Rigid particles suspensions

D Avino, G., Maffettone, P.L., Hulsen, M.A., Peters, G.M.W., Numerical simulation of planar elongational flow of concentrated rigid particle suspensions in a viscoelastic fluid, J. Non-Newtonian Fluid Mech. 150 (2008) 65. 

Suppose we have a physical system with small rigid particles immersed in an atomic solvent. We assume that the densities of the solvent and the colloid material are roughly equal. Then the particles will not settle to the bottom of their container due to gravity. As theorists, we have to model the interactions present in the system. The obvious interaction is the excluded-volume effect caused by the finite volume of the particles. Experimental realizations are suspensions of sterically stabilized PMMA particles, (Fig. 4). Formally, the interaction potential can be written as... 

Newtonian fluids containing a high concentration of rigid particles can show non-Newtonian flow behaviour with increasing shear rate, due to a break up of agglomerates in the shear field [4]. For many pseudoplastic fluid suspensions the... 

We may note that, for a nonzero internal viscosity, the system of equations for the moments is found to be open the equations for the second-order moments contain the fourth-order moments, etc. This situation is encountered in the theory of the relaxation of the suspension of rigid particles (Pokrovskii 1978). Incidentally, for 7 —> 00, equation (F.28) becomes identical to the relaxation equation for the orientation of infinitely extended ellipsoids of rotation (Pokrovskii 1978, p. 58). 

In all but the most basic cases of very dilute systems, with microstructural elements such as rigid particles whose properties can be described simply, the development of a theory in a continuum context to describe the dynamical interactions between structure and flow must involve some degree of modeling. For some systems, such as polymeric solutions, we require modeling to describe both polymer-solvent and polymer-polymer interactions, whereas for suspensions or emulsions we may have an exact basis for describing particle-fluid interactions but require modeling via averaging to describe particle-particle interactions. In any case, the successful development of useful theories of microstructured fluids clearly requires experimental input and a comparison between experimental data and model... 

Such a generic derivation was first effected by Landau and Lifshitz (1959) in the absence of any complicating factors. Included in these complicating factors are inertia, which necessitates the introduction of Reynolds stresses, as well as interfacial tension, present when the suspension is composed of droplets rather than rigid particles. Batchelor s (1970) analysis incorporates such factors. 

This agrees with the mobility expression for a rigid particle with a radius b in concentrated suspensions obtained in previous papers [5, 12]. 

The effective viscosity of a suspension of particles of types other than rigid particles has also been theoretically investigated. Taylor [22] proposed a theory of the electroviscous effect in a suspension of uncharged liquid drops. This theory has been extended to the case of charged liquid drops by Ohshima [17]. Natraj and Chen [23] developed a theory for charged porous spheres, and Allison et al. [24] and Allison and Xin [25] discussed the case of polyelectrolyte-coated particles. 

Suspension polymerizalion is also known as pearl or bead polymerization. Ki-netically, suspension polymerizations are water-cooled bulk reactions. Monomer droplets with dissolved initiator are dispersed in water. As the polymerization proceeds the droplets become transformed into sticky, viscous monomer-swollen particles. Eventually, they become rigid particles with diameters in the range of about (50-500) 10 cm. The final reaction mixture typically contains 25-50% of polymer dispersed in water. The viscosity of the system remains fairly constant during the reaction and is determined mainly by the continuous water phase. 

Jeffrey, D. J. and Acrivos, A. 1976. The rheological properties of suspensions of rigid particles. Am. Inst. Chem. Engineers.. 7. 22 417-432. 

X C2. Although these ratios are not consistent with data for PBG solutions, they are consistent with measurements of the Frank constants of suspensions of the tobacco mosaic virus (Lee and Meyer 1986), which is an extremely rigid particle. Thus, discrepancies between theory and measurements for PBG are most likely due to the partial flexibility of PBG molecules. 

E. J. Hindi and L. G. Leal, Constitutive equations in suspension mechanics. Part 2. Approximate forms for a suspension of rigid particles affected by Brownian rotations. J. Fluid Mech. 76, 187-208 (1976). 

Thurston GB, Bowling DJ. The frequency dependence of the Kerr effect for suspensions of rigid particles. J Colloid Interface Sci 1969 30 34-45. 

Kamis, A., Goldsmith, H.I., and Mason, S.G. (1966). The Kinetics of Flowing Dispersions 1. Concentrated Suspensions of Rigid Particles, J. Colloid and Interface Science 22, 531-553. 

The subject matter will frequently be concerned with situations where the fluid contains a dispersed phase that cannot be considered a component—for example, macromolecules, rigid particles, or droplets. In these cases the continuum approximation is assumed to hold within the suspending fluid and the dispersed phase. The concentration of the rigid or fluid dispersed phase will encompass both dilute and concentrated suspensions. 

K5a. Karnis, A., Goldsmith, H. L., and Mason, S. G., The kinetics of flowing dispersions I. Concentrated suspensions of rigid particles. J. Coll. Interface Set. (in press). 

Fane et al. (1982) discussed the possibility of UF flux enhancement by particulates. It was found that rigid particles larger than 1 pm could enhance flux. Cohesive and compressible particles, even if large, would cause flux reduction. Milonjic et al. (1996) filtered hematite suspensions and found that increased pressure and stirring lead to a increased flux. Chudacek and Fane (1984) measured deposit layers of several pm on UF membrane by macrosolutes and silica colloids. 

Let us now consider the simplest case, a system consisting of diluted suspension of spheres, such as a latex particle suspension. As with any rigid body, there will be only two dynamic modes a translation of the center of mass and the rotation around the center of mass. For a sphere, the only mode that participates in the fluctuations in concentration will be the translational one since a rotation around the center of mass will have no effect on mass transfer in a scattering volume element. This means that in the case of a sphere, we can factorize t) as... 

The consequences of these restrictive assumptions have been reviewed by Dukhin " and others, who noted that the neglect of the outer regions leads to error in assessing the source of the polarization. The diffuse outer, and not the inner ionic double layer is according to Dukhin, the major source of polarization. The details of this comparison were recently reviewed. So much for the effective electrical polarization of rigid spherical bodies surrounded by an ionic double layer, i.e., for a large class of colloidal particle suspensions. 

A careful account of the problem can be found in Ref. [95]. Ohshima et al. [96] first found a numerical solution of the problem, valid for arbitrary values of the zeta potential or the product Ka. In the same paper, they dealt with the problem of finding the sedimentation potential and the DC conductivity of a suspension of mercury drops. The problems are solved following the lines of the electrophoresis theory of rigid particles previously derived by O Brien and White [18]. The liquid drop is assumed to behave as an ideal conductor, so that electric fields and currents inside the drop are zero, and its surface is equipotential. The main difference between the treatment of the electrophoresis of rigid particles and that of drops is that there is a velocity distribution of the fluid inside the drop, Vj, governed by the Navier-Stokes equation with zero body force (in the case of electrophoresis), and related to the velocity outside the drop, v, by the boundary conditions ... 

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