Civil Engineering Reference
In-Depth Information
In the situation of greatest interest, the three forces in the Navier -Stokes equation
have the same order of magnitude, and the characteristic values are related by
O
p
c
L
O
η
c
υ
c
l
2
ρ
0
c
ω
c
υ
c
=
=
(5.71)
This leads to
O(ε
−
2
)
Rt
L
=
O(ε
−
2
)
S
L
=
O(
1
)
Q
L
=
(5.72)
The dimensionless quantities satisfy the following equations
ε
2
η
∗
v
∗
+
ε
2
(λ
∗
+
η
∗
)
∇
·
υ
∗
)
p
∗
=
jω
∗
ρ
0
υ
∗
∇
(
−∇
(5.73)
jω
∗
ξ
∗
+
ρ
0
∇
·
υ
∗
=
0
(5.74)
υ
∗
/
s
=
0
(5.75)
In Equation (5.66) the relative variations of pressure, temperature, and density, are of
the same order of magnitude
O
ξ
ρ
0
O
τ
T
0
O
p
P
0
=
=
(5.76)
leading to
O(ρ
0
c
p
τ)
=
O(p)
(5.77)
In Equation (5.65) the only dimensionless number which must be estimated is
ωρ
0
c
p
τ
c
L
2
κτ
c
=
|
jωρ
0
c
p
τ
|
|
N
L
=
(5.78)
κτ
|
The thermal skin depth given by Equation (4.59) is of the same order of magnitude
as the viscous skin depth, and is of the same order of magnitude as the pore size, leading
to
κ
ωρ
0
c
p
=
O(l
2
)
,andto
O(ε
−
2
)
N
L
=
(5.79)
The dimensionless quantities satisfy the following equations
ε
2
(κ
∗
τ
∗
)
jω
∗
(ρ
0
c
p
τ
∗
−
p
∗
)
=
(5.80)
P
0
ξ
∗
τ
∗
T
0
p
∗
=
ρ
0
+
(5.81)
τ
∗
/
s
=
0
(5.82)
