Similarity solutions linear diffusion equation

For illustration, we present in Fig. 3.3.1a,b the results of a numerical solution of the original system (3.3.19) (Curve 1) for e = 10-2, 10-3, 7=1 together with a plot of the leading term (3.3.48) (Curve 2). We also present for comparison a plot of 1 erfcx (Curve 3), the similarity solution for the linear diffusion equation with the boundary and initial conditions analogous... 

In other words, the kinetics of smoothing out of Z(r,t) fluctuations is governed by a simple linear diffusion equation which solution and, in particular, the Green function, are well known. Similarly to (2.1.43), (2.1.45) we can now write down... 

The theory has been verified by voltammetric measurements using different hole diameters and by electrochemical simulations [13,15]. The plot of the half-wave potential versus log[(4d/7rr)-I-1] yielded a straight line with a slope of 60 mV (Fig. 3), but the experimental points deviated from the theory for small radii. Equations (3) to (5) show that the half-wave potential depends on the hole radius, the film thickness, the interface position within the hole, and the diffusion coefficient values. When d is rather large or the diffusion coefficient in the organic phase is very low, steady-state diffusion in the organic phase cannot be achieved because of the linear diffusion field within the microcylinder [Fig. 2(c)]. Although no analytical solution has been reported for non-steady-state IT across the microhole, the simulations reported in Ref. 13 showed that the diffusion field is asymmetrical, and concentration profiles are similar to those in micropipettes (see... 

The mathematical formulations of the diffusion problems for a micropippette and metal microdisk electrodes are quite similar when the CT process is governed by essentially spherical diffusion in the outer solution. The voltammograms in this case follow the well-known equation of the reversible steady-state wave [Eq. (2)]. However, the peakshaped, non-steady-state voltammograms are obtained when the overall CT rate is controlled by linear diffusion inside the pipette (Fig. 4) [3]. 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a parabolic partial differential equation. For steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in semi-infinite domains using the Laplace transform technique, a similarity solution technique and Maple. In addition, we describe how numerical similarity solutions can be obtained for nonlinear partial differential equations in semi-infinite domains. 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

Stone2 has summarized a generalization of the solution for this simple linear problem, due to Pattle3 and Pert4, which is extremely useftd in the analysis of thin-film problems. This is the development of similarity solutions for the (/-dimensional symmetric diffusion equation with a diffusivity that depends on the concentration,... 

We can obtain the concentration profiles simply by rescaling the velocity profiles, using Eqs. (4.5.14) and (4.5.15) therefore we do not explicitly discuss the similarity solution of the boundary layer diffusion equation. However, in terms of the similarity variable given above and from the series expressions for the velocity profile near the wall, the equation is reducible to the linear form... 

Equation [7.2.53], as opposed to [7.2.52], is non-linear and cannot be solved analytically for arbitrary function D = D(k(T))). Note that in the case of the planar non-linear diffusion, if the self-similarity conditions are satisfied, the problem has analytical solution for particular forms of the dependencies D = D(k) (e.g., linear, exponential, power-law, etc ). However, the resulting relationships are rather cumbersome. The approximate solution of the problem was derived in the case Ja 1, using the perturbation method ... 

In this equation the ) (17) function assumes a similar role to the f(x,y, 17) function in the previous example. The reader is referred back to Section 12.5 for a discussion of this equation and the one dimensional time dependent solution. In the present chapter in Section 13.9 an example of linear diffusion into a two dimensional surface was presented. For that example, a triangular mesh array was developed and shown in Figure 13.15. The present example combines the nonlinear diffusion model of Section 12.5 with the FE mesh of Figure 13.15 to demonstrate a second nonlinear PDE solution using the FE approach. The reader should review this previous material as this seetion builds upon that material. 

Similar treatment of the Knox equation does not predict that values of H(min) should be independent of the solute diffusivity neither does it predict that (uopt) should vary linearly with solute diffusivity. Consequently, the relationships shown in Figures 5... 

The above model was solved numerically by writing finite difference approximations for each term. The equations were decoupled by writing the reaction terms on the previous time steps where the concentrations are known. Similarly the equations were linearized by writing the diffusivities on the previous time step also. The model was solved numerically using a linear matrix inversion routine, updating the solution matrix between iterations to include the proper concentration dependent diffusivities and reactions. 

For d 4 singular term (d — l)(d — 3)/(4 j2) does not allow to find the solution at r/ = 0. It has simple interpretation in systems with so large space dimensionalities no variable rj = r/fo exists there. Similar to d = 3 in the linear approximation, for d 4 we can find the stationary solutions, Y (r, oo) = y0(r). For them the reaction rate K(oo) = Kq — const and the classical asymptotics n(t) oc Ya, ao = 1 hold. Therefore, for a set of kinetic equations derived in the superposition approximation the critical space dimension could be established for the diffusion-controlled reactions. 

For systems that exhibit slow anomalous transport, the incorporation of external fields is in complete analogy to the existing Brownian framework which itself is included in the fractional formulation for the limit a —> 1 The FFPE (19) combines the linear competition of drift and diffusion of the classical Fokker-Planck equation with the prevalence of a new relaxation pattern. As we are going to show, also the solution methods for fractional equations are similar to the known methods from standard partial differential equations. However, the temporal behavior of systems ruled by fractional dynamics mirrors the self-similar nature of its nonlocal formulation, manifested in the Mittag-Leffler pattern dominating the system equilibration. 

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