General boundary conditions

A general boundary condition for liquid flow at sdid surfaces, Namre 389 (1997) 360-362. 

Table 7.2 gives tabulated values of the error function and related functions in the solution of other semi-infinite conduction problems. For example, the more general boundary condition analogous to that of Equation (7.27), including a surface heat loss,... 

Van der Laan (V4) extended this for much more general boundary conditions that took into account the different dispersion in the entrance and exit sections. These boundary conditions were originally introduced by Wehner and Wilhelm (W4). They assumed that the total system could be divided into three sections an entrance section from X = — co to X = 0 (designated by subscript a), the test section from X = 0 to X = Xe (having no subscript), and the exit section from X= X to +< (designated by subscript b), each section having different dispersion characteristics. This is illustrated in the second sketch in Table II. 

Since the step response (F curve) is the time integral of the pulse response (C curve), the Laplace transform of the step response for the general boundary conditions as used by van der Laan will be given by dividing equation (27) by the transform variable, p,... 

Thus the response to a periodic injection for very general boundary conditions can be found by substituting p = +iu into Eq. (34). The results for the general case would be very complicated so, as an illustration of the form of the periodic response, we will consider only the simplest case a doubly infinite system. For such a system, = H/ = Dii,, and Eq. (34) reduces to... 

These equations were used, for example, by Ebach and White (El). The periodic response for more general boundary conditions could be found from Eq. (34) by the same method, but the results would of course be much more complicated. 

This is the value for general boundary conditions, and it will reduce to the equations given by the above authors for each of their special cases. 

Equation (171) turned out to be the same result that had been obtained using the simpler boundary conditions assuming no diffusion in the fore and after sections. In other words, the solution of Eq. (168) with the general boundary conditions gives the same result as with the simpler boundary conditions. Wehner and Wilhelm used their analytical solutions for a first order reaction to show that this indeed was true Eqs. (169)... 

In order to solve the system of Eqs. 11.1, we need to specify the initial and the boimdary conditions of the problem. The general boundary conditions were discussed in Chapter 6, Section 6.2.1. Simpler boundary conditions were introduced and justified in Chapter 2, Section 2.1.4, and in the discussion of the single-component problem. In isocratic elution, the column contains no sample component but is filled with the pure mobile phase (when the mobile phase is a mixture and one of its components may compete with the sample components, we have a more complex problem, discussed in Chapter 13)... 

P. A.Thompson and S. M. Troian, A general boundary condition for liquid flow at solid surfaces. Nature (London) 389, 360-2 (1997) (and references therein). 

Inside the drop, we require that the velocity and pressure fields be bounded at the origin [which is a singular point for the spherical coordinate system that we will use to solve (7 199)]. Finally, at the drop surface, we must apply the general boundary conditions at a fluid interface from Section L of Chap. 2. However, a complication in using these boundary conditions is that the drop shape is actually unknown (and, thus, so too are the unit normal and tangent vectors n and t and the interface curvature V n). As already noted, we can expect to solve this problem analytically only in circumstances when the shape of the drop is approximately (or exactly) spherical, and, in this case, we can use the method of domain perturbations that was first introduced in Chap. 4. In this procedure, we assume that the shape is nearly spherical, and develop an asymptotic solution that has the solution for a sphere as the first approximation. An obvious question in this case is this When may we expect the shape to actually be approximately spherical ... 

For problems that involve boundaries or interfaces where we need to satisfy more general boundary conditions, involving either the tangential velocity component or viscous stresses, a more general theory is needed instead of (12-95). The most commonly used generalization is due to Brinkman, though this is valid only for highly porous materials... 

In general, boundary conditions are imposed at the end of each stage of (2.41) or the leapfrog time-step. Finally, in the case of absorbing boundary conditions, all derivatives are computed by the implicit algorithm across the entire domain including its interior and the absorber. Then, each system is updated by (2.41). 

In general, boundary conditions are difficult to specify and oftentimes difficult to incorporate into the numerical scheme. Typical boundary conditions used are given in [92, 97, 129, 130, 136]. Boundary conditions for the mass continuity Eq. (22) specify a zero electron density at the wall, or an electron flux equal to the local thermal flux multiplied by an electron reflection coefficient. The ion diffusion flux is set to... 

Boundary Conditions for the RTE. The solution of the radiative transfer equation in a given geometry is subject to boundary conditions, which give the radiation intensity distribution on the boundaries. The boundary intensity is comprised of two components (1) contribution due to emission at the boundary surfaces and (2) contribution due to diffuse and specular reflection of radiation intensity incident on the boundaries. The radiation incident on the boundary is due to intensity emitted from all volume and surface elements in the medium. In mathematical terms, the general boundary condition on any surface element is written as [1,6] ... 

The boundary conditions on and where the potential is applied are called essential or Dirichlet conditions. Since in theory no difference has to be made between anode and cathode, we introduce the notation Fg to designate them both (F = F U F2) and the generalized boundary conditions are written... 

Boundary Element Analysis of Rotationally Symmetric Systems under General Boundary Conditions". Boundary Element Methods in Engineering, CISM Course, Udine, 1983. 

Lorenzani S (2011) Higher order slip accmding to the linearized Boltzmann equation with general boundary conditions. Philos Trans R Soc Lond Ser A 369(1944) 2228-2236... 

The general boundary condition for conservation of mass of some species in the interfacial region is derived in Chapter 6. For the present case of an insoluble surfactant, and in the absence of surface diffusion, we anticipate that the first two terms of Equation 5.32 should suffice with T replaced by A ... 

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