The purpose of this tutorial is to introduce the main concepts behind normal and anomalous diffusion. Starting from simple, but well known experiments, a series of mathematical modeling tools are introduced, and the relation between them is made clear. First, we show how Brownian motion can be understood in terms of a simple random walk model. Normal diffusion is then treated (i) through formal...

These summarize methods for solving the diffusion equation. 1 Parabolic equations The diffusion equation is ∂φ ∂t = ∂ ∂x k ∂φ ∂x (1) This can describe thermal diffusion (for example, as part of the energy equation in compressible flow), species/mass diffusion for multispecies flows, or the viscous terms in incompressible flows. In this form, the diffusion coefficient (or conductivity), k, can b...

Conformal mapping has been applied mostly to harmonic functions, i.e. solutions of Laplace’s equation. In this paper, it is noted that some other equations are also conformally invariant and thus equally well suited for conformal mapping in two dimensions. In physics, these include steady states of various nonlinear diffusion equations, the advection-diffusion equations for potential flows, and...

Conformal mapping has been applied mostly to harmonic functions, i.e. solutions of Laplace’s equation. In this paper, it is noted that some other equations are also conformally invariant and thus equally well suited for conformal mapping in two dimensions. In physics, these include steady states of various nonlinear diffusion equations, the advection-diffusion equations for potential flows, and...

We consider convective systems in a bounded domain, in which viscous fluids described by the Stokes system are coupled using the Boussinesq approximation to a reaction-advection-diffusion equation for the temperature. We show that the resulting flows possess relaxation-enhancing properties in the sense of [12]. In particular, we show that solutions of the nonlinear problems become small when th...

Calcium [Ca] is an important second messenger which plays an important role in signal transduction. There are several parameters that affect its concentration profile like buffer source etc. The effect of stationary immobile buffer on Ca concentration has been incorporated which is a very important parameter needed to be taken into account in order to make the model more realistic. Interdepende...

Conformal mapping has been applied mostly to harmonic functions, i.e. solutions of Laplace’s equation. In this paper, it is noted that some other equations are also conformally invariant and thus equally well suited for conformal mapping in two dimensions. In physics, these include steady states of various nonlinear diffusion equations, the advection–diffusion equations for potential flows, and...

We present a new monotone finite volume method for the advection–diffusion equation with a full anisotropic discontinuous diffusion tensor and a discontinuous advection field on 3D conformal polyhedral meshes. The proposed method is based on a nonlinear flux approximation both for diffusive and advective fluxes and guarantees solution non-negativity. The approximation of the diffusive flux uses...

It is shown: 1) that in two-dimensional, incompressible, viscous flows the vorticity-area distribution evolves according to an advection-diffusion equation with a negative, time dependent diffusion coefficient and 2) how to use the vorticity-streamfunction relations, i.e., the so-called scatter-plots, of the quasi-stationary coherent structures in order to quantify the experimentally observed c...

An upper bound on the mixing efficiency is derived for a passive scalar under the influence of advection and diffusion with a body source. For a given stirring velocity field, the mixing efficiency is measured in terms of an equivalent diffusivity, which is the molecular diffusivity that would be required to achieve the same level of fluctuations in the scalar concentration in the absence of st...