Advection Type Research Articles

Self-Coagulation Theory and Related Comet- and Semi-Circle-Shaped Structures in Electronegative and Gaseous Discharging Plasmas in the Laboratory

In this work, the two-dimensional fluid models for two types of inductively coupled plasma, Ar/O2 and Ar/SF6, are numerically solved by the finite element method. Four interesting phenomena revealed by the simulations are reported: (1) comet-shaped and semi-circle-shaped structures in Ar/O2 and Ar/SF6 plasmas, respectively; (2) blue sheaths that surround the two structures; (3) the collapse and dispersion of semi-circle-shaped structures of certain Ar/SF6 plasma cations and anions when they are observed separately; and (4) the rebuilding of coagulated structures by minor cations in the Ar/SF6 plasma at the discharge center. From the simulation detail, it was found that the cooperation of free diffusion and negative chemical sources creates the coagulated structure of anions, and the self-coagulation theory is therefore built. The advective and ambipolar types of self-coagulation are put forth to explain the co-existence of blue sheath and internal neutral plasma, among which the advective type of self-coagulation extends the Bohm’s sheath theory of cations to anions, and the ambipolar type of self-coagulation originates from the idea of the ambipolar diffusion process, and it updates the recognition of people about the plasma collective interaction. During the ambipolar self-coagulation, each type of Ar/SF6 plasma cations and anions is self-coagulated, and the coagulated plasma species are then modeled as mass-point type (or point-charge type, more precisely). When the charge amounts of two point-charge models of plasma species with the same charge type are equal, the expelling effect caused by the Coulomb’s force of them leads to the collapse or dispersal of heavily coagulated species. The simulation shows that the lighter the species is, the easier it self-coagulates and the more difficult its coagulation is broken, which implies the inertia effect of density quantity. Moreover, the collapse of cation coagulation creates the spatially dispersed charge cloud that is not shielded into the Debye’s length, which indicates the anti-collective behavior of electronegative plasmas when they are self-coagulated. The rebuilt coagulated structure of minor Ar/SF6 plasma species at the discharge center and the weak coagulation of electrons in the periphery of the main coagulated structure that is under the coil are caused by the monopolar and spontaneous (non-advective) type of self-coagulation. The analysis predicts an intensity order of physically driven coagulation force, chemical self-coagulation force, and ambipolar self-coagulation force. The popular coagulated structure of the electronegative ICP sources is urgently needed to validate the experiment.

Lie-group modeling and simulation of a spherical robot, actuated by a yoke–pendulum system, rolling over a flat surface without slipping

The present paper aims at introducing a mathematical model of a spherical robot expressed in the language of Lie-group theory. Since the main component of motion is rotational, the space SO(3)3 of three-dimensional rotations plays a prominent role in its formulation. Because of friction to the ground, rotation of the external shell results in translational motion. Rolling without slipping implies a constraint on the tangential velocity of the robot at the contact point to the ground which makes it a non-holonomic dynamical system. The mathematical model is obtained upon writing a Lagrangian function that describes the mechanical system and by the Hamilton minimal-action principle modified through d’Alembert virtual work principle to account for non-conservative control actions as well as frictional reactions. The result of the modeling appears as a series of non-holonomic Euler–Poincaré equations of dynamics plus a series of auxiliary equations of reconstruction and advection type. A short discussion on the numerical simulation of such mathematical model complements the main analytic-mechanic development.

PECULIARITIES OF WATER AND SEDIMENT RUN-OFF IN A RAVINE AT THE BEGINNING OF THE FLOOD

The formation of water and sediment run-off in a ravine in different natural zones with humid climate begins before the main flood event. The type of spring (advective or solar type) determines the volume of water and sediment run-off, which is formed depending on the volume of snow masses in the ravine and in its catchment area. The density parameters of the snow mass, formed depending on the alternation of thaws and snow compaction under the influence of wind, determine the nature of the water flow through the snow mass or under the snow. At the same time, with the advective type of spring, the water flow does not produce any erosive work. The solar type of spring corresponds to active erosion of the sides and top of the ravine with a slight impact on the thalweg. The purpose of this work is to show the role of snow masses in the catchment and in the ravine and their influence on the formation of water and sediment run-off in the pre-water period with different types of spring. The work was carried out on the basis of observations during field work in different regions of the country and showed that the flow of water in the ravine at the initial stage of high water depends on the conditions for the formation of snow masses in the ravine. It has been established that snow masses accumulated during winter have an ambiguous effect on the water flow in the pre-water area and largely depend on specific weather conditions and anthropogenic impact on the gully catchment.

Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

In this paper, approaches to the numerical recovering of the initial condition in the inverse problem for a nonlinear singularly perturbed reaction–diffusion–advection equation are considered. The feature of the formulation of the inverse problem is the use of additional information about the value of the solution of the equation at the known position of a reaction front, measured experimentally with a delay relative to the initial moment of time. In this case, for the numerical solution of the inverse problem, the gradient method of minimizing the cost functional is applied. In the case when only the position of the reaction front is known, the method of deep machine learning is applied. Numerical experiments demonstrated the possibility of solving such kinds of considered inverse problems.

Singularly Perturbed Stationary Diffusion Model with a Cubic Nonlinearity

We consider a multidimensional singularly perturbed stationary diffusion model with a cubic nonlinearity. For models of this type, a modified asymptotic method of boundary functions, which extends the classical asymptotic analysis methods to the case of multidimensional problems, and the asymptotic method of differential inequalities, which is based on the comparison principle, are used to study the existence of asymptotically Lyapunov stable solutions with internal layers as stationary solutions of the corresponding parabolic problems. Sufficient conditions are established for the existence of such solutions in the form of some conditions on the coefficients of the equation, an asymptotic approximation to the solution of an arbitrary accuracy order with coefficients is constructed in closed form, and the formal constructions are justified. This result can be used for creating efficient numerical algorithms for direct and coefficient inverse problems for stationary equations of the reaction–diffusion–advection type as well as for constructing test examples. Heat and mass transfer problems occurring in chemical industry are pointed out as possible applications of our results.

Use of asymptotic analysis for solving the inverse problem of source parameters determination of nitrogen oxide emission in the atmosphere

The possibilities of using asymptotic analysis for solving the inverse problem of restoring the parameters of the source of nitrogen oxide industrial emissions into the atmosphere are demonstrated. The asymptotic analysis allows to reduce the subproblem for a three-dimensional singularly perturbed equation of the reaction–diffusion–advection type to a much simpler problem for numerical solution. This allows to significantly increase the efficiency of the numerical solution of the original inverse problem. Numerical experiments demonstrate the effectiveness of the proposed approach.

Asymptotically Stable Stationary Solutions of the Reaction–Diffusion–Advection Equation with Discontinuous Reaction and Advection Terms

We study the Lyapunov asymptotic stability of the stationary solution of the spatially one-dimensional initial–boundary value problem for a nonlinear singularly perturbed differential equation of the reaction–diffusion–advection (RDA) type in the case where the advection and reaction terms undergo a discontinuity of the first kind at some interior point of an interval. Sufficient conditions are derived for the existence of a stable stationary solution with a large gradient near the point of discontinuity. An asymptotic method of differential inequalities is used to prove the existence and stability theorems. The resulting stability conditions can be employed to create mathematical models and develop numerical methods for solving “stiff” problems arising in various applications, for example, when simulating combustion processes and in nonlinear wave theory.

An algorithm for solving the Burgers–Huxley equation using the Elzaki transform

In this research work, we propose an algorithm which involves the coupling of a new integral transform called the Elzaki transform and the well-known homotopy perturbation method on the Burgers–Huxley equation which is a type of nonlinear advection–diffusion partial differential equation. The Burgers–Huxley equation which models reaction mechanisms, diffusion transports and nerve ion propagation which is applicable in traffic flows, acoustics, turbulence theory, hydrodynamics and generally mechanics is the fusion of the well-known Burgers equation and the Huxley equation. We proffer an analytical solution in the form of a Taylor multivariate series of displacement x and time t using the proposed Elzaki homotopy transformation perturbation method (EHTPM) to three cases of the Burgers–Huxley equation. This solution converges rapidly to a closed form which is the same as the exact solutions obtained using the normal analytical methods from the existing literatures. The exact results and that of our proposed EHTPM when compared via tables and 3D plots show an excellent agreement devoid of errors.

The Taylor‐least‐squares time integrator scheme applied to tracer equations of a sea ice model

AbstractThe viscous‐plastic sea ice model based on [2] describes the motion of sea ice for scales of several thousand kilometers. The numerical model for the simulation of sea ice circulation and evolution over a seasonal cycle includes the consideration of the sea ice thickness and sea ice concentration. Transient advection equations describe the physical behavior of both thickness and concentration with the velocity of the sea ice as the coupling field. Recent research on a finite element implementation of the sea ice model is devoted to formulations based on the (mixed) Galerkin variational approach, compare to [1] and [6] for instance. Here, particular treatments are necessary regarding the stabilization of the complex numerical scheme, especially for the first‐order advection equation. It is therefore suggested to utilize the mixed least‐squares finite element method (LSFEM), which is well established in the branch of, e.g., fluid mechanics. A significant advantage of the method is its applicability to first‐order systems. Thus, this method results in stable and robust formulations also for not self‐adjoint operators like the tracer equations. Moreover, in [7], the authors provide a promising higher‐order time integration scheme for transient advection equations denoted as Taylor‐least‐squares scheme, which is investigated in this article. The presented least‐squares finite element formulation is based on the unsteady sea ice equations including two tracer equations of transient advection type. The numerical problem of a Box test case is investigated.

Global dynamics of a Lotka–Volterra competition–diffusion–advection system in heterogeneous environments

We study a Lotka–Volterra type reaction–diffusion–advection system, which describes the competition for the same resources between two aquatic species undergoing different dispersal strategies, as reflected by their diffusion and/or advection rates. For the non-advective case, this problem was solved by Dockery et al. [9], and recently He and Ni [14] provided a further classification on the global dynamics for a more general model. However, the key ideas developed in [9,14] do not appear to work when advection terms are involved. By assuming the resource function is decreasing in the spatial variable, we establish the non-existence of co-existence steady state and perform sufficient analysis on the local stability of two semi-trivial steady states, where new techniques were introduced to overcome the difficulty caused by the non-analyticity of stationary solutions as well as the diffusion–advection type operators. Combining these two aspects with the theory of monotone dynamical systems, we finally obtain the global dynamics, which suggests that the competitive exclusion principle holds in most situations.

Definition of a temporal distribution index for high temporal resolution precipitation data over Peninsular Spain and the Balearic Islands: the fractal dimension; and its synoptic implications

Precipitation on the Spanish mainland and in the Balearic archipelago exhibits a high degree of spatial and temporal variability, regardless of the temporal resolution of the data considered. The fractal dimension indicates the property of self-similarity, and in the case of this study, wherein it is applied to the temporal behaviour of rainfall at a fine (10-min) resolution from a total of 48 observatories, it provides insights into its more or less convective nature. The methodology of Jenkinson & Collison which automatically classifies synoptic situations at the surface, as well as an adaptation of this methodology at 500 hPa, was applied in order to gain insights into the synoptic implications of extreme values of the fractal dimension. The highest fractal dimension values in the study area were observed in places with precipitation that has a more random behaviour over time with generally high totals. Four different regions in which the atmospheric mechanisms giving rise to precipitation at the surface differ from the corresponding above-ground mechanisms have been identified in the study area based on the fractal dimension. In the north of the Iberian Peninsula, high fractal dimension values are linked to a lower frequency of anticyclonic situations, whereas the opposite occurs in the central region. In the Mediterranean, higher fractal dimension values are associated with a higher frequency of the anticyclonic type and a lower frequency of the advective type from the east. In the south, lower fractal dimension values indicate higher frequency with respect to the anticyclonic type from the east and lower frequency with respect to the cyclonic type.

A multigrid perspective on the parallel full approximation scheme in space and time

SummaryFor the numerical solution of time‐dependent partial differential equations, time‐parallel methods have recently been shown to provide a promising way to extend prevailing strong‐scaling limits of numerical codes. One of the most complex methods in this field is the “Parallel Full Approximation Scheme in Space and Time” (PFASST). PFASST already shows promising results for many use cases and benchmarks. However, a solid and reliable mathematical foundation is still missing. We show that, under certain assumptions, the PFASST algorithm can be conveniently and rigorously described as a multigrid‐in‐time method. Following this equivalence, first steps towards a comprehensive analysis of PFASST using blockwise local Fourier analysis are taken. The theoretical results are applied to examples of diffusive and advective type.

Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments

We study the dynamics of a SIS epidemic model of reaction–diffusion–advection type. The persistence of infected and susceptible populations and the global stability of the disease free equilibrium are established when the basic reproduction number is greater than or less than or equal to one, respectively. We further consider the effects of diffusion and advection on asymptotic profiles of endemic equilibrium: When the advection rate is relatively large comparing to the diffusion rates of both populations, then two populations persist and concentrate at the downstream end. As the diffusion rate of the susceptible population tends to zero, the density of the infected population decays exponentially for positive advection rate but linearly when there is no advection. Our results suggest that advection can help speed up the elimination of disease.

Toward fault-tolerant parallel-in-time integration with PFASST

We introduce and analyze different strategies for the parallel-in-time integration method PFASST to recover from hard faults and subsequent data loss. Since PFASST stores solutions at multiple time steps on different processors, information from adjacent steps can be used to recover after a processor has failed. PFASST's multi-level hierarchy allows to use the coarse level for correcting the reconstructed solution, which can help to minimize overhead. A theoretical model is devised linking overhead to the number of additional PFASST iterations required for convergence after a fault. The potential efficiency of different strategies is assessed in terms of required additional iterations for examples of diffusive and advective type.

Trends in the aerosol load properties over south eastern Italy

The long-term (2003-2013) variations in columnar aerosol properties at Lecce, a site representative of the central Mediterranean, have been analysed for trend assessment. The study focuses on aerosol optical thickness (AOT) at 340, 440, 500 and 1020 nm and Ångström exponent (AE) for the pair 440-870 nm, retrieved from a sun photometer operating within the Aerosol Robotic Network (AERONET). A non-parametric trend analysis of the monthly mean, median and upper and lower tails (90th and 10th percentiles) suggests that the aerosol load has decreased during the study period, while the mean particle size remained unchanged. The characteristic advections reaching the study site were found by clustering analysis of back trajectories at 500, 1500 and 3000 m. Despite the strong influence they have on aerosol load and particle size, neither of the trends in advection routes could explain the tendencies found in the columnar aerosol properties. However, trends in aerosol data by advection type allow understanding the overall trends. Aerosol properties under flows with high residence time over continental Europe present differences according to the specific residing area. More specifically, no trend is found when flows arrive from Ukraine and the Balkans, while under advections from north-western/central Europe there are downward trends in the background levels and a reduction of the fine fraction. Negative trends are also found under flows with high residence time over the Mediterranean and northern Africa, again with differences according to the residing area.

Performance characterization of passive micromixer with dual opposing strips on microchannel walls

Micromixers are an extremely important component in integrated microfluidic systems; however, achieving uniform mixing in laminar flow with a low Reynolds number can be exceedingly difficult. This study developed a chaotic advection type micromixer using dual opposing strips on microchannel walls, which generated two-layer vortices, enlarged the contact interface, and increased diffusion flux between the mixing liquids, ultimately leading to superior mixing performance. Numerical simulation was employed in the design of the micromixer and a micromilling machine was used in conjunction with PDMS casting for the fabrication of the microfluidic devices. Experiments conducted at flow velocities of 1mm/s, 5mm/s, and 10mm/s produced the following results: (1) 90% mixing performance in a channel length of 15mm at the three flow velocities, (2) an increase in flow velocity deteriorated the mixing performance because the geometric groove features reverse the mixing performance achieved by previous features, (3) misalignment between the top and bottom micro strips detracted from mixing performance.

A numerical solution of two-dimensional fox-rabies dynamics of advection type

This article analyses a mode of two-dimensional fox-rabies dynamics with advection terms. The fox population is divided into two species: susceptible S and infective I. Finite-difference techniques are employed to compute the numerical solutions of this model. Iterative schemes with second-order accuracy are derived and their convergence is investigated by using the maximum principle analysis. An illustrative numerical example with graphical reis given to highlight the efficiency of the derived numerical schemes.

Study on the Flow Field of Air Floating Pool of Pressurized Dissolved-Air Floatation Water Treatment System

This paper used the FLUENT software to analyze a flow field of the advection type air-float pool, the simulation analysis is applied to study of partition plates, water retaining plate and floatation pool length depth ratio and other key parameters which influence the flotation pool flotation effect .The research results provide the basis for the structure design of improving the flotation pool. It has a certain reference value for the same type of flotation pool design.

A unified term for directed and undirected motility in collective cell invasion

In this paper, we develop mathematical models for collective cell motility. Initially we develop a model using a linear diffusion–advection type equation and fit the parameters to data from cell motility assays. This approach is helpful in classifying the results of certain cell motility assay experiments. In particular, this model can determine degrees of directed versus undirected collective cell motility. Next we develop a model using a nonlinear diffusion term that is able to capture in a unified way directed and undirected collective cell motility. One goal of this work is to demonstrate that the forms of collective cell motility seen in the scratch assays and possibly other systems of interest need not reference external and more complicated migratory signals such as chemotaxis, but rather could be based on quorum sensing alone, collectively represented as density-dependent diffusivity. As an application we apply the nonlinear diffusion approach to a problem in tumor cell invasion, noting that neither chemotaxis or haptotaxis are present in the system under consideration in this article.

On one-dimensional fox-rabies dynamics of advection type

A one-dimensional model of fox-rabies of two nonlinear partial differential equations of hyperbolic type is studied. Finite difference techniques are applied to compute the numerical solutions of the initial/boundary value problem. The convergence of the resulting schemes, which have a second order accuracy in space and time, is investigated. The method is tested for different values of advection rate; numerical and graphical results showed that the method is consistent with the dynamic behaviour of fox-rabies.