Weak flows

The following qualitative picture emerges from these considerations in weak flow where the molecular coils are essentially undeformed, the polymer solution should behave approximately as a Newtonian fluid. In strong flow of a highly dilute polymer solution where the macroscopic velocity field can still be approximated by the Navier-Stokes equation, it should be expected, nevertheless, that in the immediate proximity of a chain, the fluid will be slowed down because of the energy intake to stretch the molecular coil thus, the local velocity field may deviate from the macroscopic description. In the general case of polymer flow,... 

Flows that produce an exponential increase in length with time are referred to as strong flows, and this behavior results if the symmetric part of the velocity gradient tensor (D) has at least one positive eigenvalue. For example, 2D flows with K > 0 and uniaxial extensional flow are strong flows simple shear flow (K = 0) and all 2D flows with K < 0 are weak flows. 

Khakhar, D. V., and Ottino, J. M., Fluid mixing (stretching) by time periodic sequences for weak flows. Phys. Fluids 29(11), 3503-3505 (1986a). 

In a weak flow field, Eq. (64) can be rewritten in a form similar to that for the direct n appearing in Leslie and Ericksen s phenomenological theory [160—163] for nematic systems. Thus, we have... 

A corrected and more general analysis of the primary electroviscous effect for weak flows, i.e., for low Pe numbers (for small distortions of the diffuse double layer), and for small zeta potentials, i.e., f < 25 mV, was carried out by Booth in 1950. The result of the analysis leads to the following result for the intrinsic viscosity [rj] for charged particles in a 1 1 electrolyte ... 

However, viscometric measurements of dilute polymer solutions in a steady flow are inadequate for this purpose although, as already indicated, viscosity is related to molecular rotation. This has been demonstrated by Zimm s theory ). Zimm considered the kinetics of the motion and deformation of a kinetically flexible polymer chain in a weak mechanical field with harmonic velocity gradient g at frequency v. It has been found that under steady and weak flow conditions... 

As alluded to in the introduction to this entry, the LE theory was conceived for small molecule LCs while molecular theories are intended for LCPs. LC molecules retain their equilibrium orientation distribution. LCPs are susceptible to disturbances to their distribution function T (m) its temporal relaxation gives rise to molecular viscoelasticity, while its spatial gradient produces distortional elasticity. A natural question is whether the molecular theories reduce properly to the continuum LE theory in the limiting case of an undisturbed orientation distribution. This situation arises in the weak flow limit where the flow is weak De < . 1) and spatial distortions are small ([V l [Pg.2962]

The idea that the LE theory applies to weak flows, while the molecular theories to strong flows provides a convenient framework for interpreting experimental... 

In the following year, the averaged current at TW8 (within the crater in the interior of the small bay and close to site TW2) was directed toward northwest within the surface layer down to 7 m. Further down the weak flow indicates the influence of the crater, the upper edge of which is 11 m deep. 

Case II refers to situations where the particle-wall interactions are purely repulsive. The particles are separated from the wall by a thin layer of solvent, even in the absence of any motion. Slip is thus possible for very slow flows, indicating that the sticking yield stress is vanishingly small. The residual film thickness for weak flows corresponds to a balance between the osmotic forces and the short-range repulsive forces, independently of any elastohydrodynamic contribution. This is clearly reflected in Fig. 16c, d, where we observe that the particle facet is nearly flat and symmetric. Since tire pressure in the leading and rear regions of the facet are equal and opposite, the lift force is very small. The film thickness, which is set by the balance of the short-range forces, is constant so that the stress/velocity relationship is linear. 

If particles are known to be spherical in shape and nondeformable in the relatively weak flow fields associated with Brownian motion (this may be expected in the case of synthetic latex particles, many proteins, and viruses and probably also holds for certain emulsion particles with rigid ordered interfaces, the Stokes radius will closely correspond to the hard sphere radius R, related to Rg through Rg = 3/5 R and may also be similar to that observed in the electron microscope Rem. The value of Rg should, however, on detailed inspection be greater than the radii measured by the latter methods because it includes bound solvent molecules. The discrepancy can be used to estimate the degree of solvation 81 grams solvent/gram of the particle through the relation ... 

Figures 3a and 3b show the calculated flow fields for the 260 mm immersion case. The predominant flow direction in the slag was upwards from the interface to the top and back down in a loop. This flow was driven by both natural convection and the buoyancy generated by the carbon monoxide gas produced by the electrode-slag reactions. Hence, it was strongest around the electrodes, oftheordra-of 0.1 m/s, and decreased to die ordra-ofO.OI m/s away from the electrodes. In addition to the principal recirculatory flow loop around the electrodes, a second weaker recirculatory loop formed below the electrodes due to the temperature difference between the electrode and the interface. By contrast, the bullion was relatively quiescent, except for a weak flow loop near the side wall, apparently in a counterintuitive direction (Figure 3b). More work is required to establish whether this loop is a real physical phenomenon or an artefact of the mesh geometry.

But according to de Gennes (69) such a transition is unlikely in a weak flow its experimental evidence in strong flows has been claimed by several authors from pressure losses measurements in short capillaries or in slits [Quibrahim (22), Ambari (70)] and from flow birefringence measurements in extensional flows [Pope et al. (71), Cressely et al. (72)]. It must be... 

It is not easy to measure internal friction processes. For example, the contributions of eqs. (VI. 103, VI. 104) do not show up in the static viscosity for weak flows. This can be understood as follows. To discuss the viscosity increment 5t) due to our dilute coils, we may choose any type of (weak) shear flow. It is then convenient (as noted first hy Kramers ) to choose a longitudinal shear flow, such as the one shown in eq. (VI.6S). In this situation the molecule does not rotate but simply stretches to a certain equilibrium length r (for a given shear rate s). Internal friction is involved only if the chain varies its length (or, equivalently, its conformations). In the present case the length is constant, and there is no dissipation associated with internal friction. 

Wakeman [8] also performed some experiments with larger particles at very low concentrations, which enabled the motions of individual particles to be observed. Figure 2.2 shows a glass bead of approximately 100 pm in diameter that is rolling along a woven mesh with a weak flow of liquid through the mesh and a cross flow above the mesh. As also observed by Maddey and Sherman [7], rolling particles came to rest when they reached a stable position. 

The two types of flow which have been most studied are shearing, a weak flow, and elongation, a strong flow. The former may be generalized to include Poiseuille and Couette flows. 

There are significant differences in the behavior of polymeric fluids in these two types of deformation, and each type of deformation has a different effect on the orientation of macromolecules. For example, uniaxial and planar extensional flows impart significant molecular orientation in polymers during flow compared to shear flows. On the other hand, biaxial extensional flow is a weak flow and does not lead to a strong degree of molecular orientation. Furthermore, the rheological response can be significantly different for a polymer in extensional flow versus shear flow. We demonstrate these differences later in this chapter. 

In the form above the equations are coupled nonlinear algebraic equations, and one would have to use Newton s method to solve these equations. However, they can be simplified by the fact that for shear flow (this is called a weak flow in the continuum mechanics literature) tr t is small and the term sk tr t/jjo approaches zero. Hence, Z(ti 1.0 (where means approximately). We also note that r y = Tyx, because the stress tensor is symmetric. Solving the above equations we find that... 

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