Asymptotic Analysis and Singular Perturbation Theory

To nondimensionalize these equations with respect to (U, L, ρ), we introduce dimensionless
variables
and find that
Here,
u ∗ = u
U , p∗ = p
ρU 2 , x∗ = x
L , t∗ = Ut
L ,
u ∗ t ∗ + u∗ · ∇ ∗ u ∗ + ∇ ∗ p ∗ = ε∆ ∗ u ∗ ,
∇ ∗ · u ∗ = 0.
ε = ν 1
=
UL Re .
The boundary conditions correspond to a flow of speed 1 past a sphere of radius
1. Thus, assuming that no other parameters enter into the problem, the drag
computed from the solution of these equations depends only on ε, as obtained from
the dimensional analysis above.
Dimensional analysis leads to continuous scaling symmetries. These scaling symmetries
are not the only continuous symmetries possessed by differential equations.
The theory of Lie groups and Lie algebras provides a systematic method for computing
all continuous symmetries of a given differential equation [18]. Lie originally
introduced the notions of Lie groups and Lie algebras precisely for this purpose.
Example 1.13 The full group of symmetries of the one-dimensional heat equation
ut = uxx
is generated by the following transformations [18]:
u(x, t) ↦→ u(x − α, t),
u(x, t) ↦→ u(x, t − β),
u(x, t) ↦→ γu(x, t),
u(x, t) ↦→ u(δx, δ 2 t),
u(x, t) ↦→ e −εx+ε2 t u(x − 2εt, t),
2
1 −ηx x
u(x, t) ↦→ √ exp u
1 + 4ηt 1 + 4ηt 1 + 4ηt ,
t
,
1 + 4ηt
u(x, t) ↦→ u(x, t) + v(x, t),
where (α, . . . , η) are constants, and v(x, t) is an arbitrary solution of the heat
equation. The scaling symmetries involving γ and δ can be deduced by dimensional
arguments, but the symmetries involving ε and η cannot.
For further discussion of dimensional analysis and self-similar solutions, see [1].
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