The Oseen tensor

Appendix 3.m The Oseen tensor Defining the Fourier transform as 

Since the tensor H(r) depends on the vector r only, it can be written in terms of the scalars A and B and the unit vector r parallel to r, as 

The integrals are easily evaluated by introducing the coordinates t = t-f and I = it r to give 

Noise and Stochastic Processes. Dover Publishing Co., New York (1954) represents a collection of classic papers. 

Batchelor, G. K., An Introduction to Fluid Dynamics, Chap. 4. Cambridge Univ. Press (1970). 

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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],... 

We can take the Rouse term l/ ke 02rm/0m2 (ke = 3kBT//2) entropic spring constant) into consideration formally, if we define the element Tnm of the Oseen tensor as Tnm = E/ . The equation of motion (13) thus becomes... 

The equations of motion (75) can also be solved for polymers in good solvents. Averaging the Oseen tensor over the equilibrium segment distribution then gives = l/ n — m Y t 1 = p3v/rz and Dz kBT/r sNY are obtained for the relaxation times and the diffusion constant. The same relations as (80) and (82) follow as a function of the end-to-end distance with slightly altered numerical factors. In the same way, a solution of equations of motion (75), without any orientational averaging of the hydrodynamic field, merely leads to slightly modified numerical factors [35], In conclusion, Table 4 summarizes the essential assertions for the Zimm and Rouse model and compares them. 

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... 

When we use the Fourier representation of the Oseen tensor G, Ay is given by... 

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]. 

Equating this right-hand side to that of eqn. (208) and again inverting to use the Oseen tensor, T... 

The viscosity of the medium is t, and 1 is the unit tensor. (The Oseen tensor is the Green s function for the Navier-Stokes equation under the conditions that the fluid is incompressible, convective effects can be neglected, and inertial effects coming from the time derivative can be neglected.)... 

We note that Eq. (1.8) is still applicable to this case. Due to the asymmetry the formulas discussed in Section 4 are not valid for rod-like molecules. However, we observe that the Oseen tensor assumes the following form (41)... 

If we introduced the diagonalization approximation of the Oseen tensor we would have obtained... 

In comparison, we note that our results involve a function F(2Ao). We can conclude that the diagonalization approximation of the Oseen tensor amounts to introducing an average interaction parameter. Our rigorous results show that the approximation gives results with a smaller characteristic parameter. 

This conclusion is very natural cind important. Although we have considered in this section only rod-like molecules to reach this conclusion it seems that the diagonalization approximation of the Oseen tensor will cause similar errors even in the case of other types of polmer. 

The transport phenomena of rod-like molecules have been treated in terms of the Rouse model by Bloomfield and Zimm (42). However, the diagonalization approximation in the Oseen tensor was used and, in addition, the evaluation was made only to s = Y2. s being a parameter introduced in the expression... 

The idea of Kirkwood (25) is combined with the Rouse model by Pyun and Fixman (14). The theory allows a uniform expansion of the bond length by a factor a such as introduced by Flory. The nondiagonal term of the Oseen tensor is considered but only to the first order by a perturbation method. Otherwise, their theory is identical to Zimm s theory in Hearst s version in the treatment of the integral equation (14). 

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... 

Zimin, each frictional element is assumed to be a point and the hydro-dynamic interactions between these elements and the solvent are described by the Oseen tensor (23,35). This method is derived from solution of the Navier-Stokes equation assuming the existence of point resistances (34). Although frictional elements of finite size were used in the calculation of translational friction coefficients by Edwards and Oliver (35,36), they have not been applied to the intrinsic viscosity or to dynamic mechanical properties to date. 

Calculation of Complex Modulus. In the Zimm theory, the Oseen tensor is approximated by its average value over the equilibrium configuration ... 

Fixman Method. Fixman proposed a method to solve a more general formulation than that of Zimm (41-46). This method avoids the approximation due to preaveraging of the Oseen tensor. Moreover, long range interaction (or the excluded volume) effects can be explicitly taken into account. The formulation is not much different from Zimm s and most of Eqs. (2.1)-(2.10) are used without any modification except... 

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