Civil Engineering Reference
In-Depth Information
be related to the viscous effects by a model worked out by Stinson (1991). The simplified
linear equations that satisfy the velocity
υ
, the pressure
p
, and the acoustic temperature
τ
of the fluid in a pore, i.e. the variation of the temperature in an acoustic field, are
ρ
0
∂
∂t
=−∇
p
+
η
υ
(4.1)
ρ
0
c
p
∂
∂t
=
κ
∇
∂p
∂t
2
τ
+
(4.2)
where
is the shear viscosity (the volume viscosity is neglected),
κ
is the thermal con-
ductivity,
c
p
is the specific heat per unit mass at constant pressure. For air in standard
conditions,
η
2
.
610
−
2
wm
−
1
k
−
1
. The pressure is consid-
ered constant on a cross- section of the tube. The boundary conditions at the air - frame
interface are
1
.
84 10
−
5
kg m
−
1
s
−
1
,
η
=
κ
=
υ
=
0
(4.3)
τ
=
0
(4.4)
(
0
,
0
,υ
3
)
parallel to the
x
3
axis and whose magnitude only depends on
x
1
. The variation of this velocity with
x
1
produces viscous stress. Due to the viscosity, the air is subjected to a shear force parallel
to the
x
3
axis, and proportional to
∂υ
3
/∂x
1
. More precisely, the right-hand side of plane
is subjected from the left-hand side of
to a stress
T
parallel to
X
3
. The projection
of
T
on the
x
3
axis is
A volume of air is represented in Figure 4.1, with a velocity
υ
η
∂υ
3
(x
1
)
∂x
1
T
3
(x
1
)
=−
(4.5)
The resulting force due to the stresses at
and
for a layer of unit lateral area is
d
F
parallel to the
x
3
axis such that
η
∂υ
3
(x
1
)
∂x
1
η
∂υ
3
(x
1
+
x
1
)
∂x
1
d
F
3
=−
+
(4.6)
Per unit volume of air, this force is equal to
η
∂
2
υ
3
∂x
1
X
3
=
(4.7)
Π
Π′
X
3
v(x
1
)
+∆
v
v(x
1
)
x
1
∆
x
1
X
1
X
2
Figure 4.1
A simple velocity field in a viscous fluid.
