Hydrodynamics interactions

Marlow and Rowell discuss the deviation from Eq. V-47 when electrostatic and hydrodynamic interactions between the particles must be considered [78]. In a suspension of glass spheres, beyond a volume fraction of 0.018, these interparticle forces cause nonlinearities in Eq. V-47, diminishing the induced potential E. 

In dilute solutions, tire dependence of tire diffusion coefficient on tire molecular weight is different from tliat found in melts, eitlier entangled or not. This difference is due to tire presence of hydrodynamic interactions among tire solvent molecules. Such interactions arise from tire necessity to transfer solvent molecules from tire front to tire back of a moving particle. The motion of tire solvent gives rise to a flow field which couples all molecules over a... 

In dilute polymer solutions, hydrodynamic interactions lead to a concerted motion of tire whole polymer chain and tire surrounding solvent. The folded chains can essentially be considered as impenneable objects whose hydrodynamic radius is / / is tire gyration radius defined as... 

At finite concentration, tire settling rate is influenced by hydrodynamic interactions between tire particles. For purely repulsive particle interactions, settling is hindered. Attractive interactions encourage particles to settle as a group, which increases tire settling rate. For hard spheres, tire first-order correction to tire Stokes settling rate is given by [33]... 

Thus one must rely on macroscopic theories and empirical adjustments for the determination of potentials of mean force. Such empirical adjustments use free energy data as solubilities, partition coefficients, virial coefficients, phase diagrams, etc., while the frictional terms are derived from diffusion coefficients and macroscopic theories for hydrodynamic interactions. In this whole field of enquiry progress is slow and much work (and thought ) will be needed in the future. 

Groot R D, T J Madden and D J Tildesley 1999. On the Role of Hydrodynamic Interactions in Bloc Copolymer Microphase Separation. Journal of Chemical Physics 110 9739-9749. 

Dispersion of a soHd or Hquid in a Hquid affects the viscosity. In many cases Newtonian flow behavior is transformed into non-Newtonian flow behavior. Shear thinning results from the abiHty of the soHd particles or Hquid droplets to come together to form network stmctures when at rest or under low shear. With increasing shear the interlinked stmcture gradually breaks down, and the resistance to flow decreases. The viscosity of a dispersed system depends on hydrodynamic interactions between particles or droplets and the Hquid, particle—particle interactions (bumping), and interparticle attractions that promote the formation of aggregates, floes, and networks. 

Hindered Settling When particle concentration increases, particle settling velocities decrease oecause of hydrodynamic interaction between particles and the upward motion of displaced liquid. The suspension viscosity increases. Hindered setthng is normally encountered in sedimentation and transport of concentrated slurries. Below 0.1 percent volumetric particle concentration, there is less than a 1 percent reduction in settling velocity. Several expressions have been given to estimate the effect of particle volume fraction on settling velocity. Maude and Whitmore Br. J. Appl. Fhys., 9, 477—482 [1958]) give, for uniformly sized spheres,... 

These models are designed to reproduce the random movement of flexible polymer chains in a solvent or melt in a more or less realistic way. Simulational results which reproduce in simple cases the so-called Rouse [49] or Zimm [50] dynamics, depending on whether hydrodynamic interactions in the system are neglected or not, appear appropriate for studying diffusion, relaxation, and transport properties in general. In all dynamic models the monomers perform small displacements per unit time while the connectivity of the chains is preserved during the simulation. 

Before turning to dynamics, we should hke to point out that, because no solvent is explicitly included, the Rouse model [37,38] (rather than the Zimm model [39]) results in the dilute limit, as there is no hydrodynamic interaction. The rate of reorientation of monomers per unit time is W, and the relaxation time of a chain scales as [26,38]... 

Appearance of hydrodynamic interaction around particles of filler... 

Equation (23) predicts a dependence of xR on M2. Experimentally, it was found that the relaxation time for flexible polymer chains in dilute solutions obeys a different scaling law, i.e. t M3/2. The Rouse model does not consider excluded volume effects or polymer-solvent interactions, it assumes a Gaussian behavior for the chain conformation even when distorted by the flow. Its domain of validity is therefore limited to modest deformations under 0-conditions. The weakest point, however, was neglecting hydrodynamic interaction which will now be discussed. 

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 ... 

It is usual to define a screening length ( H) as the distance at which hydrodynamic interaction becomes negligible (Fig. 8) [44]. 

The non-free draining character of flexible polymer chains was considered in the Zimm model [48], In this model, the effect of hydrodynamic interaction at the location of bead i is taken into account by an additional fluid velocity term vj ... 

Due to the gradual decay of the hydrodynamic interaction (Eq. 27), the extra velocity component at bead i results from the motion of all the remaining beads and it is presumed that vj depends linearly on the hydrodynamic forces acting on these beads ... 

The Zimm model predicts correctly the experimental scaling exponent xx ss M3/2 determined in dilute solutions under 0-conditions. In concentrated solution and melts, the hydrodynamic interaction between the polymer segments of the same chain is screened by the host molecules (Eq. 28) and a flexible polymer coil behaves much like a free-draining chain with a Rouse spectrum in the relaxation times. 

As coils become more expanded, the hydrodynamic interaction decreases this is reflected in the Zimm relaxation time which approaches the Rouse value in good solvent. 

Since the hydrodynamic interaction decreases as the inverse distance between the beads (Eq. 27), it is expected that it should vary with the degree of polymer chain distortion. This is not considered in the Zimm model which assumes a constant hydrodynamic interaction given by the equilibrium averaging of the Oseen tensor (Eq. 34). 

The steady states of such systems result from nonlinear hydrodynamic interactions with the gas flow field. For the convex flame, the flame surface area F can be determined from the relation fSl = b zv, where Sl is the laminar burning velocity, the cross-section area of the channel, and w is the propagation velocity at the leading point. 

The values for the lipid molecules compare well (althoughJgiey are still somewhat larger) with the experimental value of 1.5x10 cm /s as measured with the use of a nitroxide spin label. We note that the discrepancy of one order of magnitude, as found in the previous simulation with simplified head groups, is no longer observed. Hence we may safely conclude that the diffusion coefficient of the lipid molecules is determined by hydrodynamic interactions of the head groups with the aqueous layer rather than by the interactions within the lipid layer. The diffusion coefficient of water is about three times smaller than the value of the pure model water thus the water in the bilayer diffuses about three times slower than in the bulk. 

Measurement of the Hydrodynamic Interaction Force Acting between Two Trapped Particies 121... 

Zimm [34] extended the bead-spring model by additionally taking hydrodynamic interactions into account. These interactions lead to changes in the medium velocity in the surroundings of each bead, by beads of the same chain. It is worth noting that neither the Rouse nor the Zimm model predicts a shear rate dependency of rj. Moreover, it is assumed that the beads are jointed by an ideally Hookean spring, i.e. they obey a strictly linear force law. 

First approaches to approximating the relaxation time on the basis of molecular parameters can be traced back to Rouse [33]. The model is based on a number of boundary assumptions (1) the solution is ideally dilute, i.e. intermolecular interactions are negligible (2) hydrodynamic interactions due to disturbance of the medium velocity by segments of the same chain are negligible and (3) the connector tension F(r) obeys an ideal Hookean force law. 

Mixing and dispersion of viscous fluids—blending in the polymer processing literature—is the result of complex interaction between flow and events occurring at drop length-scales breakup, coalescence, and hydrodynamic interactions. Similarly, mixing and dispersion of powdered solids in viscous liquids is the result of complex interaction between flow and... 

In this section, we consider flow-induced aggregation without diffusion, i.e., when the Peclet number, Pe = VLID, where V and L are the characteristic velocity and length and D is the Brownian diffusion coefficient, is much greater than unity. For simplicity, we neglect the hydrodynamic interactions of the clusters and highlight the effects of advection on the evolution of the cluster size distribution and the formation of fractal structures. 

Batchelor, G. K., and Green J. T., The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. J. Fluid Mech. 56, 375-400 (1972). 

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