Oseen interaction tensor

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. 

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). 

Our theory may be imderstood better if compared with the KR theory. Their theory has been developed along the observations discussed in Section 1. We note that Ff of Eq. (1.4) which depends on all the segments is replaced in their theory by a one body force determined by the ordering number of a segment irrespective of its location. For this reason it was necessary to replace the Oseen hydrodynamical interaction tensor by its average. 

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. 

The components of represent stochastic displacements and are obtained using the multivariate Gaussian random number generator GGNSM from the IMSL subroutine library (30). p ° is the initial hydrodynamic interaction tensor between subunits iJand j. Although the exact form of D. is generally unknown, it is approximated here using the Oseen tensor with slip boundary conditions. This representation has been shown to provide a reasonable and simple point force description of the relative diffusion of finite spheres at small separations (31). In this case, one has... 

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. 

Now Yoshizaki and Yamakawa°° have extended the calculation to third-order terms, but with the Oseen tensor pre-averaged. In this way a precise lower bound for Og was obtained, close to that obtained by Auer and Gardner using Kirkwood-Riseman theory. The paper by Bixon and Zwanzig performs an infinite order calculation based upon the PF treatment. Using a numerical method, they obtain g 2.76 x 10 , between the Zimm and PF values. For flexible polymers (as for rigid rods) the pre-averaging of the hydrodynamic interaction tensor thus introduces only a small error the effect on the spectrum of relaxation times is more dramatic cf. columns 3 and 4 of Table 2), and the relaxation time of the slowest mode (proportional to 1/A/) is more than twice as slow. This difference should be detectable experimentally. 

HO. is known as the Oseen hydrodynamic interaction tensor . In a common... 

Tests of the validity of the Kirkwood-Riseman picture, inquiring directly if diffusing objects actually have cross-diffusion tensors that match their supposed hydrodynamic interactions, have recently been accomplished Crocker used videomicroscopy and optical tweezers to study the correlated Brownian motions of a pair of 0.9 xm polystyrene spheres, thereby determining their cross-diffusion ten-sors(3). Crocker found that the diffusion tensors are accurately described by the hydrodynamic interaction tensors, exactly as Kirkwood and Riseman had assumed. An optical trap experiment by Meiners and Quake observed the motions of two Brownian particles, further confirming the validity of the Oseen approximation for hydrodynamic interactions(4). 

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... 

HOs is known as the Oseen hydrodynamic interaction tensor. In a common situation there exists a certain flow field, vo r) and a suspended particle moves at first with the liquid. Application of an external force, f, on the par-... 

In equation (51) the idea of hydrodynamic interaction was introduced, but this effect was neglected in subsequent equations. To include this effect it is necessary to insert an expression for, say, v l. A commonly used expression is vi = in which 12 is the Oseen-Burgers tensor,... 

Since the hydrodynamic interaction decreases as the inverse distance between the beads (Eq. 27), it is expected that it should vary with the degree of polymer chain distortion. This is not considered in the Zimm model which assumes a constant hydrodynamic interaction given by the equilibrium averaging of the Oseen tensor (Eq. 34). 

If the Brownian particles were macroscopic in size, the solvent could be treated as a viscous continuum, and the particles would couple to the continuum solvent through appropriate boundary conditions. Then the two-particle friction may be calculated by solving the Navier-Stokes equations in the presence of the two fixed particles. The simplest approximation for hydrodynamic interactions is through the Oseen tensor [54],... 

A model that can take these findings into account is based on the idea that the screening of hydrodynamic interactions is incomplete and that a residual part is still active on distances r > H(c) [40,117]. As a consequence the solvent viscosity r s in the Oseen tensor is replaced by an effective... 

Such a decomposition of the diffusion coefficient has previously been noted by Pattle et al.(l ) Now we must evaluate >. The time-integrated velocity correlation function Aj j is due to the hydrodynamic interaction and can be described by the Oseen tensor. The Oseen tensor is related to the velocity perturbation caused by the hydrodynamic force, F. By checking units, we see that A is the Oseen tensor times the energy term, k T, or... 

The physical nature of this phenomenon is related to the presence of hydrodynamic interactions described by the Oseen tensor [22, 25]. The role of the finely porous medium in classical electroosmosis is played in this case by the gel which can be roughly considered as a collection of pores of size where is the mesh size of the gel [22]. 

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