Theories and Equations for Suspensions

The fourth chapter presents some constitutive theories and equations for suspensions. Suspension rheology normally deals with the flow behavior of two-phase systems in which one phase is solid particles like fillers but the other phase is water, organic liquids or pol)oner solutions. Literature on suspension rheology does not include flow characteristics of filled polymer systems. Neverttieless, ttiis chapter needs to be included as the foimdations for understanding ttie basics of filled polymer rheology stem from the flow behavior of suspensions. In fact, most of the constitutive theories and equations that are used for filled polymer systems are borrowed firom those that were initially developed for suspension rheology. 

Constitutive theories and equations for suspensions C Rod-shaped particles... 

Chapter 4 deals witii constitutive theories and equations for suspensions and lays down the foundations for understanding the basics of filled polymer rheology. Starting from the simplest dilute... 

This equation is identical to the Maxwell [236,237] solution originally derived for electrical conductivity in a dilute suspension of spheres. Hashin and Shtrikman [149] using variational theory showed that Maxwell s equation is in fact an upper bound for the relative diffusion coefficients in isotropic medium for any concentration of suspended spheres and even for cases where the solid portions of the medium are not spheres. However, they also noted that a reduced upper bound may be obtained if one includes additional statistical descriptions of the medium other than the void fraction. Weissberg [419] demonstrated that this was indeed true when additional geometrical parameters are included in the calculations. Batchelor and O Brien [34] further extended the Maxwell approach. 

Note that the above study is performed for a simple system. There exists a large body of literature on the study of diffusion in complex quasi-two-dimensional systems—for example, a collodial suspension. In these systems the diffusion can have a finite value even at long time. Schofield, Marcus, and Rice [17] have recently carried out a mode coupling theory analysis of a quasi-two-dimensional colloids. In this work, equations for the dynamics of the memory functions were derived and solved self-consistently. An important aspect of this work is a detailed calculation of wavenumber- and frequency-dependent viscosity. It was found that the functional form of the dynamics of the suspension is determined principally by the binary collisions, although the mode coupling part has significant effect on the longtime diffusion. 

Theoretical trends in the study of suspensions employ concepts and techniques originally developed in connection with theories of liquids, for example, equation hierarchies, closure problems, and Monte Carlo methods. In marked contrast with the definitive achievements reviewed in the previous section, the present section outlines a field currently under active development. 

Electrokinetic equations describing the electrical conductivity of a suspension of colloidal particles are the same as those for the electrophoretic mobility of colloidal particles and thus conductivity measurements can provide us with essentially the same information as that from electrophoretic mobihty measurements. Several theoretical studies have been made on dilute suspensions of hard particles [1-3], mercury drops [4], and spherical polyelectrolytes (charged porous spheres) [5], and on concentrated suspensions of hard spherical particles [6] and mercury drops [7] on the basis of Kuwabara s cell model [8], which was originally applied to electrophoresis problem [9,10]. In this chapter, we develop a theory of conductivity of a concentrated suspension of soft particles [11]. The results cover those for the dilute case in the limit of very low particle volume fractions. We confine ourselves to the case where the overlapping of the electrical double layers of adjacent particles is negligible. 

This theory takes into account the micro-rotational effects due to rotation of molecules. This becomes important with polymers or polymeric suspensions. The physical model assigns a substructure to each continuum particle. Each material volume element contains microvolume elements which can translate, rotate, and deform independently of the motion of the microvolime. In the simplest case, these fluids are characterised by 22 viscosity coefficients and the problem is formulated in terms of a system of 19 equations with 19 unknowns. The equations for a 2-D case were solved numerically and compared to experimental results. It is concluded that the model based on the micropolar fluid theory gives a better fit than the Navier - Stokes equations. However, it seems that the difference is small. 

Joseph DD, Lundgren TS, Jackson R, SaviUe DA (1990) Ensemble Averaged and Mixture Theory Equations for Incompressible Fluid-Particle Suspensions. Int J Multiphase Flow 16 (l) 35-42... 

The discussion which follows is divided into four main parts. In Part I we establish the starting equations for the kinetic theory of dumbbell suspensions and obtain some results of a general nature. In Part II we summarize the results for a wide variety of steady-state and unsteady-state shearing flows of rigid dumbbell suspensions. In Part III elonga-tional flows are discussed. Then in Part IV some other flows are considered. Finally in Part V we discuss the properties of mixtures as well as several additional topics. 

Considerable discrepancies exist between the theory and practice of centrifugation. Complex variables not accounted for in Equations 18.10 and 18.11, such as concentration of the suspension, nature of the medium, and characteristics of the centrifuge, will affect the sedimentation properties of a mixed population of particles. Moreover, the frictional coefficient, f in the case of an asymmetrical molecule (e.g. a protein such as myocin) can be several times the frictional coefficient (i.e. f0) of a sphere. This results in particles sedimenting at a slower rate. Equation 18.10 can, therefore, be modified to give Equation 18.12 ... 

Kinetic Theory and Rheology of Dumbbell Suspensions d) The Diffusion Equation for the Distributim Function... 

The main problem in extending the microstructural theories to high Peclet number and volume fraction is related to the formulation of the many-body interactions. Recently, based on the Smoluchowski equation, Nazockdast and Morris (2012) developed a theory for concentrated hard-sphere suspensions under shear. The theory resulted in an integro-differential equation for the pair distribution function. It was used to capture the main features of the hard sphere structure and to predict the rheology of the suspension, over a wide range of volume fraction (<0.55) for 0 < Pe < 100 (Nazockdast and Morris 2012). 

Consider next the differences between conservation Equations 4.1-4.3 and their analogous equations in the kinetic theory of gases. In the mass conservation equation for the suspension dispersed phase, the averaged particle volume flux appears as follows ... 

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