Diffusion equation superposition

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

A related technique is the current-step method The current is zero for t < 0, and then a constant current density j is applied for a certain time, and the transient of the overpotential 77(f) is recorded. The correction for the IRq drop is trivial, since I is constant, but the charging of the double layer takes longer than in the potential step method, and is never complete because 77 increases continuously. The superposition of the charge-transfer reaction and double-layer charging creates rather complex boundary conditions for the diffusion equation only for the case of a simple redox reaction and the range of small overpotentials 77 [Pg.177]

Several general methods are available for solving the diffusion equation, including Boltzmann transformation, principle of superposition, separation of... 

Equations 3-45a to 3-45d, in conjunction with the following superposition principle, are powerful in deriving solutions for the diffusion equation with infinite medium. 

This result is known as the principle of superposition. The principle is useful in solving diffusion equations with the same boundary conditions, but different initial conditions, or with the same initial conditions but different boundary conditions, or other more general cases. Suppose we want to find the solution to the diffusion equation for the following initial condition ... 

The above derivation has not made use of the initial and boundary conditions yet, and shows only that A may take any constant value. The value of A can be constrained by boundary conditions to be discrete Ai, A2,..., as can be seen in the specific problem below. Because each function corresponding to given A is a solution to the diffusion equation, based on the principle of superposition, any linear combination of these functions is also a solution. Hence, the general solution for the given boundary conditions is... 

The one-dimensional diffusion equation in isotropic medium for a binary system with a constant diffusivity is the most treated diffusion equation. In infinite and semi-infinite media with simple initial and boundary conditions, the diffusion equation is solved using the Boltzmann transformation and the solution is often an error function, such as Equation 3-44. In infinite and semi-infinite media with complicated initial and boundary conditions, the solution may be obtained using the superposition principle by integration, such as Equation 3-48a and solutions in Appendix 3. In a finite medium, the solution is often obtained by the separation of variables using Fourier series. 

The superposition principle can be used to combine solutions for linear partial differential equations, like the diffusion equation. It is stated as follows ... 

The superposition of two solutions therefore also solves the diffusion equation with superposed boundary and initial conditions. 

The equations derived above, describing the A + B —> B reaction kinetics in terms of the correlation functions g and g2, have the form of the nonlinear generalised multi-dimensional diffusion equation. Ignoring the multidimensionality of the operator terms in (5.2.11), these equations could be formally considered as similar to the basic non-linear equations for the A + B — 0 reaction (Section 5.1). Equations studied in both Sections 5.1 and 5.2 are derived with the help of the Kirkwood superposition approximation, the use of which leads to several equations for the correlation functions of similar and dissimilar reactants. 

When supposing that hydrogen in the probe is contained in traps with different binding energies u,. and corresponding diffusion constant D , and hydrogen concentrations one can use the superposition principle, due to linearity of diffusion equation (5). Then the total flux of hydrogen from the probe q(t) can be expressed by the sum ... 

Predicting fast and slow rates of sorption and desorption in natural solids is a subject of much research and debate. Often times fast sorption and desorption are approximated by assuming equilibrium portioning between the solid and the pore water, and slow sorption and desorption are approximated with a diffusion equation. Such models are often referred to as dual-mode models and several different variants are possible [35-39]. Other times two diffusion equations were used to approximate fast and slow rates of sorption and desorption [31,36]. For example, foraVOCWerth and Reinhard [31] used the pore diffusion model to predict fast desorption, and a separate diffusion equation to fit slow desorption. Fast and slow rates of sorption and desorption have also been modeled using one or more distributions of diffusion rates (i.e., a superposition of solutions from many diffusion equations, each with a different diffusion coefficient) [40-42]. 

The final non-shearing flow to be considered is the superposition of steady shearing flow with transverse small-amplitude oscillations. Here vx — kz, vy — Ske K°e z, and vz = 0. The diffusion equation for this flow is obtained from Eq. (3.9) with the first line of the right side omitted, and Eqs. (3.11) and (3.12) with kxz = k, Kyz = 9tf. kV , and all other... 

For each n = 1,2,..., F (r ) also satisfies the original diffusion equation. By the superposition principle for linear equations,... 

To obtain a quantitative description of this experiment, one might consider first the result of the forward step, then use the concentration profiles applicable at r as initial conditions for the diffusion equation describing events in the reversal step. In the case outlined above, the effects of the forward step are well known (see Section 5.4.1), and this direct approach can be followed straightforwardly. More generally, reversal experiments present very complex concentration profiles to the theoretician attempting to describe the second phase, and it is often simpler to resort to methods based on the principle of superposition (37, 38). We will introduce the technique here as a means for solving the present problem. 

The final non-shearing flow to be considered is the superposition of steady shearing flow with transverse small-amplitude oscillations. Here v =Kz, = and v = 0. The diffusion equation for this... 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

In Eq. (4.13) NT is the total number of internal degrees of freedom per unit volume which relax by simple diffusion (NT — 3vN for dilute solutions), and t, is the relaxation time of the ith normal mode (/ = 1,2,3NT) for small disturbances. Equation (4.13), together with a stipulation that all relaxation times have the same temperature coefficient, provides, in fact, the molecular basis of time-temperature superposition in linear viscoelasticity. It also reduces to the expression for the equilibrium shear modulus in the kinetic theory of rubber elasticity when tj = oo for some of the modes. 

When the diffusion profile is time-dependent, the solutions to Eq. 4.18 require considerably more effort and familiarity with applied mathematical methods for solving partial-differential equations. We first discuss some fundamental-source solutions that can be used to build up solutions to more complicated situations by means of superposition. 

The analysis of the diffusion-controlled computer simulations confirms once more conclusions drawn above for the static reactions of immobile particles. In particular, the superposition approximation gives the best lower bound estimate of the kinetics reaction, n = n(i). Divergence of computer simulations and analytical theory being negligible for equal concentrations become essential for large depths and when one of reactants is in excess. The obtained results allow us to use the superposition approximation for testing the applicability of simple equations of the linear theory in those cases when computer simulations because of some reasons cannot be performed. Examples will be presented in Chapter 6. 

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. 

A comparison with the correlation dynamics of the A + B —> 0 reaction, equations (5.1.33) to (5.1.35), shows their similarity, except that now several terms containing functionals J[Z have changed their signs and several singular correlation sources emerged. The accuracy of the superposition approximation in the diffusion-controlled and static reactions was recently confirmed by means of large-scale computer simulations [28]. It was shown to be quite correct up to large reaction depths r = 3 studied. 

Here the first term arises from the diffusive approach of reactants A into trapping spheres around B s it is nothing but the standard expression (8.2.14). The second term arises due to the direct production of particles A inside the reaction spheres (the forbidden for A s fraction of the system s volume). Unlike the Lotka-Volterra model, the reaction rate is defined by an approximate expression (due to use of the Kirkwood superposition approximation), therefore first equations (8.3.9) and (8.3.10) of a set are also approximate. 

Let us consider as the basic equations those of the diffusion-controlled stochastic Lotka model which are derived in the superposition approximation, thus neglecting terms having a small parameter 5(t). 

Equation (4.50) for the particular case of equal diffusion coefficients and rK 0 was deduced by Kambara [22] by applying the Superposition Principle. 

Under these conditions (see Eqs. (4.199)-(4.202)), it can be easily demonstrated that the Superposition Principle can be applied and the diffusion differential equation and the boundary value problem of this process, independently of the electrode geometry, are simplified to... 

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