Solvent velocity field

For dense suspensions of spherieal particles, an espeeially aecurate method ealled Stokesian dynamics has been developed by Bossis and Brady (1989). In Stokesian dynamics, one solves a generalized form of Eq. (1-40), in which the simple Stokes law for the drag force on sphere i, = — (x/ — v ), is replaced by a more accurate tensor expression that accounts for the hydrodynamic interactions—that is, the disturbances to the solvent velocity field produced by the relative motions of the other spheres. The Stokesian dynamics method accounts for hydrodynamic interactions among widely separated spheres by a multipole expansion, as well as for closely spaced ones by a lubrication approximation. Results from this method appear in Figs. 6-8 and 8-8. 

In Eq. (67). the first term corresponds to the fluid conductivity with uncharged walls. The second term is due to the difference between the concentrations of the ionic species, which have different mobilities. The third one results from the convection of the local charge disequilibrium since this disequilibrium and the solvent velocity field are proportional to this is a second-order contribution. However, since expression (36) for j/ is only first-order accurate, the last term in (67) accounts only partly for the contribution. 

To imderstand d5mamics and transport properties, a set of (nonlinear) Langevin equations describing the Edwards model and the solvent velocity field is proposed as a minimal model. Comparison of the outcome of the model with experimental results has revealed that this minimal model is reasonable (47) but closely studied, less than minimal. The importance of the direct chain-chain friction must not be underestimated (48). With this additional elfect, the model seems minimal, but a more careful study may be needed. 

In the linearized form, Eqs. 41 and 43 describe the time dependence of the displacement field u and the pressure field p. The Tanaka-Fillmore Eq. 27 can be obtained from these equations under the assumption of the special case that the polymer deformation velocity field and the solvent velocity field are in exactly opposite directions for every point in the whole space. The cooperative diffusion coefficient then has the form... 

These conclusions are of obvious and significant consequence to the dynamics of macromolecular solutions. The issues involved can be addressed in the following way a relatively weak, sharply localized force f (r,t) is applied to the solvent component of a macromolecular solution. The solvent velocity field that develops in response to this force may be expressed in the manner... 

Hydrodynamic interaction is a long-range interaction mediated by the solvent medium and constitutes a cornerstone in any theory of polymer fluids. Although the mathematical formulation needs somewhat elaborate methods, the idea of hydrodynamic interaction is easy to understand suppose that a force is somehow exerted on a Newtonian solvent at the origin. This force sets the surrounding solvent in motion away from the origin, a velocity field is created which decreases as ... 

Fig. 8. Velocity field caused by a point force in a pure solvent and in...

Here v(r) is the velocity field at any space point r, p is the pressure, po is the shear viscosity of the solvent containing salt ions and counterions, and F(r) is the force arising from any potential field in the solution. As an example, if electrical charges are present in the solution under an externally imposed uniform electrical field E, F(r) is given by... 

The first and most rigorous theory on the viscoelasticity was developed by Kirkwood (25). In this theory it is considered that the chain elements counteract the hydrodynamic torques through their micro-brownian motion (26). Then the statistical orientation of the macromolecule in the velocity field of the solvent contributes terms linear as well as non-linear in the rate of shear to the stress and thus to the Newtonian part of the intrinsic viscosity. This nonlinear term exhibits a phase lag when a sinusoidal hydrodynamical force is applied. 

Figure 33-18 Three components A, B, and C are shown compressed against the accumulation wall in FFF to different amounts because of different interactions with the external field. When the flow is begun, component A experiences the lowest solvent velocity because it is the closest to the wall. Component B protrudes more into the channel, where it experiences a higher flow velocity. Component C, which interacts the least with the field, experiences the highest solvent flow velocity and thus is displaced the most rapidly by the flow.

We saw in Chapter 10 that the boundary-layer structure, which arises naturally in flows past bodies at large Reynolds numbers, provides a basis for approximate analysis of the flow. In this chapter, we consider heat transfer (or mass transfer for a single solute in a solvent) in the same high-Reynolds-number limit for problems in which the velocity field takes the boundary-layer form. We saw previously that the thermal energy equation in the absence of significant dissipation, and at steady state, takes the dimensionless form... 

Many theoretical models have been developed to describe the electrical polarizability of polyelectrolytes [2-4,13-40], The problem is, however, extremely complex and difficult, even if simple models are assumed for the geometry of the polyions [37]. This is because many fields are involved concentrations of small ions, the electrical potential, and the solvent velocity, to be determined as functions of space and time, which are coupled with each other through essentially nonlinear equations. Our numerical approach, however, need not introduce, as in most of the theories, somewhat ad hoc approximations such that counterions are classified into free and bound ions, only the latter contributing to the polarizability, nor neglect interactions between counterions. 

Consideration of another major modification that has been applied to the flexible chain model seems pertinent at this point. It has long been appreciated that the velocity field of the solvent would be perturbed deep inside a coiled polymer molecule. It is clear that this effect is not considered in the above treatment because the viscous drag is given as psXi in equation (3-51) irrespective of whether Xi happens to be inside the coiled molecule or on its surface. Thus one might expect the Rouse formulation to be most applicable to polymer-solvent systems in which the elongated conformations of polymer chains predominate. For such conformations, there would be little shielding of one part of a molecule by another part of the same molecule. This is the case in... 

The problem is to determine the effect of the excluded volume of particles of radius on the Taylor dispersion coefficient (Eq. 4.6.27). In so doing, note that in the dispersion coefficient, a should be replaced hy a - a, and U should not represent the average translational speed but rather the magnitude of the difference between the largest and smallest velocities in the flow field. The largest particle velocity remains equal to the centerline solvent velocity but due to the excluded volume, the smallest velocity is not zero but is the value of the solvent velocity at a distance from the wall. This represents a reduction in the velocity difference and, therefore, in the dispersion coefficient. Accordingly, U should be multi-... 

The above work was extended by Ranade and Mashelkar [47] by considering the dissolution of a spherical polymeric particle in a convective field. The transport equations written were very similar to the ones used by Devotta et al. [46]. In addition, the solvent velocity, Ui, was given as... 

First let us consider how the velocity profile caused by a point force is affected when a small number of polymers are present in solution. For simplicity we consider the steady state in the velocity field, though this assumption is not essential." Let be the point force acting at the origin. For pure solvent, the velocity perturbance is given by... 

The approximate experimental determination of xl), is based on measurement of the velocity of a charged particle in a solvent subjected to an applied voltage. Such a particle experiences an electrical force that initiates motion. Since a hydrodynamic frictional force acts on the particle as it moves, a steady state is reached, with the particle moving with a constant velocity U. To calculate this electrophoretic velocity U theoretically, it is, in general, necessary to solve Poisson s equation (Equation 3.19) and the governing equations for ion transport subject to the condition that the electric field is constant far away from the particle. The appropriate viscous drag on the particle can be calculated from the velocity field and the electrical force on the particle from the electrical potential distribution. The fact that the sum of the two is zero provides the electrophoretic velocity U. Actual solutions are complex, and the electrical properties of the particle (e.g., polarizability, conductivity, surface conductivity, etc.) come into play. Details are given by Levich (1962) (see also Problem 7.8). 

In the foregoing discussion, the hydrodynamic interaction has not been taken into account it impUes that the mth segment of a inacromolecule moves in the field of solvent velocities which is perturbed by the motion of the nth chain segment (for details see (de Gennes, 1976b, 1979 Gothb et al, 1986)). 

The solvent is characterized by a velocity field, The force exerted... 

---