Zero-flux boundary condition

The concentration at the wall, a(7), is found by applying the zero flux boundary condition. Equation (8.14). A simple way is to set a(I) = a(I — 1) since this gives a zero first derivative. However, this approximation to a first derivative converges only 0(Ar) while all the other approximations converge O(Ar ). A better way is to use... 

As in the full-field formulation, we assigned a zero flux boundary condition, i.e. j = 0 at the outer boundary of the domain as well as on the axis of symmetry ahead of the crack tip (Fig. 5b). Also, along the crack surface, we assumed the NILS hydrogen concentration CL to be in equilibrium... 

The last issue that remains to be addressed is whether the MBL results are sensitive to the characteristic diffusion distance L one assumes to fix the outer boundary of the domain of analysis. In the calculations so far, we took the size L of the MBL domain to be equal to the size h - a of the uncracked ligament in the pipeline. To investigate the effect of the size L on the steady state concentration profiles, in particular within the fracture process zone, we performed additional transient hydrogen transport calculations using the MBL approach with L = 8(/i — a) = 60.96 mm under the same stress intensity factor Kf =34.12 MPa /m and normalized T-stress T /steady state distributions of the NILS concentration ahead of the crack tip are plotted in Fig. 8 for the two boundary conditions, i.e. / = 0 and C, =0 on the outer boundary. The concentration profiles for the zero flux boundary condition are identical for both domain sizes. For the zero concentration boundary condition CL = 0 on the outer boundary, although the concentration profiles for the two domain sizes L = h - a and L = 8(/i - a) differ substantially away from the crack tip. they are very close in the region near the crack tip, and notably their maxima differ by less than... 

Since the surface is not crossed by any gradient lines, it is referred to as the surface of zero flux. As further discussed below, the virial theorem is satisfied for each of the regions of space satisfying the zero-flux boundary condition. 

For radial concentration profiles, a quadratic representation may not be adequate since application of the zero flux boundary conditions at r, = cp0 and r, = 1.0 leads to d2 = d3 = 0. Thus a quadratic representation for the concentration profiles reduces to the assumption of uniform radial concentrations, which for a highly exothermic system may be significantly inaccurate. 

Fig. 16. Bifurcation diagram of temporal dissipative structures, c (maximal amplitude of the oscillation minus the homogeneous steady-state value) is sketched versus B for a two-dimensional system with zero flux boundary conditions. The first bifurcation occurs at B = Bn and corresponds to a stable homogeneous oscillation. At B, two space-dependent unstable solutions bifurcate simultaneously. They become stable at B a and Bfb. Notice that as it is generally the case Bfa Bfb.

Fig. 17. Temporal dissipative structure after various time intervals during the period of oscillation. The reaction medium is a circle with zero flux boundary conditions. The lines correspond to isoconentrations. A =2, B = 5.4, Dt = 8 10 3, D2 = 4- 1G"3. Curves of equal concentration for Y are represented by full or broken lines when the concentration is, respectively, larger or smaller than the unstable steady state. The radius of the circle r0 = 0.5861. 

Fig. 10.2. A typical non-uniform stationary-state profile. Note the vanishing spatial derivative at the end walls (r = 0 and r = a0) appropriate to zero-flux boundary conditions.

Murray (1982) has confirmed this pattern of behaviour empirically for a variety of two-variable models with zero-flux boundary conditions such as those considered here. In general, the dominant mode increases in wave number n as the size of the reaction zone y increases, but decreases as the ratio of diffusivities increases—as shown in Fig. 10.6. 

Bader 1975) that the condition for the satisfaction of the principle of stationary action is that each subsystem lj be bounded by a surface Si satisfying a zero-flux boundary condition of the form... 

Since the zero-flux boundary condition (eqn (8.109)) is also satisfied by an atom, that is, by a quantum subsystem, the atomic action and Lagrangian integrals vanish as they do for a total closed system. Indeed, one may view the vanishing of the action integral over some total system as being the result of the action integral vanishing separately over the space of each atom in the system. 

Equating the results given in eqns (8.190) and (8.191), one obtains an expression for the virial theorem of an open system which satisfies the zero-flux boundary condition (8.109)... 

The possibility developed in this treatise of a well-defined physical partitioning scheme Qi(f) ( = 1, 2,. . ., n at any time t, as afforded by the zero-flux boundary condition... 

Boundary conditions for this simulation assume that there is no flux of protein across the cellular membrane. In situations where the cell and stimulus is symmetric, zero flux boundary conditions can also be used to take advantage of this symmetry. For example, in (17) 3. three-dimensional model of the cell used symmetry in the x, y, and z directions was developed. This allowed simulations to run on only one-eighth of the cell, saving both computer time and memory. 

It is through a generalization of Schwinger s principle that one obtains a prediction of the properties of an atom in a molecule. The generalization is possible only if the atom is defined to be a region of space bounded by a surface which satisfies the zero flux boundary condition, a condition repeated here as equation 15,... 

For the loading step, Eq. 4 is subject to the following boundary conditions C, = Cjo at the sample reservoir and zero flux boundary conditions, n.VC, = 0, at all the other boundaries across the chip, where n is the unit normal to the surface and C,o is an initially applied concentration for the ith species. The initial conditions for Eq. 4 are C = C,o at the sample reservoir and C, = 0 at all the other places across the chip. For the dispensing step, the boundary conditions are similar to that for the loading step however, the initial conditions are the concentration distribution of the loading step when it reaches steady state, C = C/ ioading ... 

The above equation is subject to the zero flux boundary conditions, n.Vn, = 0 at both solid walls and reservoirs. As one can see that the Nemst-Planck equation is coupled with the Poisson equation and the N-S equation, which are very difficult to solve simultaneously. Another difficulty arises when all the species of... 

Combinations of Dirichlet and Neumann boundary conditions are used to solve the electronic and protonic potential equations. Dirichlet boundary conditions are applied at the land area (interface between the bipolar plates and the gas diffusion layers). Neumann boundary conditions are applied at the interface between the gas charmels and the gas diffusion layers to give zero potential flux into the gas charmels. Similarly, the protonic potential field requires a set of potential boimdary condition and zero flux boundary condition at the anode catalyst layer interface and cathode catalyst layer interface respectively. 

Table I compares the experimental and calculated Kell s lor lour critical conllguratlons. The calculated values were obtained using the AIM-6, one-dlmenslonal dillusion code with a 16-group ciws-sectlon library obtained Irom Los Alamos. The method ol computation required agreement between KeU s comiHited in slab and cylindrical geometry. Equivalence in Kelt between reflected cores and bare cores (with zero flux boundary conditions) provided energy independent transverse buckling terms,...

In order to solve a partial differential equation like eq. (6.1), we need to know the boundary conditions. In most situations of interest—for example, in a Petri dish with an impermeable wall—the appropriate condition to use is the zero-flux boundary condition ... 

The above equation is subject to the zero flux boundary conditions, Vnj = 0 at both solid walls and reservoirs. 

We now analyze equation (3.1) in detail. Our purpose is to show that the interaction of steady modes may lead to a bifurcation to time-periodic solutions. The problem is motivated by the observation that polyhedral flames on a Bunsen burner may sometimes rotate about its vertical axis. Similar bifurcation problems were recently discussed in the context of convective instabilities [9]. Here we analyse the simplest case of a two-dimensional (z ,y) flame subject to zero-flux boundary conditions in the y-direction ... 

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