Two-fluid Hydrodynamic Equations Research Articles

Lattice Boltzmann model for binary mixtures.

An a priori derivation of the lattice Boltzmann equations for binary mixtures is provided by discretizing the Boltzmann equations that govern the evolution of binary mixtures. The present model leads to a set of two-fluid hydrodynamic equations for the mixture. In existing models, employing the single-relaxation-time approximation, the viscosity and diffusion coefficients are coupled through the relaxation parameter tau, thus limited to unity Prandtl number and Schmidt number. In the present model the viscosity and diffusion coefficient are independently controlled by two relaxation parameters, thus enabling the modeling of mixtures with an arbitrary Schmidt number. The theoretical framework developed here can be readily applied to multiple-species mixing.

Landau-Khalatnikov two-fluid hydrodynamics of a trapped Bose gas

Starting from the quantum kinetic equation for the noncondensate atoms and the generalized Gross--Pitaevskii equation for the condensate, we derive the two-fluid hydrodynamic equations of a trapped Bose gas at finite temperatures. We follow the standard Chapman--Enskog procedure, starting from a solution of the kinetic equation corresponding to the complete local equilibrium between the condensate and the noncondensate components. Our hydrodynamic equations are shown to reduce to a form identical to the well-known Landau-Khalatnikov two-fluid equations, with hydrodynamic damping due to the deviation from local equilibrium. The deviation from local equilibrium within the thermal cloud gives rise to dissipation associated with shear viscosity and thermal conduction. In addition, we show that effects due to the deviation from the diffusive local equilibrium between the condensate and the noncondensate (recently considered by Zaremba, Nikuni, and Griffin) can be described by four frequency-dependent second viscosity transport coefficients. We also derive explicit formulas for all the transport coefficients. These results are used to introduce two new characteristic relaxation times associated with hydrodynamic damping. These relaxation times give the rate at which local equilibrium is reached and hence determine whether one is in the two-fluid hydrodynamic region.

Electron kinetic effects in standing shear Alfvén waves in the dipolar magnetosphere

The electron kinetic response to an electric current driven by a standing shear Alfvén wave (SAW) is considered for the case of a dipolar geometry. The parallel electric field is found from the electron gyrokinetic equation along with the SAW dispersion coefficient. Electron trapping in the dipolar magnetic field significantly reduces the parallel electric conductivity and in this way increases the amplitude of the parallel electric field and SAW dispersion. It is demonstrated that the two-fluid hydrodynamic equations used by many authors significantly underestimate the electron response and, consequently, the magnitude and location of the parallel electric field under conditions where the electron bounce frequency is larger than the SAW frequency. This is especially important in a plasma with density depressions near the Earth’s polar magnetosphere.

Two-Fluid Hydrodynamics in Trapped Bose Gases and in Superfluid Helium

A review is given of recent theoretical work on the superfluid dynamics of trapped Bose gases at finite temperatures, where there is a significant fraction of non-condensate atoms. One can now reach large enough densities and collision cross-sections needed to probe the collective modes in the collision-dominated hydrodynamic region where the gas exhibits characteristic superfluid behavior involving the relative motions of the condensate and non-condensate components. The precise analogue of the Landau-Khalatnikov two-fluid hydrodynamic equations was recently derived from trapped Bose gases, starting from a generalized Gross-Pitaevskii equation for the condensate macroscopic wavefunction and a kinetic equation for the non-condensate atoms.

First and second sound in a uniform Bose gas

We have recently derived two-fluid hydrodynamic equations for a trapped weakly interacting Bose gas. In this paper we use these equations to discuss first and second sound in a uniform Bose gas. These results are shown to agree with the predictions of the usual two-fluid equations of Landau when the thermodynamic functions are evaluated for a weakly interacting gas. In a uniform gas, second sound mainly corresponds to an oscillation of the superfluid (the condensate) and is the low-frequency continuation of the Bogoliubov-Goldstone symmetry-breaking mode.

Free expansion of a low-density laser plasma

Expansion of a laser plasma, formed by multistage selective ionisation of metal vapours in the atomic vapour laser isotope separation (AVLIS) method, is considered. The main feature of such a plasma is that the mean free path of particles in the plasma is much greater and the Debye radius is much smaller than the transverse dimensions of the ionisation region. A numerical solution of equations of two-fluid hydrodynamics is used to show that, when expansion becomes steady, the average-mass velocity is equal to about half the most probable velocity of the particles which have the temperature of electrons and the mass of ions. Plasma expansion becomes steady in a time approximately 1.5 times longer than the time of flight across the initial transverse size of the ionisation region at the steady expansion velocity. It is therefore reasonable to expect efficient ion extraction by free expansion in laser isotope separation by the AVLIS method.

Radiation Hydrodynamics of Accreting Magnetic White Dwarfs

AbstractWe solved the stationary one-dimensional two-fluid radiation hydrodynamic equations including cyclotron radiation for a wide range of mass flow rates. Here, we discuss the implications for accretion phenomena on the white dwarfs in AM Her binaries.

Contact and pressure balance structures in two-fluid cosmic-ray hydrodynamics

The role of cosmic-ray-modified contact discontinuities and pressure balance structures in two-fluid cosmic-ray hydrodynamics in one Cartesian space dimension are investigated by means of analytic and numerical solution examples, as well as by weakly nonlinear asymptotics. The fundamental wave modes of the two-fluid cosmic-ray hydrodynamic equations in the long-wavelength limit consist of the backward and forward propagating cosmic-ray-modified sound waves, with sound speed dependent on both the cosmic-ray and thermal gas pressures; the contact discontinuity; and a pressure balance mode in which the sum ofthe cosmic ray and thermal gas pressure perturbations is zero. The pressure balance mode, like the contact discontinuity is advected with the background flow. The interaction of the pressure balance mode with the contact discontinuity is investigated by means of the method of multiple scales. The thermal gas and cosmic-ray pressure perturbations satisfy a linear diffusion equation, and entropy perturbations arising from nonisentropic initial conditions for the thermal gas are frozen into the fluid. The contact discontinuity and pressure balance eigenmodes both admit nonzero perturbations in the thermal gas, whereas the cosmic-ray-modified sound waves are isentropic. The total entropy perturbation is shared between the contact discontinuity and pressure balance eigenmodes, and examples are given in which there is a transfer of entropy between the two modes. In particular, N-wave type density disturbances are obtained which arise as a result of the entropy transfer between the two modes. A weakly nonlinear geometric optics perturbation expansion is used to study the long timescale evolution of the short-wavelength entropy wave and the thermal gas sound waves in a slowly varying, large-scale background flow. The weakly nonlinear geometric optics expansion is also used to generalize previous studies of squeezing instability for short-wavelength sound waves in the two fluid model, by including a weakly nonlinear wave steepening term that leads to shock formation, as well as the effect of long time and space dependence of the background flow. Implications of cosmic-ray-modified pressure balance structures and contact discontinuities in models of the interaction of traveling interplanetary shocks and compression and rarefraction waves with the solar wind termination shock are briefly discussed.

Molecular hydrodynamics of degenerate weakly nonideal bose gas. I. Generalized equations of two-fluid hydrodynamics

A degenerate weakly nonideal Bose gas is investigated at temperatures near zero. Relations connecting the irreducible Green's functions are used to obtain exact expressions for the two-time temperature-dependent Green's functions. In the case of weak nonideality an expression possessing interpolation properties with respect to the frequency and momentum is obtained for the density—density Green's function. At low frequencies, the results of two-fluid hydrodynamics are reproduced. At wavelengths less than the mean free path the energy of the elementary excitations and the damping obtained in the paper agree with the results of perturbation theory with respect to the coupling constant. An expression for the operator of the superfluid velocity is obtained.

Time evolution of the mass exchange in grazing heavy-ion collisions

On the basis of a macroscopical approach to the description of two interpenetrating quantal objects, the equations of two-fluid hydrodynamics for the cohesion stage of deeply inelastic heavy-ion collisions are formulated. The elasticity of the ions is analyzed in peripheral mass exchange reactions at intermediate energies. The system of closed equations of Newtonian mechanics, which simultaneously describes the motion of the ions along classical trajectories as well as the mass time evolution during the interaction period are derived and solved. The role of mass exchange in the friction force is discussed.

Behavior of the thermal conductivity of dilute 3He-4He mixtures in the superfluid phase.

We present a new solution to the Khalatnikov two-fluid hydrodynamic equations which suggests that thermal conduction experiments in superfluid $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures measure the diffusive conductivity. This implies that for small $^{3}\mathrm{He}$ concentrations the measured conductivity approaches a constant. The singular behavior predicted by Khalatnikov's original solution can be eliminated using a constraint imposed by the tight coupling of concentration and temperature gradients. Microscopically, we find that diffusive mass transport in bulk superfluid is limited by the ability of the diffusing atoms to get rid of the thermodynamic diffusion energy they carry, leading to an apparent diffusivity which can be much smaller that the isothermal mass diffusivity. We also describe a preliminary experiment to observe two effects predicted by our analysis. In agreement with our predictions, we find that (1) the thermal relaxation time in dilute mixtures is very long and (2) the temperature gradient in a superfluid mixture is caused by $^{3}\mathrm{He}$ mass diffusion. The latter observation gives rise to a new technique for measuring the effective thermal conductivity which avoids the main difficulties encountered by the previous method.

Kinetic theory of binary gas mixtures with large mass disparity

The Boltzmann equations for a binary mixture of gases are considered in the asymptotic limit when their molecular weight ratio and the light gas Knudsen number are small quantities. A first mass-ratio expansion reduces the cross-collision operator of the light gas Boltzmann equation to a Lorentz form, uncoupling its kinetic behavior from that of the heavy gas. The light gas distribution function is then determined to first order in the Knudsen number, independently of the degree of nonequilibrium characterizing the heavy gas, whose influence is felt only through its hydrodynamic quantities. All transport coefficients arising are determined variationally for arbitrary interaction potentials using Sonine polynomial expansions as trial functions. A remarkable feature of this analysis is that it yields binary transport information (i.e., diffusion and thermal diffusion coefficients) from considering only the Boltzmann equation for the light gas. A second mass expansion reduces the cross-collision operator of the heavy gas equation to a Fokker–Planck form. The corresponding coefficients involve integrals over the light gas distribution function determined previously and are evaluated explicitly in terms of the hydrodynamic quantities and transport coefficients of the light gas. The heavy gas distribution function can be determined by solving a Fokker–Planck equation at dilutions large enough to make heavy–heavy collisions negligible, or by a new Knudsen number expansion when the molar fraction of the heavy gas is of order 1. In this latter case, the heavy gas kinetic behavior is independent of the light gas, being characterized by the same transport coefficients of the pure heavy gas. The problem is then reduced to a set of two-fluid hydrodynamic equations.

Transport coefficients in a dilute but condensed Bose gas

The kinetic equations for a dilute superfluid developed previously are used to derive the Landau-Khalatnikov two-fluid hydrodynamic equations. Explicit expressions for the associated transport coefficients are given for very low temperatures and for moderately low temperatures.

Correlation Properties of Charges in Plasma Streams and the Possibility of Superconductivity

The fields from test bodies and individual charges in charged particle streams are studied by means of the equations of two-fluid hydrodynamics for charged liquids. In the presence of a current the charges of a plasma are not screened completely. In the case of a small drift velocity compared to the dispersion of the charge velocities the screening is not complete because of the forces of magnetic interaction which is a relativistic effect. In the opposite limit of a high drift velocity the interaction of charges at large distances has the character of the exchange of plasmons. The potential of pair interactions between individual charges in particle streams is calculated. This interaction is anisotropic in space; in the direction perpendicular to the velocity of relative motion it is attractive at large distances. If the electromagnetic self-contraction of the current channel together with the radiation cooling leads to the degeneration of electrons, the electron subsystem may pass to the superconducting state. The estimates using the temperature vertex function show that the transition temperature Tc may be of the order of some hundred degrees. At low electron density Tc is about 10-3 of the Fermi energy, and falls off exponentially at higher plasma densities.

Transport theory for a weakly interacting condensed Bose gas

The Landau-Khalatnikov two-fluid hydrodynamic equations are derived for a dilute, weakly interacting, condensed Bose gas on the basis of a microscopic theory. Explicit expressions for the transport coefficients in the linearized equations are given for very low temperatures and for moderately low temperatures below the $\ensuremath{\lambda}$ point.

Wave propagation in the presence of a spatially uniform external periodic magnetic field in two-temperature plasma

Starting from the two-fluid model hydrodynamic equations, a dispersion relation is obtained for wave propagation through a two-temperature plasma perpendicular to the direction of the spatially uniform external magnetic field B0cosω0t and several excitation conditions are deduced.

A Numerical Simulation of Cooling Coronal Flare Plasma

view Abstract Citations (34) References (40) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A Numerical Simulation of Cooling Coronal Flare Plasma Doschek, G. A. ; Boris, J. P. ; Cheng, C. C. ; Mariska, J. T. ; Oran, E. S. Abstract We have simulated the cooling of coronal flare plasma (Te > 107 K) using a numerical model of a vertical magnetic flux tube containing an idealized flare chromosphere, transition region, and corona. The model solves the set of one-dimensional, two-fluid hydrodynamic equations. The cooling of the flux tube is calculated for a specific case beginning with an initial atmosphere in hydrostatic equilibrium and a maximum temperature of about 18 × 106 K. The behavior of temperature, density, and velocity is calculated as a function of height as the system cools. Early in the cooling, energy is transported by conduction into the transition region and chromosphere where it is radiated away. Later, the transition region-corona interface moves upward into the tube at velocities of about 20 km s-1, while the chromosphere cools and the coronal component cools by both conduction and radiation. Coronal downflow velocities of about 60 km s-1 are evident during this phase. The expected spectral line emission from the system in X-ray lines of Fe XXV, Fe XXIV, Fe XXII, O VIII, and O VII is also calculated and compared to recent observational results. Some observational results can be explained as a consequence of simple cooling of flare flux tubes. The expected spectral line emission from certain transition region lines is also briefly considered. The dependence of our results on flare size is discussed, and our results are compared with similar previous work. Publication: The Astrophysical Journal Pub Date: July 1982 DOI: 10.1086/160086 Bibcode: 1982ApJ...258..373D full text sources ADS |

Collective Modes in Quantum Lattice or Three-Dimensional XY Model. II

An external field is applied to the XY model which was studied in a previous paper. With the help of Mori's memory function formalism, two types of collective modes are obtained. One of those, which was previously pointed out to correspond to the first sound in superfluid helium, survives at the critical temperature Tc. The other is a new mode, which disappears as a result of symmetry restored above Tc. This mode comes about owing to the coupling between the Goldstone mode and the energy fluctuation due to an external field, and is regarded to correspond to the second sound in superfluid helium. The linearized two-fluid hydrodynamic equations for superfluid helium are obtained in the context of the XY model, in which the detailed correspondence to the superfluid helium is clarified.

Solar transition region response to variations in the heating rate

The response of a numerical model for the upper chromosphere, transition region, and corona to variations in the energy input has been examined. The numerical model solves the set of one-dimensional two-fluid hydrodynamic equations in a simple vertical magnetic flux tube. The atmosphere responds to both the increase and decrease in energy deposition by smoothly readjusting the temperature gradient and the amount of material in the region of peak radiating efficiency to radiate away energy being deposited. At no time during this readjustment is a departure from a thin laminar transition region structure seen. In addition, a time-dependent description of the nonequilibrium ionization of all of the ionization stages of oxygen has been included. This calculation is coupled with the self-consistent calculations of the dynamical variables. It is found that the nonequilibrium ionization balance calculations for both heating and cooling small loops in the quiet sun predict relative ionic abundances which differ substantially from those which would be predicted by an equilibrium calculation

Gauge Wheel of SuperfluidHe4

The assumption is explored that the equilibrium order parameter of a superfluid rotates with a rotating container. Since rotations and changes of phase are then intimately related in inhomogeneous configurations, effects like the /sup 3/He-A ''gauge wheel'' of Liu and Cross should arise in any superfluid. Such effects may be difficult to observe, but it is shown that their existence in superfluid /sup 4/He is directly implied by nothing more than the conventional equations of two-fluid hydrodynamics.