This course considers slightly compressible fluid flow in porous media. The differential equation governing the flow can be derived by performing a mass balance on the fluid within a control volume. 1.1 One-dimensional Case. First consider a one-dimensional case as shown in Figure 1: A. ∆x z y x.

Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the diffusion coefficient, D. Fick's first law can be used to derive his second law which in turn is identical to the diffusion equation.

The convection–diffusion equation is a parabolic partial differential equation that combines the diffusion and convection (advection) equations. It describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection.

The solution to the 1D diffusion equation is: ( ,0) sin 1 x f x L u x B n n =∑ n = ∞ = π Initial condition: = ∫ L n xdx L f x n L B 0 ( )sin 2 π As for the wave equation, we find :

This equation is the 1D diffusion equation. It is occasionally called Fick’s second law. In many problems, we may consider the diffusivity coefficient D as a constant.

Diffusion in liquids is much slower that in gases due to the high density of the solvent and the greater chance for interactions between the solute and solvent. The value of D for a solute in a liquid depends on temperature, viscosity of liquid and size and shape of solute.

The following steps comprise the finite volume method for one-dimensional steady state diffusion - Divide the domain into equal parts of small domain. Place nodal points at the center of each small domain. Create control volumes using these nodal points.

Both advection and diffusion move the pollutant from one place to another, but each accomplishes this differently. Diffusion goes both ways (regardless of a stream direction). Diffusion D∂2c/∂x2 has a second-order derivative, which means that if x is replaced by –x the term does not change sign (symmetry).

2 The Diffusion equation The one-dimensional diffusion equation is a parabolic second-order partial differential equation of the form 𝜙 𝑡 − 2𝜙 𝑥2 =0 (1) where 𝜙= 𝜙(𝑥,𝑡) is the density of the diffusing material at spatial location 𝑥 and time 𝑡, and the parameter is the diffusion coefficient.

the diffusion equation will provide guidance in choosing appropriate algorithms for viscous fluid flow (Chaps. 15-18). In this chapter the one-dimensional diffusion equation will be used as a vehicle for developing explicit and implicit schemes. Attention will be given to the stability and accuracy of the various schemes.