Suspension of rigid spheres

Apart from chemical composition, an important variable in the description of emulsions is the volume fraction, outer phase. For spherical droplets, of radius a, the volume fraction is given by the number density, n, times the spherical volume, 0 = Ava nl2>. It is easy to show that the maximum packing fraction of spheres is 0 = 0.74 (see Problem XIV-2). Many physical properties of emulsions can be characterized by their volume fraction. The viscosity of a dilute suspension of rigid spheres is an example where the Einstein limiting law is [2]... 

In general, these formulas hold well for < 0.10, while empirical models using higher-order polynomials in can be used to fit almost any viscosity data. An equation that is often used for emulsions and concentrated suspensions of rigid spheres is that by Krieger and Dougherty ... 

Krieger, I.M., and Dougherty, T.J. (1959). A mechanism for non-Newtonian flow in suspensions of rigid spheres. Trans. Soc. Rheology 3, 137-152. 

Einstein,2 for a suspension of rigid spheres of Tadius r which is small compared with the distance between them and the radius of a capillary, found for the relative viscosity (> o=viscosity of suspending liquid) ... 

It can, in fact, be proven that a dilute suspension of rigid spheres will always be Newtonian at the first 0(C) correction to the bulk stress, with an effective viscosity given by (7 196) of the form... 

In this extension of the Tsai-Halpin equation (Brassell and Wischmann, 1974), K is a generalized Einstein (1905, 1906, 1911) coefficient equal to 2.5 for a suspension of rigid spheres in a matrix having a Poisson s ratio equal to 0.5, and given approximately for other values of v in the reference cited. Note that K = A + I [see equation (12.5)]. The constant B is defined as in equation (A-6) and ij/ is given by functions such as... 

Theoretical study for filled systems originates with Einstein s well-known treatment of the viscosity of a dilute suspension of rigid spheres [50,51] as... 

Housiadas and Tanner (2009), following the approach of Greco et al. (2005), have used a perturbation analysis to obtain the analytical solution for the pressure and the velocity field up to 0 (pDe) of a dilute suspension of rigid spheres in a weakly viscoelastic fluid, where

volume fraction of the spheres and De is the Deborah number of the viscoelastic fluid. The analytical solution was used to calculate the bulk first and second normal stress in simple shear flows and the elongational viscosity. The main results are... 

Housiadas KD, Tanner RI (2009) On the rheology of a dilute suspension of rigid spheres in a weakly viscoelastic matrix fluid. J Non-Newtonian Huid Mech 2009 162 88-92 Hughes TJR, Liu WK, Zimmermann TK (1981) Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput Methods Appl Mech Eng 29 329-349 Huilgol RR (2006) On the derivation of the symmetric and asymmetric Hele-Shaw flow equations for viscous and viscoplastic fluids using the viscometric fluidity function. J Non-Newtonian Fluid Mech 138 209-213... 

Roscoe R (1952) The viscosity of suspensions of rigid spheres. Br J Appl Phys 3 267-269 Rosen BW, Hashin Z (1970) Effective thermal expansion coefBcients and specific heats of composite materials. Int J Eng Sci 8 157-173... 

Various workers have proposed theoretical models to predict the flow behavior of polymer blends. Einstein studied the shear flow behavior of a suspension of rigid spheres in Newtonian fluids. Taylor (21, 22) extended this concept to include dispersions of one liquid in another liquid based on their shear viscosities and also accounted for circulation in the droplets. According to his model, for a conq>onent 2 which is dispersed in a conq>onent 1, the blend viscosity is given by the following equation ... 

Rutgers, I.R. (1962) Relative viscosity of suspensions of rigid spheres in Newtonian liquids, Rheol. Acta, 2, 202-10 (1962) Relative viscosity and concentration, Rheol Acta, 2,305-48. 

Some experimental viscosity results on dilute suspensions of rigid spheres x, glass 5 fim in zinc iodide glycerin (Manley and Mason, 1954) O. polystyrene aqueous latices, 0.42, 0.87 /xm (Saunders, 1961) v. low shear rate, and A, high shear rate limits for nonaqueous polystyrene latices, 0.16-0.43 urn (Krieger, 1972). 

In comparison to suspensions of rigid spheres, the overwhelming additional effect with axisymmetric particles is orientation. Obviously the orientation of a nonspherical particle with respect to the flow will greatly affect the velocity field around it and thus the particle stress, Tp in eq. 10.2.8. For example, if the particle is a rod with its long axis aligned in the flow direction, the alteration of the... 

We have shown in the preceding section that the rheological properties of particulate-filled molten thermoplastics and elastomers depend on many factors (1) particle size (t/p), (2) particle shape (a), (3) volume fraction of filler (f)), and (4) applied shear rate (y) or shear stress a). The situation becomes more complicated when interactions exist between the particulates and polymer matrix. There is a long history for the development of a theory to predict the rheological properties of dilute suspensions, concentrated suspensions, and particulate-filled viscoelastic polymeric fluids. As early as 1906, before viscoelastic polymeric fluids were known to the scientific community, Einstein (1906,1911) developed a theory predicting the viscosity of a dilute suspension of rigid spheres and obtained the following expression for the bulk (effective) viscosity of a suspension ... 

Viscosity can depend strongly on fillers added to provide a range of mechanical, transport, electrical, magnetic, or other physical properties. The well-known Einstein relationship, for example, provides the viscosity of a dilute suspension of rigid spheres, obtaining the bulk or effective viscosity of a suspension as (Han 2007) ... 

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