The way most of us learn scientific disciplines is by the use of well established formulas (something like the ideal gas law relating the pressure, temperature and volume, PV=nRT). It is likely true that many scientific disciplines are vitally dependent on such "pure facts", however it is also true that any active discipline will consist of many findings that are considerably more controversial. After years of repetitive learning, many students find this unsettling. Fluid mechanics has a few well known formulas found in elementary physics texts (i.e. Bernoulli's equation), but for the most part, the "flowing" nature of the problems requires differential equations for a valid description. This is because differential equations relate the value of a function to its rate of change with time. While the laws of fluid mechanics in the form of the Navier-Stokes equations are partial differential equations (the quantities one wants to solve for are functions of more than one variable) here I discuss some general aspects of differential equations.

Perhaps
the
simplest
differential
equation
is
d/dt A(t)
=
k
A(t),
the
solution
of
which
defines
the
family
of
exponentials,
A(t)=
C
exp(kt).
Exponentials
are
generally
taught
quite
early
in
the
mathematics
curriculum,
but
their
peculiar
properties
only
truly
make
sense
when
treated
as
the
single
example
of
a
function
for
which
the
derivative
is
proportional
to
the
function
itself.
This
is
the
first
way
one
can
use
differential
equations:
to
define
a
new
family
of
functions
that
one
does
not
have
a
formula
for
ahead
of
time.
The
differential
equation
governing
exponentials,
like
many
other
simple
DEs
(the
simple
harmonic
oscillator,
for
one),
is
**
linear**.
This
means
if
A_{1}(t)
and
A_{2}(t)
are
solutions,
then
A_{1}(t)+A_{2}(t)
is
also
a
solution.
Much
like
linear
functions
(lines)
are
the
best
understood
functions
of
one
variable,
linear
equations
(be
they
differential
or
not)
are
the
best
understood
equations.
Indeed
it
is
not
that
far
off
the
mark
to
say
that
two
hundred
years
of
applied
mathematics
was
largely
concerned
with
deriving
and
solving
linear
equations.
It
may
be
surprising
that
a
great
many
physical
phenomena
can
be
described
by
linear
equations
(most
notably
Quantum
Mechanics
is
a
fundamentally
linear
theory).
We
note
that
as
computation
has
grown
both
in
power
and
availability,
more
and
more
nonlinearity
has
entered
into
the
discourse
of
modern
applied
mathematics.

At its most basic nonlinearity means we can no longer take two solutions and add them to produce a third. However, this is a purely mathematical road block, and nonlinearity would be a footnote were it not for the following two points:

- Non-linearity enriches the possible dynamics of a system. On the one hand it can lead to disorder, as in the notion of "chaos" popularised in the late 1980s. A chaotic system can have two repetitions of the same experiment start very close to one another, but drift apart very quickly (indeed exponentially). However nonlinearity can also serve to preserve order. For linear waves an initial shape breaks up into a great many small ripples of varying lengths, each represented by a sine or cosine. For nonlinear waves, however, it is possible that an initial shape breaks up into a train of highly stable waves (called solitons) that propagate without changing form and interact as if they were particles (think pool balls) instead of waves.
- Non-linearity is a natural part of physical theories. In a flowing fluid the analogue of Newton's second law tells us how the velocity changes at a point. Some of the change must be due to the momentum carried by fluid that will arrive at the point an instant from now. In turn, this momentum, by definition depends on the velocity of the fluid. Of course the mathematics here is complicated (AMATH 231 is a good start, with AMATH 361 finishing the job).

In physical systems the balance between order and disorder is much more robust than in simplified mathematical systems. Thus for example, a wind has a persistent direction even it has smaller variations (that cause a flag to flutter, for example). Physical systems tend to be forced with typical timescales (nights are cooler than days, spring is wetter than summer, the earth's orbit varies on a time scale of tens of thousands of years causing ice age cycles). Thus non-linearity in physics is often a balancing act between nearly random aspects and stable patterns.