Neuman condition

At an axi-symmetric boundary Neuman conditions are used for all the fields, except for the normal velocity component which is zero because the flow direction turns at this point. The assumption of cylindrical axi-symmetry in the computations prevents lateral motion of the dispersed gas phase and leads to an unrealistic radial phase distribution [73, 66[. Krishna and van Baten [73] obtained better agreement with experiments when a 3D model was applied. However, experience showed that it is very difficult to obtain reasonable time averaged radial void profiles even in 3D simulations. 

The applicable mathematical form of the flux equation is model dependent. For example, the first type model consists of differential equations (DEs). They are developed to yield concentration profiles in the sediment layers as well as the flux. These DEs typically use Equation 4.1 as a boundary condition. The solutions to these DEs require one or more of the following boundary condition categories the Dirichlet condition, the Neuman condition, or a third condition. The first two types are the most common these require mathematical functions containing gradients of the dependent variable (i.e., Cw) as well as functions of the dependent variable itself. For these diffusive-type fluxes, the transport parameter is a diffusion coefficient such as Dg. Several other transport parameters are commonly used and represent diffusion in air and the biodiffusion or bioturbation of soil/sediment particles. 

Boundary conditions can be of several types. The most common boundary conditions are specifications of the value of the solution on the boundary nodes, the Dirichlet condition, or of the normal derivative on the boundary nodes, the Neuman condition. A more general set of boundary conditions known as mixed boundary conditions involve both the value and the normal derivative. A typical mixed boundary condition is of the form du du... 

Neuman boundary condition, when the derivatives of the generic variable on the boundary are known and this yields an extra equation. 

The exact form of the matrices Qi and Q2 depends on the type of partial differential equations that make up the system of equations describing the process units, i.e., parabolic, elliptic, or hyperbolic, as well as the type of applicable boundary conditions, i.e., Dirichlet, Neuman, or Robin boundary conditions. The matrix G contains the source terms as well as any nonlinear terms present in F. It may or may not be averaged over two successive times corresponding to the indices n and n + 1. The numerical scheme solves for the unknown dependent variables at time t = (n + l)At and all spatial positions on the grid in terms of the values of the dependent variables at time t = nAt and all spatial positions. Boundary conditions of the Neuman or Robin type, which involve evaluation of the flux at the boundary, require additional consideration. The approximation of the derivative at the boundary by a finite difference introduces an error into the calculation at the boundary that propagates inward from the boundary as the computation steps forward in time. This requires a modification of the algorithm to compensate for this effect. 

Neuman boundary conditions are used for the particle phase. These are generally defined for all z and t ... 

For the bulk gas phase Dirichlet boundary conditions were specified at the reactor inlet for all the variables, and Neuman boundary condition were employed at the outlet for all the variables except for the pressure which is specified by a Dirichlet boundary condition. 

Similar conditions exist if a drop of liquid is placed on the surface of a solid which is soluble in this liquid. Due to diffusion through the liquid, the system may come to a true equilibrium described by the Neuman equation ... 

The second type of boundary condition (or Neuman type) is defined by the concentration gradient on the boimderies ... 

It has been suggested that the mineral portions of bone and tooth tissue are calcium-deficient hydroxyapatites (Posner and Perloff, 1957 Neuman and Neuman, 1953). The possible presence of hydrogen bonds in these materials may provide a much needed parameter for an estimation of the stoichiometry of these biological hydroxyapatites under varying conditions (Posner et al, 1960). 

The effective material symmetry of the composite body is given by the overlap of the crystallographic symmetry elements of all phases (Neuman s principle) together with the geometric mixing rule for phases. In the finite composite bo, also the boimdaty conditions could contribute to the effective material symmetry. Anisostropy of the material properties could be enhanced as well as lowered by the geometry conditions for the composite phases. Composite properties could be tailored to the wide variety of required parameters for specific application needs. 

Diaz-Ortiz A, Diez-Barra E, de la Hoz A, Prieto R Moreno A, Langa F, Prange T, Neuman A (1995) Facial selectivity in cycloadditions of a chiral ketene acetal under microwave irradiation in solvent-free conditions. Configurational assignment of the cycloadducts by NOESY experiments and molecular mechanics calculations. J Oig Chem 60 4160-4166... 

Note that a Dirichlet boundary condition is specified for the left-hand side of the domain, i.e.0o = l.InordertoimplementtheNeumannboundary conditionontheright-handside of the domain (the no-heat-flux condition on the tip), we can extend the computational domain from =i x /i for i = 0,1, 2,..., n to i =0, 1, 2,..., n + 1. For this reason, tile dependent variable is also calculated at the fictitious boundary outside the domain, at fs. The Neuman boundary condition, i.e. the derivative at tiie right-hand side of the... 

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