Grand resistance matrix

As a consequence of Lorentz reciprocal theorem (see Happel and Brenner, 1965) the grand resistance matrix R(xJV, e ) possesses many internal symmetries, greatly reducing the number of its independent elements. Another important feature of R is that it depends only on the instantaneous configuration ( N, eN) of the particulate phase. 

In a series of papers, Felderhof has devised various methods to solve anew one- and two-sphere Stokes flow problems. First, the classical method of reflections (Happel and Brenner, 1965) was modified and employed to examine two-sphere interactions with mixed slip-stick boundary conditions (Felderhof, 1977 Renland et al, 1978). A novel feature of the latter approach is the use of superposition of forces rather than of velocities as such, the mobility matrix (rather than its inverse, the grand resistance matrix) was derived. Calculations based thereon proved easier, and convergence was more rapid explicit results through terms of 0(/T7) were derived, where p is the nondimensional center-to-center distance between spheres. In a related work, Schmitz and Felderhof (1978) solved Stokes equations around a sphere by the so-called Cartesian ansatz method, avoiding the use of spherical coordinates. They also devised a second method (Schmitz and Felderhof, 1982a), in which... 

The partitioned grand resistance matrix in Eq. (7.13) is a function only of the instantaneous geometrical configuration of the particulate phase. This consists of the fixed particle shapes together with the variable relative particle positions and orientations. As such, geometrical symmetry arguments (where such symmetry exists) may be used to reduce the number of independent, nonzero scalar components of the coefficient tensors in Eq. (7.13) for particular choices of coordinate axes (e.g., principal axis systems). 

As a consequence of the linearity of the governing equations, the grand resistance matrix formulation is again applicable, yielding... 

Computation of the grand resistance matrix R for each possible particle configuration provides a major numerical challenge (see Section VIII). The various methods cited in Section II for dealing with the many-body problem are potentially useful in this context. Detailed calculations must be performed for a number of accessible configurations, and the configurational evolution determined by interpolation in order to effect the requisite time averaging. 

Incorporated into the jr matrix is the intrinsic resistance of the entire particle system. It is called the grand resistance matrix. In view of the symmetry relations... 

The first commercial blend of this type is Cylon . Here PVC is the matrix, and PA (that melts below 215°C ) the dispersed phase. The two resins were compatibilized using the well known PVC plasticizer — Elvaloy (a terpolymer of ethylene, carbon monoxide and acrylics). These soft to semi-rigid alloys were commercialized for wire coating, automotive applications and blow molding [Grande, 1997 Hofmann, 1998]. They are flame, abrasion and chemicals resistant, easy to process, and tough. 

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