Roscoe equation

Hence, if the ash fusion test shows a short slag, the influence of the solids can be respected by the Einstein-Roscoe equation [103] ... 

Sato et al. " measured the viscosities of some binary and ternary alkali carbonates. Since melt creep must be prevented, a highly sintered alumina crucible was used instead of a gold-plated nickel crucible. Homogeneity of a mixture sample was achieved by gas bubbling. A laser beam is combined with a computer-assisted time counter to obtain the logarithmic decrement. Roscoe s equationi3i has been used for calculation of the viscosity, while it has been claimed by Abe et al. that the viscosities calculated from Roscoe s equation are 0.6-1.5% lower than those from more rigorous equations. 

FIG. 4.11 The effect of particle crowding on viscosity. The solid line is drawn according to Equation (48) the points are experimental results. (Data from R. Roscoe, Br. J. Appl. Phys., 3, 267(1952).)... 

Munera and Guzman [56] obtained new explicit noncyclic solutions for the three-dimensional time-dependent wave equation in spherical coordinates. Their solutions constitute a new solution for the classical Maxwell equations. It is shown that the class of Lorenz-invariant inductive phenomena may have longitudinal fields as solution. But here, these solutions correspond to massless particles. Hence, in this framework a photon with zero rest mass may be compatible with a longitudinal field in contrast to that Lehnert, Evans, and Roscoe frameworks. But the extra degrees of freedom associated with this kind of longitudinal solution without nonzero photon mass is not clear, at least at the present state of development of the theory. More efforts are needed to clarify this situation. 

D. F. Roscoe, Maxwell s Equations as a Consequence of the Orthogonality between Irreducible Two Index Representation of the Poincare, preprint, 1997. 

Another equation for concentrated suspensions is that of Brinkman [391] (sometimes also attributed to Roscoe [392]) ... 

A modification of this equation for dispersions with a monodisperse particle distribution was proposed by Roscoe [46] ... 

Figure 1 shows a comparison, published by Mori and Ototake [13], of the experimental dependences of viscosity on concentration of dispersions of solid particles based on the data of Vand [34], Robinson [12], Orr and Blocker [5], Dalla Valle and Orr [17] with the theoretical equations based on the hydrodynamic approach used by Einstein (1), Simha (30), Vand (31), Roscoe (44) and the phenomenological equation of Mori and Ototake (14). A more complicated form of the theoretical dependence, naturally makes it possible to describe experimental results over a wider range, but for concentrated dispersions most of theoretical equations remain inapplicable. 

Several closed-form equations have also been proposed. Using Einstein s equation (eq 10), Roscoe (42) and Brinkman (43) independently developed the following equation for the higher concentration suspension ... 

Equation 7.2 has been also solved for dilute suspension of anisometric particles [Hinch and Leal, 1972], elastic spheres [Goddard and Miller, 1967 Roscoe, 1967], and emulsions [Oldroyd, 1953, 1955 Barthes-Bisel and Chhim, 1981]. 

Roscoe, D. F., Numerical Solution of the Navier-Stokes Equations for a Three-Dimensional Laminar Flow in Curved Pipes using Finite Difference Methods, J. Engg. Math., 12, 303-323 (1978). 

The physical meaning of the quantities a(A) and K X) can be stated thus the reciprocal of the Napierian coefficient of absorption a(A) corresponds to the thickness of the medium for which the incident intensity is divided by the base of the Napierian logarithm (i.e., e = 2.718281828...), while the reciprocal of the Bunsen-Roscoe coefficient of absorption K X) corresponds to the thickness of the medium for which the incident intensity is divided by ten. In most practical applications the decadic absorption coefficient is used, but the Napierian absorption coefficient is introduced more naturally in theoretical equations. 

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