Curvilinear model

FIG. 2 Collision models (a) traditional rectilinear model, and (b) curvilinear model that considers hydrodynamic forces between particles. (Adapted from Ref. 3.)... 

All of these calculations have been made with the curvilinear model for the differential sedimentation kernel. Flow through an aggregate should increase the kernel. Using the rectilinear kernel is a way to test the effect that such enhanced flow might have (38, 46). As should be expected, it causes a substantial decrease in the maximum algal population. The effect of increased... 

An important improvement in Smoluchowski s approach was to consider hydrodynamic interactions between two particles as they approach each other. These interactions are of two types and result in curvilinear models. First are deviations from rectilinear flow paths that occur as two particles approach each other. Second is the increasing hydrodynamic drag that occurs as two particles come into close proximity. 

Illustrative Cases. Three cases are illustrated in Figure 9, marked by the circles labeled A, B, and C. Case A refers to classical experiments by Swift and Friedlander (27) on the coagulation of monodisperse latex particles (diameter = 0.871 pm) in shear flow and in the absence of repulsive chemical interactions. Considering a velocity gradient of 20 s 1, HA is 0.0535, log HA is — 1.27, and dfdj is 1.0 for these experimental conditions. The circle labeled A in Figure 9 marks these conditions and indicates that the hydrodynamic corrections to Smoluchowsla s model predict a reduction of about 40% in the aggregation rate by fluid shear. The experimental measurements by Swift and Friedlander showed a reduction of 64%. This observed reduction from Smoluchowski s rectilinear model was therefore primarily physical or hydro-dynamic and consistent with the curvilinear model. 

Curvilinear Model More than Two Input Chromatograms... 

At the lower end of Table 3.4 the characteristic values % -slope S and intercept In(koo) for the linear model are calculated directly and the curvature factor set to zero. The three values for the curvilinear model according to Eq. (3.16) are determined iteratively using the Excel Solver and copied into the table. 

The most successful theoretical framework in which the accumulating data has been understood is the tube model of de Gennes, Doi and Edwards [2]. We visit the model in more detail in Sect. 2, but the fundamental assumption is simple to state the topological constraints by which contingent chains may not cross each other, which act in reality as complex many-body interactions, are assumed to be equivalent for each chain to a tube of width a surrounding and coarse-graining its own contour (Fig. 2). So, motions perpendicular to the tube contour are confined while those curvilinear to it are permitted. The theory then resembles a dynamic version of rubber elasticity with local dissipation, and with the additional assumption of the tube constraints. 

There are a number of ways to model calibration data by regression. Host researchers have attempted to describe data with a linear function. Others ( 4,5 ) have chosen a higher order or a polynomial method. One report ( 6 ) compared the error in the interpolation using linear segments over a curved region verses using a curvilinear regression. Still others ( 7,8 ) chose empirical or spline functions. Mixed model descriptions have also been used ( 4,7 ). 

It is not unusual to encounter a problem that is not conveniently posed in one of the common coordinate systems (i.e., cartesian, cylindrical, or spherical). As an illustration consider the flow behavior for the system shown in Fig. 5.20. The analysis seeks to understand the details of the flow field and pressure drop in the narrow conical gap between the movable flow obstruction and the conical tube wall. Intuitively one can anticipate that the flow may have a relatively simple behavior, with the flow parallel to the gap. However, such simplicity can only be realized when the flow is described in a coordinate system that aligns with the gap. An orthogonal curvilinear coordinate system can be developed to model this problem. 

In the case of polynomial or curvilinear regression, as given by the model ... 

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