Hydrodynamic equations Oseen

This only applies to values of r which are not too large compared with A, since otherwise the inertia terms in the hydrodynamic equations begin to play a part. Compare C. W. Oseen, Hydrodynamik, Leipzig 1927. 

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],... 

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. 

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

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. 

The radial hydrodynamic component (y component) of the force is denoted by fj, and represents the net externally applied hydrodynamic force on the particle resulting from the particle being driven toward (or away from) the collector by the external flow (undisturbed or disturbed) plus any negative resistive lubrication force arising from a close approach of the particle to the collector. The attractive molecular London force acting along the line of centers is denoted by (Ad denotes adhesion). Because of the linearity of the Stokes-Oseen equation, the velocity fields and associated forces may be superposed. 

The radial component of the hydrodynamic force is denoted as Fsu The net hydrodynamic force is a sum of the external force acting on the particle from the liquid flowing around the obstacle, and the force of viscous resistance of the liquid film dividing surfaces of particle and cylinder. The external force can push the particle closer to or pull it away from the obstacle s surface. Note that the force of viscous resistance is negative. Next, denote as Fad the molecular force of the Van der Waals attraction. This force is directed along the perpendicular line from the particle to the symmetry axis of the cylinder. Since the Navier-Stokes equations in the Oseen s approximation are linear, the forces and velocity fields induced by them are additive. 

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). 

Turrently, hydrodynamic interactions between suspended particles cannot be included in a DDFT. However, it is well known that, e.g., the rheology of suspensions cannot be explained without taking these into account. Hydrodynamic interactions in a simple approximation based on Oseen tensors have been included in the Fokker-Planck equation (Eq. 3), and the equivalent of Eq. 4 has been derived and discussed [15, 16]. However, this equation contains three-point and two-point correlations in a form such that the sum rule in Eq. 5 cannot be used. 

The hydrodynamic coupling tensor Xik given by the Oseen or the Navier-Stokes equations for Newton s law... 

In the Zimm model (see Fig. 2A) the hydrodynamic interactions are included by employing the Oseen tensor Him the tensor describes how the mth bead affects the motion of the /th bead. This leads to equations of motion that are not Unear anymore and that require numerical methods for their solution. In order to simplify the picture, the Oseen tensor is often used in its preaveraged form, in which one replaces the operator by its equiUb-rium average value [5]. For chains in -solvents, this leads for the normal modes to equations similar to the Rouse ones, the only difference residing in the values of the relaxation times. An important change in behavior concerns the maximum relaxation time Tchain> which in the Zimm model depends on N as and implies a speed-up in relaxation compared to the Rouse model. Accordingly, the zero shear viscosity decreases in the Zimm model and scales as Also, in the Zimm model the diffusion coefficient... 

In the Zimm theory, the flow perturbations and the co-operative hydrodynamic interactions between segments are treated using the Oseen tensor, pre-averaged for simplification. Pyun and Fixman (PF) avoided this approximation by a perturbation solution of the Kirkwood diffusion equation up to second order. One of the consequences was that [equation (3)] was re-evaluated (see Table 1). 

The components E and were estimated as the solution of the stationary Navier-Stokes equations for liquid flow around a single fibre. The author then used what they claimed was Oseen s approximation to describe the velocity field components and Eg, as well as the hydrodynamic pressure P ... 

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,... 

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