Chapter 2: FAQs 217
viscosity. This parameter appears in the Navier–Stokes
equation which, together with the Euler equation for
conservation of mass, governs the flow of fluids. And
this means the flow of air around an aircraft, and the
flow of glass. These equations are generally difficult to
solve. In university lectures they are solved in special
cases, perhaps special geometries. In real life during the
design of aircraft they are solved numerically. But these
equations can often be simplified, essentially approxi-
mated, and therefore made easier to solve, in special
‘regimes.’ The two distinct regimes are those of high
Reynolds number and low Reynolds number. WhenRe
is large we have fast flows, which are essentially invis-
cid to leading order. Assuming thatRe1 means that
the Navier–Stokes equation dramatically simplifies, and
can often be solved analytically. On the other hand if
we have a problem whereRe 1thenwehaveslow
viscous flow. Now the Navier–Stokes equation simpli-
fies again, but in a completely different way. Terms
that were retained in the high Reynolds number case
are thrown away as being unimportant, and previously
ignored terms become crucial.
Remember we are looking at what happens when a
parameter gets small, well, let’s denote it by. (Equiv-
alently we also do asymptotic analysis for large para-
meters, but the we can just define the large parameter
to be 1/
.) In asymptotic analysis we use the following
symbols a lot:O(·),o(·)and∼. These are defined as
follows.
We say that f( )=O(
g( ))
as → 0if lim
→ 0f( )
g( )is finite.We say that f( )=o(
g( ))
as → 0if lim
→ 0f( )
g( )→ 0.