Biomedical Engineering Reference
In-Depth Information
15
Solution of the one-dimensional
convection-diffusion equation by
means of the Finite Element Method
15.1
Introduction
This chapter extends the formulation of the previous chapter for the one-
dimensional diffusion equation to the time-dependent convection-diffusion equa-
tion. Although a good functioning of the human body relies on maintaining a
homeostasis or equilibrium in the physiological state of the tissues and organs,
it is a dynamic equilibrium. This means that all processes have to respond to
changing inputs, which are caused by changes of the environment. The diffusion
processes taking place in the body are not constant, but instationary, so time has
to be included as an independent variable in the diffusion equation. Thus, the
instationary diffusion equation becomes a
partial
differential equation.
Convection is the process whereby heat or particles are transported by air or
fluid moving from one point to another point. Diffusion could be seen as a process
of transport through immobilized fluid or air. When the fluid itself moves, particles
in that fluid are dragged along. This is called convection and also plays a major
role in biomechanics. An example is the loss of heat because moving air is passing
the body. The air next to the body is heated by conduction, moves away and carries
off the heat just taken from the body. Another example is a drug that is released
at some spot in the circulation and is transported away from that spot by means of
the blood flow. In larger blood vessels the prime mechanism of transportation is
convection.
15.2
The convection-diffusion equation
Assuming that the source term
f
=
0, the unsteady one-dimensional convection-
diffusion equation can be written as
∂
u
∂
t
+
c
∂
u
∂
x
,
v
∂
u
∂
∂
x
∂
x
=
(15.1)
with
u
a function of both position
x
and time
t
:
u
=
u
(
x
,
t
) .
(15.2)
