Poisson radial form

For a perfect elastic body, when axial pressure is applied, the radial pressure is of the same magnitude and related to the axial force via a constant. This constant is known as Poisson s ratio. As illustrated in Figure 4A, the compression and decompression phases of the cycle are straight lines which exhibit no hysteresis and a zero intercept. However, few if any pharmaceutical materials behave as pure elastic bodies. Usually, some amount of force is needed to eject the formed compact from the die. The ejection force (EF) is need to over come the friction between the tablet and die-wall, and for a given coefficient of friction the higher the residual die-wall force the higher the EF that is needed (19-21). 

According to a nonlinear Poisson-Boltzmann analysis, the initial radial decay of f from a cylindrical chain segment is much steeper than predicted by a Debye-Hiickel analysis see Figure 3 (28). The steep decay lessens at large distance, and eventually adopts an asymptotic functional form compatible with a Debye-Hiickel approximation. However, to superimpose across this distant region the predictions of the Poisson-Boltzmann analysis onto those formidated... 

The free counterions form an electrical double layer in which the counterion concentrations around each micelle decrease in a Poisson-Boltzmann distribution into the aqueous phase. Figure 6 illustrates the double layer and the radial distributions of counterions at different salt concentrations obtained by solving the Poisson-Boltzmann equation. Note that the thickness of the double layer depends on the ion salt concentration. The graph also illustrates a two-site model for ion distribution used in the pseudophase models to describe measured ion distributions in solutions of ionic association colloids, that is, counterions are either bound or free (see below). However, explanations based only on coulombic interactions between headgroups and counterions fail to account for commonly observed trends in ion-specific effects, for example, the Hofmeister... 

Before we discuss in detail the numerical discretization scheme used for the Poisson equation, which by the way is very similar to the discretization of the radial Schrodinger equation given in Eq. (9.120) [491], we sum up some of its general features. We consider the general form of the radial Poisson equation,... 

In equation (5.80) y = fracture surface energy, G = shear modulus, i> = Poisson s ratio, and L is the slip length along which shear acts. When radial cracks are formed along with median cracks, L < 1.4a where a is, as usual, the half diagonal of the indent impression. Hagan s equations for critical load and the critical flaw length such a load produces are... 

Consider a long cylindrical fuel rod of height H and radius R, surrounded by closely fitting cladding of thickness c (see Fig. 6.6). The heat production rate q " in the rod is assumed to be uniform so that the temperature variation is essentially a radial one, and the Poisson equation reduces to the one-dimensional form in cylindrical coordinates... 

One of the most important yet simplest systems is that of a single-charged surface immersed in a bulk z z electrolyte because it serves as a reference model for colloidal and biomolecular systems. In this section we will obtain a general expression for the asymptotic form of the Poisson-Boltzmann potential for a radially-symmetric particle (a cylinder or sphere) and show that it reduces to that for a plane in the limit of large particle radius. We then show how the asymptotic result can be modified slightly to correct the value of the potential at the particle surface. 

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