General Conservation Laws Research Articles

Bound-Preserving Framework for Central-Upwind Schemes for General Hyperbolic Conservation Laws

Bound-Preserving Framework for Central-Upwind Schemes for General Hyperbolic Conservation Laws

CONSERVATION LAWS OF CHEMICAL REACTIONS IN A CLOSED NONISOTHRUM INCOMPLETE MIXING REACTOR

In chemical reactions occurring stationary or non-stationary in closed isothermal ideal mixing reactors (CSTR), the classical linear stoichiometric laws of conservation are always satisfied, reflecting the constancy of atoms of one type or another during the entire reaction. In real conditions, taking into account possible deviations from ideal mixing conditions or violations of the constancy of temperature conditions, various macroscopic factors (mass transfer, heat transfer) can have a noticeable influence on the course of the reaction. To study the kinetics of such processes, a more general mathematical model of an incomplete mixing reactor (IMR) is used. It is of interest to study the influence of diffusion and convection on the patterns of stationary modes of non-isothermal chemical reactions associated with the existence of more general (non-stoichiometric) conservation laws (kinetic invariants). It is more difficult to detect such invariants (especially temperature ones) in IMRs than in CSTR, and at present they are practically not studied. In this work, for chemical reactions occurring in a closed non-isothermal IMR according to the kinetic law of mass action, linear concentration and temperature conservation laws are established, which take into account the complicating processes of diffusion and convection. These conservation laws relate the stoichiometry of the reaction mechanism to the concentrations of reagents, the temperature of the reaction medium and the intensity of macrokinetic processes, which allows them to be used for a more reasonable identification of linear and nonlinear mechanisms of chemical reactions when solving the inverse problem of chemical kinetics. The new kinetic conservation laws presented in the work can be discovered a priori and verified experimentally. For citation: Kol'tsov N.I. Conservation laws of chemical reactions in a closed nonisothrum incomplete mixing reactor. ChemChemTech [Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol.]. 2024. V. 67. N 7. P. 111-118. DOI: 10.6060/ivkkt.20246707.6976.

A novel continuum dislocation density field-based crystal plasticity theory

In this work, a novel dislocation density field-based crystal plasticity formulation, that incorporates up-scaled continuum dislocation density fields to represent all possible characters of the dislocation density, is presented. The continuum dislocation field theory, formulated assuming large strain kinematics, is based on an all-dislocation concept, whereby individual dislocation density types are described as vector fields. The evolutionary behaviour of the dislocation density fields is defined in terms of a glide component derived from the classical general conservation law for dislocation density fields proposed originally by Kroner and Acharya, a rotation one given by the curl of the dislocation velocity vector as proposed originally by Ngan and co-workers, and a statistically stored dislocation source/sink component.The proposed formulation has been numerically implemented within the finite element method using a modified up-wind stabilisation approach, which results on eight extra independent nodal degrees of freedom per slip system, associated with the upscaled dislocation densities of pure edge and screw character. Furthermore, the required boundary conditions are simple: initial total dislocation density and dislocation flux or velocity, without the need for higher order boundary conditions. Full details of the numerical implementation of the crystal plasticity theory are given.The theory is then used to investigate classical boundary value problems involving 1D and 2D dislocation pile-ups, and the development of boundary layers in a deforming single crystal strip under constrained conditions. The main general understanding that emerges from the case studies is that, as either the dislocation pile-ups or the boundary layers develop with deformation, geometrically necessary dislocations are generated and lead to an overall behaviour that is size-dependent. Furthermore, it is shown that, in the absence of sources or sinks, which are statistically stored in nature, that total number of dislocations in the pile-up problems is preserved. The numerical predictions of the 1D pile-up and strip simple shear problems are in good agreement with published solutions.

Effect of Cattaneo-Christov heat and mass flux in Carreau-Yasuda tri-nanofluid

The Cattaneo-Christov heat and mass flux theory extends classic models to provide more accurate predictions in situations with instantaneous heat and mass transfer. Herein, an enhancement of heat and mass transfer in Carreau-Yasuda (CY) tri-nanofluid (TNF) over a stretched surface is modeled and studied via generalized conservation laws subjected to Cattaneo-Christov heat and mass flux theory. Silicon carbide (SiC), Aluminum Oxide (Al2O3), and Silver (Ag) are recruited as active nanoparticles in ethylene glycol (C2H6O2), and hired as CY fluid. Based on Finite Element Method (FEM), the code is created and validated for the formulated problems by comparing the current results to the published benchmarks. The computational findings showed that the thickness of the thermal and diffusion boundary layer decreases by increasing the thermal (δ) and diffusion relaxation (δc) parameters. The numerical analysis showed that CY-TNF exerts the largest shear stress on the wall compared to CY mono nanofluid (NF), and CY hybrid nanofluid (HNF). If CY-TNF has to flow over the specific surface, then adequate surface strength must be ensured. Moreover, comparing enhancement in thermal properties of CY-NF, CY-HNF, and CY-TNF revealed that the CY-TNF performs better than CY-HNF and CY-NF. Owing to the highest heat and mass flux, CY-TNF has an optimized ability to transport heat and mass, thus proving to be an efficient fluid for all processes requiring optimized heat and mass flux.

The Simulation of the Nonlinear Consolidation of Porous Media

In this paper, the general formulation of the problem of coupled deformation of a porous deformable medium with a fluid flowing through the pores is formulated, mathematically investigated and numerically implemented within the framework of physical and geometric nonlinearity. We present the formulation of the problem in velocities of solid phase displacements and the rate of pore pressure change in differential and variational forms. A phenomenological approach was used to formulate the mechanical model. The equations of the coupled consolidation model were derived from the general conservation laws of continuum mechanics using spatial averaging over a representative volume element. The consolidation model took into account the change in the porosity and permeability of the medium during deformation. The equations of filtration and porosity change, originally presented in Euler approach, were reformulated in Lagrangian coordinates of the solid phase using the relative fluid velocity according to ALE (Arbitrary Lagrangian - Eulerian) approach. The Gâteaux differentiation technique was used to linearize the variational equilibrium equations. For spatial discretization of the saddle system of equations, the finite element method (FEM) was used: quadratic serendipity elements for approximating the equilibrium equations and Brick type elements for approximating the filtration equation. To solve the system of equilibrium and filtration equations, a generalization of the implicit scheme with internal iterations at each time step by the Uzawa method was used. The results of numerical simulation of elastoplastic deformation of a water-saturated soil under load with fluid outflow are presented. To simulate the constitutive relations of elastoplastic deformation of soil under short-term loads, a generalization of S.S. Grigoryan's model to large deformations is proposed. The calculations were carried out in our own program code. The developed consolidation model can be used to simulate the formation of tracking ruts and unevenness of natural roads, as well as to calculate the uneven settlement of engineering structures.

An efficient invariant-region-preserving central scheme for hyperbolic conservation laws

Due to the Riemann solver free and avoiding characteristic decomposition, the central scheme is a simple and efficient tool for numerical solution of hyperbolic conservation laws (Nessyahu and Tadmor, J. Comput. Phys., 87(2):314-329,1990). But the theoretical Courant number CFL in order to preserve the invariant region of the numerical solution is very small, and there is lack of the stability proof for nonlinear systems. By adding a limiter on the reconstructed slope without requiring clipping condition, we enlarge the value of the CFL to admit larger time step. Then a widely applicable stability proof, which is suitable for general hyperbolic conservation laws, is given by writing the evolved solution as convex combinations in terms of the Lax-Friedrichs scheme. Some numerical experiments are carried out to verify the robustness.

Lax-Wendroff flux reconstruction method for hyperbolic conservation laws

The Lax-Wendroff method is a single step method for evolving time dependent solutions governed by partial differential equations, in contrast to Runge-Kutta methods that need multiple stages per time step. We develop a flux reconstruction version of the method in combination with a Jacobian-free Lax-Wendroff procedure that is applicable to general hyperbolic conservation laws. The method is of collocation type, is quadrature free and can be cast in terms of matrix and vector operations. Special attention is paid to the construction of numerical flux, including for non-linear problems, resulting in higher CFL numbers than existing methods, which is shown through Fourier analysis and yielding uniform performance at all orders. Numerical results up to fifth order of accuracy for linear and non-linear problems are given to demonstrate the performance and accuracy of the method.

Generalized characteristics for finite entropy solutions of Burgers’ equation

We prove the existence of generalized characteristics for weak, not necessarily entropic, solutions of Burgers’ equation ∂tu+∂xu22=0,whose entropy productions are signed measures. Such solutions arise in connection with large deviation principles for the hydrodynamic limit of interacting particle systems. The present work allows to remove a technical trace assumption in a recent result by the two first authors about the L2 stability of entropic shocks among such non-entropic solutions. The proof relies on the Lagrangian representation of a solution’s hypograph, recently constructed by the third author. In particular, we prove a decomposition formula for the entropy flux across a given hypersurface, which is valid for general multidimensional scalar conservation laws.

Generalized compacton equation, conservation laws and exact solutions

Compactons were introduced as solutions of nonlinear dispersive partial differential equations. There exists a variety of third-order equations that support compacton solutions. We consider a generalized third-order nonlinear K(fm,gn) equation that generalizes Rosenau–Hyman K(m,n) equation, extended Rosenau–Pikovsky K(cosmu,cosnu) equation, logarithmic KdV equation, generalized Gardner, and several other third-order nonlinear dispersive equations. In this paper, we obtain conservation laws of the K(fm,gn) equation in the most generalized form of f(u) and g(u). Several exact solutions of various equations of the K(fm,gn) form are derived from these generalized conservation laws by using the double reduction theory.

Non-biodegradable objects may boost microbial growth in water bodies by harnessing bubbles.

Given the ubiquity of bubbles and non-biodegradable wastes in aqueous environments, their transport through bubbles should be widely extant in water bodies. In this study, we investigate the effect of bubble-induced waste transport on microbial growth by using yeasts as model microbes and a silicone rubber object as model waste. Noteworthily, this object repeatedly rises and sinks in fluid through fluctuations in bubble-acquired buoyant forces produced by cyclic nucleation, growth and release of bubbles from object's surface. The rise–sink movement of the object gives rise to a strong bulk mixing and an enhanced resuspension of cells from the floor. Such spatially dynamic contaminant inside a nutrient-rich medium also leads to an increment in the total microbe concentration in the fluid. The enhanced concentration is caused by strong nutrient mixing generated by the object's movement which increases the nutrient supply to growing microbes and thereby, prolonging their growth phases. We confirm these findings through a theoretical model for cell concentration and nutrient distribution in fluid medium. The model is based on the continuum hypothesis and it uses the general conservation law which takes an advection–diffusion growth form. We conclude the study with the demonstration of bubble-induced digging of objects from model sand.

Invariant analysis and conservation laws of the time-fractional [formula omitted]-family peakon equations

In this paper, we further extend the theories of Lie symmetry group and conservation law to study the time-fractional b-family peakon equations. The main distinction of the equations with the usual time-fractional partial differential equations is the mixed derivative of Riemann-Liouville time-fractional derivative and integer-order x-derivative. Thus we first give a prolongation formula of the infinitesimal generator for the case of mixed derivative, then after finding the Lie symmetries, we use them to transform the equations into fractional and integer-order ordinary differential equations respectively. Some exact solutions and power series solutions are constructed. Finally, a general conservation law formula is given based on the idea of nonlinear self-adjointness and some nontrivial conservation laws of the equations are presented.

A Comparative Study from Spectral Analyses of High-Order Methods with Non-Constant Advection Velocities

The response of numerical methods to flow properties dynamic changes is a key ingredient of numerical modeling calibration. In this context, a range of spectral analyses and canonical fluid mechanics problems are explored to compare the responses of the high-order spectral difference scheme and of the flux reconstruction methods. Spatial eigen-analysis (Hu et al. in J Comput Phys 151(2):921–946, 1999; Hu and Atkins in J Comput Phys 182(2):516–545, 2002), based on spatially evolving oscillations, is used to get useful insights for problems with inflow/outflow boundary conditions (i.e., non-periodic), while non-modal analysis (Fernandez et al. in Comput Methods Appl Mech Eng 346:43–62, 2019. https://doi.org/10.1016/j.cma.2018.11.027 ) focusses on the short-term dynamics of the discretised system. These two approaches are also extended to the general scalar conservation law with non-constant advection velocity. All these tools are used to obtain more insight about the expected behavior of the mentioned numerical methods for typical engineering applications. On this regard, despite the lack of a mathematical proof of stability in the non-linear case, the selected numerical schemes, for which an extensive literature is available for applications to Euler and Navier–Stokes equations, will be assumed to be stable. Findings are then verified considering one-dimensional linear advection, and two- and three-dimensional flows modeled with the compressible Euler equations. Implications in the accurate control of numerical dissipation for turbulent flows is also addressed. Within the context of high-order methods, this work complement the former temporal spectral analyses of the discontinuous Galerkin approach discretising the linear advection equation.

The QUICK scheme is a third‐order finite‐volume scheme with point‐valued numerical solutions

AbstractIn this paper, we resolve the ever‐present confusion over the Quadratic Upwind Interpolation for Convective Kinematics (QUICK) scheme: it is a second‐order scheme or a third‐order scheme. The QUICK scheme, as proposed in the original reference (B. P. Leonard, Comput. Methods. Appl. Mech. Eng., 19, (1979), 59‐98), is a third‐order (not second‐order) finite‐volume scheme for the integral form of a general nonlinear conservation law with point‐valued solutions stored at cell centers as numerical solutions. Third‐order accuracy is proved by a careful and detailed truncation error analysis and demonstrated by a series of thorough numerical tests. The QUICK scheme requires a careful spatial discretization of a time derivative to preserve third‐order accuracy for unsteady problems. Two techniques are discussed, including the QUICKEST scheme of Leonard. Discussions are given on how the QUICK scheme is mistakenly found to be second‐order accurate. This paper is intended to serve as a reference to clarify any confusion about third‐order accuracy of the QUICK scheme and also as the basis for clarifying economical high‐order unstructured‐grid schemes as we will discuss in a subsequent paper.

Dynamically Consistent NSFD Discretization of Some Productive-Destructive Population Models Satisfying Conservations Laws

In this article, two general construction methods of nonstandard finite difference (NSFD) models are considered for productive-destructive models that also satisfy conservation laws: one for productive-destructive (PD) and the other for conservative systems. It is observed that the general NSFD method for PD systems may not result in numerical models for such systems that are dynamically consistent with respect to the conservation laws. This is illustrated through two examples, with one satisfying a direct conservation law and the other a generalized conservation law. Alternative NSFD schemes that are dynamically consistent with respect to the conservation laws are constructed for these examples using the general method for conservative systems.

Numerical Method for Non-Linear Conservation Laws: Inviscid Burgers Equation

This paper deals with the Burgers equation which is the most common model used in the nonlinear conservation laws. Here the theoretical aspect of conservation law is discussed by using inviscid Burgers equation. At first, we introduce the general non-linear conservation law as a partial differential equation and its solution procedure by the method of characteristic. Next, we present the weak solution of the problem with entropy condition. Taking into account shock wave and rarefaction wave, the Riemann problem has also been discussed. Finally, the finite volume method is considered to approximate the numerical solution of the inviscid Burgers equation with continuous and discontinuous initial data. An illustration of the problem is provided by some examples. Moreover, the Godunov method provides a good approximation for the problem.

The bubble-induced population dynamics of fermenting yeasts.

Bubble-induced transport is a ubiquitous natural and industrial phenomenon. In brewery, such transport occurs due to gas bubbles generated through anaerobic fermentation by yeasts. Two major kinds of fermentation viz. top (ale) and bottom (lager) fermentation, display a difference in their yeast distributions inside a sugar broth. The reason for this difference is believed to be yeast-bubble adhesion arising due to surface hydrophobicity of the yeast cell wall; however, the physical mechanism is still largely a mystery. In this report, through in vivo experiments, we develop a novel theoretical model for yeast distribution based on the general conservation law. This work clarifies that bubble-induced diffusion is the dominant transport mechanism in bottom-fermentation by lagers whereas, yeast-bubble adhesion plays a leading role in transporting ales in top-fermentation, thereby corroborating the centuries-old belief regarding distribution difference in yeast population in two kinds of fermentation.

A computational procedure for exact solutions of Burgers’ hierarchy of nonlinear partial differential equations

This paper considers the exact solution of Burgers’ hierarchy of nonlinear evolution equations. We construct the general nth conservation law of the hierarchy and prove that these expressions may be transformed into ordinary differential equations. In particular, a coordinate transformation leads to the systematic reduction of the conservation law properties of the Burgers’ hierarchy. Such an approach yields a nonlinear equation, where a second transformation is derived to linearize the expression. Consequently, this approach describes a procedure for finding the exact solutions of the hierarchy. A formula of the nth solution is provided, and to demonstrate its application, we discuss the solution to several members of the nonlinear hierarchy.

Theoretical Foundation of Rapid Distortion Theory on Transversely Sheared Mean Flows

The focus of this paper is on Rapid Distortion Theory on transversely sheared mean flows, which is often used to investigate turbulence-solid surface interactions. The main purpose of the paper is to bring together and present in a consistent fashion a general theory that has been developed in several different papers that have been published in the Journal of Fluid Mechanics. The equations for the unsteady pressure and velocity flections (which decouple from the entropy fluctuations) are rewritten in terms of a gauge function in order to obtain expressions that involve two arbitrarily convected quantities. A pair of very general conservation laws are used to derive upstream boundary conditions that relate these quantities to the actual physical variables. The entropy fluctuations can be determined after the fact once the solutions for the pressure and velocity fluctuations are known. The result involves a third arbitrary convected quantity that is equal to the entropy fluctuations at upstream infinity and can, therefore, be specified as an additional upstream boundary condition. A secondary purpose of the paper is to summarize a number of applications of the theory that have also appeared in the literature and show how they compare with an experiment.

Spatial scales and locality of magnetic helicity

Context. Magnetic helicity is approximately conserved in resistive magnetohydrodynamic models. It quantifies the entanglement of the magnetic field within the plasma. The transport and removal of helicity is crucial in both dynamo development in the solar interior and active region evolution in the solar corona. This transport typically leads to highly inhomogeneous distributions of entanglement. Aims. There exists no consistent systematic means of decomposing helicity over varying spatial scales and in localised regions. Spectral helicity decompositions can be used in periodic domains and is fruitful for the analysis of homogeneous phenomena. This paper aims to develop methods for analysing the evolution of magnetic field topology in non-homogeneous systems. Methods. The method of multi-resolution wavelet decomposition is applied to the magnetic field. It is demonstrated how this decomposition can further be applied to various quantities associated with magnetic helicity, including the field line helicity. We use a geometrical definition of helicity, which allows these quantities to be calculated for fields with arbitrary boundary conditions. Results. It is shown that the multi-resolution decomposition of helicity has the crucial property of local additivity. We demonstrate a general linear energy-topology conservation law, which significantly generalises the two-point correlation decomposition used in the analysis of homogeneous turbulence and periodic fields. The localisation property of the wavelet representation is shown to characterise inhomogeneous distributions, which a Fourier representation cannot. Using an analytic representation of a resistive braided field relaxation, we demonstrate a clear correlation between the variations in energy at various length scales and the variations in helicity at the same spatial scales. Its application to helicity flows in a surface flux transport model show how various contributions to the global helicity input from active region field evolution and polar field development are naturally separated by this representation. Conclusions. The multi-resolution wavelet decomposition can be used to analyse the evolution of helicity in magnetic fields in a manner which is consistently additive. This method has the advantage over more established spectral methods in that it clearly characterises the inhomogeneous nature of helicity flows where spectral methods cannot. Further, its applicability in aperiodic models significantly increases the range of potential applications.

Linear and nonlinear spin-orbital coupling in golden-angle spiral quasicrystals.

The appealing characteristics of quasi-crystalline nanostructure offer tremendous possibilities to tailor the transmission of the angular momenta. Moreover, the second harmonic generation existing in nonlinear nanostructures also exhibits remarkable potential in the fundamental and applied research areas of the angular momenta conversion. By systematically studying the general angular momenta conservation law, we show that the high-dimensional angular momenta transformation and spin-orbital coupling are realized by the nonlinear sunflower-type quasicrystals, which feature the high-fold rotational symmetry and possess an increasing degree of rotational symmetry in Fourier space. Interestingly, since the sequential Fibonacci numbers are essentially encoded in the distinctive nonlinear sunflower-type patterns, the high-fold angular momenta transformation regularly occurs at both linear and nonlinear wavelengths. The investigations of fundamental physics for the unique quasi-crystals reveal scientific importance for manipulating the angular momenta of nonlinear optical signals, which plays a key role in the promotion and development of modern physics.