General Conservation Laws Research Articles

Post-Newtonian conservation laws in rigid quasilocal frames

In recent work we constructed completely general conservation laws for energy (McGrath et al 2012 Class. Quantum Grav. 29 215012) and linear and angular momentum (Epp et al 2013 Class. Quantum Grav. 30 195019) of extended systems in general relativity based on the notion of a rigid quasilocal frame (RQF). We argued at a fundamental level that these RQF conservation laws are superior to conservation laws based on the local stress–energy–momentum tensor of matter because (1) they do not rely on spacetime symmetries and (2) they properly account for both matter and gravitational effects. Moreover, they provide simple, exact, operational expressions for fluxes of gravitational energy and linear and angular momentum. In this paper we derive the form of these laws in a general first post-Newtonian (1PN) approximation, and then apply these approximate laws to the problem of gravitational tidal interactions. We obtain formulas for tidal heating and tidal torque that agree with the literature, but without resorting to the use of pseudotensors. We describe the physical mechanism of these tidal interactions not in the traditional terms of a Newtonian gravitational force, but in terms of a much simpler and universal mechanism that is an exact, quasilocal manifestation of the equivalence principle in general relativity. As concrete examples, we look at the tidal heating of Jupiter’s moon Io and angular momentum transfer in the Earth–Moon system that causes a gradual spin-down of the Earth and recession of the Moon. In both examples we find agreement with observation.

Simulating cavitating liquid jets using a compressible and equilibrium two-phase flow solver

In internal combustion engines the injection of high-pressure liquid fuel into a low-pressure gas through a nozzle passage is an important process to atomize the liquid and achieve optimal fuel–air mixing. A Computational Fluid Dynamics (CFD) model is developed in the present work to simulate the internal- and external-nozzle flow fields in an integrated way. The model assumes that the flow within and near the nozzle is continuous, and an Eulerian flow solver is developed using the general conservation laws of fluid dynamics. Differences in the thermodynamic states of the liquid and gas phases are modeled with a Stiffened Gas Equation of State (EOS). A practical phase equilibrium solver is developed, and is implemented into the Eulerian flow solver to predict phase changes in the flows – in particular, cavitation of the liquid within the injector nozzle passage. The combined equilibrium solver is applied to single-component and two component flows with one component being non-condensable air. A number of test problems are simulated to verify the numerical methods and validate the proposed models. These include two-phase shock tube problems, a converging–diverging nozzle flow problem, a submerged liquid jet problem, and a cavitating liquid jet problem.

Four proofs of Gittins’ multiarmed bandit theorem

We study four proofs that the Gittins index priority rule is optimal for alternative bandit processes. These include Gittins’ original exchange argument, Weber’s prevailing charge argument, Whittle’s Lagrangian dual approach, and Bertsimas and Nino-Mora’s proof based on the achievable region approach and generalized conservation laws. We extend the achievable region proof to infinite countable state spaces, by using infinite dimensional linear programming theory.

L (L )-boundedness and convergence of DG(p) solutions for nonlinear conservation laws with boundary conditions

We prove the $L^\infty(L^\infty)$-boundedness of a higher-order shock-capturing streamline-diffusion DG-method based on polynomials of degree $p\geq 0$ for general scalar conservation laws. The estimate is given for the case of several space dimensions and for conservation laws with initial and boundary conditions.

Momentum in general relativity: local versus quasilocal conservation laws

We construct a general relativistic conservation law for linear and angular momentum for matter and gravitational fields in a finite volume of space that does not rely on any spacetime symmetries. This work builds on our previous construction of a general relativistic energy conservation law with the same features (McGrath et al 2012 Class. Quantum Grav. 29 215012). Our approach uses the Brown and York (1993 Phys. Rev. D 47 1407–19) quasilocal stress–energy–momentum tensor for matter and gravitational fields, plus the concept of a rigid quasilocal frame (RQF) introduced in (Epp et al 2009 Class. Quantum Grav. 26 035015; 2012 Classical and Quantum Gravity: Theory, Analysis, and Applications (Nova Science)). The RQF approach allows us to construct, in a generic spacetime, frames of reference whose boundaries are rigid (their shape and size do not change with time), and that have precisely the same six arbitrary time-dependent degrees of freedom as the accelerating and tumbling rigid frames we are familiar with in Newtonian mechanics. These RQFs, in turn, give rise to a completely general conservation law for the six components of momentum (three linear and three angular) of a finite system of matter and gravitational fields. We compare in detail this quasilocal RQF approach to constructing conservation laws with the usual local one based on spacetime symmetries, and discuss the shortcomings of the latter. These RQF conservation laws lead to a deeper understanding of physics in the form of simple, exact, operational definitions of gravitational energy and momentum fluxes, which in turn reveal, for the first time, the exact, detailed mechanisms of gravitational energy and momentum transfer taking place in a wide variety of physical phenomena, including a simple falling apple. As a concrete example, we derive a general relativistic version of Archimedes’ law that we apply to understand electrostatic weight and buoyant force in the context of a Reissner–Nordström black hole.

Discrete flow mapping: transport of phase space densities on triangulated surfaces

Energy distributions of high-frequency linear wave fields are often modelled in terms of flow or transport equations with ray dynamics given by a Hamiltonian vector field in phase space. Applications arise in underwater and room acoustics, vibroacoustics, seismology, electromagnetics and quantum mechanics. Related flow problems based on general conservation laws are used, for example, in weather forecasting or in molecular dynamics simulations. Solutions to these flow equations are often large-scale, complex and high-dimensional, leading to formidable challenges for numerical approximation methods. This paper presents an efficient and widely applicable method, called discrete flow mapping , for solving such problems on triangulated surfaces. An application in structural dynamics, determining the vibroacoustic response of a cast aluminium car body component, is presented.

Conservation laws in gravitational theories with general nonminimal coupling

We use the Lagrange-Noether methods to derive the conservation laws for models in which matter interacts nonminimally with the gravitational field. The nonminimal coupling function can depend arbitrarily on the gravitational field strength. The obtained result generalizes earlier findings. The generalized conservation laws provide the basis for the derivation of the equations of motion for the nonminimally coupled test bodies.

Symmetry Analysis and Conservation Laws for the Hunter—Saxton Equation

In this paper, the problem of determining the most general Lie point symmetries group and conservation laws of a well known nonlinear hyperbolic PDE in mathematical physics called the Hunter-Saxton equation (HSE) is analyzed. By applying the basic Lie symmetry method for the HSE, the classical Lie point symmetry operators are obtained. Also, the algebraic structure of the Lie algebra of symmetries is discussed and an optimal system of one-dimensional subalgebras of the HSE symmetry algebra which creates the preliminary classification of group invariant solutions is constructed. Particularly, the Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained. Mainly, the conservation laws of the HSE are computed via three different methods including Boyer's generalization of Noether's theorem, first homotopy method and second homotopy method.

Numerical and dimensional investigation of two-phase countercurrent imbibition in porous media

In this paper, we introduce a numerical solution of the problem of two-phase immiscible flow in porous media. In the first part of this work, we present the general conservation laws for multiphase flows in porous media as outlined in the literature for the sake of completion where we emphasize the difficulties associated with these equations in their primitive form and the fact that they are, generally, unclosed. The second part concerns the 1D computation for dimensional and non-dimensional cases and a theoretical analysis of the problem under consideration. A time-scale based on the characteristic velocity is used to transform the macroscopic governing equations into a non-dimensional form. The resulting dimensionless governing equations involved some important dimensionless physical parameters such as Bond number Bo, capillary number Ca and Darcy number Da. Numerical experiments on the Bond number effect is performed for two cases, gravity opposing and assisting. The theoretical analysis illustrates that common formulations of the time-scale forces the coefficient Da1/2/Ca to be equal to one, while formulation of dimensionless time based on a characteristic velocity allows the capillary and Darcy numbers to appear in the dimensionless governing equation which leads to a wide range of scales and physical properties of fluids and rocks. The results indicate that the buoyancy effects due to gravity force take place depending on the location of the open boundary.

Quasilocal conservation laws: why we need them

We argue that conservation laws based on the local matter-only stress–energy–momentum tensor (characterized by energy and momentum per unit volume) cannot adequately explain a wide variety of even very simple physical phenomena because they fail to properly account for gravitational effects. We construct a general quasilocal conservation law based on the Brown and York total (matter plus gravity) stress–energy–momentum tensor (characterized by energy and momentum per unit area), and argue that it does properly account for gravitational effects. As a simple example of the explanatory power of this quasilocal approach, consider that, when we accelerate toward a freely-floating massive object, the kinetic energy of that object increases (relative to our frame). But how, exactly, does the object acquire this increasing kinetic energy? Using the energy form of our quasilocal conservation law, we can see precisely the actual mechanism by which the kinetic energy increases: it is due to a bona fide gravitational energy flux that is exactly analogous to the electromagnetic Poynting flux, and involves the general relativistic effect of frame dragging caused by the object’s motion relative to us.

SPIN FLUIDS IN BIANCHI-I f(R)-COSMOLOGY WITH TORSION

We study Weyssenhoff spin fluids in Bianchi type-I cosmological models, within the framework of torsional f(R)-gravity; the resulting field equations are derived and discussed in both Jordan and Einstein frames, clarifying the role played by the spin and the non-linearity of the gravitational Lagrangian f(R) in generating the torsional dynamical contributions. The general conservation laws holding for f(R)-gravity with torsion are employed to provide the conditions needed to ensure the preservation of the Hamiltonian constraint and the consequent correct formulation of the associated initial value problem. Examples are eventually given.

General constrained conservation laws. Application to pedestrian flow modeling

We generalize the results on conservation laws with local flux constraint obtained in [1, 9] to general flux functions and nonclassical solutions arising for example in pedestrian flow modeling. We first define the constrained Riemann solver and the entropy condition, which singles out the unique admissible solution. We provide a well posedness result based on wave-front tracking approximations and Kruzhkov doubling of variable technique. We then provide the framework to deal with nonclassical solutions and we propose a front-tracking finite volume scheme allowing to sharply capture classical and nonclassical discontinuities. Numerical simulations illustrating the Braess paradox are presented as validation of the method.

Primordial vorticity and gradient expansion

The evolution equations of the vorticities of the electrons, ions and photons in a pre-decoupling plasma are derived, in a fully inhomogeneous geometry, by combining the general relativistic gradient expansion and the drift approximation within the Adler–Misner–Deser decomposition. The vorticity transfer between the different species is discussed in this novel framework and a set of general conservation laws, connecting the vorticities of the three-component plasma with the magnetic field intensity, is derived. After demonstrating that a source of large-scale vorticity resides in the spatial gradients of the geometry and of the electromagnetic sources, the total vorticity is estimated to lowest order in the spatial gradients and by enforcing the validity of the momentum constraint. By acknowledging the current bounds on the tensor to scalar ratio in the (minimal) tensor extension of the ΛCDM paradigm, the maximal comoving magnetic field induced by the total vorticity turns out to be, at most, of the order of 10−37 G over the typical comoving scales ranging between 1 and 10 Mpc. While the obtained results seem to be irrelevant for seeding a reasonable galactic dynamo action, they demonstrate how the proposed fully inhomogeneous treatment can be used for the systematic scrutiny of pre-decoupling plasmas beyond the conventional perturbative expansions.

General Fractal Conservation Laws Arising from a Model of Detonations in Gases

We consider a model of cellular detonations in gases. They consist in conservation laws with a non-local pseudo-differential operator whose symbol is asymptotically |ξ|λ, where 0 < λ ≤ 2; it can be decomposed as the λ/2 fractional power of the Laplacian plus a convolution term. After defining the notion of entropy solution, we prove the well-posedness in the L∞ framework. In the case where 1 < λ ≤ 2 we also prove a regularizing effect. In Appendix, we show that the assumptions made to perform the mathematical study are satisfied by the considered physical model of detonations.

On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws

This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix. The resulting scheme is much simpler than the original one and is applicable to general hyperbolic conservation laws, while retaining the attractive features of the original solver: the method is entropy-satisfying, differentiable and complete in the sense that it attributes a different numerical viscosity to each characteristic field, in particular to the intermediate ones, since the full eigenstructure of the underlying hyperbolic system is used. To illustrate the potential of the proposed scheme we show applications to the following hyperbolic conservation laws: Euler equations of compressible gas-dynamics with ideal gas and real gas equation of state, classical and relativistic MHD equations as well as the equations of nonlinear elasticity. To the knowledge of the authors, apart from the Euler equations with ideal gas, an Osher-type scheme has never been devised before for any of these complicated PDE systems. Since our new general Riemann solver can be directly used as a building block of high order finite volume and discontinuous Galerkin schemes we also show the extension to higher order of accuracy and multiple space dimensions in the new framework of PNPM schemes on unstructured meshes recently proposed in [9].

Breakup modeling of a liquid jet in cross flow

We propose a novel breakup model to simulate the catastrophic breakup regime in a supersonic cross flow. A developed model has been extended from an existing Kelvin-Helmholtz/Rayleigh-Taylor (K-H/R-T) hybrid model. A new mass reduction rate equation, which has critical effects on overall spray structure, is successfully adopted, and the breakup length, which is an important parameter in existing model, is replaced by the breakup initiation time. Measured data from the supersonic wind tunnel with a dimension of 762×152×127 mm was employed to validate the newly developed breakup model. A nonaerated injector with an orifice diameter of 0.5 mm is used to inject water into a supersonic flow prescribed by the momentum flux ratio of the liquid jet to free stream air, q0. The conservation-element and solution-element (CE/SE) method, a novel numerical framework for the general conservation law, is applied to simulate the supersonic compressible flow. The spray penetration height and average droplet size along with a spray penetration axis are quantitatively compared with data. The shock train flow structures induced by the presence of a liquid jet are further discussed.

An Equation-Solving Solution Gradient Strategy for High-Resolution and Compact Discretization of Hyperbolic Conservation Laws

The present study extends our previous formulation in simulating steady incompressible Navier-Stokes equations to solving unsteady compressible flows. It is designed by taking the solution gradient as an additional computational variable to build a high-resolution and compact discretization. Essential ingredients concerning the derivation procedure are detailed in the frame of general hyperbolic conservation laws. Numerical analyses on modeled problems are performed to reveal its stability criterion and formal accuracy. Several one- and two-dimensional problems are solved numerically, and the computational results are compared with those acquired by existing schemes to demonstrate their relative efficiency. It is found that the present formulation will be a useful tool to simulate general hyperbolic conservation laws.

A High Order Sharp-Interface Method with Local Time Stepping for Compressible Multiphase Flows

AbstractIn this paper, a new sharp-interface approach to simulate compressible multiphase flows is proposed. The new scheme consists of a high order WENO finite volume scheme for solving the Euler equations coupled with a high order path-conservative discontinuous Galerkin finite element scheme to evolve an indicator function that tracks the material interface. At the interface our method applies ghost cells to compute the numerical flux, as the ghost fluid method. However, unlike the original ghost fluid scheme of Fedkiw et al. [15], the state of the ghost fluid is derived from an approximate-state Riemann solver, similar to the approach proposed in [25], but based on a much simpler formulation. Our formulation leads only to one single scalar nonlinear algebraic equation that has to be solved at the interface, instead of the system used in [25]. Away from the interface, we use the new general Osher-type flux recently proposed by Dumbser and Toro [13], which is a simple but complete Riemann solver, applicable to general hyperbolic conservation laws. The time integration is performed using a fully-discrete one-step scheme, based on the approaches recently proposed in [5,7]. This allows us to evolve the system also with time-accurate local time stepping. Due to the sub-cell resolution and the subsequent more restrictive time-step constraint of the DG scheme, a local evolution for the indicator function is applied, which is matched with the finite volume scheme for the solution of the Euler equations that runs with a larger time step. The use of a locally optimal time step avoids the introduction of excessive numerical diffusion in the finite volume scheme. Two different fluids have been used, namely an ideal gas and a weakly compressible fluid modeled by the Tait equation. Several tests have been computed to assess the accuracy and the performance of the new high order scheme. A verification of our algorithm has been carefully carried out using exact solutions as well as a comparison with other numerical reference solutions. The material interface is resolved sharply and accurately without spurious oscillations in the pressure field.

General Covariant Conservation Law of Energy-Momentum in f(R) Gravity

In the light of the local Lorentz transformations and the general Noether theorem, a new formulate of the general covariant energy-momentum conservation law in f(R) gravity is obtained, which does not depend on the coordinative choice.

On the hierarchy of conservation laws in a cellular automaton

Conservation laws in cellular automata (CA) are studied as an abstraction of the conservation laws observed in nature. In addition to the usual real-valued conservation laws we also consider more general group-valued and semigroup-valued conservation laws. The (algebraic) conservation laws in a CA form a hierarchy, based on the range of the interactions they take into account. The conservation laws with smaller interaction ranges are the homomorphic images of those with larger interaction ranges, and for each specific range there is a most general law that incorporates all those with that range. For one-dimensional CA, such a most general conservation law has—even in the semigroup-valued case—an effectively constructible finite presentation, while for higher-dimensional CA such effective construction exists only in the group-valued case. It is even undecidable whether a given two-dimensional CA conserves a given semigroup-valued energy assignment. Although the local properties of this hierarchy are tractable in the one-dimensional case, its global properties turn out to be undecidable. In particular, we prove that it is undecidable whether this hierarchy is trivial or unbounded. We point out some interconnections between the structure of this hierarchy and the dynamical properties of the CA. In particular, we show that positively expansive CA do not have non-trivial real-valued conservation laws.