Transport - Environmental Modeling - page 70

functions uðsÞ¼C exp for all values of C . Such solutions are called general

solutions and contain one or more integration constants , like C in the example. In
order to restrict the solution space, additional conditions have to be specified. The
additional requirements are usually formulated as boundary and initial conditions .
The mathematical formulation based on differential equations is completed by
these conditions.
The number of conditions, required to deliver a unique solution, is mainly
determined by the order of the differential equation. For first order equations
(which contain only first derivatives) one condition is needed, while second order
equations require two conditions. The term initial condition usually refers to the
variable time t and a condition at

ðsÞ

0. The term boundary condition refers to
a space variable x , y or z and a condition at the boundary of the model region. In
the just mentioned example s ¼ t , the initial condition u ( t ¼

t ¼

0) ¼ u 0 leads to the

unique solution uðtÞ¼u 0 exp . Typical for 1D steady states in sediment layers

is s ¼ z and the boundary condition u ( z ¼

ðtÞ

0) ¼ u 0 at the sediment-water interface.

The unique solution uðzÞ¼u 0 exp is then valid, representing an exponentially

declining profile of the unknown variable.
A fundamental classification distinguishes three types of boundary conditions.
A first type boundary condition or Dirichlet type 8 condition specifies the value of
the unknown dependent variable at the boundary. There is a concentration value to
be given in a mass transport problem and a temperature value in a heat transport
problem.
In a second type boundary condition or Neumann 9 type condition, the derivative
of the variable is specified. As this gradient is proportional to diffusive flux, one can
interpret these conditions best as a specified diffusive flux. In mass transport the
concentration gradient is to be given, while in heat transport the temperature
gradient is prescribed.
A prominent role plays the condition with a vanishing gradient. According to
Fick's Law or Fourier's Law there is no diffusive flux then. Often the condition is
simply referred to as ' no-flow' condition; but it should be kept in mind that
a vanishing gradient still allows advective flux. If there is a non-zero velocity
component across the boundary, then there usually is heat or mass flux across
that boundary even when the so called 'no-flow' condition is declared. Thus, it is
more precise to use the characterization 'no-diffusive flow' instead of 'no-flow.'
Only a zero velocity normal to the boundary and a vanishing gradient together
guarantee no flux.
The third type, Cauchy 10 -or Robin boundary condition, is the general condition
as it requires a relationship between the gradient and a given value of the variable:

ðzÞ

8 Peter Gustav Lejeune Dirichlet (1805-1859), German mathematician.
9 Carl Gottfried Neumann (1832-1925), German mathematician.
10 Augustin Louis Cauchy (1789-1857), French mathematician.