Deformable spheres

Note that for large r dr the drop becomes rigid and the coefficient of f goes to S / 2, the Einstein result. In the other limit, a gas bubble, the coefficient is 1. 

For steady shear, eq. 10.2.24 gives a positive first normal stress coefficient 

For steady uniaxial extension Frankel and Acrivos (1970) report 

R uced viscosity versus volume fraction for several emulsions with vaiying vis cosity ratios. Ada from Nawab and Mason (1958). 

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 

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For nonadhering bodies in contact in the presence of capillary condensation, the previous result for rigid solids is found to apply more generally to systems of small, hard, but deformable spheres in contact in vapor near saturation ... 

Spheroidal particles can be treated analytically, and allow study of shapes ranging from slightly deformed spheres to disks and needles. Moreover, a spheroid often provides a useful approximation for the drag on a less regular... 

Ol). Results for thin disks are obtained in the limit as 0. Approximate relationships, obtained by treating the spheroid as a slightly deformed sphere (H3, SI), are also given. The drag ratio may conveniently be expressed as the ratio of the resistance of the spheroid to that of the sphere with the same equatorial radius a ... 

Equation (4-40) is equivalent to the approximate results for spheroids given in Table 4.1 Figs. 4.3 and 4.4 demonstrate that the approximation is useful even for grossly deformed spheres. 

Equation (6.25) makes it possible in this case to interpret a dilute solution of macromolecules as a suspension of solid non-deformable spheres with a radius close to the mean square radius of inertia. 

Normally, hyperbolic invariant manifolds persist under perturbation [22]. If we are in the setting where the form of Eq. (1) must hrst be obtained by applying Normal Form theory, then we are restricted to a sufficiently small neighborhood of the equilibrium point. In this case the nonlinear terms are much smaller than the linear terms. Therefore, the sphere present in the linear problem becomes a deformed sphere for the nonlinear problem and still has (2n — 2)-dimensional stable and unstable manifolds in the (2n — l)-dimensional energy surface since normal hyperbolicity is preserved under perturbations. 

The approximately constant value of the volumes of the VD polyhedra allows us to consider atoms as soft (easy-to-deform) spheres of constant volumes. Approaching of the two atoms due to their chemical interaction is accompanied by mutual deformations of their spheres (Fig. 3f), which, in the end, leads to the transformation of the spheres into VD polyhedra. The shapes of the VD polyhedra are controlled by the arrangement of atoms in the stmcture that can thus be considered as a close packing of soft spheres. 

Under unidirectional loading, the shape of the deformed sphere of isotropic material is an ellipsoid. This deformed shape must also be attained by a cell of rubber-toughened epoxy because the overall material is isotropic. This shape would not be attained from application of the load alone constraints... 

To the first order in e the center of reaction coincides with the centroid of the deformed sphere. 

As an example of the application of these relations, consider the slightly deformed sphere obtained by setting... 

For the slightly deformed sphere described by Eq. (64) Ripps and Brenner (R5a) give the following expression for the symbolic dyadic force operator ... 

For the case where the interfacial forces dominate those due to viscosity, Taylor (T2) found that the surface of the droplet is described by the equation of the slightly deformed sphere s... 

In polar form the equation of this slightly deformed sphere is... 

R5a. Ripps, D. L., and Brenner, H., The Stokes resistance of a slightly deformed sphere II. Intrinsic resistance operators for an arbitrary initial flow (to be published) see also Ripps, D. L., Invariant differential operators for slow viscous flow past a particle. Ph.D. Dissertation, New York University, New York, 1966. 

For two-phase systems Einstein (34) in 1906 was the first to obtain the viscosity for a very dilute suspension of solid spheres the resulting stress expression is Newtonian. However, it has been only within the past decade that nonlinear viscoelastic expressions for the stress tensor in dilute suspensions have been obtained deformable spheres (35), ellipsoids (36), emulsions (37, 38), For a survey of activities in this field, see the summary by Barthes-Biesel and Acrivos (39),... 

In the coordinate system of the eigenvectors of h and e, the equation of the deformed sphere becomes ... 

All calculations are based on mechanical molecular models the atoms are more or less deformable spheres, the bonds are rigid springs. Attraction and repulsion are calculated separately as a function of the torsion angle. A typical exercise, for example, proceeds from the total energy of an ethane molecule, which is made up of five components (Figure 4-4) ... 

Consequently analytical methods are mostly confined to creeping flows. Roughly, there are two types of problems that can be solved. The first of these deals with interfaces that show small deviations from simple geometric forms, as for instance the case of a slightly deformed sphere settling in an infinite fluid. The second type constitutes cases where interfacial position changes, but only very slowly. Then its variation can be neglected to the first approximation and the lubrication theory approximation or the slender body approximation applied. It should be noted that both the above methods yield approximate solutions. 

The cases of a sphere and slightly deformed sphere in a uniform flow field are considered first in Sections 4 and 5. The mathematical method used conventionally in these problems is the regular asymptotic expansion. The reader is introduced to this method. In Section 6, the dip coating problem under the lubrication theory approximation is examined. (The closely related slender body approximation is outlined in Problem 7.5.) A more sophisticated method of matched asymptotic expansions is used to solve this problem and its main features... 

Taylor and Acrivos (1964) made the solution satisfy the boundary conditions on the surface r = a, but satisfied the normal stress boundary condition on the surface of a slightly deformed sphere a(l + < )) by using a Taylor series expansion to express any functiony(r) at the interface as j a) + Since... 

In their AIM discussion on 120 alkali halide perovskites Luana et aZ. discussed the shape of the ions and showed how the topological description contained the classical picture in terms of slightly deformed spheres. They enthusiastically concluded from this and previous work that AIM supplied a rigorous foundation for important historical concepts like ionicity, index of coordination, coordination polyhedra or atomic/ionic volume in a solid. 

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