General Hydrodynamic Equations Research Articles

Electrokinetic phenomena at grafted polyelectrolyte layers

During the last decades the electrokinetic theory of Smoluchowski (Z. Phys. Chem. 92 (1918) 129) was extended to be applicable for soft surfaces (grafted polyelectrolyte layers (PL), biological and artificial membranes, etc.) by either using the Debye approximation or numerical solutions. In the theory of Ohshima (Colloids Surf. A 103 (1995) 249) the nonlinearized Poisson–Boltzmann (PB) equation for thick and uniform PL is solved analytically and a general hydrodynamic equation is derived in an integral form. These advantages in the theory of Ohshima provided a base for the further development of a generalized electrokinetic theory for soft surfaces. In his theory the final equation for the electroosmotic (electrophoretic) velocity is specified for the case of the complete dissociation of ionic sites within PL. Accordingly, the equation may be used only if the difference between p K and pH is very large. However, it turned out that an analytical solution of the nonlinearized PB equation for thick PL is possible for any degree of dissociation. This was achieved using the approximation of excluded coions if the absolute value of the reduced Donnan potential is larger than 2 and due to the simplification in the case of weak dissociation, when the absolute value of the reduced Donnan potential is less than 2. Combining this generalized double layer (DL) theory for PL and the theory of Ohshima enables to obtain an analytical equation for electroosmosis for the general case of any degree of dissociation. This equation creates for the first time a theoretical base for the interpretation of electrokinetic fingerprinting (EF) for the characterization of soft surfaces.

Boundary Conditions in the Theory of Generalized Hydrodynamic Equations

Kinetic boundary conditions (in particular, slip conditions) are used in the theory of rarefied gases; these boundary conditions are a corollary of the Boltzmann equation and make it possible to use the Navier-Stokes equations for describing flows of moderately rarefied gases. It is demonstrated that the slip conditions may be derived as a result of solution of generalized hydrodynamic equations (GHE). In other words, the GHE may be used for an adequate description of the kinetic wall layer.

Kinetics and hydrodynamics of long-wave fluctuations under external random force

A universal phenomenological approach to study evolution of long-wave fluctuations caused by the external random force is developed. The Furutzu–Novikov theorem is generalized for non-Gaussian processes. General kinetic and hydrodynamic equations that include the effect of long-wave fluctuations are derived for the Gaussian distribution of the random force. The nonequilibrium entropy is determined in the presence of fluctuations in the system and the H-theorem is proved. It is shown that the Obukhov–Kolmogorov turbulence spectra can be found from the hydrodynamic equations of long-wave fluctuations obtained in this paper.

Generalized hydrodynamics for a Poiseuille flow: Theory and simulations

From the complete Boltzmann's equation we obtain general hydrodynamic equations for the laminar stationary Poiseuille flow driven by an acceleration of gravity $g$. This theoretical framework implies highly nonlinear transport equations. The hydrodynamic equations are perturbatively solved up to sixth order using a small adimensional parameter $\mathcal{F}$ proportional to $g$. The predictions are compared with our own simulational results obtaining very good agreement. A second and small adimensional parameter that naturally enters the formalism is a Knudsen number Kn proportional to the ratio between the mean free path and the width of the Poiseuille channel and it serves to understand the role of the finite size effects. It will be seen in particular that there is a heat flux with a normal component ${q}_{y}$ and a heat flux ${q}_{x}$ parallel to the isotherms and that their ratio is inversely proportional to the Reynolds number: ${q}_{x}{/q}_{y}\ensuremath{\sim}\mathcal{F}/\mathrm{Kn}\ensuremath{\sim}1/\mathrm{Re}$.

Modélisation physique et par éléments finis de l'évolution des profils de vitesses en aval des courbes dans un contexte hydrodynamique

This paper presents an experimental and a numerical analysis of velocity profiles evolution after curves. A particular attention has been focused on the influence of bend succession on secondary currents dynamics. A three-dimensional finite element model has been developed based on the general hydrodynamic equations assuming fluid incompressibility and hydrostatic pressure. Taylor-Galerkine and Petrov-Galerkine schemes have been implemented to damp out numerical noise and, consequently, stabilize the flow. Parallel to the numerical modelling of velocity profiles, a measurement campaign of velocities and deviation angles has been conducted in the laboratory. The simulated transversal velocities have been compared with the experimental velocities measured at different sections. A good agreement was observed confirming the validity of the formulation and the reliability of the approach.

Influence of hydrodynamic models on the prediction of submicrometer device characteristics

The influence of different hydrodynamic models on the prediction of submicron device characteristics is studied using the General Hydrodynamic Equation solver. We analyze the simulation results of various structures of Silicon-on-insulator (SOI) Metal-Oxide-Semiconductor Field-Effect-Transistors (MOSFETs) and silicon germanium (SiGe) Heterostructure Bipolar Transistors (HBTs). When the size of devices is reduced to the submicron regime, the current drive of a SOI-MOSFET is enhanced mainly by the nonlocal effect. However, the current drive of a HBT is reduced by the dominant thermal back-diffusion effect. The sensitivity of predicted device characteristics on various transport parameters is also discussed.

On the Self-Similar Solutions of Generalized Hydrodynamics Equations and Nonlinear Wave Patterns

Solutions of the system of dynamical equations of state and equations of the balance of mass and momentum are studied. The system possesses families of periodic, quasiperiodic and soliton-like invariant solutions. Self-similar solutions of this generalized hydrodynamic system are studied. Various complicated regimes, arising as a result with terms desribing relaxing and dissipative properties of the medium are described.

Impulsive stimulated light scattering from glass-forming liquids. I. Generalized hydrodynamics approach

The generalized hydrodynamics equations, including nonlocal behavior in time, are used to calculate density response functions for impulsive stimulated light scattering (ISS) from glass-forming liquids. Single-exponential relaxation dynamics are considered first, and then the results are generalized to account for complex relaxation. Thermal diffusion, acoustic, and relaxation modes are described. The time dependence of both acoustic and relaxation modes yields the structural relaxation dynamics. In addition, the relative amplitudes of the modes in ISS data yield the Debye–Waller factor fq, called the nonergodicity parameter in mode-coupling theory (MCT) of glass-forming liquids, in the limit of low wave vector q. This permits testing of MCT predictions of a square-root cusp in fq→0(T) at a crossover temperature Tc. The information which can be obtained through ISS is compared in theoretical and practical terms to that obtainable through low-frequency light scattering spectroscopy.

Mechanics of collisional motion of granular materials. Part 1. General hydrodynamic equations

Collisional motion of a granular material composed of rough inelastic spheres is analysed on the basis of the kinetic Boltzmann–Enskog equation. The Chapman–Enskog method for gas kinetic theory is modified to derive the Euler-like hydrodynamic equations for a system of moving spheres, possessing constant roughness and inelasticity. The solution is obtained by employing a general isotropic expression for the singlet distribution function, dependent upon the spatial gradients of averaged hydrodynamic properties. This solution form is shown to be appropriate for description of rapid shearless motions of granular materials, in particular vibrofluidized regimes induced by external vibrations.The existence of the hydrodynamic state of evolution of a granular medium, where the Euler-like equations are valid, is delineated in terms of the particle roughness, β, and restitution, e, coefficients. For perfectly elastic spheres this state is shown to exist for all values of particle roughness, i.e. − 1≤β≤1. However, for inelastically colliding granules the hydrodynamic state exists only when the particle restitution coefficient exceeds a certain value em(β)< 1.In contrast with the previous results obtained by approximate moment methods, the partition of the random-motion kinetic energy of inelastic rough particles between rotational and translational modes is shown to be strongly affected by the particle restitution coefficient. The effect of increasing inelasticity of particle collisions is to redistribute the kinetic energy of their random motion in favour of the rotational mode. This is shown to significantly affect the energy partition law, with respect to the one prevailing in a gas composed of perfectly elastic spheres of arbitrary roughness. In particular, the translational specific heat of a gas composed of inelastically colliding (e = 0.6) granules differs from its value for elastic particles by as much as 55 %.It is shown that the hydrodynamic Euler-like equation, describing the transport and evolution of the kinetic energy of particle random motion, contains energy sink terms of two types (both, however, stemming from the non-conservative nature of particle collisions) : (i) the term describing energy losses in incompressibly flowing gas; (ii) the terms accounting for kinetic energy loss (or gain) associated with the work of pressure forces, leading to gas compression (or expansion). The approximate moment methods are shown to yield the Euler-like energy equation with an incorrect energy sink term of type (ii), associated with the ‘dense gas effect’. Another sink term of the same type, but associated with the energy relaxation process occurring within compressed granular gases, was overlooked in all previous studies.The speed of sound waves propagating in a granular gas is analysed in the limits of low and high granular gas densities. It is shown that the particle collisional properties strongly affect the speed of sound in dense granular media. This dependence is manifested via the kinetic energy sink terms arising from gas compression. Omission of the latter terms in the evaluation of the speed of sound results in an error, which in the dense granular gas limit is shown to amount to a several-fold factor.

Self-consistent solution for the generalized hydrodynamics model of suspension dynamics: Comparison of theory with rheological and optical measurements.

A self-consistent theory for the dynamics and rheology of suspensions of Brownian colloids is presented completely in terms of physicochemical suspension properties. The theory uses the Rogers-Young closure [Phys. Rev. A 30, 999 (1984)] of the Ornstein-Zernike equation for the equilibrium structure and the generated hydrodynamics theory developed by Hess and Klein [Adv. Phys. 33, 173 (1983)] for the dynamic properties. Direct hydrodynamic interactions are neglected. Mode-mode coupling is used to close the generalized hydrodynamics equations and provide a self-consistent set of equations for the dynamics. Two closures for the three-particle vertex function, a two-body approximation and convolution, are derived and compared. All accessible linear viscoelastic properties of the suspension are then calculated from the dynamics of the intermediate-scattering function. Numerical solutions are obtained for Yukawa particles through the viscoelastic approximation. Comparisons with dynamic-light-scattering measurements of the cumulants and the intermediate-scattering function of dilute but strongly correlated suspensions demonstrate the accuracy of the self-consistent solution with the two-body approximation for low particle concentrations. Comparisons of the mechanical properties of concentrated, strongly correlated suspensions demonstrate the accuracy of the self-consistent solutions with the convolution approximation. The results are interpreted in terms of the cage-melting model for colloid dynamics.

Incompressible navier‐stokes and euler limits of the boltzmann equation

AbstractWe consider solutions of the Boltzmann equation, in a d‐dimensional torus, d = 2, 3,magnified imageFor macroscopic times τ = t/ϵN, ϵ « 1, t ≧ 0, when the space variations are on a macroscopic scale x = ϵN−1r, N ≧ 2, x in the unit torus. Let u(x, t) be, for t ≦ t0, a smooth solution of the incompressible Navier Stokes equations (INS) for N = 2 and of the Incompressible Euler equation (IE) for N > 2.We prove that (*) has solutions for t ≦ t0 which are close, to O(ϵ2) in a suitable norm, to the local Maxwellian [p/(2πT)d/2]exp{−[v − ϵu(x,t)]2/2T} with constant density p and temperature T. This is a particular case, defined by the choice of initial values of the macroscopic variables, of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations. For N ≧ 3 these equations correspond to variable density IE while for N = 2 they involve higher‐order derivatives of the density.

On the effects of an isomerization chemical reaction upon the sound velocity

The more general hydrodynamic equations (including viscosity, heat conduction and diffusion effects) for a reactive system, in which the isomerization reaction A+A⇄B+B occurs, are used to find the influence of the reaction upon the sound velocity in the extreme cases of fast and slow reactions. A Monte Carlo simulation, based on the Boltzmann equation, has been carried out in order to verify the theoretical expressions of the sound velocity.

Thermal convection and nonlinear effects of a superfluid3He-4He mixture in a porous medium

The convective instability of superfluid3He-4He mixtures in porous media is investigated. The general hydrodynamic equations are derived and reduced to a single nonlinear equation for a scalar field. The superfluid mixtures in a porous medium have a constant4He chemical potential and behave essentially like a classical fluid in a porous medium. Two-fluid effects are calculated both at the onset of steady convection and the subsequent boundary of instability. The shift of critical Rayleigh number is about 1% or less at the onset of convection, but can be as large as 20% or more at the instability boundary for some regions aroundT ≈ 1 K. This two-fluid shift is quite large compared to the corresponding 0.001% shift at the onset of convection for bulk superfluid3He-4He mixtures.

NUMERICAL SOLUTION OF TRANSIENT HEAT TRANSPORT THROUGH HELIUM II

Abstract Transient heat transport is analyzed theoretically by solving the general hydro-dynamic equations of superfluid helium. The system that completely describes the heat transport process consists of two coupled equations, whose unknown quantities are the temperature T and the normal fluid velocity VN, under assumptions that are precisely noted. The transport process is both propagative and diffusive: the temperature front, due to a heat flux disturbance, propagates at the second sound velocity but decreases along the helium channel by a diffusion phenomenon. These two effects exist simultaneously, but it shown that, according to the pulse duration or amplitude, one of these phenomena can be accentuated. In particular, the higher the heat pulse amplitude, the more pronounced the diffusion.

Numerical Solution of Transient Heat Transport Through Helium II

Abstract Transient heat transport is analyzed theoretically by solving the general hydro-dynamic equations of superfluid helium. The system that completely describes the heat transport process consists of two coupled equations, whose unknown quantities are the temperature T and the normal fluid velocity VN, under assumptions that are precisely noted. The transport process is both propagative and diffusive: the temperature front, due to a heat flux disturbance, propagates at the second sound velocity but decreases along the helium channel by a diffusion phenomenon. These two effects exist simultaneously, but it shown that, according to the pulse duration or amplitude, one of these phenomena can be accentuated. In particular, the higher the heat pulse amplitude, the more pronounced the diffusion.

Finite Element Solution for General Fluid Motion

General hydrodynamic equations of the same form as the Navier-Stokes and continuity equations are numerically solved by a variational method based on a quasi-linear variational principle. The finite element decomposition results in a simple linear matrix system that is readily solved by standard first order numerical methods. A pseudo-iterative procedure is used to accommodate the basic nonlinearity of the equations of motion.

General Hydrodynamic Equations from the Linear Boltzmann Equation

A derivation of hydrodynamic equations from the linearized Boltzmann equation is presented which is based upon Zwanzig's projection operator method. This is obtained in the simple compact form of linearized Burnett equations with frequency and wavelength dependent transport coefficients, for which explicit expressions are given for well-behaved intermolecular potentials. The effects of the initial distribution function may be included, although for long times the initial state terms disappear. This disappearance of the initial state terms implies that for long times the general hydrodynamic equations and the Chapman-Enskog hydrodynamic equations agree. The latter are exact for all times for a special initial state. As an application of the theory, an expression is found for the dispersion equation for sound propagating in a gas, that holds for any well-behaved intermolecular potential.

DYNAMICAL STUDY OF AN H II REGION

ABSTRACT SABATINo SOFIAt Institute for Space Studies Goddard Space Flight Center, New York and Departmen 't of Astronomy University of South Florida Reccived March 18, 1969 A study of the "equilibrium region" left behind the expanding ionization front in an H ii region is performed in order to investigate the influence of the usually neglected dynamical effects in the structure of the region. To that effect, we consider a 40,000' K star surrounded by an H II region with an initial electron temperature of 10,0x' K, and an initial electron density of 100. The gas is allowed to heat up, ccol down, and move according to general hydrodynamic equations. It is found that after about 15,000 years the gas has relaxed to the same temperature found by the static treatment, and that while small velocities (due to the gravitational attraction of the star) and density differences have developed, they are not significant. In consequence, a static treatment for the "equilibrium region" is perfectly adequate, unless one is looking for transient effects of a time scale below about 15,000 years.