Laplacian diffusion equation

Actually, this is not really diffusion-XimiiQd, but rather Laplacian growth, since the macroscopic equation describing the process, apart from fluctuations, is not a diffusion equation but a Laplacian equation. There are some crucial differences, which will become clearer below. In some sense DLA is diffusion-limited aggregation in the limit of zero concentration of the concentration field at infinity. 

The right-hand side of eqn. (9), which is the diffusion equation or Fick s second law, involves two spherically symmetric derivatives of p(r, t). In the general case of three-dimensional space, lacking any symmetry, it can be shown that the Laplacian operator... 

Those who are familiar with elementary quantum mechanics should recognize that the differential operator in Eq. (7.3.3), that is, the angular part of the Laplacian operator, is — 12 where I is the dimensionless orbital angular momentum operator of quantum mechanics (see Dicke and Witke, 1960). Thus the rotational diffusion equation can be written as... 

Reaction-diffusion equations may be written in the form of (3.2.2) if p is interpreted as a Laplacian operator multiplied by the matrix D ... 

All the simulations reported in this chapter are fully resolved. A first-order Euler method is used for time-stepping the equations. The Laplacian terms in the reaction-diffusion equations are approximated with a 9-point finite-difference formula, which to leading order, eliminates the underlying 4-fold symmetry of square grid [13]. (For spiral waves in excitable media, anisotropies in the grid have a far greater effect on solutions than do anisotropies in the boundaries.)... 

The designations employed in equation (5.6) are as follows D is the EEP diffusion coefficient in its own gas V is the Laplacian operator N is the concentration of EEPs in a gaseous phase N is the concentration of parent gas K is the rate constant of EEP de-excitation by own gas v is the rate constant of EEP radiative de-excitation ro is the cylinder radius v is the heat velocity of EEPs x, r are coordinates traveling along the cylinder axis and radius, respectively. 

For an incompressible liquid (i.e. a liquid with an invariant density which implies that the mass balance at any point leads to div v = 0) the time dependency of the concentration is given by the divergence of the flux, as defined by equation (13). Mathematically, the divergence of the gradient is the Laplacian operator V2, also frequently denoted as A. Thus, for a case of diffusion and flow, equation (10) becomes ... 

In order to go beyond the simple description of mixing contained in the IEM model, it is possible to formulate a Fokker-Planck equation for scalar mixing that includes the effects of differential diffusion (Fox 1999).83 Originally, the FP model was developed as an extension of the IEM model for a single scalar (Fox 1992). At high Reynolds numbers,84 the conditional scalar Laplacian can be related to the conditional scalar dissipation rate by (Pope 2000)... 

In this form one sees an analogy in the vorticity equation to the other transport equations— a substantial-derivative description of advective transport, a Laplacian describing the diffusive transport, and possibly a source term. It is interesting to observe that the vorticity equation does not involve the pressure. Since pressure always exerts a normal force that acts through the center of mass of a fluid packet (control volume), it cannot alter the rotation rate of the fluid. That is, pressure variations cannot cause a change in the vorticity of a flow field. 

This is known as Pick s second law it states that the time derivative of the concentration at a given point in space is equal to the diffusion coefficient times the Laplacian of the concentration. In deriving this equation it has been assumed that the diffusion coefficient is independent of concentration. 

In classical, continuum theories of diffusion-reaction processes based on a Fickian parabolic partial differential equation of the form, Eq. (4.1), specification of the Laplacian operator is required. Although this specification is immediate for spaces of integral dimension, it is less straightforward for spaces of intermediate or fractal dimension [47,55,56]. As examples of problems in chemical kinetics where the relevance of an approach based on Eq. (4.1) is open to question, one can cite the avalanche of work reported over the past two decades on diffusion-reaction processes in microheterogeneous media, as exemplified by the compartmentalized systems such as zeolites, clays and organized molecular assemblies such as micelles and vesicles (see below). In these systems, the (local) dimension of the diffusion space is often not clearly defined. 

Divide the entire mass transfer equation by the scahng factor for diffusion (i.e ,mixCAo/i )- This is an arbitrary but convenient choice. Any of the r + 2 dimensional scaling factors can be chosen for this purpose. When the scaling factor for the diffusion term in the dimensional mass transfer equation is divided by ,mixCAo/T, the Laplacian of the molar density contains a coefficient of unity. When the remaining r - -1 scaling factors in the dimensional mass transfer equation are divided by i,mixCAo/T, the dimensionless mass transfer equation is obtained. Most important, r -h 1 dimensionless transport numbers appear in this equation as coefficients of each of the dimensionless mass transfer rate processes, except diffusion. Remember that the same dimensionless number appears as a coefficient for the accumulation and convective mass transfer rate processes on the left-hand side of the equation. 

The magneto-hydrodynamic effects are important when fluid dynamics is strongly coupled with the magnetic field. It demands the diffusion of the magnetic field to be less than convection, which means the curl in equation (3-254) should exceed the Laplacian. If the space scale of plasma is L, the following parameters are required ... 

Here, 8 = DJD is the ratio of the diffusion coefficients, = d /d z is the one-dimensional Laplacian, and % is the initial concentration the reactant. These equations describe a front propagating in the positive 4 direction, where the boundary conditions are given by... 

This is the constitutive equation for a solution of Hookean dumbbells in which concentration gradients are present it includes terms up through third order. The inclusion of a term involving the Laplacian of the stress tensor was first suggested by El-Kareh and Leal [27] to account for diffusion of macromolecules across streamlines. Various other equations containing the Laplacian term have appeared m the literature these have been compared by Bens and Mavrantzas [30]. 

Standard techniques of vector analysis allow to equate the heat flow into the volume V to the heat flow across its surface. This operation leads to the linear and homogeneous Fourier differential equation of heat flow, given as Eq. (3). The letter k represents the thermal diffusivity in m s, which is equal to the thermal conductivity k divided by the density and specific heat capacity. The Laplacian operator is + d dy + d ld-z, where x, y, and z are the space coordinates. 

The differential equation of Laplace (V c = 0) describes the formation of fractal distribution of solid matter that results from very different processes (Figure 7.5). Diffusion-limited aggregation (c is the concentration), electrogalvanic deposition (c is the electric potential), and viscous invasion (c is the local pressure) are three Laplacian processes that produce similar fractal distributions. They all imply a strong positive feedback and have the same mathematics. The first two are of significance for materials synthesis. 

Up to now we have considered the diffusion-reaction equation only in a one-dimensional spatial setting. If a correct formulation is to be given in space, the second derivative term d cjd must be replaced by the Laplacian cf cjdy + (f cjdy + = Ac, taking into account the... 

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