Drag theory

The effect of solute atoms on grain boundary migration cannot adequately be described by standard impurity drag theories. A more satisfactory agreement is obtained by taking an interaction ofthe impurities in the boundary into account. 

Using the concept of Rh has enabled a number of mathematical models to be derived for the permeability of a porous material based on its porosity (see Table 8.3). The models were derived from the following three differing theoretical approaches drag theory, cell theory, and capillary theory. 

Drag theory considers the physical basis of permeability in terms of the frictional resistance of the walls of the material s pores reducing the free flow of fluid through the material, the degree of resistance being dependent on the viscosity of the fluid.+ The drag on the fluid by a pore wall is estimated from the Navier-Stokes equations, and the sum of aU the resistances of the pore walls is assumed to be the total resistance of the material to the fluid flow. This total resistance would be the reciprocal of the specific permeability coefficient, k, of Equation [8.6]. 

The term e refers to volume fraction liquid, (a) Experimental settling rate, (b) Drag theory, (c) Stokes law using physical properties of suspension for p and p. d) Empirical — hydraulic radius, (e) Empirical — hydraulic radius for flocculated systems. 

A finite time is required to reestabUsh the ion atmosphere at any new location. Thus the ion atmosphere produces a drag on the ions in motion and restricts their freedom of movement. This is termed a relaxation effect. When a negative ion moves under the influence of an electric field, it travels against the flow of positive ions and solvent moving in the opposite direction. This is termed an electrophoretic effect. The Debye-Huckel theory combines both effects to calculate the behavior of electrolytes. The theory predicts the behavior of dilute (<0.05 molal) solutions but does not portray accurately the behavior of concentrated solutions found in practical batteries. 

MacKay, D. (1977). A critical survey of receptor theories of drag action. In Kinetics of drug action, edited by J. M. Van Rossum, pp. 255—322. Springer-Verlag, Berlin. 

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 Spite of the existence of numerous experimental and theoretical investigations, a number of principal problems related to micro-fluid hydrodynamics are not well-studied. There are contradictory data on the drag in micro-channels, transition from laminar to turbulent flow, etc. That leads to difficulties in understanding the essence of this phenomenon and is a basis for questionable discoveries of special microeffects (Duncan and Peterson 1994 Ho and Tai 1998 Plam 2000 Herwig 2000 Herwig and Hausner 2003 Gad-el-Hak 2003). The latter were revealed by comparison of experimental data with predictions of a conventional theory based on the Navier-Stokes equations. The discrepancy between these data was interpreted as a display of new effects of flow in micro-channels. It should be noted that actual conditions of several experiments were often not identical to conditions that were used in the theoretical models. For this reason, the analysis of sources of disparity between the theory and experiment is of significance. 

Celata et al. (2006) studied experimentally the drag in glass/fused silica microtubes with inner diameter ranging from 31 to 259 jam for water flow with Re > 300. The drag measurements show that the friction factor for all diameters agrees well with predictions of conventional theory A = 64/Re (for the smallest diameter 31 pm, the deviations of experimental points from the line A = 64/Re do not exceed... 

A transition linearly coupled to the phonon field gradient will experience, from the perturbation theory perspective, a frequency shift and a drag force owing to phonon emission/absorption. Here we resort to the simplest way to model these effects by assuming that our degree of freedom behaves like a localized boson with frequency (s>i. The corresponding Hamiltonian reads... 

The hydrodynamic region has received considerable attention over the years. Equations (2-63) and (2-64) follow the buoyancy-drag force balance theory. If we... 

Furthermore, the closures for the fluid—particle drag and the particle-phase stresses that we discussed were all derived from data or analysis of nearly homogeneous systems. In what follows, we refer to the TFM equations with closures deduced from nearly homogeneous systems as the microscopic TFM equations. The kinetic theory based model equations fall in this category. 

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