Tensor resistivity

Schon, J.H., Mollison, R.A., Georgi, D.T., 1999. Macroscopic electrical anisotropy of laminated reservoirs a tensor resistivity saturation model. In Presented at the SPE Annual Technical Conference and Exhibition. Paper SPE 56509. 

Quadrupole interaction energy tensor X Residual resistivity (solid state) Pr... 

Hydrodynamic and frictional effects may be described by a Cartesian mobility tensor which is generally a function of all of the system coordinates. In models of systems of beads (i.e., localized centers of hydrodynamic resistance) with hydrodynamic interactions, is normally taken to be of the form... 

In this section, we use the Cartesian force of Section VI to derive several equivalent expressions for the stress tensor of a constrained system of pointlike particles in a flow field with a macroscopic velocity gradient Vv. The excess stress of any system of interacting beads (i.e., point centers of hydrodynamic resistance) in a Newtonian solvent, beyond the Newtonian contribution that would be present at the applied deformation rate in the absence of the beads, is given by the Kramers-Kirkwood expression [1,4,18]... 

The use of these resistance tensors is developed in detail by Happel and Brenner (H3). While enabling compact formulation of fundamental problems, these tensors have limited application since their components are rarely available even for simple shapes. Here we discuss specific classes of particle shape without recourse to tensor notation, but some conclusions from the general treatment are of interest. Because the translation tensor is symmetric, it follows that every particle possesses at least three mutually perpendicular axes such that, if the particle is translating without rotation parallel to one of these axes, the total... 

Harper and Chang (H4) generalized the analysis for any three-dimensional body and defined a lift tensor related to the translational resistances in Stokes flow. Lin et al (L3) extended Saffman s treatment to give the velocity and pressure fields around a neutrally buoyant sphere, and also calculated the first correction term for the angular velocity, obtaining... 

Figure 3 Resistivity of a single crystal Bi-Sr-Ca-Cu-O. (a) Averaged a,b plane resistivity and the ratio of c to a,b plane resistivity vs temperature, (b) Three components of the resistivity tensor. Ref. 4.

It would be interesting to investigate (in both the plane and spherical cases as well) the very difficult nonlinear problem of a fluid whose resistivity depends on the magnetic field. Here the resistivity should be considered as a tensor relating the electric field vector to the current vector. 

This means that the resistivity and thermal conductivity tensors are no longer symmetric. For example,... 

We shall assume that the dumbbell is situated in the stream of viscous fluid characterised by the mean velocity gradient tensor Vij. According to (2.8), the resistance force for every particle of the dumbbell can be written as... 

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). 

The crystal symmetry of most organic metals is low, often only monoclinic, and the principal axes of the conductivity tensor (cr) are not precisely defined. However, in practice the conductivity along the chain direction (cr,) is particularly high, usually 300 to 2500 (Q-cm) 1 at room temperature, while that in one direction perpendicular to the chains (07) is very low [2]. Therefore, these two directions must be very close to the principal axes of ex. The third perpendicular direction has intermediate conductivity (07). So in the situation, where > 07, which is quite common, the principal axes of the conductivity and resistivity (p) tensors are known reasonably well. When measuring these quantities on a single crystal, care must be taken either to ensure that the current distribution is uniform, or alternatively, special methods such as those of Montgomery [11] or van der Pauw [12] must be used. Some insight into these problems can be obtained by consideration of the equivalent isotropic sample [11,13]. 

The definitions of effective diffusivity tensors are key parameters in the solution of the transport equations above. For an isotropic medium, the effective diffusivity is insensitive to the detailed geometric structure, and the volume fraction of the phases A and B influences the effective diffusivity. When the resistance to mass transfer across the cell membrane is negligible, the isotropic effective diffusivity, Ds e = Dg eI may be obtained from Maxwell s equation... 

In general, for single crystal samples the resistivity is anisotropic and will be represented by a second rank tensor of the form... 

This tensor is symmetric and requires a maximum of six independent parameters for low symmetry samples. The number of parameters is reduced for more symmetric samples. In general, polymeric samples will either be isotropic, with one resistivity parameter, or have axial symmetry generated by mechanical processes, see Section 1.3.6, with two resistivity parameters. Montgomery (1971) introduced a more general 4-point probe method suitable for the measurement of anisotropic solids. This method utilises a sample in the form of a rectangular prism with electrodes attached at the corners, illustrated in Fig. 5.19(c). Measurement of the voltage/current ratios for opposite pairs of electrodes,... 

Now we consider the resistive force characterizing the movement of the particle along the streamline expressed as the product between tensor and its normal surface A (A = m/p.s j where Sj is the apparent height of the deformed particle)... 

---