Oseen interaction

A probably more correct treatment has recently been given by Fixman (91) and by Pyun and Fixman (92). These authors avoid the preaveraging of the Oseen-interaction tensor [eq. (3.20)]. A comparison of results will be given at the occasion of the discussion of eigen values. Fortunately, Zimm s results appear to be only slightly different. 

Equivalently, a frictional force produced by a sphere moving in a fluid gives rise to a perturbation flow. The perturbation flow has been derived by Oseen (45) and Burgers (46). Following these authors, let us discuss the character of the Oseen interaction. After discussing the Oseen tensor we shall apply the interaction to the evaluation of the Stokra friction and the Einstein viscosity formula. The derivation of the latter formula given in this appendix seems to be the simplest amoi many derivations. 

As a second example of using the Oseen interaction, let us derive the Einstein viscosity formula for spherical particles. For a complete perturbation flow due to a sphere at the origin it is necessary to consider composite forces at the origin instead of a single force. Fortunately, in the case of a herical particle a doublet force gives a satisfactory result. For elongated molecules more complex forces must be introduced. 

We see that the question of the nature of the nonlocality of the friction tensor is indeed a complex one. Spatial nonlocality can arise from a variety of effects such as static solvent structural correlations, dynamic solvent effects that give rise to Oseen interactions at large distance, and contributions from the direct forces between the molecules. 

In conclusion we would like to mention the article by Altenberger and Dutch (1973) which discusses the contribution of the hydrodynamic interaction (Oseen interaction) between solute particles to the concentration dependence of the diffusion coefficient. This treatment yields Kb = 1 for small spheres if the m is introduced into the treatment. 

By the theory of Schmidt et al. [40], D 0) in eq 3.21 may be replaced by Do with a negligible error. Remarkably, eq 3.21 was derived without invoking the preaveraging approximation to the Oseen interaction tensor. Akcasu and Gurol [43] showed that the non-preaveraged D(0) is given by... 

The intrinsic viscosity [//(r)] and the friction coefficient f r)g of an unperturbed ring polymer were first calculated by Bloomfield and Zimm [69] and by Fukatsu and Kurata [70] using the Kirkwood-Riseman theory with the preaveraged Oseen interaction tensor. In the non-draining limit their calculations yield... 

The Yamakawa-Fujii theory [2, 3] was developed by using the Kirkwood-Riseman formalism with the effect of chain thickness approximately taken into account. The following remarks may be in order. The Oseen interaction tensor was preaveraged. Force points were distributed along the centroid of the wormlike cylinder (not over the entire domain occupied by the cylinder). The no-slip hydrodynamic condition was approximated by equating the mean solvent velocity over each cross-section of the cylinder to the velocity of the cylinder at that cross-section (Burgers approximate boundary condition). 

Motion of a Fluid Zimm adopted the Kirkwood-Riseman approximate form of the Oseen interaction formula to describe the force on the motion of a fluid ... 

Hydrodynamic interactions on a long length scale can be measured with two-point rheology, in which fluorescent or other beads are mixed with a polymer solution, and videomicroscopy is used to measure the Brownian displacements AR, of pairs of beads. The cross-correlations AR, ARy) determine the cross-diffusion tensors as a function of the separation between beads. For beads a fraction of a micron in size in polymer solutions and interbead distances out to 100 xm, measurements of Crocker, et al.(25), Gardel, et a/.(26), and Chen, et al.(21) agree the cross-diffusion tensor falls off with distance as /R, and has at least approximately the magnitude expected for the Oseen interaction in these viscous polymer... 

When the spheres are closer together, r < 3a, then the simulated hydrodynamic interactions match the Rotoe-Prager interaction rather better than the Oseen interaction. This confirms that the weight function does make the particles behave as volume sources, rather than points. The best fit between simulation results and Stokes flow is obtained for an effective particle radius that is roughly 0.33w, where w is the range of the weight function. So for two-point interpolation (w = 2b) the effective size is about 0.1 b, for three-point interpolation (w = 3b) it is about l.Ob,... 

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