1. Transport and mixing
1.1 The material derivative
Let
be the velocity of a fluid at the point
and time . Consider also
some scalar field
such as the temperature or density.
We are interested not only in the
partial derivative of
with respect to time,
, but also in the time derivative following the
motion of the fluid,
.
The latter is the socalled
material derivative
:
.
(1.1)
In Cartesian coordinates,
,
and
.
(1.2)
We will also be using spherical coordinates. The notation
is conventional for the
(eastward, northward, radially outward) components of the velocity field. In addition to the radial
distance from the origin, , we use the symbol
for latitude (
at the south pole,
at the
north pole) and
for longitude (ranging for zero to
).
The gradient operator in these coordi
nates is
,
(1.3)
so that
.
(1.4)
The material derivative of a vector, such as the velocity
itself, is defined just as in 1.1.
But care is required when considering the material derivative of a component of a vector if the
unit vectors of one’s coordinate system are position dependent, as they are in spherical coordi
nates. For example, the radial component of the material derivative of the velocity is not equal to
the material derivative of the radial component of the velocity; rather,
.
(1.5)
The second term on the rhs is sometimes referred to as the
metric term
.
After obtaining
expressions for
, etc., one arrives at the sphericalcoordinate expression for the acceleration
of a fluid parcel (see Problem 1.1):
V x
t
,
(
29
x
x y z
, ,
(
29
=
t
χ
x
t
,
(
29
χ
χ
t
∂
∕
∂
d
χ
dt
∕
d
χ
dt
∕
χ
x
V x
t
,
(
29
t
δ
+
t
t
δ
+
,
(
29
χ
x
t
,
(
29
–
[
]
t
0
→
δ
lim
t
δ
∕
=
∂ ∂
t
∕
V
∇
⋅
+
(
29χ
=
V
u v w
, ,
(
29
=
∇
∂
x
∂
y
∂
z
,
,
(
29
=
d
χ
dt
∕
χ
t
∂
∕
u
χ
x
∂
∕
v
∂χ
y
∂
∕
w
∂χ
z
∂
∕
+
+
∂
+
∂
=
u v w
, ,
(
29
r
θ
π
2
∕
–
π
2
∕
λ
2
π
∇
λ
ˆ
r
θ
cos
(
29
1
–
∂ ∂λ
∕
(
29
θ
ˆ
r
1
–
∂ ∂θ
∕
(
29
r
ˆ
∂ ∂
r
∕
(
29
+
+
=
d dt
∕
∂ ∂
t
∕
r
θ
cos
(
29
1
–
u
∂ ∂λ
∕
(
29
r
1
–
v
∂ ∂θ
∕
(
29
w
∂ ∂
r
∕
(
29
+
+
+
=
V
r
ˆ
d
V
dt

⋅
d
dt

r
ˆ
V
⋅
(
29
V
d
r
ˆ
dt

⋅
–
=
d
r
ˆ
dt
∕
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.
(1.6)
Another case of interest is the material derivative of an infinitesimal material line segment.
Suppose that
and
are infinitesimally close to each other, with the vector
pointing
from one point to the other, and let
be the evolution of this vector assuming that its end
points move with the flow.
Then, from the figure below,
. To make sure
that we understand this notation, suppose that the line segment is oriented vertically at the time in
question.
Then, in Cartesian coordinates,
.
We refer to the third term as stretching, as it lengthens or shortens the line segment, and the other
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 Spring '11
 Wereley
 Derivative, dt, Molecular diffusion

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