One-dimensional Approximation Research Articles

Mathematical model of fluid flow along a straight through filled with a distributed flow from above

In a one-dimensional approximation and in the absence of friction forces, a mathematical model has been developed for the steady flow of an incompressible fluid along a straight line, for example, a drainage gutter, into which a distributed flow flows from above. The boundary condition missing for solving the system of equations of momentum and continuity is determined using the principle of minimum potential energy. For a rectangular chute, equations are obtained that allow one to calculate the distributions of the layer thickness and fluid velocity along the length of the chute with a plug at one end and without a plug, slope and without slope to the horizon, depending on the intensity of the incoming flow, the size of the chute, and the density of the liquid. This model, by means of a simple recalculation, can also be extended to the flow in a trough of a different cross-sectional profile. The results of the study can be applied to the calculation of external drainage systems.

First-passage times in conical varying-width channels biased by a transverse gravitational force: Comparison of analytical and numerical results.

We study the crossing time statistic of diffusing point particles between the two ends of expanding and narrowing two-dimensional conical channels under a transverse external gravitational field. The theoretical expression for the mean first-passage time for such a system is derived under the assumption that the axial diffusion in a two-dimensional channel of smoothly varying geometry can be approximately described as a one-dimensional diffusion in an entropic potential with position-dependent effective diffusivity in terms of the modified Fick-Jacobs equation. We analyze the channel crossing dynamics in terms of the mean first-passage time, combining our analytical results with extensive two-dimensional Brownian dynamics simulations, allowing us to find the range of applicability of the one-dimensional approximation. We find that the effective particle diffusivity decreases with increasing amplitude of the external potential. Remarkably, the mean first-passage time for crossing the channel is shown to assume a minimum at finite values of the potential amplitude.

Groundwater flow in saturated riparian buffers and implications for nitrate removal.

A saturated riparian buffer (SRB) is an edge-of-field conservation practice that intercepts tile drainage and reduces nitrate flux to nearby streams by redistributing the flow as shallow groundwater. In this study, a three-dimensional, finite-difference groundwater model representative of SRBs in central Iowa was developed to assess the flow of groundwater and implications for nitrate removal during spring conditions, when flow to the SRB is highest. The model reproduces field observations of water level with Nash-Sutcliffe efficiency of 0.68, which is deemed acceptable for hydrologic models. The modeling shows that groundwater flow is three-dimensional near the distribution pipe and the stream and primarily one-dimensional in the rest of the buffer. The path the water takes in flowing toward the stream depends on where it exits the distribution pipe. When nitrate is not limiting, the potential for nitrate removal depends on the length of the path-and thus travel time-and depth because denitrification potential varies with depth. Travel time Tt can be estimated well with slight modifications to a one-dimensional approximation: Tt = 1.11Lx /vx , where Lx is the buffer width and vx is a one-dimensional approximation of the average linear velocity of groundwater. Refining knowledge of SRB function is an important step toward enhancing design for improving water quality.

Electrochemical impedance simulation of porous electrodes with variously shaped pores using 3-dimensional finite element method

Porous electrodes are widely used in many applications, such as enzyme electrodes, energy devices, and electrosynthesis. However, evaluation of porous electrodes using electrochemical impedance spectroscopy, one of the most promising methods for determining electrochemically relevant structural information, is challenging. Thus, in this study, impedance spectra were simulated using a three-dimensional, finite element method-based model for porous electrodes. The shape of the electrode pore was varied, and the effect on the resulting impedance spectrum was analyzed. The results indicated that for certain pore shapes, such as wide and shallow cylinders, a three-dimensional model was required to obtain accurate results. However, for narrow and deep cylindrical pores, a classical one-dimensional approximation was sufficiently accurate, and a 45º inclined line was observed in the impedance spectra with high frequencies, even when a three-dimensional model was employed. Moreover, wide and shallow or narrowing pores result in a higher inclination angle, whereas widening or branching pores result in a lower inclination angle. These features were also observed when charge transfer and diffusion processes were considered in terms of a mediator-type enzyme electrode reaction.

Planar Gas Expansion under Intensive Nanosecond Laser Evaporation into Vacuum as Applied to Time-of-Flight Analysis.

A computational investigation of the dynamics of gas expansion due to intense nanosecond laser evaporation into vacuum has been carried out. The problem is solved in a one-dimensional approximation, which simplifies calculations and at the same time allows one to analyze the main features of the expansion dynamics. For analysis we use three different approaches. Two of them are based on kinetic analysis via the direct simulation Monte Carlo (DSMC) method and numerical solution of the model Bhatnagar-Gross-Krook (BGK) equation. The third one focuses on derivation of an analytical continuum solution. Emphasis is placed on the analysis of the velocity distribution function and the average energy of particles passing through the time-of-flight detector on the normal to the evaporation surface, which is important for interpreting experimental measurements. The formulated problem is quite difficult as the considered flow is time-dependent, contains discontinuities in boundary conditions and involves large variations of local Knudsen numbers as well as steep gradients of the velocity distribution function. Data were obtained on the particle energy in the time-of-flight distribution for the range of regimes from the free molecular flow to continuum one. The maximum attainable average energy of particles in the time-of-flight distribution is determined. The non-monotonicity of the energy increase was found, which is explained based on analysis of the velocity distribution of particles.

Electric arc I–V modeling and related plasma spectrometry issues

This article is devoted to studying the properties of an electric arc column as a development of the authors’ early pioneer experiments. The object of modeling is a free-burning electric arc between evaporating copper electrodes in atmospheric air as the basis for the functioning of many modern technologies. It includes the determination of fundamental characteristics, such as the radial structure, and the current–voltage characteristics of the electric arc column under the assumption of plasma equilibrium. The middle cross section between the electrodes of a spheroidal shape arc is considered in order to limit the problem to a one-dimensional cylindrical approximation. It is strictly solved from the Elenbaas–Heller energy equation, with no resorting to the simplified channel model. The radial structure of the electric arc column is carefully considered with known temperature functions of electric and heat conductivities. Convenient functional approximations depending on temperature are proposed for the mentioned coefficients of copper–air plasma. The boundary between the arc column and atmospheric air is strictly located taking into account the chemical processes in the plasma of the copper–air mixture. The paper also presents some bases features of high-speed plasma spectrometry, substantiating the reliability of the obtained experimental data. As shown, the non-monotonicity of the current–voltage characteristics can arise due to the non-monotonicity of thermal conductivity as a function of temperature. Also, the loss of energy with the so-called ionization energy diffusion is insignificant in the overall energy balance of the arc. The results of the numerical simulation are compared with the experimental data.

Computational modeling of capillary perfusion and gas exchange in alveolar tissue

Gas exchange is an essential function of the respiratory system that couples fundamentally with perfusion in respiratory alveoli. Current mathematical formulations and computational models of these two phenomena rely on one-dimensional approximations that neglect the intricate volumetric microstructure of alveolar structures. In this work, we introduce a coupled three-dimensional computational model of pulmonary capillary perfusion and gas exchange that conforms to alveolar morphology. To this end, we derive non-linear partial differential equations and boundary conditions from physical principles that govern the behavior of blood and gases in arbitrary alveolar domains. We numerically solve the resulting formulation by proposing and implementing a non-linear finite-element scheme. Further, we carry out several numerical experiments to validate our model against one-dimensional simulations and demonstrate its applicability to morphologically-inspired geometries. Numerical simulations show that our model predicts blood pressure drops and blood velocities expected in the pulmonary capillaries. Moreover, we replicate partial pressure dynamics of oxygen and carbon dioxide reported in previous studies. This overall behavior is also observed in three-dimensional alveolar geometries, providing more detail associated with the spatial distribution of fields of interest and the influence of the shape of the domain. We envision that this model opens the door for enhanced in silico studies of gas exchange and perfusion on realistic geometries, coupled models of respiratory mechanics and gas exchange, and multi-scale analysis of lung function; furthering our understanding of lung physiology and pathology. Codes are available at https://github.com/comp-medicine-uc/num-model-alv-perfusion-gas-exchange.

Dispersion relation for the dust ionization and dust acoustic waves in the gas discharge complex plasma

A theoretical approach is developed for the dust ionization (DIW) and dust acoustic waves (DAW) propagating in the cloud of microparticles in the low-pressure gas discharge under microgravity conditions. The theory explores the fluid approximation for the microparticle subsystem of complex plasma combined with the kinetic equation for the ions. In the one-dimensional approximation, the wave equation is obtained whose solution defines the dispersion relation for the waves in complex plasma involving the oscillations of microparticles. Obtained dispersion relation unifies both the dust ionization and the dust acoustic waves (DIW and DAW, respectively). According to this dispersion relation, the effect of microparticles on the recombination rate leads to a number of peculiarities. Among them are existence of the minimum frequency, above which the wave propagation is possible, a weak dependence of the DIW wave number on the frequency, and a high phase velocity of DIW as compared to DAW. It is demonstrated that no instability of DIW is possible, whereas DAW can reveal instability under proper conditions. Calculation results correlate with those obtained in a recent experiment.

Quantification of Exposure Level in a Reverberation Chamber for a Large-Scale Animal Study

The US National Toxicology Program (NTP) conducted a long-term carcinogenic and toxicity study with small animals on the effects of mobile phone radiation at 900 MHz and 1900 MHz, respectively, and showed an increase in incidence of malignant schwannomas in the heart of male rats at very higher exposure levels. To verify and clarify the NTP study results, Japan and Korea are conducting a joint study using the same reverberation chamber type exposure system as in the NTP study. The purpose of this paper is to derive a quantitative relationship between the electric field strength in the reverberation chamber and the exposure level, i.e., the mean whole-body averaged specific absorption rate (WBA <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\_$</tex-math></inline-formula> SAR) of rats based on a large-scale finite difference time domain (FDTD) simulation. First, two FDTD simulation methods, one with a single plane wave incidence and the other with simultaneous plane wave incidence, were compared, and then the former method was adopted for convenience and saving calculation time. The derived quantitative relationship between the WBA <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\_$</tex-math></inline-formula> SAR and the electric field strength required to achieve the WBA <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\_$</tex-math></inline-formula> SAR consists of a one-dimensional approximation model which considers only the body mass of rats and a two-dimensional approximation model which considers not only the body mass but also the number of rats. The two approximation models cover the entire two-year exposure period, and the exposure level in the two-year long-term exposure experiment is being well controlled. Moreover, unlike dosimetry in the NTP study, an uncertainty analysis of numerical dosimetry was performed in more details, and the combined standard uncertainty in the WBA <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\_$</tex-math></inline-formula> SAR was found to be 16.1 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> .

Розрахунково-методичне забезпечення для проведення комплексних розрахунків надзвукового обтікання летальних апаратів з прямоточними повітряно-реактивними двигунами

This paper discusses the use of the authors’ fast methods and programs for the calculation of 3D supersonic flow about a flying vehicle and thermogas dynamic processes in the components of an airframe-integrated ramjet. To conduct fast comprehensive calculations, use is made of marching methods, which are two to three orders of magnitude faster than pseudoviscosity methods. 3D supersonic flows about the airframe, in the inlet section of the air intake, and in the exhaust jet are calculated using a “viscous layer” model or Godunov’s scheme for the inviscid approximation. Subsonic flows in the outlet section of the air intake and in the combustion chamber are calculated using a “narrow channel” or a quasi-one-dimensional model. The elements of the presented methods and programs that complement a previously proposed fast comprehensive model are described in more detail. A method for assigning the spatial shape of the flying vehicle surface and the ramjet duct walls is described. A simplified approach to determining the critical area of the exit nozzle in the one-dimensional approximation is proposed. The paper substantiates the advantages of marching methods over pseudoviscosity ones in the predesigning of ramjets with direct account for flow choking, which may occur in the combustion chamber or the exit nozzle. The calculated 3D flows in the individual components and the full assembly of a stylized-shape flying vehicle are presented. The main advantages of the proposed methods and programs are their comprehensiveness and fast computation speed. Their use in the calculation of 3D supersonic flow about a ramjet flying vehicle shortens the ramjet component predesigning time.

Lateral Interaction of Cylindrical Transmembrane Peptides in a One-Dimensional Approximation

Lateral Interaction of Cylindrical Transmembrane Peptides in a One-Dimensional Approximation

Droplet formation simulation using mixed finite elements

Droplet formation happens in finite time due to the surface tension force. The linear stability analysis is useful to estimate the size of a droplet but fails to approximate the shape of the droplet. This is due to a highly nonlinear flow description near the point where the first pinch-off happens. A one-dimensional axisymmetric mathematical model was first developed by Eggers and Dupont [“Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] using asymptotic analysis. This asymptotic approach to the Navier–Stokes equations leads to a universal scaling explaining the self-similar nature of the solution. Numerical models for the one-dimensional model were developed using the finite difference [Eggers and Dupont, “Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] and finite element method [Ambravaneswaran et al., “Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops,” Phys. Fluids 14, 2606–2621 (2002)]. The focus of this study is to provide a robust computational model for one-dimensional axisymmetric droplet formation using the Portable, Extensible Toolkit for Scientific Computation. The code is verified using the Method of Manufactured Solutions and validated using previous experimental studies done by Zhang and Basaran [“An experimental study of dynamics of drop formation,” Phys. Fluids 7, 1184–1203 (1995)]. The present model is used for simulating pendant drops of water, glycerol, and paraffin wax, with an aspiration of extending the application to simulate more complex pinch-off phenomena.

Accuracy of one-dimensional approximation in neutron star quasi-normal modes

Since the eigenfrequency of gravitational waves from cold neutron stars becomes a complex number, where the real and imaginary parts respectively correspond to an oscillation frequency and damping rate, one has to somehow solve the eigenvalue problem concerning the eigenvalue in two-dimensional parameter space. To avoid this bother, one sometimes adopts an approximation, where the eigenvalue is in one-dimensional parameter space. In this study, first, we show the accuracy of the zero-damping approximation, which is one of the one-dimensional approximations, for the fundamental and 1st pressure modes. But, this approximation is not applicable to the spacetime mode, because the damping rate of the spacetime mode is generally comparable to the oscillation frequency. Nevertheless, we find the empirical relation for the ratio of the imaginary part to the real part of the eigenfrequency, which is expressed as a function of the steller compactness almost independently of the adopted equations of state for neutron star matter. Adopting this empirical relation, one can express the eigenfrequency in terms of just the real part, i.e., the problem to solve becomes an eigenvalue problem with a one-dimensional eigenvalue. Then, we find that the frequencies are estimated with good accuracy even with such approximations even for the 1st spacetime mode.

Steady state diffusion in tubular structures: Assessment of one-dimensional models

Steady-state diffusion in long axisymmetric structures is considered. The goal is to assess one-dimensional approximations by comparing them with axisymmetric eigenfunction expansions. Two problems are considered in detail: a finite tube with one end that is partly absorbing and partly reflecting; and two finite coaxial tubes with different cross-sectional radii joined together abruptly. Both problems may be modelled using effective boundary conditions, containing a parameter known as the trapping rate. We show that trapping rates depend on the lengths of the finite tubes (and that they decay slowly as these lengths increase) and we show how trapping rates are related to blockage coefficients, which are well known in the context of potential flow along tubes of infinite length.

On a Strong Increase in the Amplitude of the Wave Function of a Massive Nonrelativistic Particle Incident on a Crystal (One-Dimensional Approximation)

On a Strong Increase in the Amplitude of the Wave Function of a Massive Nonrelativistic Particle Incident on a Crystal (One-Dimensional Approximation)

Improved capillary rheometry for viscous Newtonian filaments

The efficacy of the kinematic differential analysis of McCarroll et al. [“Differential analysis of capillary breakup rheometry for Newtonian liquids,” J. Fluid Mech. 804, 116 (2016)], expanding on McKinley and Tripathi [“How to extract the Newtonian viscosity from capillary breakup measurements in a filament rheometer,” J. Rheol. 44, 653 (2000)], to evaluate the surface tension to viscosity ratio for Newtonian filaments is examined. The analysis is valid during and after stretch, while the latter is traditionally applied after cessation of stretch as the midfilament radius approaches zero. Through numerical simulations, the evaluation of viscosity is investigated for two common stretch histories: (a) the ramp function and (b) the modified step strain. The challenges with stretch are twofold: rapid stretch (large capillary number) results in a nearly cylindrical filament with a rapid change near the plate that challenges the one-dimensional (1D) approximation, while slow stretch results in a nearly static solution with limited viscous information. We examine the capillary number-aspect ratio parameter space and find the ramp function with small stretching speeds optimal. Hence, the most accurate measurements are taken while the filament is being stretched.

One-dimensional modeling approaches for the piloted ignition of poly (methyl methacrylate)

An unsteady one-dimensional solid-gas model is presented for the spontaneous and the piloted ignition of poly (methyl methacrylate) exposed to thermal irradiance. The mathematical description of the igniter (electrical spark) is given special attention. The simplest approach, namely the heating spark model, describes heat transfer from a pulsating high-temperature plane source (direct heating), together with the heat released by the pre-ignition reactions (indirect heating). A high overestimation of the thermal effects of the spark is observed owing to the one-dimensional approximation. An iterative solution method, indicated as the non-heating spark model, is also implemented to eliminate direct and indirect spark heating which, however, also cuts off possible flashing phenomena. The third approach, indicated as the chemical spark model, disregards the direct spark heating but still describes pre-ignition combustion. This approach is the simplest and the most effective for providing predictions in agreement with literature measurements for both the process dynamics and the ignition delay times.

Influence of volume and aspect ratio of liquid bridges on capillary breakup rheometry

Capillary thinning of liquid bridges is routinely used for extensional rheology of Newtonian and complex fluids. Although it is expected that the volume and aspect ratio of a liquid bridge significantly influence its dynamics, the role played by these parameters in rheological characterization has not been previously studied. We perform numerical simulations of Newtonian as well as viscoelastic liquid bridges with the one-dimensional slender-filament approximation of Eggers and Dupont [“Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] and Ardekani et al. [“Dynamics of bead formation, filament thinning and breakup in weakly viscoelastic jets,” J. Fluid Mech. 665, 46–56 (2010)]. Sample volume and bridge aspect ratio control two phenomena that can adversely impact rheological characterization: the tendency to form satellite drops at the necking plane and the slowing down of capillary thinning due to the proximity (in parameter space) of the liquid-bridge stability boundary. The optimal range of these parameter values to avoid drop formation and slowdown is discussed.

Theoretical analysis of time-dependent jetting on the surface of a thin moving liquid layer

Unsteady two-dimensional problem of a thin liquid layer with prescribed time-dependent influx into the layer, position of the influx section, and the thickness of the liquid at this section is studied by methods of asymptotic analysis. The ratio of the rate of the liquid thickness variation at the influx section to the influx velocity plays a role of a small parameter of the problem. The influx parameters are such that the flow in the thin layer is inertia dominated, with gravity, surface tension, and liquid viscosity being approximately negligible. Such flows were studied with respect to several applications, some of which are listed in the Introduction. One of the applications concerns with splashing during droplet impact onto a rigid substrate and related kinematic discontinuity propagating along the spray sheet, which is produced by the spreading droplet. This type of splashing was studied by Yarin and Weiss [“Impact of drops on solid surfaces: Self-similar capillary waves, and splashing as a new type of kinematic discontinuity,” J. Fluid Mech. 283, 141–173 (1995)] within a quasi-one-dimensional approach averaging the flow velocity over the layer thickness. We also start with the one-dimensional thin-layer approximation assuming the influx flow is accelerated. Such influx conditions lead to unbounded growth of the thickness of the liquid layer at a certain location and at a certain time instant within the one-dimensional approach. The present study recovers for the first time the structure of the flow close to this singularity using methods of asymptotic analysis. To this aim, the second-order outer solution, which is valid outside the region of the unbounded flow, is derived. The second-order outer solution is used to find proper stretched inner variables and the equations governing the inner flow at the leading order. It is shown that the inner free-surface flow in the stretched variables is two-dimensional, potential, non-linear, and independent of any parameters of the original problem.

Asymptotic analysis of subglacial plumes in stratified environments

Accurate predictions of basal melt rates on ice shelves are necessary for precise projections of the future behaviour of ice sheets. The computational expense associated with completely resolving the cavity circulation using an ocean model makes this approach unfeasible for multi-century simulations, and parametrizations of melt rates are required. At present, some of the most advanced melt rate parametrizations are based on a one-dimensional approximation to the melt rate that emerges from the theory of subglacial plumes applied to ice shelves with constant basal slopes and uniform ambient ocean conditions; in this work, we present an asymptotic analysis of the corresponding equations in which non-constant basal slopes and typical ambient conditions are imposed. This analysis exploits the small aspect ratio of ice shelf bases, the relatively weak thermal driving and the relative slenderness of the region separating warm, salty water at depth and cold, fresh water at the surface in the ambient ocean. We construct an approximation to the melt rate that is based on this analysis, which shows good agreement with numerical solutions in a wide variety of cases, suggesting a path towards improved predictions of basal melt rates in ice-sheet models.