23
FLUID DYNAMICS
4. Integral form of the basic laws
The properties which define a flow are:
* density (viscosity) -
ρ
,
μ
* temperature -
Τ
* velocity -
U
:
U
x
=
U
,
U
y
=
V
,
U
z
=
W
* normal stress, pressure -
P
(
σ
x
,
σ
y
,
σ
z
)
* shear stress -
τ
xy
,
τ
xz
,
τ
yz
There are 6 unknowns:
P
,
ρ
,
T
,
U
For many flows:
ρ
=const.,
T
=const.
⇒
4 unknowns
Shear
stresses
are
proportional
to
velocity
gradients, so they don't constitute additional
unknowns.
Thus, 6 equations are required!
The tools: mechanics (Newton's)
thermodynamics
electromagnetics (Maxwell's)

24
4.1. The fundamental laws
1. Conservation of mass
(1)
2. Conservation of momentum
(3)
3. Conservation of energy
(1)
4. Equation of state
(1)
5. Entropy
(1)
4.1.1.
There are 2 approaches to derive the
theoretical model:
I - Lagrange
, history of each particle
Follow a specific particle and find its
location and properties at every instant.
Therefore,
x
(
t
),
y
(
t
), and
z
(
t
).
Then,
P
= f [
x
(
t
),
y
(
t
),
z
(
t
),
t
]
T
= g [
x
(
t
),
y
(
t
),
z
(
t
),
t
]
etc.
II - Euler
, state of the flow at a point (
x
,
y
,
z
)
P
= f [
x
,
y
,
z
;
t
] ,
T
= g [
x
,
y
,
z
;
t
]
etc.
Variations at a fixed point with time, random particles.
∂
P
/
∂
t
,
∂
T
/
∂
t
. . . . (
x
,
y
,
z
;
t
)