Task

This homework is an activity intended to apply the integration formulas given in class. The objective is to solve the error function and applying it to a problem of diffusion in fluids.

Error Function and Fick's Law

Error Function

The error function is a special and non-elementary function that is widely used in probability, statistics and diffussion processes. It is defined through the integral:

1. Using the substitution $u=t^2$ it is possible to use the described methods for impropers integrals in order to evaluate the error function. Create a routine called ErrorFunction that, given a value of x, return the respective value of the integral.

Fick's Law

In fluid physics and thermodynamics, Fick's law is a quantitative law that describes several diffusion processes for matter or heat. Situations in which there are gradients of concentration or heat, a flux that tends to homogenise the fluid arises as a consequence of random motion of constituent particles, as predicted by the second law of thermodynamics.

The Fick's law then says:

where $\vec{j}$ is the associated flux of particles of some specie or flux of heat, $D$ is the diffusion coefficient for gradients of concentrations, or the thermal conductivity in case of gradients of heat. Finally, $c$ is the studied property, concentration or temperature.

A very simple solution of this problem is obtained when studying diffusion of a set of particles in one dimension, from a boundary located at $x=0$ and a concentration fixed at $n_0$.

2. Using $n_0 = 1$ and $D = 1m^2/s$, and the previous routine for calculating the error function, plot the resulting density number of particles $n$ as a function of the position for different times (i.e. plot several curves associated with each time).