#### A.1.1. Convective Time Derivative

[For a more rigorous discussion, see
chapter 1 of the book by R.E.Meyer, "Introduction to Mathematical Fluid
Dynamics", Wiley (1971). Part I of
the book "Fluid Mechanics", Springer (1997), by J.Spurk, gives a good
introduction to the foundamentals of the subject. ]

For our purposes, a fluid can be thought of
as a collection of "fluid particles" that is dense enough to be
considered as a continuum. Operationally, this means the characteristic
lengths of the phenomena under study must be significantly larger than the
average nearest neighbor particle distances. Here, a particle means something that can be
treated mathematically as occupying a geometric point in space but large enough
physically so that fluctuations on the atomic scale can be neglected.

As a direct extension of the mechanics of
discrete particles, the basic dynamical variables are the particle positions **x**(**z**,*t*) and velocities **v**(**z**,*t*) where **z** is the label of the
particles. Similarly, any property *F* of the system will be decribed by
F(
z,t
)
. This is called the **Lagrangian** formulism. Usually,
one simply uses the positions of the particles at, say *t *= 0, to label them, i.e.,
z=x(
z,0
)=x(
0
)
. On the other hand, the natural way to describe
a property *F* of a continuum is by
means of a (field) function
f(
x,t
)
. This is called the **Eulerian** formulism.

Now, at time *t*, the position of the particle **z** can be written as

x(
t
)=x(
z,t
)=x[
x(
0
),t
]
(A.1)

Consider now a density

f(
x,t
)=f[
x(
z,t
),t
]
=F(
z,t
)
(A.1a)

The **convective**,
or **material**,** time derivative** is defined as

D
Dt
f(
x,t
)≡
(
∂F(
z,t
)
∂t
)
z

which gives the change rate of *f* seen in a frame moving with the fluid
particle **z**. Written in terms of *f*, we have

Df
Dt
=
(
∂f[
x(
z,t
),t
]
∂t
)
z

=
(
∂f(
x,t
)
∂t
)
x
+
(
∂f(
x,t
)
∂x
)
t
⋅
(
∂x(
z,t
)
∂t
)
z
(A.2)

Now,
(
∂x(
z,t
)
∂t
)
z
is simply the velocity
Dx
Dt
=v(
x,t
)
of the fluid particle **z**. Hence

Df
Dt
=
(
∂f(
x,t
)
∂t
)
x
+v⋅
(
∂f(
x,t
)
∂x
)
t

Since *f*
is arbitrary, this implies the relation

D
Dt
=
∂
∂t
+v⋅
∇
x
(A.4)