Equations Oseen

Stake s drag ignores inertial terms in the governing equations. Oseen [2] obtained the first inertial correction to the drag force in the form of... 

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. 

Oseen(3) employs just the first two terms of equation 3.6 to give ... 

The final term in Eq. (3-27) dominates at large distances from the body. Since this is the region in which inertial effects are significant, Oseen suggested that the nonlinear term v Vv be neglected. Equation (1-33) then becomes... 

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

Equation (3-45) is analogous to the Oseen correction to the Stokes drag, and is accurate to 0[Pe]." It applies for any rigid or fluid sphere at any Re, provided that Pe - 0 and the velocity remote from the particle is uniform. Figure 3.10 shows that Eq. (3-45) is accurate for Pe < 0.5. Acrivos and Taylor (A2) extended the solution to higher terms, but, as for drag, the additional terms only yield slight improvement at Pe < 1. 

Figure 4.6 compares Eqs. (4-24) and (4-25) with selected experimental and numerical results for spheroids. When plotted in this form, (C /CDst — 1) is only weakly dependent on E for A i Re less than about unity. The drag is then very close to the Oseen value, and Eqs. (4-24) and (4-25) are accurate. Above this range, the equations predict that the drag should exceed the Oseen value, whereas the reverse occurs in practice. Thus, as for spheres in Chapter 3, analytic results have little value for A Re > 1. 

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.)... 

The BBO equation delineates a sphere in linear motion at low Reynolds numbers [Basset, 1888 Boussinesq, 1903 Oseen, 1927], For a spherical particle, the BBO equation, based upon Eq. (3.37) with the replacement of the buoyancy force with the pressure gradient force, can be expressed as [Soo, 1990]... 

The dynamic behavior of discrete particles in turbulence can be described by the BBO equation [Basset, 1888 Boussinesq, 1903 and Oseen, 1927] for the slow motion of a spherical particle in a fluid at rest. This equation was extended by Tchen (1947) to the case of a fluid moving with time-dependent velocity as... 

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 ... 

Both equations (6.4) and (6.5) contain D and u and need to be expressed in terms of a single variable in order to determine Z) if m is known or m if Z) is known. Stokes neglected the terms due to inertia and obtained a very simple relationship between settling velocity and particle size for particles settling with low velocities. Several attempts at theoretical solutions for the relationship between and Re at higher velocities have been made. Oseen [29] partially allowed for inertial effects to obtain ... 

Onsager and Fuoss viscosity equation, 125 order in liquids, 1 oriented molecules, 152, 155 0rsted s piezometer, 58 orthobaric density, 48, 327 Oseen correction for falling sphere equation, 87... 

The LE theory is rather complex since it contains both viscous and elastic stresses. It can best be understood by considering viscous and elastic effects separately. If elastic effects are neglected, the LE equations reduce to Ericksens transversely isotropic fluidy while in the absence of flow the elastic stresses are just those of the Frank-Oseen theory (discussed below in Section 10.2.2). ... 

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

Operating with F on both sides of this equation and using the Oseen... 

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). 

The application of the Lorentz-Lorenz equation gives a convincing demonstration of the general similarity of the linear response in gas and liquid but its application in the liquid introduces an approximation which has not yet been quantified. A more precise objective for the theory would be to calculate the frequency dependent susceptibility or refractive index directly. For a continuum model this may lead to a polarizability rigorously defined through the Lorentz-Lorenz equation as shown in treatments of the Ewald-Oseen theorem (see, for example Born and Wolf, plOO),59 but the polarizability defined in this way need not refer to one molecule and would not be precisely related to the gas parameters. 

The tendency of LCs to resist and recover from distortion to their orientation field bears clear analogy to the tendency of elastic solids to resist and recover from distortion of their shape (strain). Based on this idea, Oseen, Zocher, and Frank established a linear theory for the distortional elasticity of LCs. Ericksen incorporated this into hydrostatic and hydrodynamic theories for nematics, which were further augmented by Leslie with constitutive equations. The Leslie-Ericksen theory has been the most widely used LC flow theory to date. 

Hence, as Oseen noted, we cannot expect the Stokes solution to provide a uniformly valid first approximation to the solution of (9 75), but instead expect that it will break down for large values of r > 0(Re ). Thus Whitehead s attempt to evaluate the second term in the expansion (9 77) was unsuccessfiil for large r. Indeed, as noted earlier in conjunction with the thermal problem, it is not so much a surprise that we cannot obtain a solution for boundary condition (9 81) for r oo, in spite of the fact that the governing equation (9 80) is not a valid first approximation to the full Navier Stokes equation except for r < 0(1 /Re). 

Now we determine the scaling constant m by substituting (9-96) and (9-99) into the Navier-Stokes equation (9-75) and requiring that the limiting form of the resulting equation for Re -> 0 contain viscous terms and at least one inertia term, as suggested by Oseen s argument. On substitution of outer variables, (9-75) becomes... 

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