Ideal Fluid Equations Research Articles

Generalized Brenier Principle and the Closure Problem of Landgren–Monin–Novikov Hierarchy for Vorticity Field

Brenier’s concept – a representation of solutions to the equations of ideal incompressible fluids in terms of probability measures on the set of Lagrangian trajectories in the case of their stochasticity, is a generalization of Arnold’s principle of least action of finding smooth solutions of Euler’s equations. In this work, the variational generalized Brenier principle (Brenier, J. Am. Math. Soc. 1989) is used to close the infinite chain of Landgren–Monin–Novikov equations for the n-point probability density functions fn of the vortex field of two-dimensional turbulence. In addition, within the framework of the statistical approach, an approximation of the variational problem with conditions at the ends posed by Shnirelman (Mat. Sat. 1985) for the Euler equation is proposed.

A Hamiltonian description of finite-time singularity in Euler's fluid equations

The recently proposed low degree-of-freedom model of Moffatt and Kimura [1,2] for describing the approach to finite-time singularity of the incompressible Euler fluid equations is investigated. The model assumes an initial finite-energy configuration of two vortex rings placed symmetrically on two tilted planes. The Hamiltonian structure of the inviscid limit of the model is obtained. The associated noncanonical Poisson bracket [3] and two invariants, one that serves as the Hamiltonian and the other a Casimir invariant, are discovered. It is shown that the system is integrable with a solution that lies on the intersection for the two invariants, just as for the free rigid body of mechanics whose solution lies on the intersection of the kinetic energy and angular momentum surfaces. Also, a direct quadrature is given and used to demonstrate the Leray form for finite-time singularity in the model. To the extent the Moffatt and Kimura model accurately represents Euler's ideal fluid equations of motion, we have shown the existence of finite-time singularity.

Cauchy Problem for Incompressible Neo-Hookean Materials

In this paper we consider the Cauchy problem for neo-Hookean incompressible elasticity in spatial dimension $$d \geqq 2$$ . The Cauchy problem can be formulated in terms of maps $$x(t,\cdot )$$ with domain a reference space $${\mathbb {R}}^d_\xi $$ , and with values in space $${\mathbb {R}}^d_x$$ . Initial data consists of initial deformation $$\phi (\xi ) = x(0,\xi )$$ and velocity $$\psi (\xi ) = \partial x(t,\xi )/\partial t |_{t=0}$$ . We consider the initial deformations of the form $$x(0, \xi ) = A \xi + \varphi (\xi )$$ , where A is a constant $$SL(d, {\mathbb {R}})$$ matrix. We assume that $$\varphi $$ and $$\psi $$ are in Sobolev spaces $$(\varphi , \psi ) \in H^{s+1}({\mathbb {R}}^d)\times H^{s}({\mathbb {R}}^d)$$ . If $$s>s_{crit}= d/2+1$$ , well-posedness is well-known. We are here interested primarily in the low regularity case, $$s \le s_{crit}$$ . For $$d = 2, 3$$ , we prove existence and uniqueness for $$s_0 < s\le s_{crit}$$ , and we can prove the well-posedness, but for a smaller range, $$s_1 < s \le s_{crit}$$ , where, if $$d = 2$$ , $$s_0 = 7/4$$ and $$s_1 = 7/4 + (\sqrt{65}-7)/8$$ , and if $$d=3$$ , then $$s_0=2$$ and $$s_1 = 1 + \sqrt{3/2}$$ . For the full range (in s) results, as indicated above, we need additional Hölder regularity assumptions on certain combinations of second order derivatives of $$\varphi $$ . A key observation in the proof is that the equations of evolution for the vorticities decompose into a first-order hyperbolic system, for which a Strichartz estimate holds, and a coupled transport system. This allows one to set up a bootstrap argument to prove local existence and uniqueness. Continuous dependence on initial data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modification based on new estimates for Riesz potentials. The results of this paper should be compared to what is known for the ideal fluid equations, where, as shown by Bourgain and Li, the requirement $$s > s_{crit}$$ is necessary.

Exact solution of partial differential equations for the creation of jet-like flows in plasmas and neutral fluids

An exact solution of two fluid ideal classical plasma equations is presented which shows that the one-dimensional jet-like axial outflow and two-dimensional magnetic field are generated simultaneously by the density and temperature gradients of both electrons and ions in cylindrical geometry. Particular profiles of density function ψ=ln n¯ (where n¯ is normalized by some constant density N0) and temperatures Tj (for j=e,i) have to be chosen to obtain an exact solution of the complicated nonlinear partial differential equations. But this is a natural analytical exact solution of the ideal two fluid plasma equations. The basic mechanism presented here explains the creation of plasma jet-like flows along with magnetic fields in astrophysical environments such as young stellar objects, active galactic nuclei, solar spicules, flares, and coronal loops. The theoretical model is also applicable to laser induced plasma where magnetic field and plasma ablation are produced simultaneously. An exact analytical solution of ideal neutral fluid equations is also presented which shows that the jet-like outflows can be generated by the density and temperature gradients in such systems.

Symmetry Analysis of Ideal Fluid Equations in Terms of Trajectories and Weber’s Potential

Symmetry Analysis of Ideal Fluid Equations in Terms of Trajectories and Weber’s Potential

Boltzmann-Fokker-Planck formalism for dark matter-baryon scattering

Linear-cosmology observables, such as the Cosmic Microwave Background (CMB), or the large-scale distribution of matter, have long been used as clean probes of dark matter (DM) interactions with baryons. It is standard to model the DM as an ideal fluid with a thermal Maxwell-Boltzmann (MB) velocity distribution, in order to compute the heat and momentum-exchange rates relevant to these probes. This approximation only applies in the limit where DM self-interactions are frequent enough to efficiently redistribute DM velocities. It does not accurately describe weakly self-interacting particles, whose velocity distribution unavoidably departs from MB once they decouple from baryons. This article lays out a new formalism required to accurately model DM-baryon scattering, even when DM self-interactions are negligible. The ideal fluid equations are replaced by the collisional Boltzmann equation for the DM phase-space distribution. The collision operator is approximated by a Fokker-Planck operator, constructed to recover the exact heat and momentum exchange rates, and allowing for an efficient numerical implementation. Numerical solutions to the background evolution are presented, which show that the MB approximation can over-estimate the heat-exchange rate by factors of ~ 2-3, especially for light DM particles. A Boltzmann-Fokker-Planck hierarchy for perturbations is derived. This new formalism allows to explore a wider range of DM models, and will be especially relevant for upcoming ultra-high-sensitivity CMB probes.

Computing the Dirichlet--Neumann Operator on a Cylinder

The computation of the Dirichlet--Neumann operator for the Laplace equation is the primary challenge for the numerical simulation of the ideal fluid equations. The techniques commonly used for 2D fluids, such as conformal mapping and boundary integral methods, fail to generalize suitably to three dimensions. In this study, we address this problem by developing a Transformed Field Expansion method for computing the Dirichlet--Neumann operator in a cylindrical geometry with a variable upper boundary. This technique reduces the problem to a sequence of Poisson equations on a flat geometry. We design a fast and accurate solver for these subproblems, a key ingredient of which is the use of Zernike polynomials for the circular cross-section instead of the traditional Bessel functions. This lends spectral accuracy to the method and allowing significant computational speed-up. We rigorously analyze the algorithm, prove its applicability to a wide class of problems, and demonstrate its effectiveness numerically.

Systems with strong damping and their spectra

We establish a new method for obtaining nonconvex spectral enclosures for operators associated with second‐order differential equations in a Hilbert space. In particular, we succeed in establishing the existence of a spectral gap, which is the first result of this kind since the seminal results of Krein and Langer for oscillations of damped systems. While the latter and other spectral bounds are confined to dampings D that are symmetric and dominated by A0, we allow for accretive D of equal strength as A0. To achieve these results, we prove new abstract spectral inclusion results that are much more powerful than classical numerical range bounds. Two different applications, small transverse oscillations of a horizontal pipe carrying a steady‐state flow of an ideal incompressible fluid and wave equations with strong (viscoelastic and frictional) damping, illustrate that our new bounds are explicit.

Systems of Hydrodynamic Type that Approximate Two-Dimensional Ideal Fluid Equations

Statistical properties of different finite-dimensional approximations of two-dimensional ideal fluid equations are studied. A special class of approximations introduced by A.M. Obukhov (systems of hydrodynamic type) is considered. Vorticity distributions over area and quasi-equilibrium coherent structures are studied. These coherent structures are compared to structures occurring in a viscous fluid with random forcing.

Sedeonic equations of ideal fluid

In the present paper, we propose the generalized equations for an ideal fluid based on space-time algebra of sixteen-component sedeons. It is shown that the dynamics of isentropic fluid can be described by sedeonic first-order wave equation for fluid potentials. The key features of the proposed formalism are illustrated on the problem of the sound waves propagation. We consider the plane wave solution of linearized sedeonic wave equation and derive the second-order relations for the sound potential analogues to the Poynting theorem in electrodynamics. The generalization of proposed sedeonic equations for the description of viscous fluid is also discussed.

Towards fluid instabilities of stationary non-Killing horizons

Flowing black holes are asymptotically locally AdS spacetimes that are stationary but have non-Killing horizons. Holographically, they are dual to a steady-state heat flow in the boundary field theory. We investigate the stability of these black holes in the limit in which they are well-described by the relativistic conformal Navier–Stokes equation. More precisely, we study the quasi-normal modes of the linearized ideal fluid equations. Though we find no unstable modes, there are an infinite number of modes at finite transverse momentum which are arbitrarily long-lived. This suggests the possibility that either non-modal effects or nonlinear interactions between these modes can give rise to new types of gravitational instabilities.

Shear Flows of an Ideal Fluid and Elliptic Equations in Unbounded Domains

We prove that, in a two‐dimensional strip, a steady flow of an ideal incompressible fluid with no stationary point and tangential boundary conditions is a shear flow. The same conclusion holds for a bounded steady flow in a half‐plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on one‐dimensional symmetry results for solutions of some semilinear elliptic equations. Some related rigidity results of independent interest are also shown in n‐dimensional slabs in any dimension n.© 2016 Wiley Periodicals, Inc.

Quasistationary solutions of self-gravitating scalar fields around collapsing stars

Recent work has shown that scalar fields around black holes can form long-lived, quasistationary configurations surviving for cosmological timescales. With this requirement, scalar fields cannot be discarded as viable candidates for dark matter halo models in galaxies around central supermassive black holes (SMBH). One hypothesis for the formation of most SMBHs at high redshift is the gravitational collapse of supermassive stars (SMS) with masses of $\sim10^5 \rm {M_{\odot}}$. Therefore, a constraint for the existence of quasi-bound states of scalar fields is their survival to such dynamic events. To answer this question we present in this paper the results of a series of numerical relativity simulations of gravitationally collapsing, spherically symmetric stars surrounded by self-gravitating scalar fields. We use an ideal fluid equation of state with adiabatic index $\Gamma=4/3$ which is adequate to simulate radiation-dominated isentropic SMSs. Our results confirm the existence of oscillating, long-lived, self-gravitating scalar field configurations around non-rotating black holes after the collapse of the stars.

Stochastic variational method as quantization scheme: Field quantization of the complex Klein–Gordon equation

Stochastic Variational Method (SVM) is the generalization of the variation method to the case with stochastic variables. In the series of papers, we investigate the applicability of SVM as an alternative field quantization scheme. Here, we discuss the complex Klein-Gordon equation. In this scheme, the Euler-Lagrangian equation for the stochastic fields leads to the functional Schroedinger equation, which in turn can be interpreted as the ideal fluid equation in the functional space. We show that the Fock state vector is given by the stationary solution of these differential equations and various results in the usual canonical quantization can be reproduced, including the effect of anti-particles. The present formulation is a quantization scheme based on commutable variables, so that there appears no ambiguity associated with the ordering of operators, for example, in the definition of Noether charges.

Octonic formulations of Maxwell type fluid equations

Although the origins of electromagnetic theory and the fluid mechanics are completely different, the compressible ideal fluid equations can be expressed in a form analogous to that of electromagnetism. In this paper, the Maxwell type fluid equations are rewritten in terms of octons. Using the analogies between electromagnetism and fluid mechanics, the compact and elegant formulations are derived. It is proved that single octonic expression describes Maxwell-type fluid equations. The octonic fluid wave equations and fluid Poynting theorem are also formulated in a similar form to electromagnetism and linear gravity.

The angle-averaged squeezed limit of nonlinear matter N-point functions

Assuming Gaussian initial conditions, we show that in a certain, angle-averaged squeezed limit the N-point function of matter is related to the response of the matter power spectrum to a long-wavelength density perturbation, P−1dnP(k|δL)/dδLn|δL=0, with n=N−2. By performing N-body simulations with a homogeneous overdensity superimposed on a flat Friedmann-Robertson-Lemaȋtre-Walker (FRLW) universe using the separate universe approach, we obtain measurements of the nonlinear matter power spectrum response up to n=3, which is equivalent to measuring the fully nonlinear matter 3- to 5-point function in this squeezed limit. The sub-percent to few percent accuracy of those measurements is unprecedented. We then test the hypothesis that nonlinear N-point functions at a given time are a function of the linear power spectrum at that time, which is predicted in an Einstein-de Sitter (EdS) universe by standard perturbation theory (SPT) and its variants that are based on the ideal pressureless fluid equations. Specifically, we compare the responses computed from the separate universe simulations and simulations with a rescaled initial (linear) power spectrum amplitude. We find discrepancies of 10% at k≃ 0.2–0.5 h Mpc−1 for 5- to 3-point functions at z=0. The discrepancy occurs at higher wavenumbers at z=2. Thus, theoretical predictions that are insensitive to the growth history, such as SPT and its variants assuming EdS, even when carried out to arbitrarily high order, are guaranteed to fail to describe matter N-point functions (N>2) around that scale.

CFD SIMULATION OF THE HORIZONTAL SETTLER WITH FIN WORK

Horizontal settlers are one of the most important elements in the technological scheme for drinking water preparing and waste water treatment. Their use is associated with the possibility to pass a sufficiently large volume of water. When remodeling or designing horizontal tanks there is an important task to evaluate their effectiveness. Calculation of the efficiency of the settler can be made by mathematical modeling. Currently used models and methods do not allow take into account the shape of the sump and the various design features. In this paper we consider the construction of a numerical model to evaluate the effectiveness of horizontal settler modified structure. The model is based on equations of motion of an ideal fluid and mass transfer equation. For numerical simulation the finite difference schemes are used. The numerical calculation is carried out on a rectangular grid. For the formation of the computational domain markers are used. The model allows calculate the clarification process in the sump using computers small and medium power. Calculation time of one variant of the problem is a few seconds. The results of a computational experiment are presented.

Computation of Three-dimensional Flexural-gravity Solitary Waves in Arbitrary Depth

Fully-localised solitary waves propagating on the surface of a three-dimensional ideal fluid of arbitrary depth, and bounded above by an elastic sheet that resists flexing, are computed. The cases of shallow and deep water are distinct. In shallow water, weakly nonlinear modulational analysis (see Milewski & Wang 6) predicts waves of arbitrarily small amplitude and these are found numer- ically. In deep water, the same analysis rules out the existence of solitary waves bifurcating from linear waves, but, nevertheless, we find them at finite amplitude. This is accomplished using a continuation method following the branch from the shallow regime. All solutions are computed via a fifth-order Hamiltonian truncation of the full ideal free-boundary fluid equations. We show that this truncation is quantitatively accurate by comparisons with full potential flow in two-dimensions.

Growth rate analysis of scalar gradients in generalized surface quasigeostrophic equations of ideal fluids

The growth rates of scalar gradients are studied numerically and analytically in a family of two-dimensional (2D) incompressible fluid equations related to the surface quasigeostrophic (SQG) equation. The active scalar is related to the stream function ψ by θ=(-△){α/2}ψ (0 ≤ α ≤ 2). A notable difference is observed in a comparison of the instantaneous growth rates in L{p} and in L{∞} norms, depending on the stage of the time evolution. The crux is the phase-shift effect of singular integral operators, which displaces the peak location of the scalar gradient from that of the strain rate. On this basis, a method of detecting such a dislocation is proposed in view of the importance of their coalescence needed for a possible blow-up. Moreover, it is found in the long-time evolution that a solution of the SQG equation (whose regularity is not known) is less singular than that of the 2D Euler equations (known to be regular) on the time interval covered by this computation. This consistently expands an earlier observation by Majda and Tabak [Physica D 98, 515 (1996).] in some detail. A 1D model problem is discussed to illustrate the present method, and extensions to the 3D case are also are briefly discussed.

LIFTS, JETS AND REDUCED DYNAMICS

We show that complete cotangent lifts of vector fields, their decomposition into vertical representative and holonomic part provide a geometrical framework underlying Eulerian equations of continuum mechanics. We discuss Euler equations for ideal incompressible fluid and momentum-Vlasov equations of plasma dynamics in connection with the lifts of divergence-free and Hamiltonian vector fields, respectively. As a further application, we obtain kinetic equations of particles moving with the flow of contact vector fields both from Lie–Poisson reductions and with the techniques of present framework.