Suspensions of sphere

Next we consider replacing the sandwiched fluid with the same liquid in which solid spheres are suspended at a volume fraction unit volume of liquid-a suspension of spheres in this case-the total volume of the spheres is also 0. We begin by considering the velocity gradient if the velocity of the top surface is to have the same value as in the case of the... 

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. 

Lord Rayleigh [310] modeled transport in a homogeneous suspension of spheres placed in a square lattice. In terms of diffusion coefficients, his solution was... 

Freed, KF Muthukumar, M, On the Stokes Problem for a Suspension of Spheres at Finite Concentrations, Journal of Chemical Physics 68, 2088, 1978. 

Jeffrey, DJ, Conduction through a Random Suspension of Spheres, Proceedings of the Royal Society of London Series A 335, 355, 1973. 

It also follows from (8.17) and (8.23) that the circular dichroism of a suspension of spheres is proportional to the difference between the extinction cross sections for left-circularly and right-circularly polarized light ... 

Generally, these behave as Newtonian fluids and, for the case of an extremely dilute suspension of spherical non-interacting particles having a density equal to that of the continuous medium, we can apply the Einstein formula for a suspension of spheres ... 

Rather than take a limit of large separations between relatively small spheres a of incremental polarizability a (it ), we can think of interactions within dilute suspensions or solutions. At relatively large separations, the shape and the microscopic details of an effectively small speck become unimportant. The only feature that is of interest is that the dilute specks ever so slightly change the dielectric and ionic response of the suspension compared with that of the pure medium. When the suspension of spheres is vanishingly dilute, esusp is simply proportional to their number density N multiplied by a(/ ) / susp — m (/ ) + Naa(i ) [see Fig. L1.42(a)]. 

Small spheres A dilute suspension of spheres of material with a radius a and volume fraction vsph = NAVA = N(4jr/3)a3 has a composite dielectric response4 ... 

Here, the sphere center is instantaneously situated at point 0 the sphere center translates with velocity U, while it rotates with angular velocity (a r is measured relative to 0 its magnitude r is denoted by r. Moreover, f = r/r is a unit radial vector. The latter solution is derivable in a variety of ways e.g., from Lamb s (1932) general solution (Brenner, 1970). [Equation (2.12) represents a superposition (Brenner, 1958) of three physically distinct solutions, corresponding, respectively, to (i) translation of a sphere through a fluid at rest at infinity (ii) rotation of a sphere in a fluid at rest at infinity (iii) motion of a neutrally buoyant sphere suspended in a linear shear flow. The latter was first obtained by Einstein (1906, 1911 cf. Einstein, 1956) in connection with his classic calculation of the viscosity of a dilute suspension of spheres, which formed part of his 1905 Ph.D. thesis.]... 

Patzold (1980) compared the viscosities of suspensions of spheres in simple shear and extensional flows and obtained significant differences, which were qualitatively explained by invoking various flow-dependent sphere arrangements. Goto and Kuno (1982) measured the apparent relative viscosities of carefully controlled bidisperse particle mixtures. The larger particles, however, possessed a diameter nearly one-fourth that of the tube through which they flowed, suggesting the inadvertant intrusion of unwanted wall effects. 

Lundgren (1972) treated the case of a suspension of spheres. He assumed the RHS of Eq. (5.2a) to be a linear functional of v(r), choosing this functional dependence to be of the form... 

The term in the square brackets in this integral is the terminal settling velocity for a dilute suspension of spheres. The size, di, is the size which would just fall the full distance h in time t and is given by... 

In this chapter, we first discuss the case of completely salt-free suspensions of spheres and cylinders. Then, we consider the Poisson-Boltzmann equation for the potential distribution around a spherical colloidal particle in a medium containing its counterions and a small amount of added salts [8]. We also deals with a soft particle in a salt-free medium [9]. 

Most food particles are not spherical in shape so that the empirical equation (Equation 2.25) that described well (Kitano et al., 1981 Metzner, 1985) the relative viscosity versus concentration behavior of suspensions of spheres and fibers... 

This rule of thumb goes back to Maxwell (1867), who said that a viscous fluid with viscosity rjo can be thought of as a relaxing solid with modulus G that relaxes in a time period r hence, r]a Gx. Another handy rule is that the characteristic modulus of a liquid is roughly equal XovksT, where v is the number of structural units per unit volume. For a suspension of spheres, v is the number of spheres per unit volume, while for a small-molecule liquid, V is the number of molecules per unit volume thus, v = pNa/M, where p is the fluid density, M is the molecule s molecular weight, and is Avogadro s number. Hence, for a small-molecule liquid with density p = Ig/cm, Af = 100 g/mol, and T = 300 K, we estimate G 2.4 x 10 Pa = 24 MPa. 

Q1 10° - O I / 7 Figure 4.22 The two relaxation time scales and versus volume fraction (j) of spheres in a dense suspension of spheres of radius 200 nm, extracted from the light-scattering data. 

Stable particle suspensions exhibit an extraordinarily broad range of rheological behavior. which depends on particle concentration, size, and shape, as well as on the presence and type of stabilizing surface layers or surface charges, and possible viscoelastic properties of the suspending fluid. Some of the properties of suspensions of spheres are now reasonably well understood, such as (a) the concentration-dependence of the zero-shear viscosity of hard-sphere suspensions and (b) the effects of deformability of the steric-stabilization layers on the particles. In addition, qualitative understanding and quantitative empirical equations... 

There are, however, important differences between the phase behavior of sticky spheres and that of small molecules, which arise from differences in the relative ranges of the attractive potentials. These differences have been explored in a wonderful set of calculations and experiments by Cast et al. (1983) and Pusey and coworkers (Uett et al. 1995) for suspensions of spheres that are made to attract each other by the polymer-depletion mechanism. In such systems, the range of the attractive potential relative to the sphere size can be varied by controlling the ratio = Aj fa of the polymer depletion-layer thickness to the sphere radius. For 0 the potential is short-ranged, like that of sticky hard spheres,... 

It is frequently desirable to be able to describe emulsion viscosity in terms of the viscosity of the continuous phase tjq) and the amount of emulsified material. A very large number of equations have been advanced for estimating suspension (or emulsion, etc.) viscosities. Most of these are empirical extensions of Einstein s equation for a dilute suspension of spheres ... 

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