Applications of the boundary conditions

Two important applications of the boundary conditions expressed by Equation (1.3) are obvious and relevant  

1 - When the coefficient of convection, h, is infinite, the following facts appear  

The value of the flux is very high, if not infinite, explaining the vertical tangent at the origin of time of the kinetics of transfer of diffusing substance  

2 - Process of absorption or of release. Equation (1.3) is of value either for the release of additives out of the polymer in the liquid or for the absorption of the liquid into the polymer. Only the relative values of the two concentrations intervene on the process, by the following relationships  

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A fiill solution of tlie nonlinear radiation follows from the Maxwell equations. The general case of radiation from a second-order nonlinear material of finite thickness was solved by Bloembergen and Pershan in 1962 [40]. That problem reduces to the present one if we let the interfacial thickness approach zero. Other equivalent solutions involved tlie application of the boundary conditions for a polarization sheet [14] or the... 

Before we proceed to our discussion of global stiffness matrix storage schemes, we will discuss the last aspect of the finite element implementation, namely, the application of the boundary conditions. As discussed earlier, the natural boundary conditions are imbedded in the finite element equation system - it is implied that every boundary node without an... 

The effect of the wall temperature discontinuity may also be mitigated by an alternative implementation of the wall boundary condition to that described in the previous section. The two equations at each interior boundary node (l,tj), j = 2,3,...,m, are dropped, together with one equation at each of (1,-1) and (1,1). These 2m equations are replaced by application of the boundary conditions at two points within each subinterval on the line y 1. The Gaussian points are not necessarily the optimal choice, but were used in the absence of any other guideline. 

Notice something very important about these results. The application of the boundary conditions has led to a series of quantized energy levels. That is, only certain energies are allowed for the particle bound in the box. This result fits very nicely with the experimental evidence, such as the hydrogen emission spectrum, that nature does not allow continuous energy levels for bound systems, as classical physics had led us to expect. Note that the energies are quantized, because the boundary conditions require that n assume only integer values. Consequently, we call n the quantum number for this system. 

W In this case different heal fluxes are specified at the two boundaries. The application of the boundary conditions gives... 

The global system of equations created by the previous FEM steps cannot be solved, pending application of the boundary conditions. 

The constants Ci and C2, obtained from application of the boundary conditions, are... 

The two integration constants are evaluated by application of the boundary conditions (10-231). The condition... 

That problem reduces to the present one if we let the interfacial thickness approach zero. Other equivalent solutions involved the application of the boundary conditions for a polarization sheet [14] or the... 

Upon application of the boundary condition / (/<,) = 0 the solution to this equation blows up. 

The solution of the equations of motion and application of the boundary conditions proceeds as before except for an additional term (dy/dx) on the right side of Equation 5.38. If fluid B is a gas, we obtain instead of Equations 5.44 and 5.45 ... 

Following application of the boundary conditions, the set of tridiagonal simultaneous equations may be solved routinely as noted above. Compton and coworkers have used the technique extensively in mechanistic analysis including ECE reactions [48-53] and coupled kinetics reactions [54-59]. The technique has also been used to assess the influence of hydrodynamics on the... 

The application of the boundary condition (5.72a) shows that the arbitrary constant A must be taken identically zero, since 7o(0 becomes arbitrarily large as f oo (see Fig. 5.12). The value of B is obtained by imposing the source condition [cf. Eq. (5.83)]. 

We compute the constants A and C by the application of the boundary condition (7.294). This yields... 

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