Dispersion, hydrodynamic macroscopic

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

Hydrodynamic Dispersion Macroscopic dispersion is produced in a capillar) even in tlie absence of molecular diffusion because of the velocity profile produced by the adherence of the fluid to tlie wall. Tlris causes fluid particles at different radial positions to move relative to one anotlier, witli tlie result tliat a series of mixing-cup samples at tlie end of tlie capillary e.xhibits dispersion. 

Effective rates of sorption, especially in subsurface systems, are frequently controlled by rates of solute transport rather than by intrinsic sorption reactions perse. In general, mass transport and transfer processes operative in subsurface environments may be categorized as either macroscopic or microscopic. Macroscopic transport refers to movement of solute controlled by movement of bulk solvent, either by advection or hydrodynamic (mechanical) dispersion. For distinction, microscopic mass transfer refers to movement of solute under the influence of its own molecular or mass distribution (Weber et al., 1991). 

A dilute polymer solution is a system where polymer molecules are dispersed among solvent molecules. An assumption common to any existing theory for flow properties of polymer solutions is that the structure of solvent molecules is neglected and the solvent is assumed to be replaced by a continuous medium of a Newtonian nature. Thus, macroscopic hydrodynamics may be used to describe the motion of the solvent. Recently, some ordering or local structure of solvent molecules around a polymer chain has been postulated as an explanation of the stress-optical coefficient of swollen polymer networks (31,32) so that the assumption of a solvent continuum may not apply. The high frequency behavior shown in Chapter 4 could possibly due to such a microscopic structure of the solvent molecules. Anyway, the assumption of the continuum is employed in every current theory capable of explicit predictions of viscoelastic properties. In the theories of Kirkwood or... 

In practice, this local scale is considered to correspond to the size of the characterization techniques of local soil properties, let s say a small laboratory column. As such the microscopic pore scale variability is no longer explicitly modelled but encoded through effective flow and transport properties at the macroscopic level. The effective macroscopic properties contain of course the signature of the lower level microscopic variability. As such macroscopic effective moisture retention function, hydraulic conductivity or hydrodynamic dispersivity is determined by microscropic pore size distribution, connectivity and tortuosity within the macroscopic sample. 

In recent years a lot of attention has been devoted to the application of electroacoustics for the characterization of concentrated disperse systems. As pointed out by Dukhin [26,27], equation (V-51) is not valid in such systems because it does not account for hydrodynamic and electrostatic interactions between particles. These interactions can typically be accounted for by the introduction of the so-called cell model, which represents an approach used to model concentrated disperse systems. According to the cell model concept, each particle in the disperse system is inclosed in the spherical cell of surrounding liquid associated only with that individual particle. The particle-particle interactions are then accounted for by proper boundary conditions imposed on the outer boundary of the cell. The cell model provides a relationship between the macroscopic (experimentally measured) and local (i.e. within a cell) hydrodynamic and electric properties of the system. By employing a cell model it is also possible to account for polydispersity. Different cell models were described in the literature [26,27]. In each case different expressions for the CVP were obtained. It was argued that some models were more successful than the others for characterization of concentrated disperse systems. Nowadays further development of the theoretical description of electroacoustic phenomena is a rapidly growing area. 

As we consider simultaneous fluid flow and heat transfer in porous media, the role of the macroscopic (Darcean) and microscopic (pore-level) velocity fields on the temperature field needs to be examined. Experiments have shown that the mere inclusion of u0 V(T) in the energy equation does not accurately account for all the hydrodynamic effects. The pore-level hydrodynamics also influence the temperature field. Inclusion of the effect of the pore-level velocity nonuniformity on the temperature distribution (called the dispersion effect and generally included as a diffusion transport) is the main focus in this section. 

The light-scattering spectrum which is related to 7 (q, /) by Eq. (3.3.3) consequently probes how a density fluctuation <5/ (q) spontaneously arises and decays due to the thermal motion of the molecules. Density disturbances in macroscopic systems can propagate in the form of sound waves. It follows that light scattering in pure fluids and mixtures will eventually require the use of thermodynamic and hydrodynamic models. In this chapter we do not deal with these complicated theories (see Chapters 9-13) but rather with the simplest possible systems that do not require these theories. Examples of such systems are dilute macromolecular solutions, ideal gases, and bacterial dispersions. ... 

In view of the very slow time decay of the time correlation functions, it is not clear to what extent the Navier-Stokes transport coefficients can be used even in three dimensions to describe phenomena that vary on a time scale of 50tc, for on this time scale there is not yet a clear separation of microscopic and macroscopic effects. However, usually the Navier-Stokes equations are applied to phenomena that vary on a much longer time scale, and then the slow decay of the correlation functions does not interfere with the hydrodynamic processes. Nevertheless, the divergences of the Burnett and higher-order transport coefficients do appear to have experimental consequences even for three-dimensional systems. In particular, it appears that the dispersion relation for the sound wave frequency wave number k can no longer be expressed as a power series in k as was done in Eq. (133) but instead that fractional powers of the form for /i = l,2,... 

Two types of flow are recognized thixotropy, defined as a decrease of apparent viscosity under shear stress, followed by a gradual recovery when the stress is removed, and its opposite, anti-thixotropy, or rheopexy. Both are related to molecular or macroscopic changes in interactions. In thixotropic liquids, the aggregate bonding must be weak enough to be broken by flow-induced hydrodynamic forces. If dispersion is fine, even slight interactions may produce thixotropic effects. When... 

To a considerable extent, rheological properties of various disperse systems are determined by their random particle fluctuations. In the dispersed phase, these fluctuations play a leading role in the formation of a system of effective stresses that determine, in turn, the observable macroscopic flow characteristics of the dispersions. For this reason, any classification of the dispersions with respect to their hydrodynamic behavior must be based on careful consideration of the main physical mechanisms that induce particulate fluctuations, and that consequently determine fluctuation properties. 

This review deals mainly with the discussion of various macroscopic hydro-dynamic, heat, and mass transfer characteristics of bubble columns, with occasional reference to the analogous processes in modified versions of bubble columns with a variety of internals. The hydrodynamic considerations include determination of parameters like flow patterns, holdup, mixing, liquid circulation velocities, axial dispersion coefficient, etc., which all exert strong influence on the resulting rates of heat and mass transfer and chemical reactions carried out in bubble columns. Different correlations developed for estimating the aforementioned parameters are presented and discussed in this chapter. 

A more interesting way to plot the calculations is in the form of certain dimensionless groups which have been discussed extensively in the literature on hydrodynamic dispersion (Fried and Cambarnous, 1971) the groups in question are D/UL and the group UL/Dq where U is the fluid superficial velocity, L is the macroscopic length of the medium, D is the calculated... 

The system described here is chemically extremely simple and easy to handle but gives rise to complex coupled physical processes. The surfactant concentration, used here as a control parameter, induces an amazing range of shapes and motion patterns. Coupled to these shape-forming processes is the emission of very small but macroscopic droplets. This system is the first example of such a sequence of highly ordered patterns induced by coupled hydrodynamic instabilities. The resulting structures show very efficient motility, internal agitation and dispersion properties. 

The outline of the material to be presented is as follows. The physical process that contributes to hydrodynamic dispersion and how the process is parameterized at the macroscopic scale is reviewed. A summary of existing knowledge on the scale dependence of dispersivity is presented. This will be followed by a discussion on how the parameter is estimated in the field using various field testing methods. Finally, two example applications will be presented to demonstrate how this process is modeled. 

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