Classical Hydrodynamic Equations Research Articles

The full viscous quantum hydrodynamic system in one dimensional space

A viscous quantum hydrodynamic system for particle density, current density, energy density, and electrostatic potential, coupled with a Poisson equation, is studied in spatial one dimensional real line. The system is self-consistent in the sense that the electric field, which forms a forcing term in the momentum and energy equations, is determined by the coupled Poisson equation. First, the existence and uniqueness of the stationary solution is proved in an appropriate Sobolev space. Then, exponential stability of the stationary solution is established by constructing an a priori estimate. Since the techniques for classical hydrodynamic equations are not applicable here due to the quantum term, the existence of a local-in-time solution is obtained by showing the existence of local-in-time solutions of a reformulated system via the iteration method.

Determination of pressure from nonlinear equations of nonhomogeneous non-newtonian fluids

In this paper, the initial-boundary problem for a system of nonlinear equations modified by p-laplacian diffusion and nonlinear damping term, in which describe the motion of a non-homogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids is considered. Usually, a pressure does not include in the definition of weak generalized solution to equations of hydrodynamics for incompressible fluids, because weak solutions consider in the solenoidal space due to the incompressibility (continuity) equation. In the classical hydrodynamic equations, after determining a velocity and a density, a pressure can be determined by using the theorem of a representation of the space L2(Ω) into the direct sum of two orthogonal subspaces. The recovering of a pressure from the nonlinear equations of hydrodynamics modified by p-laplacian and nonlinear damping term requires a new approches. Here, under a suitable conditions on the data of the problem, a pressure has been uniquelly determined from the initial-boundary value problem for the system of nonlinear modified equations describing the motion of nonhomogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids. The main functional spaces and necessary axuillary conclusions are introduced. A space of generalized weak solutions is identified. The pressure uniquelly recovered by using the known de-Ram lemma.

Uncertainty Relations in the Madelung Picture.

Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these equations formally look like classical hydrodynamic equations, they contain all the information about the quantum system. Concerning the quantum mechanical uncertainties of position and momentum, however, this is not so obvious at first sight. We show how these uncertainties are related to the phase and amplitude of the wave function in position and momentum space and, particularly, that the contribution from the phase essentially depends on the position–momentum correlations. This will be illustrated explicitly using generalized coherent states as examples.

Derivation of equations of multimoment hydrodynamics for a gas of particles with internal structure

The equations for pair distribution functions are used to derive the equations of multimoment hydrodynamics for a gas of particles with internal structure. The equations for pair functions are derived in terms of semi-classical approximation. The basic property of the pair functions is established. In conformity with basic property, these functions remain unchanged in time along the trajectory of the inertia center of pair. The basic property of the pair distribution functions reveals the existence of infinite number of principle hydrodynamic values. The equations of multimoment hydrodynamics are constructed using limited number of principle hydrodynamic values. Selected principle values specify measurable moments. The measurable moments are represented by linear combination of principle and non-principle hydrodynamic values. The general structure of constructed multimoment conservation laws is identical to the structure of appropriate multimoment conservation laws in a gas of structureless particles. Each of the multimoment conservation laws is divided into two separate equations. The first group of conservation equations corresponds to convective phenomena. The second group of conservation equations corresponds to dissipative phenomena. Derived equations of multimoment hydrodynamics are designed for interpreting the behavior of medium states, which are far removed from the state of statistical equilibrium. Classic hydrodynamics encountered the problems when interpreting the unstable phenomena. The possibility of improvement of classic hydrodynamics equations for a gas of particles with internal structure is sought on the way toward an increase in the number of principle hydrodynamic values.

About the Transition to Turbulence Through Chaotic Distortion of Vortex Shedding

The results of direct numerical integration of the Navier-Stokes equations are evaluated against experimental data for the problem of flow around a hard sphere at rest. The evaluation is performed for both the sequence of vortex shedding regimes, replacing stable modes after the loss of stability, and the regime of turbulence replacing vortex shedding modes as Reynolds number Re increases. The evaluation demonstrates the unsuitability of classic hydrodynamics equations to interpret the phenomenon of vortex shedding. Moreover, the attainment of critical value of Re is accompanied by loss of the direction of instability development. Wrong direction of instability development results in the attainment of multiperiodic, that is, essentially chaotic, solution. Insurmountable discrepancies between calculation results and experimental data show that the chaotic deterministic solution to the Navier-Stokes equation is not suitable for interpretation of turbulence. An analogy is revealed between the sequence of modes observed in flow around a sphere as Re increases and sequence of modes in shear layer behind a cylinder with paraboloidal nose recorded while moving downstream along the contour of streamlined body. The conclusions are as follows. The turbulence of shear flow is regular unstable vortex shedding regime distorted by chaotic fluctuations. Solutions to the classic hydrodynamics equations are incapable of interpreting both regular and chaotic turbulence component. Multimoment hydrodynamics seeks for decision of these problems along the way toward an increase in the number of principle hydrodynamic values.

Derivation of the Multimoment Hydrodynamics Equations for a Gas Mixture

<p class="1Body">The equations for pair distribution functions are used to derive the multimoment hydrodynamics equations for gas mixture. The gas mixture pair distribution functions are specified. The equations for pair functions are derived directly from the general statistical mechanics concepts. The basic property of the pair functions is established. In conformity with basic property, these functions remain unchanged in time along the trajectory of the center of inertia of a pair. The basic property of the pair distribution functions reveals the existence of an infinite number of principle hydrodynamic values. Multimoment hydrodynamics equations are constructed using 3L+4 principle hydrodynamic values, where is the number of gas mixture components. Just these principle values specify measurable moments. The measurable moments are represented by linear combination of principle and non-principle hydrodynamic values. The general structure of constructed multimoment conservation laws is identical to the structure of appropriate multimoment conservation laws in a gas of identical particles. Each of the multimoment conservation laws is divided into two separate equations. The first group of conservation equations corresponds to convective phenomena. The second group of conservation equations corresponds to dissipative phenomena. Derived multimoment hydrodynamics equations are designed for interpreting the behavior of unstable systems. As is shown previously, classic hydrodynamics equations are incapable of reproducing flows after they lose stability. That is, the solutions to the classic hydrodynamics equations do not find the direction of instability development correctly. The possibility of improvement of classic hydrodynamics equations for a gas mixture is sought on the way toward an increase in the number of principle hydrodynamic values.</p>

Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction

The hydrodynamic equations with quantum effects are studied in this paper. First we establish the global existence of smooth solutions with small initial data and then in the second part, we establish the convergence of the solutions of the quantum hydrodynamic equations to those of the classical hydrodynamic equations. The energy equation is considered in this paper, which added new difficulties to the energy estimates, especially to the selection of the appropriate Sobolev spaces.

Evolution of unstable system.

Scenario of appearance and development of instability in problem of a flow around a solid sphere at rest is discussed. The scenario was created by solutions to the multimoment hydrodynamics equations, which were applied to investigate the unstable phenomena. These solutions allow interpreting Stokes flow, periodic pulsations of the recirculating zone in the wake behind the sphere, the phenomenon of vortex shedding observed experimentally. In accordance with the scenario, system loses its stability when entropy outflow through surface confining the system cannot be compensated by entropy produced within the system. The system does not find a new stable position after losing its stability, that is, the system remains further unstable. As Reynolds number grows, one unstable flow regime is replaced by another. The replacement is governed tendency of the system to discover fastest path to depart from the state of statistical equilibrium. This striving, however, does not lead the system to disintegration. Periodically, reverse solutions to the multimoment hydrodynamics equations change the nature of evolution and guide the unstable system in a highly unlikely direction. In case of unstable system, unlikely path meets the direction of approaching the state of statistical equilibrium. Such behavior of the system contradicts the scenario created by solutions to the classic hydrodynamics equations. Unstable solutions to the classic hydrodynamics equations are not fairly prolonged along time to interpret experiment. Stable solutions satisfactorily reproduce all observed stable medium states. As Reynolds number grows one stable solution is replaced by another. They are, however, incapable of reproducing any of unstable regimes recorded experimentally. In particular, stable solutions to the classic hydrodynamics equations cannot put anything in correspondence to any of observed vortex shedding modes. In accordance with our interpretation, the reason for this isthe classic hydrodynamics equations themselves.

An analytical model for shale gas permeability

Based on the kinetic theory of gases and using the regularized 13-moment method, the analytical R13 AP model is introduced for predicting gas apparent permeability of nanoporous shale samples. These samples are characterized by ultratight pores and may introduce significant rarefaction effects, especially under the laboratory conditions, which cannot be accounted for in the classical hydrodynamic equations. Due to the significance of the rarefaction effects, measured values of the gas apparent permeability depend on the operating parameters, such as pressure and temperature, and gas type in addition to pore size. The R13 AP model incorporates these parameters and can predict the apparent permeability for Knudsen numbers up to unity. This model is compared with the other models and experimental results. The results of the R13 AP model match the published experimental data of flow in nanochannels. It is shown that the gas molecular weight and temperature have a significant effect on apparent permeability of the nanochannels, and the Tangential Momentum Accommodation Coefficient (TMAC) has a minimal effect for its experimental range (0.8–1). The effect of adsorption on apparent permeability of nanochannels is studied by employing the experimental Langmuir isotherms of different shale samples. It is shown that the apparent permeability does not change linearly with surface coverage and, change of permeability with surface coverage becomes more pronounced as the surface coverage increases. R13 AP model results for apparent permeability of Carbon Dioxide and Nitrogen agree with the experimental measurements performed on a Marcellus shale sample. The absolute permeability values are calculated and compared with the ones estimated by the double slippage model for both gases. Unlike the double slippage model, the R13 AP model honors the values of apparent permeability at high pressure and estimates lower absolute permeability values. For Carbon Dioxide, the value is slightly lower due to the presence of an adsorbed layer.

Kinetic Foundation for the Multimoment Hydrodynamics Equations

The equations for the pair distribution functions are derived directly from the second equation of the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. The derivation is fulfilled within the frameworks of the multiscale method. The equations for the pair distribution functions are the kinetic foundation for the multimoment hydrodynamics equations. Solutions to the equations for the pair distribution functions predetermine the possibility of constructing the hydrodynamics equations with an arbitrary number of principle hydrodynamic values specified beforehand. The tendency to increase the number of principal hydrodynamic values is caused by the necessity of interpreting the behavior of the system after the loss of stability. Solutions to the classic hydrodynamics equations constructed for only three principle hydrodynamic values are unable to predict the direction of instability evolution. Solutions to the multimoment hydrodynamics equations are capable of reproducing correctly the phenomenon of emergence and development of instability.

Hydrodynamic analogies between the equations of classical hydrodynamics and electrodynamics in electrochemistry

It is shown that the use of equations of hydrodynamics of an incompressible and compressible fluid gives similar results for a number of experimental data from the field of classical electrodynamics used in electrochemistry. The analogue of electric current in wires is a stream that creates around itself a flow of a fluid. The analogue of electric field is the acceleration of a flow, whereas the analogue of magnetic induction is the frequency of a rotational motion of the fluid. Ampere’s law in hydrodynamics describes the interaction of flows with real bodies in terms of the Zhukovsky equation. The power laws in the fluid are similar, with some distinctions, to Maxwell equations. The expansion of the equations of conservation of momentum and mass in a series in perturbations leads to wave equations also similar to the Maxwell equations for the propagation of electromagnetic waves.

Multimoment Hydrodynamics in Problem on Flow around a Sphere: Entropy Interpretation of the Appearance and Development of Instability

Multimoment hydrodynamics equations are applied to investigate the phenomena of appearance and development of instability in problem on a flow around a solid sphere at rest. The simplest solution to the multimoment hydrodynamics equations coincides with the Stokes solution to the classic hydrodynamics equations in the limit of small Reynolds number values, . Solution to the multimoment hydrodynamics equations reproduces recirculating zone in the wake behind the sphere having the form of an axisymmetric toroidal vortex ring. The solution remains stable while the entropy production in the system exceeds the entropy outflow through the surface confining the system. The passage of the first critical value is accompanied by the solution stability loss. The solution, when loses its stability, reproduces periodic pulsations of the periphery of the recirculating zone in the wake behind the sphere. The and solutions to the multimoment hydrodynamics equations interpret a vortex shedding. After the second critical value is reached, the solution at the periphery of the recirculating zone and in the far wake is replaced by the solution. In accordance with the solution, the periphery of the recirculating zone periodically detached from the core and moves downstream in the form of a vortex ring. After the attainment of the third critical value , the solution at the periphery of the recirculating zone and in the far wake is replaced by the solution. In accordance with the solution, vortex rings penetrate into each other and form the continuous vortex sheet in the wake behind the sphere. The replacement of one unstable flow regime by another unstable regime is governed the tendency of the system to discover the fastest path to depart from the state of statistical equilibrium. Having lost the stability, the system does not reach a new stable position. Such a scenario differs from the ideas of classic hydrodynamics, which interprets the development of instability in terms of bifurcations from one stable state to another stable state. Solutions to the multimoment hydrodynamics equations indicate the direction of instability development, which qualitatively reproduces the experimental data in a wide range of Re values. The problems encountered by classic hydrodynamics when interpreting the observed instability development process are solved on the way toward an increase in the number of principle hydrodynamic values.

Steady State and Transient Gain Characteristics of the Stimulated Brillouin Scattered Mode in Quantum Semiconductor Plasmas

Using the quantum hydrodynamical model, the steady state and transient gain characteristics of the stimulated Brillouin scattered mode in semiconductor plasmas have been studied. The third-order Brillouin susceptibility due to a nonlinear current density and electrostriction of the medium have been determined using the coupled mode theory. The effect of the Bohm potential on the steady state and transient Brillouin gain coefficient was studied through the quantum corrections in the classical hydrodynamic equations. It was found that the Bohm potential in the electron dynamics enhances the steady-state and transient Brillouin gain coefficients. Reduction in the threshold pump intensity of the said process has been realized as a consequence of the inclusion of quantum correction term.

About the Prospects for Passage to Instability

The results of the direct numerical integration of the Navier-Stokes equations are evaluated against experimental data for problem on a flow around bluff bodies in an unstable regime. Experiment records several stable medium states for flow past a body. Evolution of each of these states, after losing the stability, inevitably goes by periodic vortex shedding modes. Calculations based on the Navier-Stokes equations satisfactorily reproduced all observed stable medium states. They were, however, incapable of reproducing any of a vortex shedding modes recorded experimentally. The solutions to the classic hydrodynamics equations successfully reach the boundary of instability field. However, classic solutions are unable to cross this boundary. Most likely, the reason for this is the Navier-Stokes equations themselves. The classic hydrodynamics equations directly follow from the Boltzmann equation and naturally contain the error involved in the derivation of classic kinetic equation. Just the Boltzmann hypothesis, which closed kinetic equation, allowed us to con- struct classic hydrodynamics on only three lower principal hydrodynamic values. The use of the Boltzmann hypothesis excludes higher principal hydrodynamic values from the participation in the formation of classic hydrodynamics equations. The multimoment hydrodynamics equations are constructed using seven lower principal hydrodynamic values. The numerical integration of the multimoment hydrodynamics equations in the problem on flow around a sphere shows that the solutions to these equations cross the boundary and enter the instability field. The boundary crossing is accompanied by appearance of very uncommon acts in scenario of system evolution.

On the possibility of improving classic hydrodynamics equations by an increase in the number of hydrodynamic values

Any of the infinite number of invariants of a binary collision of structureless particles was shown to annihilate the collision integral written in the 12-dimensional phase space of two particles. Its own principal hydrodynamic value corresponds to each of these invariants. The six-dimensional phase space of one particle can only accommodate three lower binary collision invariants that allow the integral of collisions to be annihilated. The first equation of the Bogolybov-Born-Green-Kirkwood-Yvon hierarchy is, however, not closed, and the transition from the 6-dimensional phase space to the hydrodynamic stage of description is therefore closed. The Boltzmann hypothesis closes the kinetic equation and therefore opens up the possibility of approximate transition to hydrodynamics. Just the Boltzmann hypothesis allows classic hydrodynamics equations constructed with the use of only three lower principal hydrodynamic values to be substantiated statistically. It should be expected that the neglect of the higher principal hydrodynamic values will limit the applicability range of classic hydrodynamics to states insignificantly removed from the statistical equilibrium state. Most probably, just this limitation is responsible for obvious discrepancy between the results of direct numerical integration of the Navier-Stokes equation and the experimental data on flows that lost stability. The possibility of the improvement of classic hydrodynamics equations is sought on the way toward an increase in the number of principal hydrodynamic values.

Some limit analysis in a one-dimensional stationary quantum hydrodynamic model for semiconductors

In this paper, we study the steady-state hydrodynamic equations for isothermal states including the quantum Bohn potential. The one-dimensional equations for the electron current density and the particle density are coupled self-consistently to the Poisson equation for the electric potential. The quantum correction can be interpreted as a dispersive regularization of the classical hydrodynamic equations. In a bounded interval supplemented by the proper boundary conditions, we investigate the zero-electron-mass limit, the zero-relaxation-time limit, the Debye-length (quasi-neutral) limit, and some combined limits, respectively. For each limit, we show the strong convergence of the sequence of solutions and give the associated convergence rate.

Nonlinear dynamics of spin and charge in spin-Calogero model

The fully nonlinear dynamics of spin and charge in spin-Calogero model is studied. The latter is an integrable one-dimensional model of quantum spin-1/2 particles interacting through inverse-square interaction and exchange. Classical hydrodynamic equations of motion are written for this model in the regime where gradient corrections to the exact hydrodynamic formulation of the theory may be neglected. In this approximation variables separate in terms of dressed Fermi momenta of the model. Hydrodynamic equations reduce to a set of decoupled Riemann-Hopf (or inviscid Burgers') equations for the dressed Fermi momenta. We study the dynamics of some nonequilibrium spin-charge configurations for times smaller than the time scale of the gradient catastrophe. We find an interesting interplay between spin and charge degrees of freedom. In the limit of large coupling constant the hydrodynamics reduces to the spin hydrodynamics of the Haldane-Shastry model.

Exact solutions and stability of rotating dipolar Bose-Einstein condensates in the Thomas-Fermi limit

We present a theoretical analysis of dilute gas Bose-Einstein condensates with dipolar atomic interactions under rotation in elliptical traps. Working in the Thomas-Fermi limit, we employ the classical hydrodynamic equations to first derive the rotating condensate solutions and then consider their response to perturbations. We thereby map out the regimes of stability and instability for rotating dipolar Bose-Einstein condensates and, in the latter case, discuss the possibility of vortex lattice formation. We employ our results to propose several routes to induce vortex lattice formation in a dipolar condensate.

Strong dipolar effects in a quantum ferrofluid

Symmetry-breaking interactions have a crucial role in many areas of physics, ranging from classical ferrofluids to superfluid (3)He and d-wave superconductivity. For superfluid quantum gases, a variety of new physical phenomena arising from the symmetry-breaking interaction between electric or magnetic dipoles are expected. Novel quantum phases in optical lattices, such as chequerboard or supersolid phases, are predicted for dipolar bosons. Dipolar interactions can also enrich considerably the physics of quantum gases with internal degrees of freedom. Arrays of dipolar particles could be used for efficient quantum information processing. Here we report the realization of a chromium Bose-Einstein condensate with strong dipolar interactions. By using a Feshbach resonance, we reduce the usual isotropic contact interaction, such that the anisotropic magnetic dipole-dipole interaction between 52Cr atoms becomes comparable in strength. This induces a change of the aspect ratio of the atom cloud; for strong dipolar interactions, the inversion of ellipticity during expansion (the usual 'smoking gun' evidence for a Bose-Einstein condensate) can be suppressed. These effects are accounted for by taking into account the dipolar interaction in the superfluid hydrodynamic equations governing the dynamics of the gas, in the same way as classical ferrofluids can be described by including dipolar terms in the classical hydrodynamic equations. Our results are a first step in the exploration of the unique properties of quantum ferrofluids.

Electrokinetic transport and separations in fluidic nanochannels

This article presents a summary of theory, experimental studies, and results for the electrokinetic transport in small fluidic nanochannels. The main focus is on the effect of the electric double layer on the EOF, electric current, and electrophoresis of charged analytes. The double layer thickness can be of the same order as the width of the nanochannels, which has an impact on the transport by shaping the fluid velocity profile, local distributions of the electrolytes, and charged analytes. Our theoretical consideration is limited to continuum analysis where the equations of classical hydrodynamics and electrodynamics still apply. We show that small channels may lead to qualitatively new effects like selective ionic transport based on charge number as well as different modes for molecular separation. These new possibilities together with the rapid development of nanofabrication capabilities lead to an extensive experimental effort to utilize nanochannels for a variety of applications, which are also discussed and analyzed in this review.