One-dimensional Solution Research Articles

Variational interface element model for 2D and 3D hydraulic fracturing simulations

This paper presents an extension of the variational interface fracture model proposed in Khisamitov and Meschke (2018) to model fluid driven fracture in porous materials. The fluid saturated material is described by the theory of poroelasticity and the effective stress concept, while the coupling between fluid flow and fracture initiation and propagation is accomplished via a variational interface element formulation, using a damage variable c as an additional degree of freedom in conjunction with linear momentum and mass balance equations along the interface zones. Introducing the damage variable c, fracture propagates according to the values of the minimizers of the total potential energy expressed in terms of effective stresses. Hence, neither local crack propagation criteria nor tracking algorithms are required. Taking into account that a fracture grows quasi-statically, the discretized system of PDEs is solved by the backward Euler-scheme, ignoring the contribution from inertia forces. In the Newton–Raphson iterative solution procedure, an operator splitting algorithm is employed to solve first the poroelastic equations, and updating subsequently the damage variable c. The proposed model for hydraulic fracturing is validated first by means of one-dimensional analytical solutions in toughness as well as in viscosity dominated regimes. The performance and the applicability of the model to simulate interactions between propagating and existing fractures is demonstrated by means of 2D and 3D numerical examples.

Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space

<p style='text-indent:20px;'>The Allen–Cahn–Nagumo equation is a reaction-diffusion equation with a bistable nonlinearity. This equation appears to be simple, however, it includes a rich behavior of solutions. The Allen–Cahn–Nagumo equation features a solution that constantly maintains a certain profile and moves with a constant speed, which is referred to as a traveling wave solution. In this paper, the entire solution of the Allen–Cahn–Nagumo equation is studied in multi-dimensional space. Here an entire solution is meant by the solution defined for all time including negative time, even though it satisfies a parabolic partial differential equation. Especially, this equation admits traveling wave solutions connecting two stable states. It is known that there is an entire solution which behaves as two traveling wave solutions coming from both sides in one dimensional space and annihilating in a finite time and that this one-dimensional entire solution is unique up to the shift. Namely, this entire solution is symmetric with respect to some point. There is a natural question whether entire solutions coming from all directions in the multi-dimensional space are radially symmetric or not. To answer this question, radially asymmetric entire solutions will be constructed by using super-sub solutions.

Meshless modeling of coupled transdermal pharmacokinetics with analytical validation

An efficient localized collocation meshless computational approach is applied to the study of transdermal drug delivery (TDD). To validate the numerical technique, a one-dimensional analytical solution is first derived for the set of coupled differential equations that govern the pharmacokinetic process in the two-compartment compound diffusion model. These equations consist of one partial differential equation (PDE) and one ordinary differential equation (ODE). These equations are solved in one-dimensional space via superposition and separation of variables to arrive at the field solution for the compound concentration in the tissue. The convergence of the numerical result to the analytical result indicates the viability of the localized collocation meshless method (LCMM) as a valid approach to the solution of TDD problems. The analytical solution is used to benchmark the numerical technique for several cases with differing pharmacokinetic coefficients, demonstrating the robustness and accuracy of the meshless numerical technique. Furthermore, the highly efficient LCCM implementation is extended to address multi-dimensional TDD cases achieving stable and physically consistent results.

Shocks make the Riemann problem for the full Euler system in multiple space dimensions ill-posed

The question of (non-)uniqueness of one-dimensional self-similar solutions to the Riemann problem for hyperbolic systems of gas dynamics in the class of multi-dimensional admissible weak solutions was addressed in recent years in several papers culminating in [] with the proof that the Riemann problem for the isentropic Euler system with a power law pressure is ill-posed if the one-dimensional self-similar solution contains a shock. Then the natural question arises whether the same holds also for a more involved system of equations, the full Euler system. After the first step in this direction was made in [], where ill-posedness was proved in the case of two shocks appearing in the self-similar solution, we prove in this paper that the presence of just one shock in the self-similar solution implies the same outcome, i.e. the existence of infinitely many admissible weak solutions to the multi-dimensional problem.

Integral sentences and numerical comparative calculations for the validity of the dispersion model for air pollutants AUSTAL2000

Integral sentences and numerical comparative calculations for the validity of the dispersion model for air pollutants AUSTAL2000

Evaluating the dynamics of groundwater, lakebed transport, nutrient inflow and algal blooms in Upper Klamath Lake, Oregon, USA

Transport of nutrients to lakes can occur via surface-water inflow, atmospheric deposition, groundwater (GW) inflow and benthic processes. Identifying and quantifying within-lake nutrient sources and recycling processes is challenging. Prior studies in hypereutrophic Upper Klamath Lake, Oregon, USA, indicated that ~60% of the early summer phosphorus (P) load to the lake was internal and hypothesized to be lakebed sediment release. Dynamic nutrient transport processes were examined to better characterize the nutrient sources. One-dimensional heat transport models calibrated to observed lakebed temperatures and a cross-sectional GW flow model provided estimates of GW-inflow rates that were greatest in spring and decreased through summer. One-dimensional solute transport models calibrated to observed lakebed pore-water dissolved silica (Si) and dissolved phosphate-phosphorus (DP) concentrations indicated that nutrients were transported from the lakebed by advection, diffusion, and enhanced mixing by benthic organisms and waves, and that DP removal occurred near the lakebed interface. Estimated water, Si, DP and total-phosphorus (TP) budgets indicated that GW contributed 21% of lake water inflow and at least 26, 20 and 16% of total Si, DP and TP inflow, respectively, when conservatively assuming background GW nutrient concentrations. However, lakebed GW (LGW) is enriched in nutrients during flow through lakebed sediment and the estimated GW contribution increased to 29 (33), 49 (67) and 43% (61%) of total Si, DP and TP inflow, respectively, if 20% (50%) of GW inflow to the lake was assumed to have LGW concentrations. Net nutrient inflow to the lake was greatest in spring and coincident with the annual diatom bloom. Inflowing dissolved nutrients appear to be assimilated by diatoms during the spring and become available for the summer Aphanizomenon flos-aquae bloom when the diatoms senesce. Thus, nutrient-enriched GW inflow and nutrient recycling by successive algal blooms must be considered when evaluating internal nutrient loading to lakes.

Semilinear integro-differential equations, II: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation

This paper addresses saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator with a radially symmetric kernel $K$, and $f$ is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone $\{(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : \, |x'| = |x''|\}$, and vanish only in this set. We establish the uniqueness and the asymptotic behavior of the saddle-shaped solution. For this, we prove a Liouville type result, the one-dimensional symmetry of positive solutions to semilinear problems in a half-space, and maximum principles in "narrow" sets. The existence of the solution was already proved in part I of this work.

A two-dimensional analytical model for organic contaminants transport in a transition layer-cutoff wall-aquifer system

Cutoff walls are widely used as in-situ vertical barriers to control pollution plumes migration in aquifer. A two-dimensional analytical model for the transport of organic contaminants through a layered subsurface system with a cutoff wall is developed. The model considers a coupled advection-dispersion-adsorption-degradation process in the whole system and a non-uniform concentration distribution of a contaminant source with respect to depth. The solution was effectively validated against a one-dimensional analytical solution and a two-dimensional numerical solution. The migration behavior of contaminants in the triple-layer system and the service performance of a cutoff wall are investigated. It is demonstrated that the previous one-dimensional contaminant migration solution would underestimate the polluted ranges but overestimate the migration distance. The consideration of a non-uniformly distributed contamination source is of great importance, especially for narrowly contaminated aquifer. The biodegradation in a cutoff wall has a significant influence on the transport of an organic contaminant through the layered system when its half-life in the cutoff wall is less than 5 years, and this effect would be enhanced when the biodegradation in natural soil is also considered. The proposed model is a practical tool for performing a comprehensive design of the location and thickness of a cutoff wall.

Material systems for FM-/AFM-coupled skyrmions in Co/Pt-based multilayers

By means of systematic first-principles calculations based on density functional theory we search for suitable materials that can host antiferromagnetically coupled skyrmions. We concentrate on fcc-stacked (111)-oriented metallic $Z$/Co/Pt ($Z=4d$ series: Y$-$Pd, the noble metals: Cu, Ag, Au, post noble metals: Zn and Cd) magnetic multilayers of films of monatomic thickness. We present quantitative trends of magnetic properties: Magnetic moments, interlayer exchange coupling, spin-stiffness, Dzyaloshinskii-Moriya interaction, magnetic anisotropy, and the critical temperature. We show that some of the $Z$ elements (Zn, Y, Zr, Nb, Tc, Ru, Rh, and Cd) can induce antiferromagnetic interlayer coupling between the magnetic Co layers, and that they influence the easy magnetization axis. Employing a multiscale approach, we transfer the micromagnetic parameters determined from $ab$ $initio$ to a micromagnetic energy functional and search for one-dimensional spin-spiral solutions and two-dimensional skyrmions. We determine the skyrmion radius by numerically solving the equation of the skyrmion profile. We found an analytical expression for the skyrmion radius that covers our numerical results and is valid for a large regime of micromagnetic parameters. Based on this expression we have proposed a model that allows to extrapolate from the $ab$ $initio$ results of monatomic films to multilayers with Co films consisting of several atomic layers containing $10\,$nm skyrmions. We found thickness regimes where tiny changes of the film thickness may alter the skyrmion radius by orders of magnitude. We estimated the skyrmion size as function of temperature and found that the size can easily double going from cryogenic to room temperature. We suggest promising material systems for ferromagnetically and antiferromagnetically coupled spin-spiral and skyrmion systems.

Experimental investigation of the turbulent cascade development by injection of single large-scale Fourier modes

The current work presents an experimental investigation of the dynamic interactions between flow scales caused by repeated actions of the nonlinear term of the Navier-Stokes equation. Injecting a narrow band oscillation, representing a single Fourier mode, into a round jet flow allows the measurement of the downstream generation and development of higher harmonic spectral components and to measure when these components are eventually absorbed into fully developed turbulence. Furthermore, the dynamic evolution of the measured power spectra observed corresponds well to the measured cascaded delays reported by others. Closely matching spectral development and cascade delays have also been derived directly from a one-dimensional solution of the Navier-Stokes equation described in a companion paper. The results in the current work provide vital information about how initial conditions influence development of the shape of the spectrum and about the extent of the time scales in the triad interaction process, which should be of significance to turbulence modelers.

Uncoupling Electrokinetic Flow Solutions

The continuum-scale electrokinetic porous-media flow and excess charge redistribution equations are uncoupled using eigenvalue decomposition. The uncoupling results in a pair of independent diffusion equations for "intermediate" potentials subject to modified material properties and boundary conditions. The fluid pressure and electrostatic potential are then found by recombining the solutions to the two intermediate uncoupled problems in a matrix-vector multiply. Expressions for the material properties or source terms in the intermediate uncoupled problem may require extended precision or careful re-writing to avoid numerical cancellation, but the solutions themselves can be computed in typical double precision. The approach works with analytical or gridded numerical solutions and is illustrated through two examples. The solution for flow to a pumping well is manipulated to predict streaming potential and electroosmosis, and a periodic one-dimensional analytical solution is derived and used to predict electroosmosis and streaming potential in a laboratory flow cell subjected to low frequency alternating current and pressure excitation. The examples illustrate the utility of the eigenvalue decoupling approach, repurposing existing analytical solutions and leveraging simpler-to-derive solutions or numerical models for coupled physics.

Solute dispersion in bifurcating networks

Abstract

Solute transport model equation for mobile phase in semi-infinite porous media

An analytical solution of one-dimensional solute transport model equation for mobile phase is derived for semi-infinite heterogeneous porous media. The transport equation considers advection, longitudinal dispersion, linear isotherm, and porosity in its formulation. The dispersion is assumed to be Fickian, characterized by a constant dispersion coefficient. Temporally dependent velocity patterns are employed for the prediction of contaminant concentration. The analytical solution is compared with two reported solutions and can be used for benchmarking numerical codesand also as a preliminary predictive tool for groundwater management. Also, the obtained analytical solution of the proposed model may be helpful in the groundwater hydrology areas.

A generic velocity-free calculation method of the blasting wave in nuclear power plants for designing venting system

Venting system, mainly composed of bursting discs, is designed to protect the structural integrity of the cabins in nuclear power plants (NPPs). During a loss of coolant accident (LOCA), much gas is rapidly released into the cabin, forming a kind of pressure longitudinal wave (PLW) named “blasting wave” and challenge the mechanical strength of the envelope structure. Simulation of the blasting wave is meaningful for designing bursting discs. CFD is a generic method for cabins with arbitrary geometric features, which consumes much time and computing resource; There exists a mathematical model based on wave equation describing PLW, but it has never been applied in the field of nuclear reactor accident because of some bottlenecks. In this study, the mathematical model based on wave equation is transformed into a generic method of simulating PLW without solving velocity field, which is in the form of a velocity-free PDE problem with velocity-related initial and boundary conditions. The boundary condition is not Dirichlet, Neumann nor Robbin, and provides condition for one-dimensional analytical solution. The problem can also be solved by finite element method (FEM). Both solutions are validated by CFD simulation in our previous study and can be extended to higher dimensions. In engineering application, the analytical solution can be directly used if the cabin’s geometry is relatively simple, while FEM solution can be used on a coarse meshing if the geometry is complex. Both applications are convenient compared with CFD method.

Evaluation of Thermal Contact Resistance Between Two Solid Surfaces Using Photoacoustic Technique

Thermal contact resistance is one of the factors that complicates heat transfer between the interfaces of different components. Because heat transfer is influenced by several factors, it is difficult to estimate all the influencing factors for practical applications. In this study, the thermal contact resistance between two solid surfaces was evaluated using the photoacoustic (PA) technique. The samples consisted of a metal foil and metal cylinders, which had various surface roughnesses at the contact faces measured at the same contact pressure. A one-dimensional analytical solution of the phase lag of the PA signal showed a similar trend with the experimental results. However, there were some quantitative differences between the two groups. By comparing the numerical calculation results for the developed heat conduction model of a cylindrical coordinate system with the experimental results, we found that the in-plane distribution of thermal contact resistance at the contact face contributed to the deviation between the experimental results and the numerical results. The thermal contact resistance was evaluated by considering the in-plane distribution of the thermal contact resistance, and reasonable values of thermal contact resistance were obtained.

Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation

The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized diffusion equation, while in the second it is a generalized wave equation. The corresponding memory kernels are expressed in both cases in terms of Mittag–Leffler functions. Explicit representations for the one-dimensional fundamental solution and the mean squared displacement are provided and analyzed analytically and numerically. The one-dimensional fundamental solution is shown to be a spatial probability density function evolving in time, which is unimodal in the diffusion regime and bimodal in the propagation regime. The multi-dimensional fundamental solutions are probability densities only in the diffusion case, while in the propagation case they can have negative values. In addition, two different types of subordination principles are formulated for the two regimes. The Bernstein functions technique is extensively employed in the theoretical proofs.

Flow transiency on analytical modeling of subsurface solute transport

Groundwater flow velocity and dispersivity might be temporally or spatially variable rather than constant. In this paper, linearly-asymptotically or exponentially distance-dependent dispersivities and temporally exponential flow velocity were coupled to the conventional advection-dispersion equation. The mathematical models were established by considering the case of a coupled time-dependent velocity and scale-dependent dispersivities where one-dimensional (1D) semi-analytical solutions were obtained using the Laplace transform in a finite domain. The solution was verified by comparing it with a numerical solution, based on finite-element COMSOL Multiphysics. The impacts of different parameters of time-dependent flow velocity and scale-dependent dispersivities on breakthrough curves (BTCs) were thoroughly analyzed. The results show that a slight change of time-dependent flow velocity will lead to considerable change of BTCs, meaning that solute transport is sensitive to the temporally variable flow velocity. Secondly, a larger growth rate of the dispersivity in linear-asymptotically distance-dispersivity function can lead to a faster solute transport at early stage, but a lower concentration at late stage; as for the exponentially distance-dependent function, the growth rate of the dispersivity has the same effects on BTCs. Thirdly, it was observed that an increase in final steady velocity (or asymptotic velocity) will amplify the impacts on solute transport due to advection; as for the asymptotic dispersivity, it has similar impacts on the solute transport due to dispersion. Overall, our results show that the effects of time-dependent flow velocity and distance-dependent dispersivities are not negligible when describing solute transport process in subsurface hydrology.

Exact Solution for Tertiary Polymer Flooding with Polymer Mechanical Entrapment and Adsorption

Two-phase transport with simultaneous component adsorption and mechanical entrapment occurs in numerous suspension and nano-colloidal flows in porous media, including polymer flooding in oilfields. To the contrary of one-dimensional (1D) self-similar solution for polymer flooding that ignores the mechanical entrapment, accounting for this phenomenon breaks self-similarity. However, this non-self-similar problem allows for an exact integration. For the first time, the present work derives an exact solution for polymer injection into a rock saturated by oil and water with simultaneous adsorption and mechanical entrapment. The splitting method that uses Lagrangian coordinate as an independent variable instead of time, splits the 3 × 3 governing system into 2 × 2 auxiliary system that determines suspended and captured polymer concentrations, and the scalar equation for unknown phase saturation. The exact solution reveals two fingerprints of the component entrapment under small entrapped concentrations: stabilisation of the breakthrough concentration at the value below the injected one, and non-stabilised pressure drop growth. Three sets of laboratory polymer core-floodings exhibit a close match by the analytical model.

Modeling One-Dimensional Nonreactive Solute Transport in Open Channel Flows Under Uncertain Flow and Solute Loading Conditions

AbstractThe study of solute transport processes in open channel flows is important for water quality management and environment protection. Solute transported in open channel flows is usually uncer...

Exact solutions for multilayer functionally graded beams bonded by viscoelastic interlayer considering memory effect

This work proposes exact solutions for simply supported multilayer functionally graded (FG) beams with viscoelastic interlayer to predict its time-dependent mechanical behavior. The stresses and deformations of FG layers are described by the two-dimensional (2-D) elasticity theory. The viscoelastic property of the interlayer is simulated by the standard linear solid model, and its constitutive relation is expressed in the convolution form which considers the strain memory effect. The general solutions of FG layers involving undetermined coefficients are obtained by virtue of the Fourier series expansion. By the use of the Laplace transform method, the solutions of stresses and deformations are determined analytically. The present solutions including Fourier series possess good convergence property. The comparison studies indicate that the finite element (FE) solutions approach the present ones when each FG layer is divided into many isotropic sub-layers in FE model and the gradient factor is taken as small value. The relative error of the one-dimensional (1-D) Euler-Bernoulli solutions is slight for slender beam with small gradient factor, but it significantly increases as the beam thickness or the gradient factor grows. Finally, the influences of the material constants on the time-dependent behavior in the beam are investigated in detail.