Velocity correlation coefficients

This result is derived in the limit where velocity fluctuations in the fluid phase are caused by the presence of particles. In a system where fluid-phase velocity fluctuations are generated by multiple sources, the fluid-particle velocity correlation coefficient would move towards zero and depend on the relative magnitude of the source terms. 

The parameter reflects interactions of the cation-constituent with the solvent and with itself. 1 likewise for the anion-constituent, while 1 is a measure of the degree of coupling between the motions of cation- and anion-constituent. The 1 approach has been most actively fostered by Don Miller (TtO 111. 115). An alternative method of representation, introduced by Gerhard Hertz (116.1171, is based on velocity correlation coefficients. There are six of these for a binary electrolyte with self-diffusion coefficients being required to evaluate them. Sets of vcc values have recently been published for concentrated solutions of electrolytes which exhibit strong complexation like Cdl (118). The main problem in calculating these turned out to be a deficiency in suitable transference number information ... 

Harris KR (2010) Relations between the fractional stokes- einstein and nemst- einstein equations and velocity correlation coefficients in ionic liquids and molten salts. J Phys ChemB 114 9572-9577... 

I quantities x and y are different, then the correlation function js sometimes referred to ross-correlation function. When x and y are the same then the function is usually called an orrelation function. An autocorrelation function indicates the extent to which the system IS a memory of its previous values (or, conversely, how long it takes the system to its memory). A simple example is the velocity autocorrelation coefficient whose indicates how closely the velocity at a time t is correlated with the velocity at time me correlation functions can be averaged over all the particles in the system (as can elocity autocorrelation function) whereas other functions are a property of the entire m (e.g. the dipole moment of the sample). The value of the velocity autocorrelation icient can be calculated by averaging over the N atoms in the simulation ... 

Other orientational correlation coefficients can be calculated in the same way as tf correlation coefficients that we have discussed already. Thus, the reorientational coiTelatio coefficient of a single rigid molecule indicates the degree to which the orientation of molecule at a time t is related to its orientation at time 0. The angular velocity autocorrelatio function is the rotational equivalent of the velocity correlation function ... 

The friction coefficient determines the strength of the viscous drag felt by atoms as they move through the medium its magnitude is related to the diffusion coefficient, D, through the relation Y= kgT/mD. Because the value of y is related to the rate of decay of velocity correlations in the medium, its numerical value determines the relative importance of the systematic dynamic and stochastic elements of the Langevin equation. At low values of the friction coefficient, the dynamical aspects dominate and Newtonian mechanics is recovered as y —> 0. At high values of y, the random collisions dominate and the motion is diffusion-like. 

Fig. 14. Drag coefficient for terminal settling velocity correlation (single particle) where A represents Stokes law B, intermediate law and C, Newton s...

An idea of the scale of turbulence can be obtained by measuring instantaneous values of velocities at two different points within the fluid and examining how the correlation coefficient for the two sets of values changes as the distance between the points is increased. 

When these are close together, most of the simultaneously measured velocities will relate to fluid in the same eddy and the correlation coefficient will be high. When the points are further apart the correlation coefficient will fall because in an appreciable number of the pairs of measurements the two velocities will relate to different eddies. Thus, the distance apart of the measuring stations at which the correlation coefficient becomes very poor is a measure of scale of turbulence. Frequently, different scales of turbulence can be present simultaneously. Thus, when a fluid in a tube flows past an obstacle or suspended particle, eddies may form in the wake of the particles and their size will be of the same order as the size of the particle in addition, there will be larger eddies limited in size only by the diameter of the pipe. 

Bubble size in the circulating beds increases with Ug, but decreases with Ul or solid circulation rate (Gs) bubble rising velocity increases with Ug or Ul but decreases with Gs the ffequeney of bubbles increases with Ug, Ul or Gs. The axial or radial dispersion coefficient of liquid phase (Dz or Dr) has been determined by using steady or unsteady state dispersion model. The values of Dz and D, increase with increasing Ug or Gs, but decrease (slightly) with increasing Ul- The values of Dz and Dr can be predicted by Eqs.(9) and (10) with a correlation coefficient of 0.93 and 0.95, respectively[10]. 

The last issue we address concerns the existence of long-time tails in the discrete-time velocity correlation function. The diffusion coefficient can be written in terms of the velocity correlation function as... 

Since the diffusion coefficient is the infinite-time integral of the velocity correlation function, we have the Einstein relation, D = kBT/Q. 

Discrete-time velocity correlation function, multiparticle collision dynamics, macroscopic laws and transport coefficients, 103-104 Dissipative structures ... 

Such a decomposition of the diffusion coefficient has previously been noted by Pattle et al.(l ) Now we must evaluate >. The time-integrated velocity correlation function Aj j is due to the hydrodynamic interaction and can be described by the Oseen tensor. The Oseen tensor is related to the velocity perturbation caused by the hydrodynamic force, F. By checking units, we see that A is the Oseen tensor times the energy term, k T, or... 

In the consideration of the statistical aspects of turbulence it was found to be of utility (B6, K4, Rl) to establish the correlation in time (B6) of the velocity vectors as a function of the distance between two points in the turbulent stream. The correlation coefficient is defined by... 

An estimate of the effect of separation of the points upon the correlation coefficient is given in Fig. 3 (C7). Batchelor (B6) has been able to predict many of the basic characteristics of the correlation coefficient shown in Fig. 3 for both the transverse, and longitudinal fluctuating velocities. Much has been written about the characteristics of double, triple, and in a few cases higher correlations (K4, L5). It is beyond the scope of this discussion to consider these more refined measures of the statistical characteristics of turbulence. It suffices to indicate that at present a reasonable beginning has been made in the evaluation of the microscopic characteristics of turbulence but that much more experimental work must be carried out in order to supply the quantitative information required to make the extensive theoretical effort capable of quantitative application. 

For computing the terminal settling velocity, correlations for drag coefficient as a function of Archimedes number... 

In this equation g(t) represents the retarded effect of the frictional force, and /(f) is an external force including the random force from the solvent molecules. We see, in contrast to the simple Langevin equation with a constant friction coefficient, that the friction force at a given time t depends on all previous velocities along the trajectory. The friction force is no longer local in time and does not depend on the current velocity alone. The time-dependent friction coefficient is therefore also referred to as a memory kernel . A short-time expansion of the velocity correlation function based on the GLE gives (fcfiT/M)( 1 — (g/M)t2/(2r) + ), where r is the decay time of g(t), and it therefore does not have a discontinuous first derivative at t = 0. The discussion of the properties of the GLE is most easily accomplished by using so-called linear response theory, which forms the theoretical basis for the equation and is a powerful method that allows us to determine non-equilibrium transport coefficients from equilibrium properties of the systems. A discussion of this is, however, beyond the scope of this book. 

Significance of Correlation Coefficient—Let denote the correlation coefficient between velocities at points in an element of turbulent fluid at times differing by /, and let 0C denote the correlation coefficient between deviations of the concentration C of a suspended material from the average of the surrounding concentrations at different points in the element in the same time difference. 

Therefore, before describing the modification of the equilibrium FDT, we need to study in details the behavior of D(t). Note, however, that the integrated velocity correlation function [, Cvv(/) df takes on the meaning of a time-dependent diffusion coefficient only when the mean-square displacement increases without bounds (when the particle is localized, this quantity characterizes the relaxation of the mean square displacement Ax2 t) toward its finite limit Ax2(oo)). 

Figure 1 Structural (left column) and dynamical (right column) properties of the systems investigated. Upper left centre-of-mass radial pair distribution function gooo( ) lower left spherical harmonic expansion coefficient g2oo(r) upper right angular velocity correlation function lower right orientational correlation function. Dotted lines CO, 80 K, 1 bar thin lines CS2, 293 K, 1 bar thick lines CS2, 293 K, 10 kbar.

---