Hydrodynamic frictional coefficient

It should be noted, however, that for molecular species of geometries other than spheres, different approximations are needed to describe the hydrodynamic frictional coefficient, / [8]. Since there is a relationship between rg and the effective molecular size and shape of the molecular species, it is clear that diffusion coefficients are sensitive to the structural properties and aggregation modes of these species. Therefore, the diffusion coefficient can be used to probe binding phenomena and intermolecular interactions [10, 11]. 

Introducing the hydrodynamic friction coefficients f = /iwcw /p = /pwcw, we may express the ionic velocities in the form... 

Since it is expected that the friction coefficient in the boundary regime is also above the hydrodynamic friction coefficient, it shows that the assumptions which are made concerning the friction calculation are not sufficient yet. It is supposed to be the temperature calculation which must be introduced to... 

Scheraga-Mandelkern equations (1953), for effective hydrodynamic ellipsoid factor p (Sun 2004), suggested that [rj] is the function of two independent variables p, the axial ratio, which is a measure of shape, and Ve, the effective volume. To relate [r ] to p and Ve, introduced f, the frictional coefficient, which is known to be a direct function of p and Ve. Thus, for a sphere we have... 

The process of transition from hydrodynamic to boundary lubrication can be described qualitatively by plotting the measured friction coefficients against film thickness, which depends on the operational conditions, such as load, sliding velocity and lubricant viscosity. A typical diagram known as the "Stribeck Curve is schematically shown in Fig. 27, in which the friction coefficients are given as a function of, ... 

Practically, aU data of friction measurements on wet tracks in the speed range of hydrodynamic lubrication exist as tire skid measurements. Figure 26.10 shows the results of a braking test on wet, finely structured concrete using a smooth tire and measuring the friction coefficient as function of... 

The hydrodynamic drag experienced by the diffusing molecule is caused by interactions with the surrounding fluid and the surfaces of the gel fibers. This effect is expected to be significant for large and medium-size molecules. Einstein [108] used arguments from the random Brownian motion of particles to find that the diffusion coefficient for a single molecule in a fluid is proportional to the temperature and inversely proportional to the frictional coefficient by... 

For heterogeneous media composed of solvent and fibers, it was proposed to treat the fiber array as an effective medium, where the hydrodynamic drag is characterized by only one parameter, i.e., Darcy s permeability. This hydrodynamic parameter can be experimentally determined or estimated based upon the structural details of the network [297]. Using Brinkman s equation [49] to compute the drag on a sphere, and combining it with Einstein s equation relating the diffusion and friction coefficients, the following expression was obtained ... 

The friction coefficient of a large B particle with radius ct in a fluid with viscosity r is well known and is given by the Stokes law, Q, = 67tT CT for stick boundary conditions or ( = 4jit ct for slip boundary conditions. For smaller particles, kinetic and mode coupling theories, as well as considerations based on microscopic boundary layers, show that the friction coefficient can be written approximately in terms of microscopic and hydrodynamic contributions as ( 1 = (,(H 1 + (,/( 1. The physical basis of this form can be understood as follows for a B particle with radius ct a hydrodynamic description of the solvent should... 

Numerous turbulent mass-transfer relationships are given in Eqs. (39)-(50), Table VII. Although the most important ones in practical applications are those for channels and tubes, several other configurations also have been investigated because of their hydrodynamic interest. Generally, it is not possible to predict mass-transfer rates quantitatively by recourse to turbulent flow theory. An exception to this is for the region of developing mass transfer, where a Leveque-type correlation between the mass-transfer coefficient and friction coefficient/can be established ... 

Figure 14 The friction coefficient at 35 knots of two different antifouling technologies and a hydrodynamic smooth reference plotted vs the ageing time. Modified from Weinell et al. (2003).

Hydrodynamic properties, such as the translational diffusion coefficient, or the shear viscosity, are very useful in the conformational study of chain molecules, and are routinely employed to characterize different types of polymers [15,20, 21]. One can consider the translational friction coefficient, fi, related to a transport property, the translational diffusion coefficient, D, through the Einstein equation, applicable for infinitely dilute solutions ... 

The hydrodynamic radius is the equivalent spherical radius obtained from the Stokes law for the chain friction coefficient, Eq. (41), with substituted by R[j. It is easily verified that... 

Since intersegment hydrodynamic interaction dominates the friction coefficient, we ignore the i = j part that leads to the free-draining contribution. If we define the inverse propagator G according to... 

Under the simplifying assumption that hydrodynamic interactions may be neglected, the only new parameter that controls the dynamics is a monomeric friction coefficient (Rouse model). Then the prediction for the rate Pj is given by ... 

The average DNA helix diameter used in modeling applications such as the ones described here includes the diameter of the atomic-scale B-DNA structure and— approximately—the thickness of the hydration shell and ion layer closest to the double helix [18]. Both for the calculation of the electrostatic potential and the hydrodynamic properties of DNA (i.e., the friction coefficient of the helix for viscous drag) a helix diameter of 2.4 nm describes the chain best [19-22]. The choice of this parameter was supported by the results of chain knotting [23] or catenation [24], as well as light scattering [25] and neutron scattering [26] experiments. 

Combining the above descriptions leads to a picture that describes the experimentally observed concentration dependence of the polymer diffusion coefficient. At low concentrations the decrease of the translational diffusion coefficient is due to hydrodynamic interactions that increase the friction coefficient and thereby slow down the motion of the polymer chain. At high concentrations the system becomes an entangled network. The cooperative diffusion of the chains becomes a cooperative process, and the diffusion of the chains increases with increasing polymer concentration. This description requires two different expressions in the two concentration regimes. A microscopic, hydrodynamic theory should be capable of explaining the observed behavior at all concentrations. 

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