Radial Poisson equation

These two relations can be used to solve the radial Poisson equations,... 

Theoretical nuclear physics does not provide a unique model function for the positive charge distribution derived from quantum chromodynamics. That is why there is a certain degree of arbitrariness in the choice of such functions. In the spherically symmetric case, the radial Poisson equation... 

This second-order differential equation is known as the radial Poisson equation. The left-hand side of Eq. (9.88) is formally identical to the one used in the nonrelativistic case. 

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,... 

Physically meaningful ionic radii may be obtained from Poisson equation for anions, and from electrostatic potentials defined in the the context of DFT for cations [17,18], However, there remains the problem of being forced to use different mathematical criteria in both cases, because the electrostatic potential of anions and cations display a different functional behaviour with respect to the radial variable. 

The Poisson equation should in any case not be applied where there is a gradient of E, e.g., in radial geometry121, see footnote on p. 207. 

Within PB theory the individual counterions are replaced by a cylindrical counterion density n(r), where r denotes the radial distance from the cylinder axis. This gives rise to an electrostatic potential i//(r) satisfying the Poisson equation... 

However, the partitioning scheme just described can be used to decompose the electron density into atomic components. Projection onto Y< m functions yields multipoles attached to the atoms. The radial resolution is automatically as good as that of the integration. The Poisson equation, relating the static potential to the density, can then be solved with high accuracy for each multipole, since the decomposition has reduced... 

Handles and Baron (H5) proposed another model for the more practical range of Reynolds numbers (about 1000). They assumed that the tangential motion caused by circulation is combined with an assumed random radial motion caused by internal vibration, and determined the eddy diffusivity subsequently used in solving the appropriate Fourier-Poisson equations. They postulated radial stream lines, as shown in Fig. 11, rather than those... 

Taking into account Poisson equation relating the volume charge density p(r) to the equilibrium potential distribution, r) (recall that the geometry of the double layer is assumed to be the same as in the absence of applied field hence, P only depends on the radial coordinate r). Equation... 

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... 

It can be easily shown that this solution satisfies the Poisson equation. For a monopolar electrode, the current distribution is radial and is inversely proportional to the conductance of the medium and the distance to the source. The potential decays to zero far away from the electrode and goes to infinity on the electrode. The singularity at r = 0 can be ehminated by assuming that the electrode is spherical with finite radius a. Equation 28.13 is then valid on the surface of the electrode r = a and for r > a [Nunez, 1981]. 

Certainly, the expression for the potential is much simpler than that for the field, and this is a very important reason why we pay special attention to the behavior of this function U(p). As follows from the behavior of the gravitational field, the potential U has a maximum at the earth s center and with an increase of the distance from this point it becomes smaller, since the first derivative in the radial direction, that is, the component of the gravitational field, is negative. At very large distances from the earth the function U has a minimum and then it starts to increase, but this range is beyond our interest. In the first chapter we demonstrated that the potential of the attraction field obeys Poisson s and Laplace s equations inside and outside the earth, respectively ... 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

The radial potential distribution inside the capillary, (r), is then obtained by solving the Poisson-Boltzmann equation for cylindrical symmetry (30). The resulting potential depends on a single adjustable constant which is fixed by the boundary condition on the potential which relates the p gential gradient at r=l/2Dp to the surface charge density, J c. Then we define... 

This equation, related to Poisson s equation of Section 13.6.1, states that the mean curvature of the pressure surface is zero. Figure 5.26 shows the predicted isobars for a flow into a mould with a cut-out (computed using the steady-state heat flow analogue). The velocity vectors are perpendicular to the isobars the circular arc isobars near the gate show there is radial flow, but in sections with parallel side walls, the velocity is parallel to the walls. 

Poisson s ratio, p, of the adhesive, is the controlling parameter which governs the magnitude of the tangential stress, cr and the radial stress, cr. Equation (4) represents the relationship between these three stresses ... 

This equation shows that a,/2 can be greater than 2 only when < 0. Negative values of the Poisson s ratio have not been observed experimentally so the absolute maximum value of a, predicted by Scherer s model is therefore 22. The maximum stresses in the matrix occur at the interface between the matrix and the inclusion. The radial stress so it is also predicted to be less than... 

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