Geology Reference
In-Depth Information
I
representation, this quantity represents the veloc-
ity of the parcel
P
at time
t
. This approach is par-
ticularly useful if we want to formulate Newton's
second law of motion for a fluid particle. In fact,
if ¡(
t
) is the particle density at time
t
,then¡(
t
)
v
(
t
)
represents its
momentum density
. Therefore, in
the context of fluid dynamics, Newton's second
law states that the rate of change of the momen-
tum density, ¡
P
v
, must be equal to the net force per
unit volume,
f
, exerted on a fluid particle
P
:
dm
dt
D
¡
v
dS
(13.3)
S.
R
/
However, the total rate of mass change in
R
can be also expressed as follows:
Z
Z
dm
dt
D
d
dt
@¡
@t
dV
¡dV
D
(13.4)
R
R
Therefore, applying Gauss' theorem (see Ap-
pendix I) to (
13.3
) and equating to (
13.4
), we
obtain the identity:
Z
¡
d
v
dt
D
f
(13.1)
@¡
@t
Cr
.¡
v
/
dV
D
0
Let
q
D
q
(
x
,
y
,
z
,
t
) be a scalar field in the
Eulerian representation and consider the equiv-
alent Lagrangian variable
q
(
t
)
D
q
(
x
(
t
),
y
(
t
),
z
(
t
),
t
)
foranassignedpath
r
(
t
)
(
x
(
t
),
y
(
t
),
z
(
t
)). The
total derivative
dq
/
dt
, which represents the rate
of change of
q following the fluid
, is called the
material derivative
(or the
substantive derivative
)
of
q
. By the chain rule, we have that this quantity
can be expressed in terms of Eulerian variables:
(13.5)
R
Since this identity holds for any arbitrary re-
gion
R
, it is equivalent to:
@¡
@t
Cr
.¡
v
/
D
0
(13.6)
This equation is known as the
continuity equa-
tion
and represents a local differential form of the
mass conservation law. It should be noted that
in (
13.6
) both the density ¡ and the velocity
v
are Eulerian variables. A Lagrangian version of
this equation can be obtained differentiating the
product ¡
v
in (
13.6
) and using (
13.2
). We obtain:
dq
dt
D
d
dt
q.x.t/;y.t/;
z
.t/;t/
@q
@x
dx
dt
C
@q
@y
dy
dt
C
@q
@
z
d
z
dt
C
@q
@t
D
@q
@t
C
v
r
q
D
(13.2)
d¡
dt
C
¡
r
v
D
0
(13.7)
This formula allows to determine the rate of
change of a Lagrangian variable from the spatial
and temporal variations of the equivalent Eulerian
quantity. We are now ready to consider one of
the most important differential equations of fluid
dynamics, which is an expression of the mass
conservation law. Let us consider an arbitrary
region of space,
R
,crossedbythefluid,fixed
with respect to an Eulerian coordinate system.
Some fluid enters this volume through its surface
S
(
R
), while other fluid moves out. Clearly, at any
point on
S
(
R
) only the component of
v
along the
normal direction
n
contributes to fluid transfer in
or out of
R
. Therefore, if
m
D
m
(
t
)isthemass
of fluid in
R
at time
t
, then the outward mass flux
per unit time through
S
(
R
), which determines the
total rate of mass change,
dm
/
dt
, is given by:
Therefore, when the Lagrangian density ¡
of any fluid particle remains constant through
time, so that
d
¡/
dt
D
0, then the velocity field is
solenoidal:
r
v
D
0
(13.8)
In this instance, the fluid parcels may deform
and rotate, but they do not expand or shrink.
Consequently, the fluid is said to be
incompress-
ible
. In terms of Eulerian variables, the continuity
equation assumes now the form:
@¡
@t
C
v
r
¡
D
0
(13.9)
