This document summarizes Fick's law of diffusion. It presents the transient form of Fick's law as an equation relating the rate of change of concentration to the divergence of the diffusion flux. It then derives the weak form of this equation by multiplying by a test function and integrating over a control volume. Finally, it discretizes the weak form using finite elements, resulting in a system of equations relating the concentration at the current and next time steps.

1. Fick’s Law
Martin Jones
Diffusion is a transport phenomenon that involves mass transport or mixing that does not require bulk
motion. Fick’s law for transient phenomena is written thus,
∂φ
∂t
= · (D(x) φ) (1)
Where φ and D are concentration and diffusivity, repectively, scalar functions that depend on space and
time. And we have φ(x, 0) = φ0(x) and φ · n = 0 on the surface for our boundary conditions.
Now multiply both sides by an arbitrary test function V (x) and integrate over a volume Ω. Also,
integrating by parts on the RHS, the equation evolves to the following form, (making use first of the following
theorem):
Ω
V · (D φ)dΩ =
S
V (D φ · n)dS −
Ω
D( V · φ)dΩ
Ω
∂φ
∂t
V (x)dΩ = −
Ω
D(x) V · φdΩ (2)
Now we will discretize and write with basis functions in N dimensional space (using Einstein notation) where
N is on the order of the number of elements in the mesh,
φ(x, tk) = φk
i ψi (3)
and
V (x) = Vjψj (4)
Using the finite difference formula for the time derivative and plugging in we get,
Ω
φk+1
i − φk
i
∆t
· ψi(Vj · ψj)dΩ = −
Ω
D(x) (Vjψj) · (φk+1
i ψi)dΩ (5)
Now we may write N equations with N unknowns due to the arbitrariness of V (1 on the ith
node and zero
every where else). One gets,
1
∆t
M + K φk+1
=
1
∆t
Mφk
(6)
Where M is given by,
Mi,j =
Ω
ψiψjdΩ (7)
and K is given by,
Ki,j =
Ω
D(x) ψi · ψjdΩ (8)