Euler Equations Of Hydrodynamics Research Articles

Waves on a Stuart vortex

We consider the linear stability of Stuart vortices, an exact periodic solution of the two-dimensional inviscid Euler equations of hydrodynamics consisting of an infinite row of co-rotating vortices, to wavelike disturbances when the vortex amplitude is small but still much larger than the disturbances. An unsteady critical layer analysis is used, with outer and inner expansions away from and near the vortex cores which we match to obtain disturbance growth rates. Disturbance-vortex interactions occur inside the critical layer and are neglected elsewhere. We consider α+-1∼O(1) and α+≈1 separately, where α+ is ratio of the disturbance wavenumber to that of the periodic vortex row. We consider primarily three-dimensional (oblique) disturbances but touch upon two-dimensional disturbances in the vortex plane. There are resonance mechanisms for oblique disturbances when k+⩽1, with k+ the ratio of the streamwise disturbance wavenumber to that of the periodic vortex row. When k+=1, this is fundamental mode or Benney–Lin instability, thought to be connected to streamwise streaks seen in experiments, and when k+=1/2, this is subharmonic instability or helical pairing, thought to be connected to vortex pairing.

Variational derivation of two-component Camassa–Holm shallow water system

By a variational approach in the Lagrangian formalism, we derive the nonlinear integrable two-component Camassa–Holm system (1). We show that the two-component Camassa–Holm system (1) with the plus sign arises as an approximation to the Euler equations of hydrodynamics for propagation of irrotational shallow water waves over a flat bed. The Lagrangian used in the variational derivation is not a metric.

Isogeometric analysis of Lagrangian hydrodynamics

Isogeometric analysis of Lagrangian shock hydrodynamics is proposed. The Euler equations of compressible hydrodynamics in the weak form are discretized using Non-Uniform Rational B-Splines (NURBS) in space. The discretization has all the advantages of a higher-order method, with the additional benefits of exact symmetry preservation and better per-degree-of-freedom accuracy. An explicit, second-order accurate time integration procedure, which conserves total energy, is developed and employed to advance the equations in time. The performance of the method is examined on a set of standard 2D and 3D benchmark examples, where good quality of the computational results is attained.

Bose-Einstein condensate general relativistic stars

We analyze the possibility that due to their superfluid properties some compact astrophysical objects may contain a significant part of their matter in the form of a Bose-Einstein condensate. To study the condensate we use the Gross-Pitaevskii equation, with arbitrary non-linearity. By introducing the Madelung representation of the wave function, we formulate the dynamics of the system in terms of the continuity equation and of the hydrodynamic Euler equations. The non-relativistic and Newtonian Bose-Einstein gravitational condensate can be described as a gas, whose density and pressure are related by a barotropic equation of state. In the case of a condensate with quartic non-linearity, the equation of state is polytropic with index one. In the framework of the Thomas-Fermi approximation the structure of the Newtonian gravitational condensate is described by the Lane-Emden equation, which can be exactly solved. The case of the rotating condensate is also discussed. General relativistic configurations with quartic non-linearity are studied numerically with both non-relativistic and relativistic equations of state, and the maximum mass of the stable configuration is determined. Condensates with particle masses of the order of two neutron masses (Cooper pair) and scattering length of the order of 10-20 fm have maximum masses of the order of 2 M_sun, maximum central density of the order of 0.1-0.3 10^16 g/cm^3 and minimum radii in the range of 10-20 km. In this way we obtain a large class of stable astrophysical objects, whose basic astrophysical parameters (mass and radius) sensitively depend on the mass of the condensed particle, and on the scattering length. We also propose that the recently observed neutron stars with masses in the range of 2-2.4 M_sun are Bose-Einstein Condensate stars.

Shock wave formation in hot nuclear matter

Assuming that nuclear matter can be treated as a perfect fluid, we study the propagation of perturbations in the baryon density at high temperature. The equation of state is derived from the non-linear Walecka model. The expansion of the Euler and continuity equations of relativistic hydrodynamics around equilibrium configurations lead to the breaking wave equation for the density perturbation. We solve it numerically for this perturbation and follow the propagation of the initial pulses.

Experimentally validated multiphysics computational model of refracting shock wave lithotripter.

We will present a computational model of the focusing of an electromagnetically induced pressure pulse by an acoustic lens and subsequent shock wave formation. Linear elasticity equations for the lens are solved simultaneously with Euler hydrodynamic equations for water considered to be a compressible medium with a Tait equation of state. The adaptive mesh refinement model allows both three‐dimensional and two‐dimensional axisymmetric computations. A number of coupling approaches at the lens‐water interface are investigated by comparison to experimental results: transfer only of pressure boundary condition, coupling of displacement velocities, and a buffer linear elasticity region in the water immediately adjacent to lens. The model is validated against single‐medium measurements (water or lens material), and the complete experimental shock wave formation process. Initial results from a crack propagation model in stone simulants placed at the lithotripter focus will also be presented.

The Generalized Relativistic Kinetic and Hydrodynamic Theory—Part I

On the basis of statistical description the motion of relativistic particles without external forces is considered. The generalized relativistic non-local Euler hydrodynamic equations are derived and investigated, including the limiting cases of the theory. The theory is applied to the propagation of plane harmonic waves of small amplitudes through a relativistic gas at rest. These results lead to construction of the relativistic quantum hydrodynamics.

BREAKING WAVES IN AN IDEAL QUARK GLUON PLASMA

We study the propagation of perturbations in the energy density in a quark gluon plasma. Expanding the Euler and continuity equations of relativistic hydrodynamics around equilibrium configurations we obtain a nonlinear differential equation called the breaking wave equation. We solve it numerically and follow the time-evolution of initially localized pulses. We find that, quite unexpectedly, these pulses live for a very long time (compared to the reaction time-scales) before breaking. In practice, they mimick the Korteweg–de Vries solitons. Their existence may have some observable consequences.

Non-linear effects on the propagation of waves in hot nuclear matter

We show that, expanding the Euler and continuity equations of relativistic hydrodynamics around equilibrium configurations and then combining them, it is possible to obtain a differential equation known as breaking wave equation. Taking a localized pulse as initial condition, we solve this differential equation numerically and study the propagation of the pulse through hot nuclear matter, described by relativistic mean field models.

Non-Uniform Dependence on Initial Data of Solutions to the Euler Equations of Hydrodynamics

We show that continuous dependence on initial data of solutions to the Euler equations of incompressible hydrodynamics is optimal. More precisely, we prove that the data-to-solution map is not uniformly continuous in Sobolev H s (Ω) topology for any $${s \in \mathbb{R}}$$ if the domain Ω is the (flat) torus $${\mathbb{T}^n=\mathbb{R}^n/2\pi\mathbb{Z}^n}$$ and for any s > 0 if the domain is the whole space $${\mathbb{R}^n}$$ .

Sound waves and solitons in hot and dense nuclear matter

Assuming that nuclear matter can be treated as a perfect fluid, we study the propagation of perturbations in the baryon density. The equation of state is derived from a relativistic mean field model, which is a variant of the non-linear Walecka model. The expansion of the Euler and continuity equations of relativistic hydrodynamics around equilibrium configurations leads to differential equations for the density perturbation. We solve them numerically for linear and spherical perturbations and follow the propagation of the initial pulses. For linear perturbations we find single soliton solutions and solutions with one or more solitons followed by “radiation”. Depending on the equation of state a strong damping may occur. We consider also the evolution of perturbations in a medium without dispersive effects. In this case we observe the formation and breaking of shock waves. We study all these equations also for matter at finite temperature. Our results may be relevant for the analysis of RHIC data. They suggest that the shock waves formed in the quark gluon plasma phase may survive and propagate in the hadronic phase.

Algebraic Discretization of the Camassa-Holm and Hunter-Saxton Equations

The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the H 1 and right-invariant metrics correspondingly. There is an analogy to the Euler equations in hydrodynamics, which describe geodesic flow for a right-invariant metric on the infinite-dimensional group of diffeomorphisms preserving the volume element of the domain of fluid flow and to the Euler equations of rigid body whith a fixed point, describing geodesics for a left-invariant metric on SO(3). The CH and HS equations are integrable bi-hamiltonian equations and one of their Hamiltonian structures is associated to the Virasoro algebra. The parallel with the integrable SO(3) top is made explicit by a discretization of both equation based on Fourier modes expansion. The obtained equations represent integrable tops with infinitely many momentum components. An emphasis is given on the structure of the phase space of these equations, the momentum map and the space of canonical variables.

From Newton’s mechanics to Euler’s equations

The Euler equations of hydrodynamics, which appeared in their present form in the 1750s, did not emerge in the middle of a desert. We shall see in particular how the Bernoullis contributed much to the transmutation of hydrostatics into hydrodynamics, how d’Alembert was the first to describe fluid motion using partial differential equations and a general principle linking statics and dynamics, and how Euler developed the modern concept of internal pressure field which allowed him to apply Newton’s second law to infinitesimal elements of the fluid.

Soliton propagation in relativistic hydrodynamics

We study the conditions for the formation and propagation of Korteweg-de Vries (KdV) solitons in nuclear matter. In a previous work we have derived a KdV equation from Euler and continuity equations in non-relativistic hydrodynamics. In the present contribution we extend our formalism to relativistic fluids. We present results for a given equation of state, which is based on quantum hadrodynamics (QHD).

Can dark matter be a Bose–Einstein condensate?

We consider the possibility that the dark matter which is required to explain the dynamicsof the neutral hydrogen clouds at large distances from the galactic centre could be in theform of a Bose–Einstein condensate. To study the condensate we use the non-relativisticGross–Pitaevskii equation. By introducing the Madelung representation of thewavefunction, we formulate the dynamics of the system in terms of the continuity equationand of the hydrodynamic Euler equations. Hence dark matter can be described as anon-relativistic, Newtonian Bose–Einstein gravitational condensate gas, whosedensity and pressure are related by a barotropic equation of state. In the case of acondensate with quartic non-linearity, the equation of state is polytropic with indexn = 1. In the framework of the Thomas–Fermi approximation the structure of the Newtoniangravitational condensate is described by the Lane–Emden equation, which can be exactlysolved. General relativistic configurations with quartic non-linearity are studied, bynumerically integrating the structure equations. The basic parameters (mass and radius) ofthe Bose–Einstein condensate dark matter halos sensitively depend on the mass of thecondensed particle and of the scattering length. To test the validity of the model we fit theNewtonian tangential velocity equation of the model with a sample of rotation curves oflow surface brightness and dwarf galaxies, respectively. We find a very good agreementbetween the theoretical rotation curves and the observational data for the low surfacebrightness galaxies. The deflection of photons passing through the dark matter halos isalso analysed, and the bending angle of light is computed. The bending angleobtained for the Bose–Einstein condensate is larger than that predicted by standardgeneral relativistic and dark matter models. The angular radii of the Einstein ringsare obtained in the small angle approximation. Therefore the study of the lightdeflection by galaxies and the gravitational lensing could discriminate between theBose–Einstein condensate dark matter model and other dark matter models.

Korteweg–de Vries solitons in relativistic hydrodynamics

Abstract In the previous work, assuming that the nucleus can be treated as a perfect fluid, we have studied the propagation of perturbations in the baryon density. For a given equation of state we have derived a Korteweg–de Vries (KdV) equation from Euler and continuity equations in non-relativistic hydrodynamics. Here, using a more general equation of state, we extend our formalism to relativistic hydrodynamics.

Solitons in relativistic mean field models

Assuming that the nucleus can be treated as a perfect fluid we study the conditions for the formation and propagation of Korteweg-de Vries (KdV) solitons in nuclear matter. The KdV equation is obtained from the Euler and continuity equations in nonrelativistic hydrodynamics. The existence of these solitons depends on the nuclear equation of state, which, in our approach, comes from well known relativistic mean field models. We reexamine early works on nuclear solitons, replacing the old equations of state by new ones, based on QHD and on its variants. Our analysis suggests that KdV solitons may indeed be formed in the nucleus with a width which, in some cases, can be smaller than one fermi.

An HLLC Riemann solver for magneto-hydrodynamics

This paper extends a class of approximate Riemann solvers devised by Harten, Lax and van Leer (HLL) for Euler equations of hydrodynamics to magneto-hydrodynamics (MHD) equations. In particular, we extend the two-state HLLC (HLL for contact wave) construction of Toro, Spruce and Speares to MHD equations. We derive a set of HLLC middle states that satisfies the conservation laws. Numerical examples are given to demonstrate that the new MHD-HLLC solver can achieve high numerical resolution, especially for resolving contact discontinuity. In addition, this new solver maintains a high computational efficiency when compared to Roe’s approximate Riemann solver.

A conjecture concerning the exponential map on D μ ( M )

It is known that solutions of the Euler equations of hydrodynamics correspond to geodesics on the group of volume–preserving diffeomorphisms of a compact manifold. We conjecture that, regardless of the dimension of the manifold, the associated Riemannian exponential map on the group is nonlinear Fredholm of index zero. Such a result has been established for the Riemannian exponential maps of natural Sobolev metrics on loop spaces and loop groups.

Parallel computations of 3-D Rayleigh-Taylor instability

Results are described of computations of 3-D compressible Rayleigh-Taylor instability. The instability is an important mixing mechanism in a variety of flows such as supernovae and inertial confinement fusion. The code uses Flux Corrected Transport to solve the Euler equations of hydrodynamics. Results of the simulations have been characterized by considering differing resolutions and in terms of growth-rates, Mach number evolution, pattern formation, and (weakly) turbulent structures.