Nonuniformities continuously varying

Suppose we have to deal with a nonuniform region in which intensive properties vary continuously in space along one or more directions—for example, a tall column of gas in a gravitational field whose density decreases with increasing altitude. There are two ways we may treat such a nonuniform, continuous region either as a single nonuniform phase, or else as an infinite number of uniform phases, each of infinitesimal size in one or more dimensions. 

We first consider fibers with nonuniformities which vary continuously with distance z along the fiber. A simple example of a step-profile fiber with varying... 

Across real surfaces and interfaces, the dielectric response varies smoothly with location. For a planar interface normal to a direction z, we can speak of a continuously changing s(z). More pertinent to the interaction of bodies in solutions, solutes will distribute nonuniformly in the vicinity of a material interface. If that interface is charged and the medium is a salt solution, then positive and negative ions will be pushed and pulled into the different distributions of an electrostatic double layer. We know that solutes visibly change the index of refraction that determines the optical-frequency contribution to the charge-fluctuation force. The nonuniform distribution of solutes thereby creates a non-uniform e(z) near the interfaces of a solution with suspended colloids or macromolecules. Conversely, the distribution of solutes can be expected to be perturbed by the very charge-fluctuation forces that they perturb through an e(z).5... 

This method is also referred to as the miscible-displacement or continuous-flow method. In this method a thin disk of dispersed solid phase is deposited on a porous membrane and placed in a holder. A pump is used to maintain a constant flow velocity of solution through the thin disk and a fraction collector is used to collect effluent aliquots. A diagram of the basic experimental setup is shown in Fig. 2-6. A thin disk is used in an attempt to minimize diffusion resistances in the solid phase. Disk thickness, disk hydraulic conductivity, and membrane permeability determine the range of flow velocities that are achievable. Dispersion of the solid phase is necessary so that the transit time for a solute molecule is the same at all points in the disk. However, the presence of varying particle sizes and hence pore sizes may produce nonuniform solute transit times (Skopp and McCallister, 1986). This is more likely to occur with whole soils than with clay-sized particles of soil constituents. Typically, 1- or 2-g samples are used in kinetic studies on soils with the thin disk method, but disk thicknesses have not been measured. In their study of the kinetics of phosphate and silicate retention by goethite, Miller et al. (1989) estimated the thickness of the goethite disk to be 80 /xm. 

We will consider a surface to be the boundary layer between a solid and a vacuum, a gas, or a liquid. Generally, we think of a surface as a part of the solid that differs in composition from the average composition of the bulk of the solid. By this definition, the surface comprises not only the top layer of atoms or molecules of a solid but also a transition layer with a nonuniform composition that varies continuously from that of the outer layer to that of the bulk. Thus, a surface may be several or even several tens of atomic layers deep. Ordinarily, however, the difference in composition of the surface layer does not significantly affect the measured overall average composition of the bulk because the surface layer is generally only a tiny fraction of the total solid. From a practical standpoint, it appears best to adopt as an operational definition of a surface that volume of the solid that is sampled by a specific measurement technique. This definition recognizes that if we use several surface techniques, we may in fact be sampling different surfaces and may obtain different, albeit useful, results. 

The discussion up to this point has focused on the relationship between the curvature of an elastic substrate and the stress in a single layer or multilayer film in which the mismatch is invariant under any translation parallel to the interface. The films considered have also been continuous and of uniform thickness over the entire film-substrate interface. Within the range of small deflections, such an equi-biaxial film stress induces a spherical curvature in the substrate midplane, except very near the edge of the substrate. What is the deformation induced in the substrate if such a film does not have uniform thickness or if the mismatch stress varies with position along the interface This question is addressed in this section for the cases when the nonuniformity in mismatch stress or thickness varies periodically along the... 

CFE separator (Figure 7.3.4) is strictly nonuniform. The electrical field varies inversely with the radius (see Problem 3.1.5) the electrophoretic radial migration velocity therefore decreases as the radial coordinate of the charged solute molecules increases. Due to the very small thickness of the annular region in the rotationally stabilized CFE separator, the electrical field is, however, almost uniform. Rolchigo and Graves (1988) have modeled continuous electrophoresis processes in general when subjected to a nonuniform electrical field. 

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