Normalized diffusion flux

On the other hand, on the bounding hypersurfaces the normal diffusive flux must be null. However, this condition will result naturally from the fact that the conditional joint scalar dissipation rate must be zero-flux in the normal direction on the bounding hypersurfaces in order to satisfy the transport equation for the mixture-fraction PDF.122... 

Here, the diffusion in the direction of flow is neglected as the transport process is dominated by convection. This may not be a valid assumption for relatively short channels. The normal diffusion flux of the specie, g d, is calculated using Equation (5.26) where [Pg.139]

Figure 7.2. The variation in liquid water volume fraction /3 through the scaled free boundary layer. Liquid volumes below are immobile. The evaporative layer has a width 0 / /H) and evaporation rate of 0 Iy/Tt), where > 1 is a rate constant for the phase change, F, and / 1 is current density scaled by normalized diffusive flux.

Find the solution u(x) of the boundary value problem and the normalized diffusion flux P x) for xED... 

Note that the computed normalized diffusion flux does not approximate the real normalized diffusion flux e-uniformly. 

The above examples illustrate the fact that, in the case of singularly perturbed elliptic and parabolic equations, the use of classical finite difference schemes does not enable us to find the approximate solutions and the normalized diffusion fluxes with e-uniform accuracy. To find approximate solutions and normalized fluxes that converge e-uniformly, it is necessary to develop special numerical methods, in particular, special finite difference schemes. 

In conclusion we may state that, if we apply classical numerical methods to the solution of the above problems, we cannot expect to obtain even the qualitative characteristics of the processes accompanied by heat transfer and/or diffusion, if boundary or interior layers (i.e., heat or diffusion layers) appear. Such processes often take place, for example, in catalytic reactions, burning, detonation and so on. To study these problems, it is necessary to use numerical methods that allow us to approximate both the solutions of the problems and the normalized diffusion fluxes uniformly with respect to the perturbation parameter. 

In Section I we obtained an intuitive impression of the numerical problems appearing when one uses classical finite difference schemes to solve singularly perturbed boundary value problems for ordinary differential equations. In this section, for a parabolic equation, we study the nature of the errors in the approximate solution and the normalized diffusion flux for a classical finite difference scheme on a uniform grid and also on a grid with an arbitrary distribution of nodes in space. We find distributions of the grid nodes for which the solution of the finite difference scheme approximates the exact one uniformly with respect to the parameter. The efficiency of the new scheme for finding the approximate solution will be demonstrated with numerical examples. 

The behavior of the errors in the normalized flux is more complicated in the case of problem (2.25). It is similar to the behavior of the errors in the normalized diffusion flux in the case of boundary value problem (1.16) for an ordinary differential equation, when the difference scheme... 

Thus, by numerical experiments we verify that the approximate solution of the Dirichlet problem (2.16), found by the classical finite difference scheme (2.28), (2.27), and the computed normalized diffusion fiux converge for N, Nq respectively, to the solution of the boundary value problem and the real normalized diffusion flux for fixed e. However, we can also see that they do not converge e-uniformly. The solution of the grid problem approaches the solution of the boundary value problem uniformly in e qualitatively well. The normalized flux computed according to the solution of the difference problem does not approach e-uniformly the real normalized flux (i.e., the flux related to the solution of the boundary value problem) even qualitatively. Nevertheless, if the solution of the singularly perturbed boundary value problem is smooth and e-uniformly bounded, the approximate solution and the computed normalized flux converge e-uniformly (when N, Nq oo) to the exact solution and flux. 

In Section III we will see that the new finite difference schemes allow us to find the normalized diffusion fluxes with an e-uniform accuracy. 

US to construct approximations that are convergent to the normalized diffusion fluxes. 

In the case of problem (3.6), we are interested in the approximation of the normalized diffusion flux P(x, t). It is known from the theory of finite difference schemes that, for a fixed value of the parameter e and N, the discrete normalized diffusion flux P 3 ig)(x, t) converges to... 

In Table XXIII we give the errors in the normalized diffusion fluxes Qi024,i024(< ) and Qio24,io24( ) for problem (3.17) computed with the finite difference scheme (3.21), (3.16) for various values of the parameter s and the number of nodes N with = N. One can see from the table that, as N increases, the error G1024,1024( 5 tends to zero (while the... 

Now we return to the study of the normalized diffusion fluxes for boundary value problems with Dirichlet boundary condition. In Section II.D the e-uniformly convergent finite difference schemes (2.74), (2.76) and (2.67), (2.72) were constructed for the Dirichlet problems (2.12), (2.13) and (2.14), (2.15), respectively. For these problems, we now construct and analyze the approximations of the normalized diffusion fluxes. We consider the normalized diffusion fluxes for problem (2.14), (2.15) in the form... 

Thus, we see that the newly constructed finite difference schemes are indeed effective and that they allow us to approximate the solution and the normalized diffusion fluxes g-uniformly for both Dirichlet and Neumann boundary value problems with singular perturbations. 

Thus, for singularly perturbed diffusion equations with mixed boundary conditions and concentrated sources, we have constructed new finite difference schemes that enable us to find solutions and normalized diffusion fluxes to an e-uniform accuracy. 

The function P(x, t) is called the normalized diffusion flux. It is expressed in terms of the first-order derivative of the function (x, t), which is the solution of the boundary value problem in dimensionless form. 

In the case of problem (5.3), we want to find the solution of the boundary value problem, as well as the normalized diffusion flux and the first-order derivative with respect to t. 

P2(0, t = P2(+0, t is the normalized diffusion flux through the roller surface. The normalized diffusion fluxes in the roll and the body are determined by the relations... 

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