Dilute suspensions of spheres

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. 

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.]... 

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... 

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 ... 

This is the famous result obtained by Einstein for the viscosity of a dilute suspension of spheres.20 Although the integral (7-195) leading to this result was evaluated by use of the... 

Problem 7-22. The Viscosity of a Multicomponent Membrane. An interesting generalization of the Einstein calculation of the effective viscosity of a dilute suspension of spheres is to consider the same problem in two dimensions. This is relevant to the effective viscosities of some types of multicomponent membranes. Obtain the Einstein viscosity correction at small Reynolds number for a dilute suspension of cylinders of radii a whose axes are all aligned. Although there is no solution to Stokes equations for a translating cylinder, there is a solution for a force- and torque-free cylinder in a 2D straining flow. The result is... 

Problem 9-26. The Effective Thermal Conductivity of a Dilute Suspension of Spheres. 

Using the analogy between electrical conductivity and permittivity (Smythe 1968), Maxwell s result Equation 3.7 can be transformed into an effective dielectric constant Kb for a dilute suspension of spheres having dielectric constant Kd (permittivity j) suspended in a continuous medium of dielectric constant Kc (permittivity ej... 

BATCHELOR, G.K. 1972. Sedimentation in a dilute suspension of spheres. J. Fluid Mech. 52, 245-268. 

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... 

Using a Monte-Carlo simulation to a system where diffusion occurs on a length scale shorter than the heterogeneity of the medium, Akanni et al. (1987) have shown that the relation obtained by Maxwell for a dilute suspension of sphere can be applicable to a wider range of porosity ... 

When the suspending m ium could be described by the power-law model, Kremesec and Slattery [145] derived the following expression for the viscosity of a dilute suspension of spheres... 

The same basic approach used to calculate the constitutive equation for dilute suspensions of spheres can be applied to spheroids. The difficulty lies in calculating the particle stress tp in eq. 10.2.8. Not only is the velocity field more complex, but Xp depends on the orientation. Thus, to get the bulk value of the stress contribution of the particles, we need to integrate over all orientations, weighting by the distribution function... 

We can construct a useful and reasonably accurate theory of intrinsic viscosity of dilute polymer solutions, building directly on the Einstein result for viscosities of dilute suspensions of spheres (Chapter 10). Recalling this result in a slightly different form ... 

In the first decade of the 20 century, Einstein [27] gave his attention to the prediction of the viscosity of a dilute suspension of spheres. Einstein began with the creeping flow Navier-Stokes equation of Newtonian fluid hydrodynamics ... 

In 1922 G. B. Jeffery [38] sought to generalize Einstein s analysis for a dilute suspension of spheres to ellipsoids. Like Einstein [27], he considered the sphere to be in shear flow... 

The first approach is based on Einstein s century-old equation for the viscosity of a very dilute suspension of spheres in a Newtonian fluid, which is given by Eq. 2.80. 

Abstract. Electrical measurements of heterogeneous media composed of solid polyelectrolytes and dilute aqueous solutions (or pure water) are interpreted in terms of a simple electrical network. It is demonstrated that this model network represents an extension of Maxwell s equation for the conductance of dilute suspensions of spheres to condensed systems. Discussing past work on cation exchange resinsolution systems, it is shown that the three empirical geometrical parameters of the model explain quantitatively the change of the low-frequency 1000 Hz) conductivity of the heterogeneous mixture... 

Maxwell [ 1, vol. I, p. 440, Equation (17)] derived the following formula for the electrical conductivity, (Q cm ) of a dilute suspension of spheres of conductivity, in a solution (or other homogeneous conductor) of conductivity, in terms of the conductivity of the two phases and of the volume fraction, /, occupied by the spheres ... 

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