Hydrodynamic formulation

Holdych, D.J., D. Rovas, J.G. Georgiadis and R.O. Buckius. An Improved Hydrodynamic Formulation for Multiphase How Lattice Boltzmann Models. Int. J. Modern Phys. C 9 1393-1404 (1998). 

Adopting the constrained search procedure, a time-dependent formulation of KS theory is also possible. > 5 Here we outline the hydrodynamic formulation of time-dependent KS theory. - For an N-electron system with a periodic time-dependent potential, we define... 

Some of these questions have strict and unambiguous answers, in a mathematical model, to other answers are derived from extensive empirical material. The present paper will discuss the problems formulated above, but concerning only rheological properties of filled polymer melts, leaving out the discussion of specific hydrodynamic effects occurring during their flow in channels of different geometrical form. 

Hydrodynamic interaction is a long-range interaction mediated by the solvent medium and constitutes a cornerstone in any theory of polymer fluids. Although the mathematical formulation needs somewhat elaborate methods, the idea of hydrodynamic interaction is easy to understand suppose that a force is somehow exerted on a Newtonian solvent at the origin. This force sets the surrounding solvent in motion away from the origin, a velocity field is created which decreases as ... 

Several attempts to relate the rate for bond scission (kc) with the molecular stress ( jr) have been reported over the years, most of them could be formally traced back to de Boer s model of a stressed bond [92] and Eyring s formulation of the transition state theory [94]. Yew and Davidson [99], in their shearing experiment with DNA, considered the hydrodynamic drag contribution to the tensile force exerted on the bond when the reactant molecule enters the activated state. If this force is exerted along the reaction coordinate over a distance 81, the activation energy for bond dissociation would be reduced by the amount ... 

In a recent version of the Tobolsky and Eyring formulation, the rate of mechanochemical degradation was considered as a Thermally Activated Barrier to Scission (TABS) process. The elastic energy function f(v /) was explicitly considered in terms of the frictional hydrodynamic drag force acting over the entire macromolecule [100]. A more detailed account of this model will be presented in Sect. 5.1. 

In addition to the fact that MPC dynamics is both simple and efficient to simulate, one of its main advantages is that the transport properties that characterize the behavior of the macroscopic laws may be computed. Furthermore, the macroscopic evolution equations can be derived from the full phase space Markov chain formulation. Such derivations have been carried out to obtain the full set of hydrodynamic equations for a one-component fluid [15, 18] and the reaction-diffusion equation for a reacting mixture [17]. In order to simplify the presentation and yet illustrate the methods that are used to carry out such derivations, we restrict our considerations to the simpler case of the derivation of the diffusion equation for a test particle in the fluid. The methods used to derive this equation and obtain the autocorrelation function expression for the diffusion coefficient are easily generalized to the full set of hydrodynamic equations. 

To summarize, the hydration status of the drug molecule and other components of a pharmaceutical formulation can affect mass transport. Solubility of drug crystals in an aqueous or nonaqueous solvent may depend on the presence or absence of moisture associated with the drug. Hydration may also determine the hydrodynamic radii of molecules. This may affect the frictional resistance and therefore the diffusion coefficient of the drug molecules. Diffusion of drugs in polymeric systems may also be influenced by the percent hydration of the polymers. This is especially tme for hydrogel polymers. Finally, hydration of... 

As in Section II,A, a set of steady-state mass and energy balances are formulated so that the parameters that must be evaluated can be identified. The annular flow patterns are included in Regime II, and the general equations formulated in Section II,A,2,a, require a detailed knowledge of the hydrodynamics of both continuous phases and droplet interactions. Three simplified cases were formulated, and the discussion in this section is based on Case I. The steady-state mass balances are... 

The analysis of tubular contactors for heat transfer with phase changes in fluid-fluid systems was shown to be heavily dependent on a proper understanding of two-phase hydrodynamics. It was shown that three basic flow patterns exist within a tube, each with a different heat-transfer mechanism. The formulation of the proper mass and energy models pinpointed three key... 

Three primary problem areas exist in dating groundwater. These are (1) Formulation of realistic geochemical-hydrodynamic models needed to interpret data which are generated by field and laboratory measurements, (2) development of sensitive and accurate analytical methods needed to measure trace amounts of various stable and unstable nuclides, and (3) theoretical and field oriented studies to determine with greater accuracy the extent and distribution of the subsurface production of radionuclides which are commonly assumed to originate only in the atmosphere. 

The substantial effect of secondary breakup of droplets on the final droplet size distributions in sprays has been reported by many researchers, particularly for overheated hydrocarbon fuel sprays. 557 A quantitative analysis of the secondary breakup process must deal with the aerodynamic effects caused by the flow around each individual, moving droplet, introducing additional difficulty in theoretical treatment. Aslanov and Shamshev 557 presented an elementary mathematical model of this highly transient phenomenon, formulated on the basis of the theory of hydrodynamic instability on the droplet-gas interface. The model and approach may be used to make estimations of the range of droplet sizes and to calculate droplet breakup in high-speed flows behind shock waves, characteristic of detonation spray processes. 

This formulation results very insightful according to Equation 8.30, particles move under the action of an effective force — We , i.e., the nonlocal action of the quantum potential here is seen as the effect of a (nonlocal) quantum force. From a computational viewpoint, these formulation results are very interesting in connection to quantum hydrodynamics [21,27]. Thus, Equations 8.27 can be reexpressed in terms of a continuity equation and a generalized Euler equation. As happens with classical fluids, here also two important concepts that come into play the quantum pressure and the quantum vortices [28], which occur at nodal regions where the velocity field is rotational. 

However, the present formulation has the advantage of furnishing a mathematically rigorous foundation to the classical theory and is readily extended to other physical situations, like plasmas32 and semiconductors. Also, it allows us to give a microscopic foundation to the theory of the electrophoretic effect, which is much more delicate because it involves the difficult question of long-range hydrodynamical correlations this point will be the object of Section VI. 

The amorphous phase is not usually a desirable state for the API because the formation process is more random and difficult to control than a crystallization. A second dispersed liquid phase is usually formed just prior to freezing and may coalesce or disperse under the influence of hydrodynamic forces in the crystallizer, making the process sensitive to micro-mixing effects on scale up. Amorphous solids also have significantly lower thermodynamic stability than related crystalline material and may subsequently crystallize during formulation and storage. Because of the non-uniformity of the amorphous solid it can more easily incorporate molecules other than the API, making purification less effective. 

Release-related bioavailability problems have been encountered in the pharmaceutical development of formulations for a number of quite different chemical entities, including ciclos-porin, digoxin, griseofulvin, and itraconazole, to name but a few. A thorough knowledge of hydrodynamics is useful in... 

A reciprocal proportionality exists between the square root of the characteristic flow rate, t/A, and the thickness of the effective hydrodynamic boundary layer, <5Hl- Moreover, f)HL depends on the diffusion coefficient D, characteristic length L, and kinematic viscosity v of the fluid. Based on Levich s convective diffusion theory the combination model ( Kombi-nations-Modell ) was derived to describe the dissolution of particles and solid formulations exposed to agitated systems [(10), Chapter 5.2]. In contrast to the rotating disc method, the combination model is intended to serve as an approximation describing the dissolution in hydrodynamic systems where the solid solvendum is not necessarily fixed but is likely to move within the dissolution medium. Introducing the term... 

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