Boundary Condition at Interface

The effort to solve Eqs.(l) evidently depends on the refractive index profile. For isotropic media in a one-dimensional refractive index profile the modes are either transversal-electric (TE) or transversal-magnetic (TM), thus the problem to be solved is a scalar one. If additionally the profile consists of individual layers with constant refractive index, Eq.(l) simplifies to the Flelmholtz-equation, and the solution functions are well known. Thus, by taking into account the relevant boundary conditions at interfaces, semi-analytical approaches like the Transfer-Matrix-Method (TMM) can be used. For two-dimensional refractive index profiles, different approaches can be... 

M. Sahraoui, and M. Kaviany, Slip and No-Slip Temperature Boundary Conditions at Interface of Porous, Plain Media Conduction, Int. J. Heat Mass Transfer, (36) 1019-1033,1993. 

At the macroscopic scale, the full competition of reactant diffusion, electron and proton migration, and charge transfer kinetics unfolds. The water balance further complicates this interplay. Moreover, performance is subject to operation conditions and complex boundary conditions at interfaces to membrane and gas diffusion layer. A vast list of structural characteristics steers this interplay, including thickness, composition, pore size distributions, and wetting properties of pores. 

Apparent slip Boundary condition at interface Perfect slip... 

Initial inverse calculations of epicardial potentials were performed utilizing the eccentric spheres model of the inhomogeneous torso. The inverse-computed epicardial potentials are given by the following expression where and are expansion coefficients, determined by surface integration of the torso potential data (Geselowitz, 1960), and boundary conditions at interfaces between torso inhomogeneities. 

Sahraoui, M., Kaviany, M., 1992. Slip and no-slip velocity boundary conditions at interface of porous, plain media. Int. J. Heat Mass Transf. 35, 927-943. 

In a continuous description, anchoring terms introduce boundary conditions at interfaces. On isotropic substrates like oxidized wafers, water, or glycerol and for flat films (without thickness gradient), the only condition is on the preferred orientation with respect to the surface normal, that is, the polar angle 0. In the cases considered here, the preferred orientation at the free interface is along the normal (9 = 0, homeotropic anchoring). The preferred... 

Combining the foregoing equations, we have the boundary condition at interface to be... 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

Because the boundary condition at y = oo Cr = 0) is unaltered by the fact that the concentration at the interface is a function of time, equation 10.106 is still applicable, although the evaluation of the constant 2 is more complicated because (C )y=o is no longer constant. 

Examples of the electrophoretic mobility, m as functions of the molecular properties, e.g., solute size, charge, and shape, and solution conditions, were discussed in a previous section. For the two-phase system considered in Figure 30, the flux and equilibrium boundary conditions at the interface between the a and p phases are given by... 

Assuming strong vertical confinement (i.e., p< < kj), the modal field solutions can be separated into two distinct polarizations TE consisting of Hz, Ep, and E() and TM consisting of Ez, Hr, and //(l. The z component of the relevant field for each polarization can be described by two coefficients in each layer Aj and Bj for TM and Cj and Dj for TE. For each polarization, the boundary conditions at the interfaces between successive layers can be represented by a simpler 2x2 transfer matrix ... 

Boundary conditions At the boundary r=a, interface equilibrium between the melt and gas phases dictates that total H2O concentration at... 

The boundary condition at the sediment-water interface which relates the diffusion equations on both sides of the boundary, is given by ... 

Solution. Yes. When D varies with concentration we have shown in Section 4.2.2 that the diffusion equation can be scaled (transformed) from zt-space to 77-space by using the variable rj = x/ /4Di (see Eq. 4.19). Also, under diffusion-limited conditions where fixed boundary conditions apply at the interfaces, the boundary conditions can also be transformed to 77-space, as we have also seen. Therefore, when D varies with concentration, the entire layer-growth boundary-value problem can be transformed into 77-space. Since the fixed boundary conditions at the interfaces require constant values of 77 at the interfaces, they will move parabolically. 

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