Fluid Dynamics Equations 153
4.1.2. Interpretation of an equation in terms of the balance equation
The terms in equation [4.3] each have a specific form corresponding to physical
mechanisms which we have discussed earlier (section 2.1.3.3). Consider a partial
differential equation which can be written in the form [4.3]:
j
j
x
q
t
f
w
w
w
w
V
[4.4]
We can integrate equation [4.3] over a
geometric domain
D
to give:
dsnqdvdv
t
³³³
w
w
DD
[4.5]
We will interpret
f
as the volume density of a quantity
F,
the amount of which
present in
D
is equal to
.
¨
fdv
D
The first term
dv
f
³
w
w
D
of equation [4.4] represents
the variation rate in the amount of the quantity
F
contained in
D
.
On the right-hand side of equation [4.4], we have separated the terms which can
be written in the form of a divergent vector
q
i
expressed using data or functions of
the problem, and those which cannot be written in this form. In integrating over the
domain
D
, the terms written in the form of a divergence have been transformed into
flux integrals of the vector q
i
on the external surface 6 of the domain
D
: they can
thus be interpreted as transfers of the quantity
F
by the vector
q
i
. These transport
terms are thus the input-output of the quantity
F
in
D
, in other words quantities
gained by
D
from (or lost to) the exterior through the surface
6
.
On the other hand, if the vector
q
i
can be written in the form
ii
vfq
, we
interpret the term
j
j
x
q
w
w
as representing the convective transport of the quantity
F
by
the velocity field
v
i
, whose physical interpretation is not important here. In fact, the
vector
q
i
will most often be the sum of a variety of terms with different physical
interpretations. In general, a flux term will only transport a quantity without creating
any; in balance equations which are integrated over a large domain bounded by a
surface
6, fluxes which only have local effects will not appear
.
Terms which cannot be written in the form of the divergence of a vector having a
physical meaning consistent with the problem considered cannot be interpreted as
flux terms. Furthermore, take in this case a balance equation by integration on a