Oseen approximation

Fig. 3.3 Streamlines relative to rigid sphere at low Re (a) Stokes s solution (b) Oseen approximation.

Corresponding streamlines are shown in Figs. 3.3b and 3.4b. Like the creeping flow result, the Oseen solution predicts infinite drift. For large r the velocity is unbounded, but the divergent terms are 0[Re ] and formally beyond the range of the Oseen approximation. For r <2/Re, the stream function may be approximated as... 

Zimm 100) has extended the Rouse model to allow for intramolecular hydro-dynamic interaction, i.e., changes in medium velocity near each bead caused by the flow disturbance from other beads on the same chain. The Oseen approximation, evaluated with the beads located at their mean equilibrium positions, was used to estimate the velocity disturbances. The intensity of the disturbance depends on the parameter h ... 

Values of p22 — P33 = N2 appear to be negative and approximately 10-30% of Nj in magnitude (82). The conventional bead-spring models yield N2=0. Indeed, N2 in steady shear flow is identically zero for all free draining models, regardless of the force-distance law in the connectors (102a). Thus, finite extensibility and, by inference at least, internal viscosity do not in themselves provide non-zero N2 values. Bird and Warner (354) have recently analyzed the rigid dumbbell model with intramolecular hydrodynamic interaction, the latter represented by the Oseen approximation. In this case N2 turns out to be non-zero but positive. 

The Oseen approximation satisfies Eqs. (3.51) through (3.53) in stretched coordinates... 

It should be emphasized that the name Oseen tensor , in chemical physics attributed commonly to the tensor expressed by eq 1.75 is somewhat misleading. In hydrodynamics, the Oseen approximation refers to a second-level approximation, see Eqs. 1.71 and subsequent remarks on the approximations. Oseen introduced his second-level approximation in 1910 to get a reliable description of the velocity field at distances greater than H/Rr, where R is the radius of the flow. However, since the Reynolds numbers for the systems investigated in solution chemistry generally are particularly low, it follows that the first-level description presented here is entirely satisfying. 

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). 

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],... 

All measurements, of course, have to be made at a finite concentration. This implies that interparticle interactions cannot be fully neglected. However, in very dilute solutions we can safely assume that more than two particles have only an extremely small chance to meet [72]. Thus only the interaction between two particles has to be considered. There are two types of interaction between particles in solution. One results from thermodynamic interactions (repulsion or attraction), and the other is caused by the distortion of the laminar fiow due to the presence of the macromolecules. If the particles are isolated only the laminar flow field is perturbed, and this determines the intrinsic viscosity but when the particles come closer together the distorted flow fields start to overlap and cause a further increase of the viscosity. The latter is called the hydrodynamic interaction and was calculated by Oseen to various approximations [3,73]. Figure 7 elucidates the effect. 

The creeping flow approximation has found wide application in problems such as lubrication, injection molding, and flow through porous media. Its application to rigid and fluid particles is discussed in Chapters 3 and 4. However, a fundamental difficulty, first recognized by Oseen, arises in applying Eq. (1-33) or (1-36) to particles in unbounded media. This difficulty, and Oseen s attempt to overcome it, are discussed in Chapter 3. 

Rather than obtaining accurate solutions to Oseen s approximate equation, Proudman and Pearson (P3) suggested a technique to obtain successive approxi-... 

Equation (2.20) also assumes laminar flow (Reynolds numbers less than about 0.1), i.e., low particle velocities, and a dilute suspension of particles that are large compared with the molecules of the fluid. For Reynolds numbers greater than about 0.1 but less than 1, Oseen s law is approximately ... 

It is worth remarking that the controversial conflict between the swarm theory and the continuum theory of liquid crystals is illusory. The swarm theory was a particular hypothetical and approximative approach to the statistical mechanical problem of interpreting properties which can be well defined in terms of a continuum theory. This point is seen less clearly from Oseen s point of departure than from that of the present paper. 

The most gratifying result of the comparison is to notice that there is one term in Oseen s expression which can be omitted, namely the last, since Kn is always zero under the conditions which justify omitting the term -f sf). For many purposes we actually have a simpler result than Oseen s. In detailed application, he in fact assumed K 2 = 0, supposing this to be an approximation. 

In the preaveraged approximation for the Oseen hydrodynamic tensor [19, 20], the linear Langevin equation may be written as... 

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