrete‐time monitoring problem of continuous‐time stochastic processes with its applications to several industrial problems. A key in our model is a doubling trick of the variables, with which we can construct an algorithm to solve the problem. The cost‐efficient monitoring policy balancing between the observation cost and information loss is governed by an optimality equation of a fixed point type, which is solvable with an iterative algorithm based on the Feynman‐Kac formula. This is a new linkage between monitoring problems and mathematical sciences. We show regularity results of the optimization problem and present a numerical algorithm for its approximation. A problem having model ambiguity is presented as well. The presented model is applied to problems of environment, ecology, and energy, having qualitatively different target stochastic processes with each other.     

Well‐posedness of a generalized time‐harmonic transport equation for acoustics in flow

We study a generalized time‐harmonic transport equation, which appears in the Goldstein equations and allows us to model the acoustic radiation in a flow. We investigate the well‐posedness of this transport problem. The result will be established under the assumption of a Ω‐filling flow, which, in 2D, is simply equivalent to a flow that does not vanish. The approach relies on the method of characteristics, which leads to the resolution of the transport equation along the streamlines, and on general results of functional analysis. The theoretical results are illustrated with numerical results obtained with a Streamline Upwind Petrov‐Galerkin finite element scheme.     

Diffusion in poro‐plastic media

A model is developed for the flow of a slightly compressible fluid through a saturated inelastic porous medium. The initial‐boundary‐value problem is a system that consists of the diffusion equation for the fluid coupled to the momentum equation for the porous solid together with a constitutive law which includes a possibly hysteretic relation of elasto‐visco‐plastic type. The variational form of this problem in Hilbert space is a non‐linear evolution equation for which the existence and uniqueness of a global strong solution is proved by means of monotonicity methods. Various degenerate situations are permitted, such as incompressible fluid, negligible porosity, or a quasi‐static momentum equation. The essential sufficient conditions for the well‐posedness of the system consist of an ellipticity condition on the term for diffusion of fluid and either a viscous or a hardening assumption in the constitutive relation for the porous solid. Copyright © 2004 John Wiley & Sons, Ltd.