SECOND BOUNDARY CONDITIONS

The inclusion m G K can be proved by standard arguments. Note that the second boundary condition (5.142) and the conditions (5.143) are included in the identity (5.145). This means that it is possible to obtain these conditions by integrating by parts provided that the solution is sufficiently smooth. Actually, we can prove that the second condition (5.142) holds in the sense 77 / (F), but the arguments are omitted here. The theorem is proved. 

The second boundary condition (5.214) and the conditions (5.215) are involved in (5.218). This means that those conditions hold at any point of r, r, respectively, provided the solution v, rriij is smooth enough. The statement can be verified by integrating by parts. Theorem 5.7 is proved. 

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure bound-aiy condition and two velocity boundaiy conditions (for each velocity component) to completely specify the solution. The no sBp condition, whicn requires that the fluid velocity equal the velocity or any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-... 

This expression already resembles that in Eq. 10. To find the value of k, we use the second boundary condition, that i >(L) = 0 that is,... 

Formulae (74)-(75) show that the elimination method is stable. The values and should be known before proceeding to the applications of (70), (74) and (75). For this reason we involve here the second boundary condition (67) and relation (69) for i = N ... 

In the case of the second boundary-value problem with dv/dn = 0, the boundary condition of second-order approximation is imposed on 7, as a first preliminary step. It is not difficult to verify directly that the difference eigenvalue problem of second-order approximation with the second kind boundary conditions is completely posed by... 

Equation (11.2) remains valid as the first boundary condition in this case. The surface concentrations, c, of the reactants will remain constant, in accordance with the Nemst equation, when the electrode potential is held constant during current flow (and activation polarization is absent). Hence, the second boundary condition can be formulated as... 

Consider the case of transient diffusion at constant potential (constant surface concentration). The first boundary condition, (11.2), is preserved and the second boundary condition can be written (for any time t) as... 

To find the values of the integration constants Cj and C2, we must formulate the boundary conditions. At large distances from the central ion, the value of / is zero, hence Cj = 0. Equation (7.30) can be used as the second boundary condition. Substituting the value of / [the second term on the right-hand side of Eq. (A.8) into... 

The second boundary condition meets the requirement that there be no flow of matter out of the closed end of the pore. 

The second boundary condition assures total finite existence probability at any time the first boundary condition implies that the recombination is fully diffusion-controlled, which has been found to be true in various liquid hydrocarbons (Allen and Holroyd, 1974). [The inner boundary condition can be suitably modified for partially diffusion-controlled reactions, which, however, does not seem to have been done.]... 

The second boundary condition is not known definitely, but is consistent with reactant A not penetrating the impermeable face at z = 1. From equations 8.5-12a to c,... 

The second boundary condition arises from the continuity of chemical potential [44], which implies - under ideally dilute conditions - a fixed ratio, the so-called (Nernst) distribution or partition coefficient, A n, between the concentrations at the adjacent positions of both media ... 

Some models, however, take the form of second-order differential equations, which often give rise to problems of the split boundary type. In order to solve this type of problem, an iterative method of solution is required, in which an unknown condition at the starting point is guessed, the differential equation integrated. After comparison with the second boundary condition a new starting point is estimated, followed by re-integration. This procedure is then repeated until convergence is achieved. MADONNA provides such a method. Examples of the steady-state split-boundary type of solution are shown by the simulation examples ROD and ENZSPLIT. 

The constant A must equal zero for the potential / to fall to zero at a large distance away from the charge and the constant B can be obtained using the second boundary condition, that / = /o at r = where a is the radius of the charged particle and /o the electrostatic potential on the particle surface. Thus we obtain the result that... 

Conservation of water mass in the spherical shell of melt that surrounds the bubble at r = 5 (the second boundary condition) requires. s. ... 

The second boundary condition involves a relation between the concentrations of A and D and the potential E. The simplest relation is obtained by (initially) assuming that there is charge-transfer equilibrium at the interface (a = 0), in which case the Nemst equation (7.47) can be applied ... 

One boundary condition for Eq. (29) is p = p0 for x -> oo. The second boundary condition can be taken as p = 0 for x = Lsc, which is true for a very fast electrode reaction that extracts practically all the holes from the space-charge region. Solving Eq. (29) with these boundary conditions and then calculating the diffusion current of holes from the bulk into the space-charge region at x = Lsc, we obtain... 

Hints and Help One of the boundary conditions can be chosen at x = 0 Cw = CRiver. Due to hydrolysis aid biodegradation, benzylchloride eventually disappears completely from the groundwater. Therefore, it can be assumed that Cw = 0 for x —> This serves to formulate the second boundary condition. 

Since xs must be bounded as r approaches zero according to the first boundary condition, we must choose Ct = 0. The second boundary condition requires that C2 = l/sinh30, leaving... 

Notice that both mass and heat balance design equations (5.19) and (5.20) are second-order two-point boundary value differential equations. Therefore each one requires two boundary conditions. These boundary conditions can be derived as shown in the example that follows. 

Ci and C2 are constants, which are defined by the boundary conditions. The boundary conditions require that, at the surface, the potential is equal to the surface potential, ip x = 0) = rfo, and that, for large distances from the surface, the potential should disappear ip(x — oo) = 0. The second boundary condition guarantees that, for very large distances, the potential becomes zero and does not grow infinitely. It directly leads to C2 = 0. From the first boundary condition we get C = Hence, the potential is given by... 

As the coordinate is moving at a velocity of u, the second term in the boundary condition at x=0 in equation (5) is zero and the boundary condition at y = 0 still represents zero flux across the boundary. The second boundary condition is obtained by assuming the concentration to be continuous at y = L and extending the concentration profile to infinity. 

The second boundary condition is obtained from the micellar charge Qm. It follows from Gauss theorem that... 

Since cells in contact with the surface are no longer part of the disperse phase, the concentration c must vanish when contact is made. Requiring the cellular flux to vanish at the upper surface of the suspension leads to the second boundary condition. Thus... 

The electroneutrality condition provides a second boundary condition ... 

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