Effect of Hydrodynamic Interaction

In 3 in formulating the equations of motion for the beads, it is assumed that the solvent flow pattern is not disturbed by the presence of the beads. The derivation in 3 and 4 can be refined by taking into account the fact that the solvent velocity at bead 1 is perturbed by the motion of bead 2 and vice versa. It is the effect of this perturbation, termed hydrodynamic interaction , that is the subject of this section. 

We therefore begin by rewriting the equations of motion for the beads thus  

17 Equation numbers in square brackets refer to the corresponding equations without hydrodynamic interaction. 

Kirkwood and Riseman 41) were the first to introduce this kind of perturbation into polymer kinetic theory, based on earlier hydro-dynamic studies by Burgers (/3) and Oseen (6/). The perturbation velocity v is given by  

To find F c) for the rigid dumbbell, we form the dot product of R with 

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The non-free draining character of flexible polymer chains was considered in the Zimm model [48], In this model, the effect of hydrodynamic interaction at the location of bead i is taken into account by an additional fluid velocity term vj ... 

The effect of hydrodynamic interactions on polymer collapse has also been studied using MPC dynamics, where the polymer beads are included in the multiparticle collision step [28, 84]. Hydrodynamic interactions can be turned off by replacing multiparticle collisions in the cells by sampling of the particle velocities from a Boltzmann distribution. Collapse occurs more rapidly in the... 

A comparison with Burchard s first cumulant calculations shows qualitative agreement, in particular with respect to the position of the minimum. Quantitatively, however, important differences are obvious. Both the sharpness as well as the amplitude of the phenomenon are underestimated. These deviations may originate from an overestimation of the hydrodynamic interaction between segments. Since a star of high f internally compromises a semi-dilute solution, the back-flow field of solvent molecules will be partly screened [40,117]. Thus, the effects of hydrodynamic interaction, which in general eases the renormalization effects owing to S(Q) [152], are expected to be weaker than assumed in the cumulant calculations and thus the minimum should be more pronounced than calculated. Furthermore, since for Gaussian chains the relaxation rate decreases... 

Several theoretical tentatives have been proposed to explain the empirical equations between [r ] and M. The effects of hydrodynamic interactions between the elements of a Gaussian chain were taken into account by Kirkwood and Riseman [46] in their theory of intrinsic viscosity describing the permeability of the polymer coil. Later, it was found that the Kirdwood - Riseman treatment contained errors which led to overestimate of hydrodynamic radii Rv Flory [47] has pointed out that most polymer chains with an appreciable molecular weight approximate the behavior of impermeable coils, and this leads to a great simplification in the interpretation of intrinsic viscosity. Substituting for the polymer coil a hydrodynamically equivalent sphere with a molar volume Ve, it was possible to obtain... 

The intrinsic viscosity can be related to the overlap concentration, c, by assuming that each coil in the dilute solution contributes to the zero-shear viscosity as would a hard sphere of radius equal to the radius of gyration of the coil. This rough approximation is reasonable as a scaling law because of the effects of hydrodynamic interactions which suppress the flow of the solvent through the coil, as we shall see in Section 3.6.1.2. The Einstein formula for the contribution of suspended spheres to the viscosity is... 

A rigorous theoretical approach to the characterization of the molecular-weight dependence of relaxation times in PL has been used only for the simplest models of the polymer chain for Gaussian subchains without hydrodynamic interactions and by taking into account the effect of hydrodynamic interactions on the PL The two-time approximation (see Eq. (4.5.1))... 

We may find the effects of hydrodynamical interactions by using the Stokes relation ... 

In the curvilinear case where all hydrodynamic interactions between the two particles are considered, only those small particles in the shaded area denoted as Ac can come into contact with the larger particle. The area Ac can be determined numerically. The result is that the actual collision rate is less than the rectilinear rate, and the actual mass-transport coefficient is equal to t(Ac/Ar)PDS(ij)]. Han and Lawler (24) calculated reductions in the rectilinear transport rate by differential sedimentation ranging from about 0.3 to 0.001, so the effects of hydrodynamic interactions on this transport process can be substantial in many cases. 

Numerical results obtained by Han and Lawler (25) for the effects of hydrodynamic interactions on particle transport by fluid shear are summarized graphically in Figure 9. These results are based on the work of Adler (26). The effects of particle size, velocity gradient, and van der Waals interaction are characterized by a dimensionless group, HA, defined as follows ... 

Measurements of Dt on sols of higher concentration show the effects of hydrodynamic interactions that lead to marked reductions in Dt (Figure 9). However, the onset of interaction sets in at considerably higher concentration for sol S4 (—0.15 g/mL) compared with sols of smaller particle size (—0.05 g/mL for S2). This feature may reflect a more extended range of hydrodynamic interactions in the sols of smaller particle size because of oligomers associated with the sol particles. 

Bussell, S. J., Koch, D. L., and Hammer, D. A., The effect of hydrodynamic interactions on the diffusion of integral membrane proteins Diffusion in plasma membranes. Submitted for publication (1994a). 

Figure 4.11 Effect of hydrodynamic interactions for spherical solutes in cylindrical fiber media. Adapted from [79] parameters for Equation 4-29 were obtained by fitting numerical simulation results. Each of the lines represents the variation of F with fiber volume fraction for systems with a particular value of 7. (which varies between 0.1 and 1 in this figure).

Honig E. P., Roebersen G., Wiersema P. H., Effect of hydrodynamic interaction on the coagulation rate of... 

The effect of hydrodynamic interaction of particles on the factor of mutual diffusion of particles has been studied in paper [22]. It formulated a theoretical basis for the determination of collision frequency of particles in a turbulent flow. The effect of internal viscosity of drops on their collision frequency was studied in [23-26]. It was shown that the correct accounting for hydrodynamic interaction ensures a good agreement between the theory and the experiment. 

For Browrtian diffusion of small particles, the influence of hydrodynamic interaction on the collision frequency was studied in works [28, 29], which also mention the decrease in the collision frequency by a factor of 1.5-2. This decrease is not as large as in the case of turbulent coagulation. There are two reasons why the effect of hydrodynamic interaction on the collision frequency of particles differs so substantially in the cases of turbulent flow and Brownian motion. First, the particle size is different in these two cases (the characteristic size of particles participating in Brownian motion is smaller than that of particles in a turbulent emulsion flow). Second, the hydrodynamic force behaves differently (the factor of Browrtian diffusion is inversely proportional to the first power of the hydrodynamic resistance factor h, and the factor of turbulent diffusion - to the second power of h). 

M. Rahnama, D. L. Koch, andE. S. G. Shaqfeh, The effect of hydrodynamic interactions on the orientation distribution in a fiber suspension subject to simple shear flow, Phys. Fluids, 7,487 (1995). 

Schroeder CM, Shaqfeh ESG, Chu S (2004) Effect of hydrodynamic interactions on DNA dynamics in extensional flow simulation and single molecule experiment. Macromolecules 37 9242-9256... 

In concentrated solutions, with the increase of the polymer concentration, the screen effect of hydrodynamic interactions is enhanced due to the interpenetration of polymer chains. We can assume that the hydrodynamic screening length is close to the screening length of volume exclusion of monomers as given by... 

Kadanoff s transformation, besides static properties, provides for including the effect of hydrodynamic interactions between chain points separated by a contour length less than a into the redefinition of a. 

The coefficient of the second term k2 gives the effect of hydrodynamic interaction between two polymer chains. The interaction is mediated by the fiow of the solvent around them. The strength of the hydrodynamic interaction is usually described by the dimensionless number called the Hu ins coefficient ... 

The effect of hydrodynamic interactions on aggregation of colloidal particles may be rather essential and simulation results show that they constrain the growth of aggregates [63]. Computational simulation predicts that many-body hydrodynamic interactions between colloidal particle are able to reduce the sohd fraction required for percolation or gelation [64, 65]. The merging of clusters into condensed aggregate was observed at particle volume fracture p as low as 0.06-0.12 [64]. 

Batchelor [167] considered the effect of hydrodynamic interaction of parallel elongated particles in pure extension on the bulk stress. The governing equations were solved under the assumption that the effect on one particle by all the others could be replaced by a cylindrical boundary condition (cell model) and the geometry and concentration of the particles satisfied the following inequality. 

This result is for absence of hydrodynamic interaction cf. Section B2 below). For the effect of hydrodynamic interaction, and the extension of the rigid dumbbell to the shishkebab model, the reader is referred to the treatise by Bird and coauthors, Chapter 11. 

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