Hydrodynamic approach

In 2001, Ferrigno and Girault proposed a hydrodynamic approach to ion-transfer reactions [122]. In the same way that the friction factor for the diffusion coefficient in the Stokes-Einstein equation can be given by solving the Navier-Stokes equation for a sphere in a laminar flow, the Navier-Stokes equation was solved numerically to account for the passage of a sharp boundary between two continuum media. These data show that the friction coefficient varies from to in a continuous manner over a distance one order of magnitude larger 

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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... 

Morris, G. A. Castile, J. Smith, A. Adams, G.G. Harding, S.E. 2009. Macromolecular conformation of chitosan in dilute solution A new global hydrodynamic approach. Carbohydrate Polymers 76, 616-621. 

Similar instability is caused by the electrostatic attraction due to the applied voltage [56]. Subsequently the hydrodynamic approach was extended to viscoelastic films apparently designed to imitate membranes (see Refs. 58-60, and references therein). A number of studies [58, 61-64] concluded that the SQM could be unstable in such models at small voltages with low associated thinning, consistent with the experimental results. However, as has been shown [60, 65-67], the viscoelastic models leading to instability of the SQM did not account for the elastic force normal to the membrane plane which opposes thickness... 

H Grijseels, DJA Crommelin, CJ De Blaey. Hydrodynamic approach to dissolution rate. Pharm Weekbl 3 129-144, 1981. 

In what follows, we present in this short review, the basic formalism of TDDFT of many-electron systems (1) for periodic TD scalar potentials, and also (2) for arbitrary TD electric and magnetic fields in a generalized manner. Practical schemes within the framework of quantum hydrodynamical approach as well as the orbital-based TD single-particle Schrodinger-like equations are presented. Also discussed is the linear response formalism within the framework of TDDFT along with a few miscellaneous aspects. 

Here we focus on yet another implementation, the single-particle hydrodynamic approach or QFD-DFT, which provides a natural link between DFT and Bohmian trajectories. The corresponding derivation is based on the realization that the density, p(r, t), and the current density, j(r, t) satisfy a coupled set of classical fluid, Navier-Stokes equations ... 

The hydrodynamic approach to liquid-state dynamics is based on the assumption that many experimental observables (like the intensity in a light scattering experiment) can be rationalized by considering the dynamics of a few slow variables. The natural choice for the slow variables are the densities of the conserved quantities—that is, the number density, p(r, t), the momentum density, g(r, t) and the energy density, e(r,t). The conservation of number, momentum, and energy are expressed locally by the conservation equations... 

In Sections V and VI, a brief history of the developments of the MCT from the hydrodynamic approach (Critical Phenomena) and the renormalized kinetic theory approach has been presented. The basic concept of MCT is to use the product of the slow (hydrodynamic) variables to span the orthogonal subspace of the fast variables. 

The majority of theories describing the concentration dependence of viscosity of diluted and moderately concentrated disperse systems is based on the hydrodynamic approach developed by Einstein [1]. Those theories were fairly thoroughly analyzed in the reviews written by Frish and Simha [28] and by Happel and Brenner [29], In a fairly large number of works describing the dependence of viscosity on concentration the final formulas are given in the form of a power series of the volume concentration of disperse phase particles — [Pg.111]

Figure 1 shows a comparison, published by Mori and Ototake [13], of the experimental dependences of viscosity on concentration of dispersions of solid particles based on the data of Vand [34], Robinson [12], Orr and Blocker [5], Dalla Valle and Orr [17] with the theoretical equations based on the hydrodynamic approach used by Einstein (1), Simha (30), Vand (31), Roscoe (44) and the phenomenological equation of Mori and Ototake (14). A more complicated form of the theoretical dependence, naturally makes it possible to describe experimental results over a wider range, but for concentrated dispersions most of theoretical equations remain inapplicable. 

With replaceable gels, either an electrokinetic or a hydrodynamic approach to sample introduction is possible. The hydrodynamic injection is generally preferred when quantitation of the PCR product is desired. In this case, the sample, introduced as a plug into the capillary, is exactly the same as that of the sample vial from which it originated. Negative components (Cl", dNTPs,... 

Blavoux B., Dray M., Fehri A., Ohve P., GroeningM., Sonntag C., Hauquin J. P., Pelissier G., and Pouchan P. (1993) Palaeoclimate and hydrodynamic approach to the Aquitaine Basin deep aquifer (France) by means of environmental isotopes and noble gases. In Isotope Techniques in the Study of Past and Current Environmental Changes in the Hydrosphere and the Atmosphere. IAEA, Vienna, pp. 293-305. 

According to Amsden [82], hydrodynamic approach should be used to deal with homogeneous hydrogels, while, for heterogeneous hydrogels, obstruction are more consistent with experimental data. 

In the previous sections a model of the frequency-dependent collisional friction has been derived. Because the zero-frequency friction for a spherical particle in a dense fluid is well modeled by the Stokes-Einstein result, even for particles of similar size as the bath particles, there has been considerable interest in generalizing the hydrodynamic approach used to derive this result into the frequency domain in order to derive a frequency-dependent friction that takes into account collective bath motions. The theory of Zwanzig and Bixon, corrected by Metiu, Oxtoby, and Freed, has been invoked to explain deviation from the Kramers theory for unimolec-ular chemical reactions. The hydrodynamic friction can be used as input in the Grote-Hynes theory [Eq. (2.35)] to determine the reactive frequency and hence the barrier crossing rate of the molecular reaction. However, the use of sharp boundary conditions leads to an unphysical nonzero high-frequency limit to Ib(s). which compromises its utility. 

Banerjee and Harbola [69] have worked out a variation perturbation method within the hydrodynamic approach to the time-dependent density functional theory (TDDFT) in order to evaluate the linear and nonlinear responses of alkali metal clusters. They employed the spherical jellium background model to determine the static and degenerate four-wave mixing (DFWM) y and showed that y evolves almost linearly with the number of atoms in the cluster. 

There are two main streams in the study of rheological properties of suspensions and polymeric solutions. One is the hydrodynamic approach/ and the other is the molecular kinetic investigation. The so-called Ree-Eyring equation is a result from the latter category it is written as follows ... 

The hydrodynamic approaches assume instantaneous reaction, and apart from a dependence on density, the simplest theories assume detonation parameters to be invariant for a substance and applicable to propagation in infinite, homogeneous (isotropic) media. They give no information on the effect of size or crystal orientation, or on the detailed mechanism by which a detonation propagates. Several theories developed by Jones in the United Kingdom and by Eyring, Wood, and Cook in the United States related detonation velocity to reaction-zone length and explosive diameter, but experimental problems severely limited their validation and application to azides. 

In the magneto-hydrodynamic approach, discnssed earlier, electron and ion velocities (we and Vi) were considered equal to each other, which contradicts the crtrrent J = e e(wi — %) in quasi-neutral plasma with density n. In two-fluid magneto-hydrodynamics, the Navier-Stokes equation includes the electron s mass m, pressure pe, and velocity and also takes into account friction between electrons and ions ... 

The differences between Pitts (P) and Fuoss-Onsager (F-O) are first, the above mentioned omission by F-O of the effect of asymmetric potential on the local velocities of the solvent near the ions second, the use of the more usual boundary conditions 5.2.28b by F-O compared to the P assumption that perturbations cease to be important at r = a. Pitts, Tabor and Daly, who have analysed in detail both treatments, concluded that the discrepancy due to the different boundary conditions is small but has the effect of reducing ionic interactions in the P treatment with respect to the F-O. This is confirmed by the analysis of data with both theories. Usually P requires a smaller value of the a parameter than F-O. The third discrepancy between the theoretical treatments is in the expression of Vj, in eqn. 5.2.5, for which F-O add a term which involves the effect of the asymmetry of the ionic atmosphere upon the central ion surrounded by such atmosphere. The last difference lies in the hydrodynamic approaches and the corresponding boundary conditions. P imposes the condition that the velocity of the smoothed... 

The dynamic exchange model employs a hydrodynamic approach wherein the dynamics of the three species in the surface layer is described by a reaction-diffusion equation and the bulk water dynamics is described by a simple diffusion equation. Therefore, in this approach, the interactions are not considered explicitly... 

Jones [20] used a Smoluchowski approach to examine interacting spherical polymers. Jones predicted that, if one polymer species is dilute and labelled, the measured diffusion coefficient from QELSS is determined only by hydrodynamic interactions of the tagged polymers and their untagged matrix neighbors, and is the single-particle diffusion coefficient. The hydrodynamic approach culminated in analyses of Carter, et al. [21] and Phillies [22] of mutual and tracer diffusion coefficients, including hydrodynamic and direct interactions and reference frame issues. 

See however the interesting quasi-hydrodynamical approach of Zwanzig and Bixon/ t Strictly speaking 4>d (vi, t) is not a distribution function, since it can take negative values, and is not properly normalized. However 4>d (vi, 0 can be related to the deviations of a nonequilibrium distribution function from the total equilibrium distribution function. For this reason we will continue to refer to 4>d (V], t) and related functions as distribution functions. tWe ignore the walls, and therefore all quantities will be defined in the thermodynamic limit, N->oo, V- oo,N/V=n, fixed. 

We shall discuss here the macroscopic dynamics of liquid crystals that is an area of hydrodynamics or macroscopic properties related to elasticity and viscosity. With respect to the molecular dynamics, which deals, for example, with NMR, molecular diffusion or dipolar relaxation of molecules, the area of hydrodynamics is a long scale, both in space and time. The molecular dynamics deals with distances of about molecular size, a 10 A, i.e., with wavevectors about 10 cm , however, in the vicinity of phase transitions, due to critical behaviour, characteristic lengths of short-range correlations can be one or two orders of magnitude larger. Therefore, as a limit of the hydrodynamic approach we may safely take the range of wavevectors q 10 cm and corresponding frequencies (O c q 10 - 10 = 10"s (c is sound velocity). 

In order to calculate the relationship between r and P away from equilibrium one must take into account the dynamics of bubble growth and decay. Such a hydrodynamic approach was developed by Zeldovich [9] and Kagan [73]. There are, however, two obvious limiting cases [69]. For small departures from equilibrium, the pressure inside the embryo can be plausibly assumed to remain constant at its equilibrium value. In this case, (44) becomes... 

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