Resistivity tensor properties

Although we will not carry our analysis further, there has been, in fact, significant progress in delineating the properties of the resistance tensors beyond the general formulae (7 22). The most important and general result is the symmetry conditions... 

In addition to these general symmetry properties, considerable effort has been made to understand the relationships between symmetries in the geometry of the problem and the forms of the resistance tensors. It is beyond our present scope to discuss these relationships in a comprehensive manner the interested reader can refer to Brenner (1972) or the textbook Low Reynolds Number Hydrodynamics by Happel and Brenner (1973) for a detailed discussion of these questions.8 Here we restrict ourselves to the results for several particularly simple cases. First, if we consider the motion of a body with spherical symmetry in an unbounded fluid, with the origin of coordinates at the geometric center of the body, it can be shown that... 

To proceed formulating the momentum equation we need a relation defining the total stress tensor in terms of the known dependent variables, a constitutive relationship. In contrast to solids, a fluid tends to deform when subjected to a shear stress. Proper constitutive laws have therefore traditionally been obtained by establishing the stress-strain relationships (e.g., [11] [12] [13] [89] [184] [104]), relating the total stress tensor T to the rate of deformation (sometimes called rate of strain, i.e., giving the name of this relation) of a fluid element. However, the resistance to deformation is a property of the fluid. For some fluids, Newtonian fluids, the viscosity is independent both of time and the rate of deformation. For non-Newtonian fluids, on the other hand the viscosity may be a function of the prehistory of the flow (i.e., a function both of time and the rate of deformation). 

Zimin, each frictional element is assumed to be a point and the hydro-dynamic interactions between these elements and the solvent are described by the Oseen tensor (23,35). This method is derived from solution of the Navier-Stokes equation assuming the existence of point resistances (34). Although frictional elements of finite size were used in the calculation of translational friction coefficients by Edwards and Oliver (35,36), they have not been applied to the intrinsic viscosity or to dynamic mechanical properties to date. 

The coefficient p, is the dynamic viscosity, a fluid property that characterizes the resistance of the fluid to shearing forces and (Vv)y = dvi/dxj is the velocity gradient tensor. Using this form of the stress tensor one obtains the Navier-Stokes equation... 

A nonspherical particle is generally anisotropic with respect to its hydro-dynamic resistance that is, its resistance depends upon its orientation relative to its direction of motion through the fluid. A complete investigation of particle resistance would therefore seem to require experimental data or theoretical analysis for each of the infinitely many relative orientations possible. It turns out, however, at least at small Reynolds numbers, that particle resistance has a tensorial character and, hence, that the resistance of a solid particle of any shape can be represented for all orientations by a few tensors. And the components of these tensors can be determined from either theoretical or experimental knowledge of the resistance of the particle for a finite number of relative orientations. The tensors themselves are intrinsic geometric properties of the particle alone, depending only on its size and shape. These observations and various generalizations thereof furnish most, but not all, of the subject matter of this section. 

This constitutive property is termed rotational inertia (or moment of inertia) because, historically, it has been thought to oppose the start of the rotation of an object (which exhibits some resistance or inertia). The hat (circumflex) over the rotational inertia symbol means that it is not a scalar but an operator. Effectively, in the most general case, the inductive relation is not linear and the rotational inertia is a tensor. If the relativistic model for translational mechanics is relatively amenable, this is not the case in rotation during a translation because of the variation of radius with the velocity at high speed. 

The mechanical properties of materials involve various concepts such as hardness, stiffness, and piezoelectric constants, Young s and bulk modulus, and yield strength. The solids are deformed under the effect of external forces and the deformation is described by the physical quantity strain. The internal mechanical force system that resists the deformation and tends to return the solid to its undeformed initial state is described by the physical quantity stress. Within the elastic limit, where a complete recoverability from strain is achieved with removal of stress, stress g is proportional to strain e. The generalized Hooke s law gives each of the stress tensor components as linear functions of the strain tensor components as... 

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