Homework 2.1 (due Jan. 26)
1. (EK Ch.9, ex. 11) Suppose the diffusion coefficent of a substance is a
function of concenration, i.e. D = f (c). Suppose that Fick’s law holds, and
there are no sources/sinks. Show that the diffusion/conservation equation
becomes:
!2
∂2c
∂c
∂c
0
= D 2 + f (c)
.
∂t
∂x
∂x
2. (EK Ch.9, ex.9) For a planar flow consider a small rectangular region
of dimensions ∆x × ∆y. Carry out steps, analogous to those in class, to
derive the two dimensional version of the conservation equation.
3. Suppose there is a field
1
1
Ψ = − x2 − y 2
2
2
that attracts particles to the origin. Let the initial concentration of particles
be c = 4 (constant everywhere).
a) What is the initial flux of the flow created by this field? (take α = 1)
b) We place a bar of unit length at point (1,1) parallel to the x-axis. What
is the number of particles crossing this bar during a time interval of length
∆t? You can still assume c = 4 if ∆t is a small interval just after the field is
turned on.
1