General Equation Research Articles

Nonsmooth data error estimates for fully discrete finite element approximations of semilinear parabolic equations in Banach space

Implicit–explicit (IMEX) Euler formula together with finite element methods is proposed to solve numerically rather general semilinear parabolic differential equations with initial data in Banach space Xα, 0<α≤1. The time-space regularity of solutions to this class of equations is first investigated. Stability and error estimates are provided for this fully discrete scheme by discrete semigroup method. In the special case when X is the Hilbert space L2(Ω), these error estimates bridge the gap between the existing results on this scheme for semilinear parabolic problems with initial data in L2(Ω) and in H1(Ω) under general nonlinearity. Numerical experiments for nonlinear Allen–Cahn equations verify and complement our theoretical results.

Hypertension and diabetes, but not leptin and adiponectin, mediate the relationship between body fat and chronic kidney disease

PurposeObesity may promote kidney damage through hemodynamic and hormonal effects. We investigated the association between body mass index (BMI), total body fat (TBF) and chronic kidney disease (CKD) and whether hypertension, diabetes, leptin and adiponectin mediated these associations.MethodsIn this cross-sectional analysis of the Netherlands Epidemiology of Obesity study, 6671 participants (45–65 y) were included. We defined CKD as eGFR <60 ml/min/1.73 m2 and/or moderately increased albuminuria. The percentage of mediation was calculated using general structural equation modeling, adjusted for potential confounding factors age, sex, smoking, ethnicity, physical activity and Dutch healthy diet index.ResultsAt baseline mean (SD) age was 56 (6), BMI 26.3 (4.4), 44% men, and 4% had CKD. Higher BMI and TBF were associated with 1.08 (95%CI 1.05; 1.11) and 1.05-fold (95%CI 1.02; 1.08) increased odds of CKD, respectively. As adiponectin was not associated with any of the outcomes, it was not studied further as a mediating factor. The association between BMI and CKD was 8.5% (95%CI 0.5; 16.5) mediated by diabetes and 22.3% (95%CI 7.5; 37.2) by hypertension. In addition, the association between TBF and CKD was 9.6% (95%CI −0.4; 19.6) mediated by diabetes and 22.4% (95%CI 4.2; 40.6) by hypertension. We could not confirm mediation by leptin in the association between BMI and CKD (35.6% [95%CI −18.8; 90.3]), nor between TBF and CKD (59.7% [95%CI −7.1; 126.6]).ConclusionOur results suggest that the relations between BMI, TBF and CKD are in part mediated by diabetes and hypertension.

Subdiffusive processes in BWRs

The objective of this work is the analysis of subdiffusive processes in BWR reactors. Nuclear reactors are highly heterogeneous systems where the phenomena at the reactor scale are not possible to describe with constitutive standard laws (normal diffusion) of energy, momentum, and mass. However, there are methods to achieve it such as the volumetric average to upscale the reactor and incorporate up-scaled constitutive laws. However, it can also be proposed constitutive laws of fractional order to consider anomalous diffusion. For the neutronic processes, the neutron diffusion theory is non-appropriated to the reactor scale, because this theory is applied to the homogeneous reactor. Given the heterogeneity of these systems in describing the dynamic behaviour of BWRs with point models, it represents a challenge to the incorporation of the heterogeneous effects whose main characteristic is that the transport phenomena are subdiffusive with relaxation times (i.e., non-instantaneous) by nature. The point models called reducer-order models are important for stability analysis and to develop control strategies. In linear or nonlinear stability analysis and system control, they are carried out with reduced order models that are zero dimension, i.e., point models that depend exclusively on time. This is because existing methodologies propose mathematical expressions in the time or frequency domain for stability analysis or to establish control strategies. The challenge is for ROM models to be able to consider subdiffusive effects on neutron motion. This is achieved by various routes from the P1 theory that includes the temporal term with relaxation effects (relaxation time) or by proposing a more general constitutive equation of fractional order for the neutron current vector, as is presented in this work. Moreover, the Fractional Order Models behavior must be as a transport theory based model, as well as in those nuclear and thermohydraulic codes which are very useful but complex, for which is sometimes it is impossible to obtain analytical expressions. To incorporate these effects in this work we consider point models of fractional order, which represent the subdiffusive phenomenon when the fractional order (known as anomalous diffusion coefficient) is greater than 0 and less than 1.

Simplified structural analysis of laminated glass panels during fire exposure

Due to their high aesthetic value, energy-efficient properties, and contribution to daylighting, the demand for using glass panels in modern buildings has considerably increased over the past decades. However, ordinary glass panels are highly susceptible to cracking during a fire because of the temperature difference between the part of the glass exposed to the fire and the part protected by the frame. Damage to the glass can allow additional oxygen intake, leading to the flashover phenomenon significantly increasing fire severity. Laminated glass is superior to ordinary glass in its impact resistance, sound insulation, and ability to maintain post-breakage integrity. This paper provides a simplified method to study the effect of temperature gradients on the cracking behaviour of laminated glass panels. The temperature of the unprotected portion of the glass panel is first estimated by evaluating the mid-thickness temperature using the general heat transfer equation. Then, equations developed based on a parametric study that utilized ABAQUS are proposed to estimate the exposed and unexposed surface temperatures. This step was followed by evaluating the temperature of the protected glass portion. Subsequently, a method based on strain-equilibrium principles was developed to predict the corresponding maximum thermal stress. Comparisons with experimental and numerical work by others validated the proposed method.

The geometrization of Maxwell's equations and the emergence of gravity and antimatter

Coupling the Maxwell tensor to the Riemann-Christoffel curvature tensor is shown to lead to a geometricized theory of electrodynamics. While this geometricized theory leads directly to the classical Maxwell equations, it also extends their physical interpretation by giving charge density and the four-velocity that describe its motion geometric definitions. Introducing mass using a conserved energy-momentum tensor, all solutions to the geometricized theory of electrodynamics developed here are shown to be consistent with the emergence of gravity obeying the General Relativity field equation augmented by a term that mimics the properties of dark matter and/or dark energy. Finally, due to the symmetries of the theory, the properties and phenomenology of antimatter emerge in solutions.

Nonlinearity in the Medium for Electromagnetic Wave Propagation Induced by Quantum Mechanical Interaction of the Field With Finite-State Atoms

We consider the interaction between a classical time-varying electric field and the atomic electrical dipole moment operator of a finite state atom and calculate approximately the mixed state evolution of the atom with such an interaction Hamiltonian up to quadratic orders in the external classical electric field using Schrodinger’s equation for mixed state evolution. From this approximate expression for the state evolution, we evaluate the average atomic electric dipole moment as a linear-quadratic function of the electric field. Later on, we also take into account Brownian motion bath noise that couples to the atomic dynamics, so that the Schrodinger equation dynamics gets modified to the Lindblad dynamics of an open quantum system, i.e., the master equation. From the average atomic dipole moment, we obtain an expression for the average polarization field of atoms constituting the medium through which the classical electromagnetic wave propagates. This average polarization is a linear-quadratic causal functional of the electric field. The resulting Maxwell equations for wave propagation, taking into account this polarization current density, thus acquire a quadratic nonlinearity which can be used to explain higher harmonic frequencies at the output of an optical fiber when the input is excited by the electric field coming from a monochromatic laser. We are then also able to calculate the mean square fluctuations in the polarization field using the evolving atomic state, and hence the electric field propagating through the medium of atoms acquires quantum fluctuations which can be used to explain time-varying random shifts in the spectral lines. As mentioned above, we include in this article an extension of this problem to the case when there is white Gaussian noise corrupting the state evolution dynamics of the atom and also the case when there is a pairwise interaction between the spin dipoles of atoms at two different locations. In the former case, we derive an approximate formula for the polarization field as well as its temporal statistical correlations, and we show how this computation can be extended to the latter case too. When there is a pairwise interaction between the atomic spins, we use a version of the approximate quantum Boltzmann equation to calculate the first and second order marginals of the atomic states, which is then used to compute the average polarization as well as its spatio-temporal correlations. Temporal correlations using second marginals of the atomic states cannot be computed using classical probabilistic methods owing to Heisenberg uncertainty, so we propose a sequential measurement-based strategy involving computation of the joint probabilities of the atomic spin components at two different times by first measuring one component of the spin, allowing for state collapse, and then following it up with nonlinear time evolution from the collapsed two-particle state using the quantum Boltzmann equation, followed further by measurement of another component of the spin at the second time.

Electromagnetically driven flow in unsupported electrolyte layers: lubrication theory and linear stability of annular flow

We consider a thin horizontal layer of a non-magnetic electrolyte containing a bulk solution of salt and carrying an electric current. The layer is bounded by two deformable free surfaces loaded with an insoluble surfactant and is placed in a vertical magnetic field. The arising Lorentz force drives the electrolyte in the plane of the layer. We employ the long-wave approximation to derive general two-dimensional hydrodynamic equations describing symmetric pinching-type deformations of the free surfaces. These equations are used to study the azimuthal flow in an annular film spanning the gap between two coaxial cylindrical electrodes. In weakly deformed films, the base azimuthal flow and its linear stability with respect to azimuthally invariant perturbations are studied analytically. For relatively thick layers and weak magnetic fields, the leading mode with the smallest decay rate is found to correspond to a monotonic azimuthal velocity perturbation. The Marangoni effect leads to further stabilisation of the flow while perturbations of the solute concentration in the bulk of the fluid have no influence on the flow stability. In strongly deformed films in the diffusion-dominated regime, the azimuthal flow becomes linearly unstable with respect to an oscillatory mixed mode characterised by the combination of radial and azimuthal velocity perturbations when the voltage applied between electrodes exceeds the critical value.

Assessment of the influence of flushing on the wear and morphological properties of the tool during electrical discharge machining

The present study is an integrated approach of experimental and simulation processes to investigate the influences of the flushing on the wear, textural feature, heat dissipation, tool-tip temperature, and elemental contents of the electrical discharge machining-tool. The general heat transfer equation in cylindrical coordinates is used to explain the thermal phenomenon through the tool, where the boundary conditions are influenced by convective interactions and dimensionless numbers. The inter-electrode flushing is considered a micro-channel flow, and the used model is conceptualized from “Hazen–Poiseuille observation.” It is observed that the adopted model has a prediction error of 8.503% and can accurately explain the thermal consequences of the process. The study reveals that tool wear is influenced by flushing velocity and flushing pressure. The tool-tip temperature reduces with the Reynolds number, and effectual expelling of the debris can be ensued with a flushing of a higher Reynolds number (Re). However, the increment of Re beyond 4500 provides rapid heat dissipation, which produces extensive residual stress and creates cracks on the surface.

PT-symmetric PINN for integrable nonlocal equations: Forward and inverse problems.

Since the PT-symmetric nonlocal equations contain the physical information of the PT-symmetric, it is very appropriate to embed the physical information of the PT-symmetric into the loss function of PINN, named PTS-PINN. For general PT-symmetric nonlocal equations, especially those equations involving the derivation of nonlocal terms due to the existence of nonlocal terms, directly using the original PINN method to solve such nonlocal equations will face certain challenges. This problem can be solved by the PTS-PINN method, which can be illustrated in two aspects. First, we treat the nonlocal term of the equation as a new local component so that the equation is coupled at this time. In this way, we successfully avoid differentiating nonlocal terms in neural networks. On the other hand, in order to improve the accuracy, we make a second improvement, which is to embed the physical information of the PT-symmetric into the loss function. Through a series of independent numerical experiments, we evaluate the efficacy of PTS-PINN in tackling the forward and inverse problems for the nonlocal NLS equation, the nonlocal derivative NLS equation, the nonlocal (2+1)-dimensional NLS equation, and the nonlocal three-wave interaction systems. The numerical experiments demonstrate that PTS-PINN has good performance. In particular, PTS-PINN has also demonstrated an extraordinary ability to learn large space-time scale rogue waves for nonlocal equations.

Free propagation of resonant waves in nonlinear dissipative metamaterials

This paper deals with the free propagation problem of resonant and close-to-resonance waves in one-dimensional lattice metamaterials endowed with nonlinearly viscoelastic resonators. The resonators' constitutive and geometric nonlinearities imply a cubic coupling with the lattice. The analytical treatment of the nonlinear wave propagation equations is carried out via a perturbation approach. In particular, after a suitable reformulation of the problem in the Hamiltonian setting, the approach relies on the well-known resonant normal form techniques from Hamiltonian perturbation theory. It is shown how the constructive features of the Lie Series formalism can be exploited in the explicit computation of the approximations of the invariant manifolds. A discussion of the metamaterial dynamic stability, either in the general or in the weak dissipation case, is presented.

Resonant decay of kinetic Alfvén waves and implication on spectral cascading

A general equation describing the resonant nonlinear mode-coupling among kinetic Alfvén waves (KAWs) is derived using nonlinear gyrokinetic theory, which can be applied to study the potentially strong spectral energy transfer of KAWs. As a first application, the parametric decay of a pump KAW into two sideband KAWs is studied, with particular emphasis on cascading in the perpendicular wavenumber. It is found that, for the “co-propagating” cases with all three KAWs propagating in the same direction along the equilibrium magnetic field line, it exhibits a dual cascading character in the perpendicular wavenumber space; while for the “counter-propagating” cases with one sideband propagating in the opposite direction with respect to the pump wave, it instead, can exhibit both dual and inverse cascading behaviors. Furthermore, the implications of these findings on shear Alfvén waves regarding instability, nonlinear saturation, and charged particle transport in fusion plasmas are also discussed.

Analytical Confirmation of the Calculated Dependencies for Heat Exchange in a Plate Recuperator when Humidifying the Auxiliary Flow

The relevance of the study is related to the need to ensure maximum reduction of energy consumption while ensuring the calculated parameters of the indoor climate in buildings under the Law of the Russian Federation “On Energy Saving...” and the updated version of SP 131. The purpose of the study is to obtain an approximate analytical expression of this dependence, which makes it possible to additionally confirm the results of field and numerical experiments previously performed by the authors for this flow treatment scheme. The objective of the study is to build a simplified mathematical model of the processes of changing the heat and humidity state of the air in the recuperator, identify the main factors affecting the increasing multiplier to the coefficient of temperature efficiency of the device and obtain the necessary numerical coefficients in formulas linking the desired and initial parameters. The general equation of thermal balance and heat transfer for the heat exchanger as a whole is used, including the integral characteristics of the apparatus in a dimensionless form, as well as standard techniques of algebraic transformations. An approximate analytical expression is obtained for an increasing multiplier to the coefficient of thermal efficiency of the recuperator taking into account the additional cooling of the inflow due to the heat consumption for evaporation. It is shown that the general structure of this ratio coincides with the one found by the authors earlier by processing the results of numerical modeling of a two-dimensional temperature field in a heat exchanger, taking into account experimental data on the amount of moisture carried away, which confirms their correctness and reliability.

The General Parametric Equation of Pythagoras Theorem and The General Connectedness Theorem

In this article, the Quarter Squares Rule is used to prove that it also satisfies the Pythagoras Theorem. Using this proof, it will be shown that there are parametric equations among sides and the radius of inner circle of a right triangle. Thus, it is proved that a quadratic equation which has some definite properties is connected with a right triangle. After that, firstly, utilizing this connection, it will be reached the general parametric equation of The Pythagoras Theorem. Secondly, it is going to be identified the general connectedness theorem. Further, it is going to be given relation between golden ratio and Earth’s axial tilt angle as an interesting example.

Experimental and molecular simulation analysis of the dissolution behavior of Musk ambrette in organic solvents

The solubility of Musk ambrette in i-propanol (IPA), i-butanol (IBA), n-heptane (HPT), n-hexane (HX), cyclohexane (CYH), methyl acetate (MAC), ethyl acetate (EAC), acetone (ACE), IPA + ACE at 278.15 K to 318.15 K was determined. The mole fraction solubility of Musk ambrette in pure IPA at the temperature of 298.15 K were found to be 0.9492 × 10-2 mol·mol−1. The difference is that the solubility of Musk ambrette in ACE reaches 0.2257 mol·mol−1, which is much higher than the solvent mentioned above. The strong intermolecular hydrogen bonding formed by IPA resulted in its lowest solvation capacity for Musk ambrette. In addition, to further investigate the solubility pattern of Musk ambrette, we used 8 models (Van’t Hoff, modified Apelblat, Buchowski-Ksiazczak, Yaws, NRTL, Wilson, Jouyban-Acree and Sun model) to fit and obtain a general equation for solubility versus temperature. These results yielded a large amount of statistically significant data. Comparisons revealed that the modified Apelblat and Yaws equation showed the least deviation in forecasting solubility variations with temperature. Meanwhile, we proved that the dissolution of Musk ambrette is a heat-absorbing entropy increasing process by Van't Hoff equation calculation. ΔsolGo for the 12 solvents is in the following order: ACE < MAC < EAC < w = 0.2007 < w = 0.4025 < w = 0.6076 < CYH < w = 0.8070 < HPT < HX < IBA < IPA. ΔsolGo is smaller the easier it dissolves, which is consistent with the law of dissolution. Secondly, we performed thermal analysis and X-ray diffraction tests on Musk ambrette crystals prepared in different solvents, which showed that none of the several systems investigated led to the creation of new crystalline forms or solvatoids, further confirming the suitability of the solvents. Besides, we discuss the correspondence between solubility and the HSP, π, α, β, and CED of the solvents and thus tentatively confirm the significant influence of the solvent properties on solubility. The principle of similarity and compatibility is satisfied for most of the solvents. In particular, the CED values of the larger solvents were not favorable for the solubilization of Musk ambrette. We also analyzed the molecular surface electrostatic potential of Musk ambrette by visualization techniques. Finally, IPA and ACE and their binary solution systems were analyzed by molecular dynamics simulations. The effect of solvent intermolecular interactions on the solubilization pattern was discussed. It was further confirmed that weak interactions of solvent molecules contribute more to the dissolution of Musk ambrette. These findings suggest that the prediction of solvent solubility can be initially achieved by molecular simulation techniques. Undoubtedly, this study will furnish a valuable information for the purification of Musk ambrette.

Thermal and compositional driven convection in thin reaction fronts.

Chemical reaction fronts separate regions of reacted and unreacted substances as they propagate in liquids. These fronts may induce density gradients due to different chemical compositions and temperatures across the front. In this paper, we investigate buoyancy-induced convection driven by both types of gradients. We consider a thin front approximation where the normal front velocity depends only on the front curvature. This model applies for small curvature fronts independent of the specific type of chemical reaction. For density changes due only to heat variations near the front, we find that convection can take place for either upward or downward propagating fronts if density gradients are above a threshold. Convection can set in even if the fluid with lower density is above the higher density fluid. Our model consists of Navier-Stokes equationscoupled to the front propagation equation. We carry out a linear stability analysis to determine the parameters for the onset of convection. We study the nonlinear front propagation for liquids confined in narrow two-dimensional domains. Convection leads to fronts of steady shape, propagating with constant velocities.

Scale up validation tests for Buckingham Pi theorem applied to leaf test experiments for iron ore slurry

Bench tests were conducted on ore slurry filtration using the leaf test method, varying scales, pressures, and water/solids ratios. These results were compared with analytical outcomes derived from applying the dimensionless Buckingham pi method to the general equation of slurry filtration with incompressible cakes under constant pressure. The aim was to validate the tests by corroborating experimental findings with calculated parameters for scale-up purposes. Vacuum pressure was employed to facilitate slurry dewatering, forming a cake comprising solids from the slurry. This cake’s thickness evolved over time, introducing specific resistivity that impeded filtering efficiency. Parameters such as cake resistance and filter media resistance were computed. Filtrate volume was measured over time to determine the filtration rate. These values were then adjusted using factors obtained through dimensional analysis. These factors facilitated comparisons to validate the feasibility of employing the Buckingham pi method for predicting slurry filtration results at larger scales. The results indicated lower errors when analyzing the filtrate rate over time with a solids/water ratio of 20/80%. When evaluating cake resistance, the method yielded lower errors at a ratio of 70/30%, while results for medium resistance were inconclusive. It’s important to consider the ratio and associated errors when predicting the behavior of an industrial filter based on bench test results, aiming to achieve theoretical scaled filtration rates and cake resistances. However, it’s not advisable to rely on the Buckingham pi method for scale-up purposes when evaluating medium resistance.

Thermodynamic analysis of micropolar-casson fluid flow with PST and PHF heating condition over a curved stretching surface

In this research, we explore the flow dynamics of micropolar Casson fluids over curved surfaces, while incorporating prescribed heat and surface fluxes, as well as heat generation effects. By applying similarity transformation to the governing model, we derive individual differential equations which are then reduced to general differential equations. This transformation yields a mathematical structure conducive to problem analysis. Through the use of numerical methods, we solve ordinary differential equations to investigate the fluid dynamics and heat transfer phenomena in this specific flow configuration. The main aim of this study is to offer a comprehensive understanding of the intricate behavior displayed by micropolar Casson fluids on curved surfaces under the influence of prescribed heat and surface fluxes, along with heat generation. By employing numerical methods, this study aims to contribute to the advancement of theoretical understanding by addressing the complex interactions between fluid dynamics and thermal properties.

Free vibration of the piezoelectric laminated composite beams under the elastic foundation

Due to its self-adaptability and electric-mechanical interaction, the piezoelectric laminated composite structure has received increased attention in the engineering community, with the intricacy of the actual scenario, the structural stability based on numerous foundations has also been recognized. In this paper, a unified formulation which is based on classical laminated beam is presented to conduct free vibration analysis of piezoelectric laminated beams subjected to classical boundary conditions with the varied elastic foundations. The semi-analytical solutions of layered beams in various boundaries have been obtained based on the two-dimensional piezoelectric elastic theory and state space-differential quadrature method (SS-DQM), the solution of the general structural equation has been generated introducing the electrical boundary conditions and interlayer continuity conditions with this understanding. The different elastic foundations have been used to calculate the frequency equation of the piezoelectric laminated construction. The proposed finite element model is validated through the analysis of the piezoelectric laminated composite beam (PLCB) using the SS-DQM, and the present solutions are compared with those available in the literature to confirm their validity. A systematic parameter study for the piezoelectric laminated beams with various boundary conditions, piezoelectric parameters, lamina numbers and thickness ratios is also performed. The results of the PLCBs with various boundary constraints under the elastic foundation are presented and they may be served as a benchmark for researchers in this field.

Phase space propagation of waves in nonhomogeneous media: corrections beyond the optical geometry limit

We investigate the corrections to the optical geometry approximation for waves traveling in non homogeneous media. We model the wave propagation in dispersive and non dispersive materials in terms of the phase space Wigner–Weyl formalism. The ray tracing optical geometry limit is introduced by numerical tests. We solve the exact Wigner propagation equation for 1D non dispersive materials. We discuss the connection of the Wigner–Weyl description of waves with the particle-wave duality phenomenon in quantum mechanics.

Spontaneous imbibition of modified salinity brine into different lithologies: an improvement of comprehensive scaling used for fractured reservoir simulation

Spontaneous water imbibition into matrix blocks can be a significant oil recovery mechanism in fractured reservoirs. Many enhanced oil recovery methods, such as injection of modified salinity brine, are proposed for improving spontaneous imbibition efficacy. Many scaling equations are developed in the literature to predict spontaneous imbibition oil recovery. However, almost none of them included the impact of the diversity in ionic composition of injected and connate brines and the blending/interaction of a low salinity imbibing brine with a higher salinity connate brine. In this research, these two issues are included to propose new scaling equations for the scaling of spontaneous imbibition oil recovery by modified salinity imbibing brines. This study uses experimental data of the spontaneous imbibition of modified salinity brines into oil-saturated rock samples with different lithologies containing an irreducible high salinity connate brine. The collected tests from the literature were performed at high temperatures and on aged altered wettability cores. The results of 110 available spontaneous imbibition laboratory experiments (85, 12 and 13 tests on chalks, dolomites and sandstones, respectively) are gathered. This research initially shows the poor ability of three selected convenient scaling equations from the literature to scale imbibition recovery by modified salinity brine. Then, our newly proposed technique to find the scaling equation for spontaneous imbibition recovery by modified salinity brine, during the abovementioned conditions in limestones (Bassir et al. in J Pet Explor Prod Technol 13(1): 79–99, 2023. https://doi.org/10.1007/s13202-022-01537-7) is used in chalks, dolomites and sandstones to develop the three new scaling equations. Finally, a new general equation to scale imbibition recovery by modified salinity brine for all four lithologies is presented. Moreover, for each of the four datasets (chalk, dolomite, sandstone and all the four lithologies), the scaled data by the new equations is matched by two mathematical expressions based on the Aronofsky et al. model and the Fries and Dreyer model. These mathematical expressions can be used to develop transfer functions in reservoir simulators for a more accurate prediction of oil recovery by spontaneous imbibition of modified salinity brine in fractured reservoirs.