Generalized Euler Equations Research Articles

The Euler limit for kinetic models with Fermi–Dirac statistics

We are interested in the connection between kinetic models with Fermi-Dirac statistics and fluid dynamics. We es- tablish that moments and parameters of Fermi-Dirac distributions are related by a diffeomorphism. We obtain the macroscopic limits when the fluid is dense enough that particles undergo many collisions per unit of time. This situation is described via a small parameter e, called the Knudsen number, that represents the ratio of mean free path of particles between collisions to some characteristic length of the flow. We give the conditions that allow us to formally derive the generalized Euler equations from the Boltzmann equation by adopting the formalism proposed in (Advances in Kinetic Theory and Continuum Mechanics, Springer, Berlin, 1991, pp. 57-71). These conditions are related to the H -theorem and assume a formally consistent conver- gence for fluid dynamical moments and entropy of the kinetic equation. We also discuss the well-posedness of the obtained Euler equations by using Godunov's criterion of hyperbolicity.

COMPUTING MARKOV-PERFECT OPTIMAL POLICIES IN BUSINESS-CYCLE MODELS

Time inconsistency is an essential feature of many policy problems. This paper presents and compares three methods for computing Markov-perfect optimal policies in stochastic nonlinear business cycle models. The methods considered include value function iteration, generalized Euler equations, and parameterized shadow prices. In the context of a business cycle model in which a fiscal authority chooses government spending and income taxation optimally, although lacking the ability to commit, we show that the solutions obtained using value function iteration and generalized Euler equations are somewhat more accurate than that obtained using parameterized shadow prices. Among these three methods, we show that value function iteration can be applied easily, even to environments that include a risk-sensitive fiscal authority and/or inequality constraints on government spending. We show that the risk-sensitive fiscal authority lowers government spending and income taxation, reducing the disincentive to accumulate wealth that households face.

Non-monotonic traveling wave and computational solutions for gas dynamics Euler equations with stiff relaxation source terms

We study the existence of non-monotone traveling wave solutions and its properties for an isothermal Euler system with relaxation describing the perfect gas flow. In order to confront our results, we first apply a mollification approach as an effective regularization method for solving an ill-posed problem for an associated reduced system for the Euler model under consideration, which in turn is solved by using the method of characteristics. Next, we developed a cheap unsplitting finite volume scheme that reproduces the same traveling wave asymptotic structure as that of the Euler solutions of the continuous system at the discrete level. The method is conservative by construction and relatively easy to understand and implement. Although we do not have a mathematical proof that our designed scheme enjoys the asymptotic preserving and well-balanced properties, we were able to reproduce consistent solutions for the more general Euler equations with gravity and friction recently published in the specialized literature, which in turn are procedures based on a Godunov-type scheme and based on an asymptotic preserving scheme, yielding good verification and performance to our method.

Trajectory analysis of the rotational dynamics of molecules

A method for analysis of the rotational dynamics of molecular systems has been proposed on the basis of the calculation of the set of exact classical vibrational–rotational trajectories. It has been proposed to compose and to numerically solve the complete system of dynamic equations consisting of Hamilton’s equations and generalized Euler equations for an arbitrary system. The computer algebra system can be applied to automatize the process of derivation and subsequent solution of dynamic equations. The variation of the picture of known bifurcation in the rotational dynamics of symmetric triatomic hydride molecules with an increase in vibrational excitation has been studied within the proposed approach. It has been shown that manifestations of bifurcation completely disappear at a quite high level of vibrational excitations.

Instability of meridional axial system in f(R) gravity

We analyze the dynamical instability of a non-static reflection axial stellar structure by taking into account the generalized Euler equation in metric f(R) gravity. Such an equation is obtained by contracting the Bianchi identities of the usual anisotropic and effective stress-energy tensors, which after using a radial perturbation technique gives a modified collapse equation. In the realm of the \(R+\epsilon R^n\) gravity model, we investigate instability constraints at Newtonian and post-Newtonian approximations. We find that the instability of a meridional axial self-gravitating system depends upon the static profile of the structure coefficients, while f(R) extra curvature terms induce the stability of the evolving celestial body.

A Generalized Euler Equation to Predict Theoretical Head of Turbomachinery

The transient performance of centrifugal pumps during transient operating periods, such as startup and stopping, has been drawn much attention in the past years due to the growing engineering needs. However, the prediction of the transient theoretical head of the impeller has not been satisfactorily established yet. In present study, a new generalized Euler equation for incompressible flow in turbomachinery is developed, which includes influences of the blade thickness and shape. Then, the proposed theoretical equation is compared with an existing one as well as the experimental data of two centrifugal volute pumps during startup periods. The results indicated that the proposed equation is feasible to estimate the transient theoretical head and the steady head rise. In addition, it is also found from both the theory and the experiments that the transient theoretical head is much higher than the steady one in the initial stage of startup period.

Existence of solutions for models of shallow water in a basin with a degenerate varying bottom

We prove the existence of solutions for the great lake equations. These equations are obtained from the three-dimensional Euler equations in a basin with a free upper surface and a spatially varying bottom topography by taking a low aspect ratio, i.e., low wave speed and small wave amplitude expansion. These equations are rewritten in an abstract form by considering generalized Euler equations as in Levermore et al. (Indiana Univ Math J 45:479–510, 1996). This paper is an extension of Levermore et al. (Indiana Univ Math J 45:479–510, 1996), where the varying bottom was assumed to be nondegenerate. Here, we discuss the degenerate case and obtain similar results as in Levermore et al. (Indiana Univ Math J 45:479–510, 1996).

Modeling, Asymptotic Analysis, and Simulation of an Energy Tower

A new model for the description of an energy tower is presented. For the modeling of the transient flow in an energy tower and the related power production a one-dimensional approach is taken. Although many simplifying assumptions are imposed, this approach is thought to incorporate most of the main physical effects. The model is derived from the general one-dimensional Euler equations of gas dynamics for a compressible humid air gas mixture. Low Mach number asymptotics allow us to handle the related problems. Numerical simulations are performed. Optimal values for the spray rate and the height of the spraying region are identified. The model gives results which are in accordance with our physical understanding. For lack of an energy tower prototype, a quantitative comparison is not possible at the moment.

A Local Entropy Minimum Principle for Deriving Entropy Preserving Schemes

The present work deals with the establishment of stability conditions of finite volume methods to approximate weak solutions of the general Euler equations to simulate compressible flows. In order to ensure discrete entropy inequalities, we derive a new technique based on a local minimum principle to be satisfied by the specific entropy. Sufficient conditions are exhibited to satisfy the required local minimum entropy principle. Arguing these conditions, a class of entropy preserving schemes is thus derived.

The homogenized equations of motion for an activated elastic lamination in plane strain

AbstractThe equations of motion for an elastic laminar spatial‐temporal composite are investigated. It is assumed that the composite is binary, that is, it is assembled of two original constituents capable of changing (in space‐time) their material density, as well as their material stiffness. The condition of plane strain was then imposed on the composite. The paper begins by attempting to evaluate the materials' average Lagrangian (action density). In doing so, it immediately becomes apparent that expressions are needed for average momentum and stress. Both quantities are found to depend linearly on average strain and average velocity. After calculating the general Euler equations of motion, isotropy was assumed, and two additional forces (one being of a Coriolis nature) were found in the averaged equations of elastodynamics due to the presence of simultaneous change in both inertial and elastic properties of the original material constituents. The appearance of these two forces is a consequence of both dynamics and plane strain; the Coriolis type force disappears in the case of one dimensional strain that arises when longitudinal dynamic disturbances propagate along an elastic bar.

GPU acceleration for general conservation equations and its application to several engineering problems

Presented is a general method for conservation equations called SHLL (split HLL) applied using Graphics Processing Unit (GPU) acceleration. The SHLL method is a purely vector-split approximation of the classical HLL method (Harten et al., 1983 [1]) which assumes the presence of local wave propagation in the algebraic derivation of fluxes across cell surfaces. The conventional HLL flux expression terms are interface-split and wave propagation velocities estimated (where required) based on local conditions. Due to the highly local nature of the SHLL fluxes, the scheme is very efficiently applied to GPU computation since the flux, initialisation and update phases of the computation are all vectorized processes. The SHLL scheme is applied to GPU computating using Nvidia’s CUDA package. Numerical schemes are presented for solutions to the general transport (convection–diffusion) equation, Euler Equations and Shallow Water Equations with results presented for several benchmark gas and shallow water flow engineering problems. Computational times are compared between high-end GPU (Nvidia C1060) and CPU (Intel Xeon 3.0 GHz, 32 MB cache) systems with reported speedups of over 67 times when applied to two dimensional simulations with multi-million cell numbers.

Observation of Generalized Euler Equations

AbstractWe investigate conditons for generalized double bracket flows to be Morse–Smale.The results yield necessary and sufficient conditions for observability in finite time. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Time-Consistent Public Policy

In this paper we study how a benevolent government that cannot commit to future policy should trade off the costs and benefits of public expenditure. We characterize and solve for Markov-perfect equilibria of the dynamic game between successive governments. The characterization consists of an inter-temporal first-order condition (a “generalized Euler equation”) for the government, and we use it both to gain insight into the nature of the equilibrium and as a basis for computations. For a calibrated economy, we find that when the only tax base available to the government is capital income—an inelastic source of funds at any point in time—the government still refrains from taxing at confiscatory rates. We also find that when the only tax base is labour income the Markov equilibrium features less public expenditure and lower tax rates than the Ramsey equilibrium.

Generalized quasi-geometric discounting

This paper derives the ‘generalized Euler equation’ for an agent with multi-period deviations from geometric discounting. The functional equation that describes optimal consumption-savings decisions involves manipulation of future selves and indirect manipulation of more distant selves through intervening selves.

Magnetic hydrodynamics with asymmetric stress tensor

In this paper we study equations of magnetic hydrodynamics with a stress tensor. We interpret this system as the generalized Euler equation associated with an Abelian extension of the Lie algebra of vector fields with a nontrivial 2-cocycle. We use the Lie algebra approach to prove the energy conservation law and the conservation of cross-helicity.

Cayley kinematics and the Cayley form of dynamic equations

The Cayley transform and the Cayley–transform kinematic relationships are an important and fascinating set of results that have relevance in N –dimensional orientations and rotations. In this paper these results are used in two significant ways. First, they are used in a new derivation of the matrix form of the generalized Euler equations of motion for N –dimensional rigid bodies. Second, they are used to intimately relate the motion of general mechanical systems to the motion of higher–dimensional rigid bodies. This approach can be used to describe an enormous variety of systems, one example being the representation of general motion of an N –dimensional body as pure rotations of an ( N + 1)–dimensional body.

Hamel coefficients for the rotational motion of anN–dimensional rigid body

Many of the kinematic and dynamic concepts relating to rotational motion have been generalized for Ndimensional rigid bodies. In this paper a new derivation of the generalized Euler equations is pr...

The Kantorovich macro-or-mesoscopic refined solution for the heterogeneous functionally gradient material complex structure

This paper is a piece of research on the complex structure of functionally gradient materials, which is an applicable triangular cantilever plate structure locally fixed and supported by its round revolving axis. Combined with the generalized Euler equation and the generalized boundary conditions, Kantorovich method and the principle of the two independent variables generalized calculus of variations are adopted to establish the bending governing equation of plates to work out the solution. In comparison with the previous work on the problem, this paper, taking into account three generalized mechanical factors and FGM macro-or-mesoscopic heterogeneity, proposes a new concept of translating the issue of theoretical initial value into the problem of semi-analytical boundary value to obtain the refined solution and then researches the joint effect of grads stress fields. Thereby a refined version of Kantorovich macro-or-mesoscopic solution is developed.

An invariant formulation of the potential integration method for the vortical equation of motion of a material point

A relativistic procedure for deriving the kinetic part of the generalized Euler equation is proposed as an argument to justify the application of the vortical equation of motion to the solution of classical discrete dynamics problems. An invariant formulation of the potential integration method for the vortical equation of motion is given for a definite class of two-dimensional motions. To demonstrate the efficiency of the method, a number of well-known theorems on the dynamics of a materialpoint are proved. A new result of the study is the fact that zero-energy hyperelliptic motions are related to the field of 'multiplicative' type forces.

TESTING FOR SUNSPOTS IN THE FOREIGN EXCHANGE MARKET

It has been shown that the nonfundamental uncertainty called sunspots matters in many areas of the economy. Noting the fact that nationwide capital movement and the speculative demand for foreign currencies are rapidly increasing, this paper conducts empirical tests on the sunspot exchange rate model. The empirical result shows that the sunspot equilibrium exchange rate deviating from Purchasing Power Parity (PPP) and interest Rate Parity (IRP) is consistent with the real data. More importantly, the Generalized Method of Moments (GMM) over-identification test is shown to support the evidence that more general Euler equations are in favor of our sunspot equilibrium exchange rate model. [C52, E44]