Multicomponent Equations Research Articles

On the Cauchy problem and peakons of a two-component Novikov system

We study a two-component Novikov system, which is integrable and can be viewed as a two-component generalization of the Novikov equation with cubic nonlinearity. The primary goal of this paper is to understand how multi-component equations, nonlinear dispersive terms and other nonlinear terms affect the dispersive dynamics and the structure of the peaked solitons. We establish the local well-posedness of the Cauchy problem in Besov spaces B with 1 ⩽ p, r ⩽ + ∞, s > max{1 + 1/p, 3/2} and Sobolev spaces Hs(ℝ) with s > 3/2, and the method is based on the estimates for transport equations and new invariant properties of the system. Furthermore, the blow-up and wave-breaking phenomena of solutions to the Cauchy problem are studied. A blow-up criterion on solutions of the Cauchy problem is demonstrated. In addition, we show that this system admits single-peaked solitons and multi-peaked solitons on the whole line, and the single-peaked solitons on the circle, which are the weak solutions in both senses of the usual weak form and the weak Lax-pair form of the system.

An Integrable Matrix Camassa-Holm Equation**Supported by National Natural Science Foundation of China under Grant Nos. 11771186 and 11671177

We present an integrable sl(2)-matrix Camassa-Holm (CH) equation. The integrability means that the equation possesses zero-curvature representation and infinitely many conservation laws. This equation includes two undetermined functions, which satisfy a system of constraint conditions and may be reduced to a lot of known multi-component peakon equations. We find a method to construct constraint condition and thus obtain many novel matrix CH equations. For the trivial reduction matrix CH equation we construct its N-peakon solutions.

Development of multi-component generalized sphere function based gas-kinetic flux solver for simulation of compressible viscous reacting flows

In this paper, a multi-component generalized sphere function based gas-kinetic flux solver is developed for simulation of compressible viscous reacting flows. This work is inspired by the existing simplified gas-kinetic schemes, which use the circular or sphere function to develop single-component gas-kinetic flux solvers. The present solver applies the finite volume method to discretize the multi-component Navier-Stokes equations and evaluate the numerical flux at the cell interface by using the local solution of Boltzmann equation. In order to unify the existing circular and sphere functions, a generalized sphere function is derived from a reduced Maxwellian distribution function by assuming that all the particles are concentrated on an N-dimensional sphere. The present solver is then developed by integrating the generalized sphere function on the sphere surface. To obtain a multi-component solver, the mass fraction from both sides of the cell interface is used to compute the densities of different species. Considering the different physical properties of the species, the internal energy is computed by enthalpy, and the temperature at the cell interface is obtained by Newton iteration. In addition, to control the numerical dissipation, which is relevant to the grid aspect ratio and the chemical reaction, an improved switch function is introduced. Several benchmark problems are simulated to validate the present solver. It is shown that the developed flux solver has a satisfied performance for simulation of multi-component compressible viscous reacting flows.

Development of multicomponent lattice Boltzmann flux solver for simulation of compressible viscous reacting flows.

In this paper, the multicomponent lattice Boltzmann flux solver (LBFS) is developed for simulation of two-dimensional compressible viscous reacting flows. This work is based on the existing LBFS for simulation of single-component compressible flows. The present solver applies the finite volume method to discretize the multicomponent Navier-Stokes equations and evaluates the numerical flux at the cell interface by local solution of the lattice Boltzmann equation. To evaluate numerical flux, the original non-free parameter D1Q4 model in the existing LBFS is extended to the multicomponent counterpart in which the total density at the cell interface is computed directly by summing the density distribution functions, and the densities of different species are calculated from the mass fractions at the left and right sides of cell interface. The internal energy is evaluated from the enthalpy which considers the different physical properties of the species, and the temperature at the cell interface is obtained by Newton iteration. In addition, an improved switch function which takes into account the reacting effects and aspect ratio of the grid is introduced to control the numerical dissipation. Several benchmark problems are simulated to validate the present multicomponent LBFS. It is shown that the present solver is carbuncle-free for the unfavorable aspect ratio grid in the test cases here and has a satisfied performance for simulation of multicomponent compressible viscous reacting flows.

The conservative splitting domain decomposition method for multicomponent contamination flows in porous media

In the paper, a new conservative splitting decomposition method (S-DDM) is developed for computing nonlinear multicomponent contamination flows in porous media over multi-block sub-domains. On each block-divided sub-domain, we take three steps to solve the coupled nonlinear system of water-head equation and multicomponent concentration equations in each time interval. The interface Darcy's velocity and the interface global concentration fluxes are first predicted by the semi-implicit flux schemes, while the solutions of water-head and multicomponent concentrations, Darcy's velocity and global concentration fluxes in the interiors of sub-domains are computed by one-directional splitting implicit solution-flux coupled schemes on staggered meshes, and finally the interface Darcy velocity and global concentration fluxes are corrected by the interior solutions. The significance of our scheme is that while it keeps the advantages of the non-overlapping domain decomposition and the splitting technique, it preserves mass on the whole domain of domain decompositions. Numerical experiments are presented to illustrate the excellent performance of our proposed conservative S-DDM approach for computing nonlinear multicomponent contamination flows in groundwater. The developed algorithm of the conservative S-DDM works efficiently over multiple block-divided sub-domains, which can be applied in simulation of large scale multicomponent contamination flows in parallel computing.

Kinetic flux vector splitting scheme for solving non-reactive multi-component flows

This paper is about multi-component flow. There is no doubt that multi-component flow has a wide range of applications, specially in aerospace it plays a vital role during reentry of space ship into earth's atmosphere thats why it cannot be neglected for a proper vehicle design. In this paper one- and two-dimensional homogenous multi-component flow models are numerically investigated by using a high resolution splitting scheme and this scheme is known as Kinetic Flux Vector Splitting scheme. This scheme preserves positivity conditions and resolves shocks, rarefaction and contact discontinuity. The scheme is based on splitting of flux functions. Moreover Runge-Kutta time stepping technique with MUSCL-type initial reconstruction is used to guarantee higher order accurate solution. This work is first done by Qamar and Warnecke (2004) for the homogeneous multi-component flow equations using central scheme, here we investigate the same work using kinetic flux vector splitting scheme (KFVS) and compared the results with central scheme to verify the efficiency of studied scheme.

LPSE: A 3-D wave-based model of cross-beam energy transfer in laser-irradiated plasmas

The new component of the “laser–plasma simulation environment” (LPSE) described here is a practical numerical model that solves the coupled vector equations for the propagation of nearly monochromatic, coherent, electromagnetic waves in inhomogeneous unmagnetized plasmas. It operates efficiently in the numerically challenging semiclassical regime where characteristic plasma scale lengths are many times greater than the wavelength of the light and solutions are highly oscillatory. Solutions can be obtained in one, two, or three spatial dimensions and time. The model includes the effects of nonlinear coupling of electromagnetic waves to the low-frequency plasma perturbations (i.e., ion-acoustic response) that are responsible for stimulated Brillouin scattering. Induced plasma perturbations are assumed to be imposed on a prescribed large-scale inhomogeneous background that includes spatially varying plasma density and flow. Our code is directly relevant to the problem of cross-beam energy transfer in laser-driven inertial confinement fusion. It may also be applicable in other areas where eikonal solutions of multicomponent wave equations (or coupled wave equations) are insufficient, such as optical scattering from ultrasound, electron dynamics in quantum devices or in nanoscale light–matter interactions.

Strict Equivalence between Maxwell–Stefan and Fast-Mode Theory for Multicomponent Polymer Mixtures

The applicability of theories describing the kinetic evolution of fluid mixtures depends on the underlying physical assumptions. The Maxwell-Stefan equations, widely used for miscible fluids, express forces depending on coupled fluxes. They need to be inverted to recover a Fickian form which is generally impossible analytically. Moreover, the concentration dependence of the diffusivities has to be modelled, e.g. by the multicomponent Darken equation. Cahn-Hilliard type equations are preferred for immiscible mixtures, whereby different assumptions on the coupling of fluxes lead to the slow-mode and fast-mode theories. For two components, these were derived from the Maxwell-Stefan theory in the past. Here, we prove that the fast-mode theory and the generalized Maxwell-Stefan theory together with the multicomponent Darken equation are strictly equivalent even for multicomponent systems with very different molecular sizes. Our findings allow to reduce the choice of a suitable theory to the most efficient algorithm for solving the underlying equations.

Parallel reservoir simulators for fully implicit complementarity formulation of multicomponent compressible flows

The numerical simulation of multicomponent compressible flow in porous media is an important research topic in reservoir modeling. Traditional semi-implicit methods for such problems are usually conditionally stable, suffer from large splitting errors, and may accompany with violations of the boundedness requirement of the numerical solution. In this study we reformulate the original multicomponent equations into a nonlinear complementarity problem and discretize it using a fully implicit finite element method. We solve the resultant nonsmooth nonlinear system of equations arising at each time step by a parallel, scalable, and nonlinearly preconditioned semismooth Newton algorithm, which is able to preserve the boundedness of the solution and meanwhile treats the possibly imbalanced nonlinearity of the system. Some numerical results are presented to demonstrate the robustness and efficiency of the proposed algorithm on the Tianhe-2 supercomputer for both standard benchmarks as well as realistic problems in highly heterogeneous media.

Extended Z-Invariance for Integrable Vector and Face Models and Multi-component Integrable Quad Equations

In a previous paper, the author has established an extension of the Z-invariance property for integrable edge-interaction models of statistical mechanics, that satisfy the star-triangle relation (STR) form of the Yang-Baxter equation (YBE). In the present paper, an analogous extended Z-invariance property is shown to also hold for integrable vector models and interaction-round-a-face (IRF) models of statistical mechanics respectively. As for the previous case of the STR, the Z-invariance property is shown through the use of local cubic-type deformations of a 2-dimensional surface associated to the models, which allow an extension of the models onto a subset of next nearest neighbour vertices of $\mathbb{Z}^3$, while leaving the partition functions invariant. These deformations are permitted as a consequence of the respective YBE's satisfied by the models. The quasi-classical limit is also considered, and it is shown that an analogous Z-invariance property holds for the variational formulation of classical discrete Laplace equations which arise in this limit. From this limit, new integrable 3D-consistent multi-component quad equations are proposed, which are constructed from a degeneration of the equations of motion for IRF Boltzmann weights.

High-speed video microscopy and numerical modeling of bubble dynamics near a surface of urinary stone.

Ultra-high-speed video microscopy and numerical modeling were used to assess the dynamics of microbubbles at the surface of urinary stones. Lipid-shell microbubbles designed to accumulate on stone surfaces were driven by bursts of ultrasound in the sub-MHz range with pressure amplitudes on the order of 1 MPa. Microbubbles were observed to undergo repeated cycles of expansion and violent collapse. At maximum expansion, the microbubbles' cross-section resembled an ellipse truncated by the stone. Approximating the bubble shape as an oblate spheroid, this study modeled the collapse by solving the multicomponent Euler equations with a two-dimensional-axisymmetric code with adaptive mesh refinement for fine resolution of the gas-liquid interface. Modeled bubble collapse and high-speed video microscopy showed a distinctive circumferential pinching during the collapse. In the numerical model, this pinching was associated with bidirectional microjetting normal to the rigid surface and toroidal collapse of the bubble. Modeled pressure spikes had amplitudes two-to-three orders of magnitude greater than that of the driving wave. Micro-computed tomography was used to study surface erosion and formation of microcracks from the action of microbubbles. This study suggests that engineered microbubbles enable stone-treatment modalities with driving pressures significantly lower than those required without the microbubbles.

Dressing Method for the Multicomponent Short-Pulse Equation

We use the Zakharov—Shabat dressing method to solve the multicomponent short-pulse equation. We obtain dressed solutions of this equation using a Riemann—Hilbert problem. The dressed solutions of the Lax pair and the multicomponent short-pulse equation are expressed in terms of Hermitian projectors. We show that the dressed solutions are related to quasideterminant solutions, and we write K-soliton solutions in terms of quasideterminants. In explicit form, we obtain one- and two-soliton solutions.

Dressing method for the multicomponent short-pulse equation

Метод одевания Захарова-Шабата используется для решения многокомпонентного уравнения коротких импульсов. Одетые решения этого уравнения получаются с помощью задачи Римана-Гильберта. Одетые решения пары Лакса и многокомпонентного уравнения коротких импульсов выражены через эрмитовы проекторы. Показано, что одетые решения также связаны с квазидетерминантными решениями, и $K$-солитонные решения записаны в терминах квазидетерминантов. В явном виде получены одно- и двухсолитонные решения.

A New Model of Displacement Efficiency of Gas Hydrate Considering the Influence of Temperature

The model of natural gas hydrate displacement failed to accurately analysis the influence degree of temperature, and it is not accurate enough to calculate the volume of gas under high pressure owing to the influence of compressibility factor, which leading to a result that the model has great error in calculating the displacement efficiency. In this paper, a new model of displacement efficiency affected by temperature is derived in the design of displacement experiment based on the law of conservation of mass, Multicomponent gas equation of state, Multicomponent thermodynamic phase equilibrium principle at the first. Next, several methods for calculating the compressibility factor of natural gas at different temperatures and pressures are compared by using Visual Basic software, and the Setzmann equation with the smallest error is selected. Finally, compared with the new model, the traditional model as well as the experimental data, the results show that the new model is closer to the experimental data. At the same time, the displacement efficiency of natural gas hydrate at different temperatures is analyzed. It is suggested that higher displacement efficiency can be obtained by reasonably increasing the temperature of the displacement reaction system.

Analytical Cartesian solutions of the multi-component Camassa-Holm equations

Here, we give the existence of analytical Cartesian solutions of the multi-component Camassa-Holm (MCCH) equations. Such solutions can be explicitly expressed, in which the velocity function is given by u = b (t) + A(t)x and no extra constraint on the dimension N is required. The advantage of our method is that we turn the process of analytically solving MCCH equations into algebraically constructing the suitable matrix A(t). As the applications, we obtain some interesting results: 1) If u is a linear transformation on x ∈ ℝN , then p takes a quadratic form of x. 2) If A = f(t)I + D with DT = −D, we obtain the spiral solutions. When N = 2, the solution can be used to describe “breather-type” oscillating motions of upper free surfaces. 3) If \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A = ({\ extstyle{{{{\\mathop \\alpha \\limits^. }_i}} \\over {{\\alpha _i}}}})N \ imes N,$$\\end{document}, we obtain the generalized elliptically symmetric solutions. When N = 2, the solution can be used to describe the drifting phenomena of the shallow water flow.

Coupling scheme of multicomponent sectional equations and Mason equations via transition rate matrix of hygroscopic growth applied to international standard problem No. 44

Most fission products are released into the containment as aerosol particles during severe accidents. In the conventional codes, two sets of equations are used to analyze the aerosol behaviors in the containment. The first set of equations are multicomponent sectional equations, solutions of which are given with the mass concentrations in each discretized section. The multicomponent sectional equations require very large number of sections to remove the numerical diffusion when hygroscopic growth occurs with other aerosol dynamics simultaneously. For efficient calculations, Mason Equations are used to analyze the hygroscopic growth. The solutions of the equations, on the other hand, are given with the radius of an individual particle. Therefore, it is essential to couple the two equations in rigorous manner for efficient analyses on the aerosol behaviors when hygroscopic growth and other aerosol dynamics occur simultaneously. In this paper, we propose a new formulation for the coupling scheme of multicomponent sectional equations and Mason equations. In the formulation, the two equations are coupled via a transition rate matrix obtained by interpolation of the mass distribution within a section for each component in multicomponent sectional equations and the solutions of Mason equations. In order to validate the formulation, it is applied to the simplified GDE to compare with the analytic solutions. The formulation is also applied to international standard problem No. 44 for the validation with experimental data. Computation time for the formulation is also compared with those of the conventional method. With negligible computation burden as additional, the formulation would analyze the aerosol behaviors efficiently, accurately, and stably when hygroscopic growth occurs with other aerosol dynamics simultaneously.

Reduced Pre-Lie Algebraic Structures, the Weak and Weakly Deformed Balinsky–Novikov Type Symmetry Algebras and Related Hamiltonian Operators

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky–Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler–Kostant–Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky–Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky–Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky–Novikov algebras, including their fermionic version and related multiplicative and Lie structures.

Application of an equation of state incorporating association to alcohols up to decanol

The accurate prediction of fluid phase equilibria is essential to the design of any process in chemical industry. Most of the simulators used to model multiphase multicomponent phase equilibria use traditional cubic equations of state, like Peng-Robinson and Soave-Redlich-Kwong. However, self-associating substances, such as alcohols, are not precisely modeled with these equations of state. Different approaches have been proposed to overcome this limitation. In this work, we use the technique that separates the compressibility factor in two parts, a physical and a chemical one, and we compare the results with the Peng-Robinson equation of state and the Cubic Plus Association equation of state against experimental data for alcohols up to decanol.The parameters are estimated adjusting the proposed EOS results to two independent data set measurements, saturation pressures and saturated liquid molar volumes of the pure components, through multi-objective optimization. The results present excellent agreement for all alcohols modeled.

The influence of bromide-based ionic liquids on solubility of {LiBr (1) + water (2)} system. Experimental (solid + liquid) phase equilibrium data. Part 1

In this work, nine ILs are investigated as anti-crystallization additives to (LiBr+water) binary systems conventionally used as a working pair in absorption refrigeration technology. The solubility of {LiBr (1)+IL (2)+water (3)} ternary systems within wide temperature and composition range was determined by the dynamic method. In this work, the following ILs: 1-ethyl-1-methylmorpholinium bromide, [C1C2MOR][Br], 1-butyl-1-methyl-morpholinium bromide, [C1C4MOR][Br], 1-hexyl-1-methylmorpholinium bromide, [C1C6MOR][Br], 1-butyl-3-methylimidazolium bromide, [C1C4IM][Br], 1-ethyl-1-methyl-piperidinium bromide, [C1C2PIP][Br], 1-butyl-1-methylpiperidinium bromide, [C1C4PIP][Br], 1-butyl-1-methyl-pyrrolidinium bromide, [C1C4PYR][Br], 1-butylpyridinium bromide, [C4Py][Br] and tri(ethyl)-butylammonium bromide, [N2,2,2,4][Br] were investigated. The experimental (solid+liquid) phase equilibria have been determined for different IL to LiBr mass fractions from w2=(0.1 to 0.3). For all of the tested system, the transition point between lithium bromide dihydrate and monohydrate form was observed. Additionally, the simple model based on the multicomponent Redlich-Kister equation and a reference state of pure chemical compounds was used to successfully describe solubility of lithium bromide in aqueous systems containing an ionic liquid. The influence of ILs cation's core on the crystallization temperature of aqueous lithium bromide system was discussed and the best IL as an anti-crystallization additive has been proposed.

Theoretical and experimental analysis of precipitation and solubility effects in lithium-sulfur batteries

A rate-dependent nucleation theory is developed to analyze the dependence of the capacity of lithium-sulfur (Li-S) batteries on discharge rates. The theory is based on expressing the nucleation rate of solid products in terms of the surface oversaturation of the precipitating species and rewriting the classical Kolmogorov-Avrami model in the form of a differential system of equations. By coupling this system of equations with the multicomponent transport equations we are able to explain qualitatively well the decrease of the specific capacity with increasing discharge rates and the characteristics of the discharge curves under variable discharge currents in Li-S batteries. Special attention is given to accurately determine the values of model parameters from experimental data, including the solubility constants of the solid products and the mobilities and diffusivities of the ions. The anodes of the batteries used in our analysis were made using Li foils and the cathodes were fabricated using freestanding carbon nanotube foams, on which solid sulfur was uniformly distributed to maximize the dissolution rate. The experimental data combined with the theoretical analysis show that the capacity limitation in our batteries is due to surface passivation by the final reaction products. In order to increase the specific capacity of Li-S batteries under normal discharge rates we conclude that it is better to avoid the solid precipitation of intermediate lithium polysulfides (LiPS) by using electrolytes with a high solubility of LiPS. In addition, it is better to use catalyst particles in the cathode that would promote the nucleation rate at specified locations. The number of catalyst particles should not be too small to avoid electrolyte oversaturation or too large to avoid the rapid passivation of the carbon surface with solid products.