Transport: Non-diffusive, flux conservative initial value problems and ...

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5.2 Non-diffusive initial value problems and the material
derivative
As a representative problem we will consider conservation of mass for a nondiffusive,
stable tracer in one dimension.
∂ρc
∂t
+ ∂ρcV
∂x
= 0 (5.2.1)
Using either equation (5.1.1) or in the special case that ρ and V are constant, (5.2.1)
can also be written as
∂c ∂c
+ V = 0
∂t ∂x
(5.2.2)
This combination of partial derivatives is known as the material derivative and is
often written as
DV ∂ ∂
≡ + V
Dt ∂t ∂x
(5.2.3)
The material derivative (or Lagrangian derivative) has the physical meaning that it
is the time rate of change that would be experienced by a particle traveling along
at velocity V. The next two examples will try to show this.
Example 1: Solutions for constant velocity If V is constant in (5.2.2) then it can
be shown that the concentration has the general solution that c(t,x) = f(x − V t)
where f is any arbitrary function. To show this let us first define a new variable
ζ = x − V t and set c = f(ζ). Therefore by simple substitution and the chain-rule
∂c
∂t
df ∂ζ
=
dζ ∂t
= −V df
dζ
∂c df ∂ζ df
= =
∂x dζ ∂x dζ
(5.2.4)
(5.2.5)
Substitution these equations into (5.2.2) shows that it is satisfied identically. But
what does it mean? It means that any arbitrary initial condition f(x0) just propagates
to the right at constant speed V . To show this just note that for any constant
value of ζ, f remains constant. However a constant value of ζ = x0 implies that
x = x0 + V t i.e. the position x simply propagates to the right at speed V .
Example 2: Non-constant Velocity and the method of characteristics It turns
out that Eq. (5.2.2) can be solved directly even if V isn’t constant because the
material derivative applies to a particle in any flow field, not just constant ones. To
show this, let us assume that we can write the concentration as
c(t,x) = c(t(τ),x(τ)) = c(τ) (5.2.6)
where τ is the local elapsed time experienced by a particle. Thus the parametric
curve l(τ) = (t(τ),x(τ)) is the trajectory in space and time that is tracked out by