Conservation Laws Research Articles

Carrollian hydrodynamics and symplectic structure on stretched horizons

The membrane paradigm displays underlying connections between a timelike stretched horizon and a null boundary (such as a black hole horizon) and bridges the gravitational dynamics of the horizon with fluid dynamics. In this work, we revisit the membrane viewpoint of a finite-distance null boundary and present a unified geometrical treatment of the stretched horizon and the null boundary based on the rigging technique of hypersurfaces. This allows us to provide a unified geometrical description of null and timelike hypersurfaces, which resolves the singularity of the null limit appearing in the conventional stretched horizon description. We also extend the Carrollian fluid picture and the geometrical Carrollian description of the null horizon, which have been recently argued to be the correct fluid picture of the null boundary, to the stretched horizon. To this end, we draw a dictionary between gravitational degrees of freedom on the stretched horizon and the Carrollian fluid quantities and show that Einstein’s equations projected onto the horizon are the Carrollian hydrodynamic conservation laws. Lastly, we report that the gravitational pre-symplectic potential of the stretched horizon can be expressed in terms of conjugate variables of Carrollian fluids and also derive the Carrollian conservation laws and the corresponding Noether charges from symmetries.

Optimizing the noise control in a two-layer conduit

In the modern world, noise pollution is a major concern due to its prevalence. This work focuses on optimizing noise control in a two-layer conduit. A conduit comprises an inlet and double-outlet zones (DOZ). The upper walls of the DOZ are lined with perforated absorbent material while the lower walls are layered with fibrous. Mathematically, the physical problem is formulated with a field wave equation together with rigid and impedance boundary conditions in the respective zones. Such governing boundary value problem (BVP) leads to the Sturm-Liouville (SL) category in which standard orthogonality relations (OR) are indispensable. The system of the linear equations of the BVP is acquired with semi-analytical Mode-Matching (MM) approach by implementing the continuity conditions of sound pressures and velocities at the matching junction with the aid of OR. These systems are truncated and solved numerically with computation learning to obtain the reflected and transmitted modal amplitudes in respective zones. Due to the lining's perforated upper walls, fibrous lower walls, and reversal in the DOZ, the analysis of the reflected and transmitted powers versus frequencies is significantly observed and is shown in graphical findings. The algebraic derivation is validated by satisfying the conservation law of power flux and matching continuity conditions for impedance, perforated, and fibrous lined boundaries.

Convergence of the numerical approximations and well-posedness: Nonlocal conservation laws with rough flux

We study a class of nonlinear nonlocal conservation laws with discontinuous flux, modeling crowd dynamics and traffic flow. The discontinuous coefficient of the flux function is assumed to be of bounded variation (BV) and bounded away from zero, and hence the spatial discontinuities of the flux function can be infinitely many with possible accumulation points. Strong compactness of the Godunov and Lax-Friedrichs type approximations is proved, providing the existence of entropy solutions. A proof of the uniqueness of the adapted entropy solutions is provided, establishing the convergence of the entire sequence of finite volume approximations to the adapted entropy solution. As per the current literature, this is the first well-posedness result for the aforesaid class and connects the theory of nonlocal conservation laws (with discontinuous flux), with its local counterpart in a generic setup. Some numerical examples are presented to display the performance of the schemes and explore the limiting behavior of these nonlocal conservation laws to their local counterparts.

Assessment of using different ozone-friendly R22 alternative refrigerants in residential air conditioners in a high-ambient temperature country

The performance of a vapor compression refrigeration system (VCRS)-based residential air conditioner operating in a high-ambient temperature (HAT) country was investigated using six zero-ODP (ozone depletion potential) refrigerants as replacements to R22. The non-flammable alternative refrigerants considered in the present research were R134a, R404A, R407C, R410A, R448A, and R507A. Using the basic conservation laws, the VCRS was modeled during steady-state operation and solved using engineering equation solver (EES) software. Coefficient of performance (COP), pressures and temperatures at compressor suction and discharge, Global Warming Potential (GWP), critical pressure and temperature, compressor pressure ratio, volumetric cooling capacity (VCC) specific cooling capacity (SCC), and refrigeration effect were utilized as assessment criteria for the alternative refrigerants considered. From these refrigerants, the highest values of suction pressure, discharge temperature, and condenser pressure were attained by R410A. In addition, the discharge temperatures for all refrigerants, except R134a, were all higher than their corresponding critical values, causing a quicker drop in the VCRS’s performance. As an alternative refrigerant, R407C showed the highest SCC of 141.0 kJ/kg followed directly by 139.2 and 138.0 kJ/kg for R410A and R448A, respectively. A reverse trend was found for VCC with respective values of 4722 and 3775 kJ/m3 for R410A and R448A. Lower volume flow rates and smaller-sized compressors are expected for higher VCC refrigerants. The same trend was found for the compressor’s specific work input and condenser’s specific heat transfer with values of (51.14, 46.82, and 45.38 kJ/kg) and (190.3, 187.8, and 183.4 kJ/kg) for R410A, R407C, and R448A, respectively. For applications in HAT countries, larger condenser’s specific heat transfer makes the refrigerant more applicable. Conversely, with respect to COP, refrigerant R134a with a value of 3.075 was the superior alternative followed by R448A and R407C with respective COPs of 3.042 and 3.011. Based on the overall assessment in terms of environmental obligation, COP, compressor input power, refrigerant flow rate required, and all the evaluations made in this research, refrigerant R448A was recommended as the most appropriate substitute to R22 which can effectively be used in residential air conditioners in a HAT country.

Lie Symmetry Analysis, Power Series Solutions and Conservation Laws of (2+1)-Dimensional Time Fractional Modified Bogoyavlenskii–Schiff Equations

In this paper, Lie symmetry analysis method is applied to the (2+1)-dimensional time fractional modified Bogoyavlenskii–Schiff equations, which is an important model in physics. The one-dimensional optimal system composed by the obtained Lie symmetries is utilized to reduce the system of (2+1)-dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to the system of (1+1)-dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative. Then the power series method is applied to derive explicit power series solutions for the reduced system. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.

Beyond Dynamics: Learning to Discover Conservation Principles

Abstract The discovery of conservation principles is crucial for understanding the fundamental behavior of both classical and quantum physical systems across numerous domains. This paper introduces an innovative method that merges representation learning and topological analysis to explore the topology of conservation law spaces. Notably, the robustness of our approach to noise makes it suitable for complex experimental setups and its aptitude extends to the analysis of quantum systems, as successfully demonstrated in our paper. We exemplify our method’s potential to unearth previously unknown conservation principles and endorse interdisciplinary research through a variety of physical simulations. In conclusion, this work emphasizes the significance of data-driven techniques in deepening our comprehension of the principles governing classical and quantum physical systems.

Data driven multi soliton solutions of the Fokas-Lenells equation using conservation laws in PINN

Data driven multi soliton solutions of the Fokas-Lenells equation using conservation laws in PINN

Cycle equivalence classes, orthogonal Weingarten calculus, and the mean field theory of memristive systems

It has been recently noted that for a class of dynamical systems with explicit conservation laws represented via projector operators, the dynamics can be understood in terms of lower dimensional equations. This is the case, for instance, of memristive circuits. Memristive systems are important classes of devices with wide-ranging applications in electronic circuits, artificial neural networks, and memory storage. We show that such mean-field theories can emerge from averages over the group of orthogonal matrices, interpreted as cycle-preserving transformations applied to the projector operator describing Kirchhoff’s laws. Our results provide insights into the fundamental principles underlying the behavior of resistive and memristive circuits and highlight the importance of conservation laws for their mean-field theories. In addition, we argue that our results shed light on the nature of the critical avalanches observed in quasi-two-dimensional nanowires as boundary phenomena.

Flow resistance of emergent rigid vegetation in steady flow

Abstract Enhanced understanding of flow resistance in open channels with emergent vegetation is essential for flood management and river ecosystem restoration. The presence of vegetation can significantly alter bed resistance, leading to a challenge in accurately predicting flow discharge, water levels, sediment transport, and bed deformation. Previous studies on vegetated flows have focused on vegetation resistance, on which the impact of vegetation has been ignored or poorly estimated. This study proposes a new analytical model, built upon the momentum conservation law, to predict flow resistance to vegetated zones in a plain bed without bed forms, explicitly quantifying bed resistance and vegetation resistance in a corollary manner. The proposed model is benchmarked against five typical sets of laboratory experiments. It is demonstrated that the present model using a modified logarithmic velocity distribution performs best, whereas that assuming a uniform velocity profile considerably overestimates the vegetation resistance and neglects the effect of vegetation on bed resistance. The ratio of bed resistance to the total resistance is shown to range between 5% and 40%, and it decreases with increasing vegetation density and decreases with water depth. Therefore, bed resistance cannot be ignored when modelling shallow water flow with sparsely distributed vegetation. It is also revealed that vegetation arrangements significantly affect flow resistance, and therefore a model incorporating the effect of vegetation arrangement performs better. Overall, the present model facilitates a viable and promising tool for quantifying flow resistance in emergent vegetated channels.

Similarity reduction, group analysis, conservation laws, and explicit solutions for the time-fractional deformed KdV equation of fifth order

Through this paper, we consider the time-fractional deformed fifth-order Korteweg–de Vries (KdV) equation. First of all, we detect its symmetries by Lie group analysis with the help of Riemann–Liouville (R-L) fractional derivatives. These symmetries are employed to convert the considered equation into a fractional ordinary differential (FOD) equation in the sense of Erdélyi-Kober (E-K) fractional operator. Also, a set of new analytical solutions for the equation under study are obtained via the power series method. We test the accuracy and effectiveness of this method by providing a numerical simulation of the obtained solution and studying the effect of [Formula: see text] which is represented graphically in 2D and 3D plots. Added to that, we prove the convergence of the power series solutions. Finally, the computation of the conservation laws is introduced in detail.

Higher-dimensional integrable deformations of the classical Boussinesq–Burgers system* *This work is supported by the National Natural Science Foundation of China (Grant Nos. 11871396, 12271433).

In this paper, the (1+1)-dimensional classical Boussinesq–Burgers (CBB) system is extended to a (4+1)-dimensional CBB system by using its conservation laws and the deformation algorithm. The Lax integrability, symmetry integrability and a large number of reduced systems of the new higher-dimensional system are given. Meanwhile, for illustration, an exact solution of a (1+1)-dimensional reduced system is constructed from the viewpoint of Lie symmetry analysis and the power series method.

On the development of steady-state and dynamic mass-constrained neural networks using noisy transient data

This paper presents the development of algorithms for mass-constrained neural network models that can exactly satisfy mass conservation laws of chemical process systems, even if the training data violates the same. As opposed to approximately satisfying mass balance constraints of a system by considering additional penalty terms in the objective function, algorithms have been developed to solve an equality-constrained optimization problem, thus ensuring the exact satisfaction of the overall mass conservation laws. For developing dynamic mass-constrained networks, hybrid series and parallel all-nonlinear static-dynamic neural network models are leveraged. The proposed algorithms for solving both the inverse and forward problems are tested by considering both steady-state and dynamic data in presence of varieties of noise characterizations. The proposed structures and algorithms are applied to the development of data-driven models of two nonlinear dynamic chemical processes, namely the Van de Vusse reactor system as well as a solvent-based post-combustion CO2 capture process.

Numerical solutions of Schrödinger–Boussinesq system by orthogonal spline collocation method

This paper is concerned with the numerical solutions of Schrödinger–Boussinesq (SBq) system by an orthogonal spline collocation (OSC) discretization in space and Crank–Nicolson (CN) type approximation in time. By using the mathematical induction argument and standard energy method, the proposed CN+OSC scheme is proved to be unconditionally convergent at the order O(τ2+h4) with mesh-size h and time step τ in the discrete L2-norm. We devise a new computation method based on the orthogonal diagonalization techniques (ODT) to realize the proposed CN+OSC scheme. In order to compare the performance of ODT, we devise an alternating direction implicit (ADI) method to compute the CN+OSC scheme for high spatial dimension SBq system. As an alternative implementation, the new method ODT not only exhibits more accurate numerical results, but also demonstrates stronger invariance preserving ability. Numerical results are reported to verify the error estimates and the discrete conservation laws.

Computational study of the performance of a solar dryer for improvement in the shelf life of the food materials.

In this study, the thermal and drying characteristics of a thin layer food sample were investigated. An indirect type, simple, efficient, and economically feasible solar dryer was fabricated and used for food preservation. However, a dynamic model of a fabricated solar dryer was also presented to gain a better insight into the drying and thermal actions. This model consists of thermal modeling of the drying chamber, solar collector, and solar-dried food sample. The law of conservation of energy was applied to evaluate the temperature at different sections of the solar dryer with respect to drying time. All listed model equations were solved in the MATLAB environment. This study helps to examine the influence of solar radiation on the collector plate temperature, drying chamber temperature, food sample temperature, and performance parameters such as thermal efficiency with respect to drying time. Model data was found in good agreement with experimental data within a 4% error. It is concluded that the drying of food material is affected by air temperature, the collector temperature, mode of heat transfer, and material characteristics such as dimension and mass of the food sample.

A class of central unstaggered schemes for nonlocal conservation laws: Applications to traffic flow models

This study introduces a new class of central unstaggered finite volume methods that are used to approximate solutions to nonlocal conservation laws. The proposed method is based on Nessyahu and Tadmor's (NT). Instead of solving Riemann's problems at the level of cell interfaces, as in the NT scheme, the approach we develop implicitly uses ghost cells while still generating the numerical solution on a single grid. We use our method with the aim of solving one-dimensional nonlocal traffic flow problems. The numerical results we present demonstrate the accuracy, high resolution, and non-oscillatory nature of the proposed method and compare very favorably with those obtained using the original NT method, demonstrating the expected simplicity of a family of unstaggered central schemes and confirming that nonlocal traffic flow models can be treated very efficiently by the suggested method.

An Effective Arbitrary Lagrangian-Eulerian-Lattice Boltzmann Flux Solver Integrated with the Mode Superposition Method for Flutter Prediction

An arbitrary Lagrangian-Eulerian lattice Boltzmann flux solver (ALE-LBFS) coupled with the mode superposition method is proposed in this work and applied to study two- and three-dimensional flutter phenomenon on dynamic unstructured meshes. The ALE-LBFS is applied to predict the flow field by using the vertex-centered finite volume method with an implicit dual time-stepping method. The convective fluxes are evaluated by using lattice Boltzmann solutions of the non-free D1Q4 lattice model and the viscous fluxes are obtained directly. Additional fluxes due to mesh motion are calculated directly by using local conservative variables and mesh velocity. The mode superposition method is used to solve for the dynamic response of solid structures. The exchange of aerodynamic forces and structural motions is achieved through interpolation with the radial basis function. The flow solver and the structural solver are tightly coupled so that the restriction on the physical time step can be removed. In addition, geometric conservation law (GCL) is also applied to guarantee conservation laws. The proposed method is tested through a series of simulations about moving boundaries and fluid–structure interaction problems in 2D and 3D. The present results show good consistency against the experiments and numerical simulations obtained from the literature. It is also shown that the proposed method not only can effectively predict the flutter boundaries in both 2D and 3D cases but can also accurately capture the transonic dip phenomenon. The tight coupling of the ALE-LBFS and the mode superposition method presents an effective and powerful tool for flutter prediction and can be applied to many essential aeronautical problems.

Numerical solution, conservation laws, and analytical solution for the 2D time-fractional chiral nonlinear Schrödinger equation in physical media

Numerical solution, conservation laws, and analytical solution for the 2D time-fractional chiral nonlinear Schrödinger equation in physical media

Symmetry-based closed-form solutions and conserved quantities of the new (2+1)-dimensional Bogoyavlensky-Konopelchenko equation in fluid mechanics

In this investigation, we explore the mathematical intricacies of the novel (2+1)-dimensional Bogoyavlensky-Konopelchenko equation, a model with practical applications in elucidating the dynamics of internal waves within deep water. The equation’s significance spans various scientific domains, including plasma physics, nonlinear optics, and fluid dynamics. Employing a comprehensive analytical approach, specifically Lie symmetry analysis, we aim to unravel the underlying complexities of this equation and obtain new analytical solutions. To further scrutinize the equation, we apply various methods, namely Kudryashov’s method, the (G′/G)-expansion method, the simplest equation technique, and the power series method, all of which have not been applied to the equation before. Through these techniques, we successfully derive solutions in diverse functional forms, encompassing rational, trigonometric, exponential, hyperbolic, and Jacobi elliptic functions. To enhance comprehension, we present our findings visually using three-dimensional and two-dimensional plots density plots via the Mathematica tool. These graphical representations effectively communicate the intricate characteristics and nuances inherent in the solutions. Our visual representations reveal a spectrum of patterns, including periodic, singular periodic, kink-shaped structures. Additionally, our investigation extends to the determination of conserved quantities associated with the new (2+1)-dimensional Bogoyavlensky-Konopelchenko equation. This involves the application of the multiplier method and Ibragimov’s theorem, two potent techniques for identifying and understanding the conservation laws governing the model.

Large time existence and asymptotic stability of the generalized solution to flow and thermal explosion model of reactive real micropolar gas

We study the long time behavior of the generalized solution of the flow and thermal explosion model of the reactive real micropolar gas. The dynamics of the chemical reaction involved and the usual laws of conservation of mass, momentum, angular momentum, and energy generate a complex governing system of partial differential equations. The fluid is nonideal and non‐Newtonian. In this work, we prove that the problem can be solved in an infinite time domain and establish the asymptotic properties of the solution. Namely, we conclude that for certain parameter values, the solution stabilizes exponentially to a steady‐state solution, while for others the stabilization occurs but at power decay rate. At the end, we conducted a few numerical tests whereby we experimentally confirmed theoretical findings about long‐term behavior of the solution.

Holographic Ricci dark energy in nonconservative unimodular gravity

The structure of unimodular gravity (UG) is invariant to a subclass of diffeomorphism, the transverse diffeomorphism, due to the unimodular condition -g=ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{-g}=\\epsilon $$\\end{document}. Consequently, there is a freedom to define how the conservation laws of the energy–momentum tensor in unimodular gravity in the cosmological context. One of the main characteristics of the complete system of equations that describe cosmological dynamics in UG is that they form an underdetermined system if the usual conservation law of the energy–momentum tensor is not considered in your structure. In this article, we propose the construction of a background cosmological model based on the description of a holographic dark energy component with a cutoff of the order of Ricci scalar in non-conservative UG. This choice means that the complete set of equations remains underdetermined, however, the new feature of this cosmological model is the appearance of an interaction between matter and dark energy. Indeed, this is a well-known characteristic of cosmological models in which we have holographic dark energy density. Consequently, we propose an ansatz to the interaction term Q=βHρm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q=\\beta H \\rho _{m}$$\\end{document}, and obtain the cosmological parameters of our model. We found a viable universe model with similar characteristics to the ΛCDM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda \ extrm{CDM}$$\\end{document} model. We performed statistical analysis of the background model using the “Cosmic Chronometer” (CC) data for H(z), and obtain as a result using AIC, and the BIC as model selection criteria that ΛCDM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda \ extrm{CDM}$$\\end{document} prevails as the best model. However, the proposed model is competitive when compared to the cosmological model ωCDM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega \ extrm{CDM}$$\\end{document}.