The outer boundary condition

As before, it is appropriate to divide the discussion of the outer boundary condition into two cases, one for survival probability (the heterogeneous problem) and one for reaction of a reactant in a vast (homogeneous) excess of the other reactant. 

In the volume of interest are two reactants A and B in solution. B is present in considerable excess over A. The typical distance between A reactants is rAA (3/47t[A] )1/3, where [A] is the number density of A (per m3), whereas that distance between B reactants is significantly less than rAA. To an excellent approximation, the distribution of B is uniform around A. As a first guess, the density of B a large distance from A can be taken as unity and the Smoluchowski [3] or Collins and Kimball [4] analysis of the diffusion of B into A used (Chap. 2, Sects. 3 and 4). If the reactants are uncharged and the rate of reaction on encounter is large, the Smoluchowski analysis shows the density of B about A [see (eqn. (16)] to be 

At long times, the approximate density of B about A is 1 — R/r and is significantly less than unity if rAA is no more than a few times R. But what is true of the B density around one A reactant is true of the B density around any other A reactant. Consider a reactant A0 which is surrounded by several other A reactants, Al5 A2, A3, all at a distance rAA apart from A0. The long-time (steady-state) density of B about Ai, or Bor about A2, etc. has the form 1 —R/r. B reactants about a distance midway 

However, the important consequences of this analysis are that the complications of the reduction in the density of B between A reactants only develops during the decay of the non-steady-state density of B towards the steady state. The average concentration of B after reaction has begun is less than the initial (or bulk) value [B]0 usually used in the Smoluchowski theory. The diffusion of B towards A is driven by the larger concentration of B at considerable distances from A than the concentration of B nearer to A. The concentration or density gradient of B at A is decreased slightly by this competition between different A reactants for B. Hence, the current B towards one A remains almost as it was in the Smoluchowski theory [eqn. (18)], viz. 

The reaction rate between A and B is, similarly, /[A] = fe[A] [B]. Because the average density of [B] must be significantly less than [B]0 after reaction has begun, then the rate coefficient, k, is a function of the density of A. Felderhof and Deutch [25] and Felderhof [460] have found the rate coefficient to be 

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To solve the diffusion equation (9) or (10) for the density p(r, f) with the random initial condition (3), the outer boundary condition (4) and the partially reflecting boundary condition (22) is straightforward. Again, the solution follows from eqn. (12), but the Laplace transform of eqn. (22) is... 

There are two initial distributions of interest, both of which satisfy the outer boundary condition (45). If B is distributed uniformly andrandomly throughout the volume of the system (such as by addition prior to the formation of A), then the random initial condition is... 

When the electron is far from the cations, it has effectively escaped and so the outer boundary condition is... 

This is a rather large expression However, because 5G and 5G can be made arbitrarily small, all the first few expressions in the square brackets are zero. The first two brackets are just eqns. (260b) and (260a), respectively. The second two are the boundary conditions on G and G (identical) with A effectively kuct/4nR2 if reaction is to be represented by a boundary condition and 0 = 0. The outer boundary condition is obtained by letting 0 = 0 or for the Green s function or homogenous problem, respectively, with A = 0. Finally, the last term contains (G 6G — G<5G )" , which is zero, since G or G, SG or 5G are zero, respectively at + co and... 

For example, the inviscid solution for flow over a flat plate is simply that the velocity is constant everywhere and equal to the velocity in the undisturbed flow ahead of the plate, say wi. In calculating the boundary layer on a flat plate, therefore, the outer boundary condition is that u must tend to u at large v. The terr large y is meant to imply outside the boundary layer , the boundary layer thickness. S, being by assumption small. 

This solution is true up to any order and it automatically satisfies the outer boundary conditions. However, this does not satisfy the wall boundary condition and one must have a boundary layer or the inner layer of say, the thickness, 6. The thickness 6, can be obtained by the distinguished limits in the inner layer by following the method given in Bender Orszag (1987). 

In addition, Strauss s work developed the means to determine the osmotic pressure for a polydisperse suspension corresponding to a lognormal size distribution. With this S5/stem a particular osmotic pressure, II(= 4cksT sinh [ezt/ (j8)/2A BT]), is assumed specifying a value of i/ (j8). The volume fraction is then calculated. For a particular particle size, a, the volume of fluid associated with the particle is determined by the specified value of t//(j8) and the boimdaiy conditions. With the outer boundary condition, dt/ /dr r=(3 = 0 and //(j8) = constant for a specific value of the osmotic pressure, the total volume firaction, can be determined by summation of the volume fraction associated with... 

We may estimate the total shift in energy for an electron gas (in a free-electron band) due to the addition of this single atom. The outer boundary condition becomes... 

Returning to the case of equal diffusion coefficients, the task Is now to solve [4.8.10 and 11], subject to the outer boundary conditions... 

As with low-flow microinfusion, the initial condition is that drug concentration in the tissue is everywhere zero, and the outer boundary condition is that the drug concentration remains zero at all times far from the cannula tip. The boundary condition at the cannula tip (at Yo) differs in that the mass outflow from the cannula is equal to the convective (not diffusive) flux at the cannula tip — that is. 

In a sense, the case where the flow is bounded is conceptually easier to treat than is the comparable unbounded flow for when the flow is bounded the disturbance created by the sphere dies away exponentially with axial distance (Tla, S16) rather than inversely with some power of the distance, as is true for the unbounded case. For this reason, the outer boundary conditions satisfied by the lower-order terms of the inner expansion can be determined without having to apply the matching principle except in a trivial sense that is, the outer boundary conditions satisfied by the lower-order terms of the inner expansion are identical to the physical outer boundary conditions. Since the force and torque on the sphere can be determined from the inner expansion alone, it is therefore not necessary to compute any terms in the outer expansion in order to obtain the leading terms in the force and torque. Thus, the analysis differs greatly from those of Rubinow and Keller (R7) and Saffman (Sla). 

The arbitrary constants A and C are obtained by applying the outer boundary condition (7.98). For the case n = 1, this yields at x = a... 

With the application of the outer boundary condition, the solution is given by the form (11.3), which we now write in terms of Xs thus... 

By changing the outer boundary condition, external resistance can be included in the diffusion model, though this makes the mathematical resolution of the model more difficult. In this case, two parameters must be identified (i) the effective diffusivity (internal resistance) and (ii) the mass transfer coefficient (external resistance). In order to reduce the number of possible solutions, the effective diffusivity identified by neglecting the external resistance can be used as the start value of diffusivity. 

The numerical results presented here are very accurate and well within the usual engineering approximations. Since the basis functions automatically satisfy the outer boundary condition(s), and they are in closed form, the computer time consumption for the satisfaction of the inner boundary condition(s) is very small. Finally, similar basis functions can be obtained for a rectangular region with cavities. However these functions will not be in closed forms. 

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