Environmental Engineering Reference
In-Depth Information
to the Boussineq approximation for an incompressible viscous fluid. The viscous
dissipation in the energy equation is considered, and the widely used exponential
or Arrhenius type relation (Kakaç
1987
) is employed to represent the temperature
dependence of the viscosity
μ
0
is the viscosity
and
B
0
is an empirical constant, both at the reference temperature
T
0
.Usingthe
vorticity (
μ(
T
)
=
μ
0
exp
[
−
B
0
(
T
−
Tc
)
], where
ʩ
=
∂
V
/
∂
X
−
∂
U/
∂
Y
) and stream function formulation (
U
=
∂ˈ/∂
Y
,
V
=
−
∂ˈ
/
∂
X
), the flow is described by the following dimensionless equations:
2
2
∂
X
2
+
∂
ˈ
ˈ
=−
,
(1)
Y
2
∂
∂
∂
2
2
∂ˈ
∂
∂
∂
X
−
∂ˈ
X
∂
(
−
ʸ
)
X
2
+
∂
exp
B
Y
=
Y
2
Y
∂
∂
Re
∂
∂
4
B
2
exp
2
2
2
(
−
B
ʸ)
∂
ˈ
∂
ʸ
∂ʸ
∂
∂ʸ
∂
∂
ˈ
+
Y
−
Re
∂
X
∂
Y
∂
X
∂
X
Y
∂
X
∂
Y
∂ʸ
∂
2
B
exp
(
−
B
ʸ)
X
∂
∂ʸ
∂
Y
∂
−
X
+
Re
∂
∂
Y
∂ʸ
∂
2
∂
2
∂ʸ
∂
B
2
exp
2
2
(
−
B
ʸ)
ˈ
−
∂
ˈ
+
−
Y
2
X
2
Re
∂
∂
X
Y
∂
∂
2
2
2
2
B
exp
(
−
B
ʸ)
ˈ
−
∂
ˈ
ʸ
∂
ʸ
+
X
2
+
Y
2
X
2
Y
2
Re
∂
∂
∂
∂
Ri
∂ʸ
∂
+
X
,
(2)
∂
2
2
∂ˈ
∂
∂ʸ
∂
X
−
∂ˈ
∂ʸ
∂
1
Pe
ʸ
∂
ʸ
Y
=
X
2
+
Y
2
Y
∂
X
∂
∂
∂
2
2
Br
exp
(
−
B
ʸ)
ˈ
−
∂
ˈ
+
.
(3)
Pe
∂
Y
2
∂
X
2
where
V
is the dimensionless
temperature. In the above equations, all velocity components (
U
in the
X
direction
and
V
in the
Y
direction) are scaled with the inflow velocity
u
0
,
U
=
(
U
,
V
) is the dimensionless velocity vector and
ʸ
v
/
u
0
;
the longitudinal and transverse coordinates are scaled with the height of the enclosure
L
,
X
=
u
/
u
0
and
V
=
=
x
/
L
and
Y
=
y
/
L
; the temperature is normalized as
ʸ
=
(
T
−
T
C
)
/(
T
H
−
T
C
)
;
u
0
, and the temperature dependent viscosity is scaled
the pressure is scaled with
ˁ
with
B
=
B
0
(
T
H
−
T
C
)
. The nondimensional parameters are the Reynolds number,
Re
=
u
0
L
/
ʽ
, the Peclet number,
Pe
=
u
0
L/
ʱ
, the Richardson number,
Ri
=
gʲ
u
0
/
k
(
T
H
−
L
/
u
0
2
, the Brinkman number,
Br
(
T
H
−
T
0
)
=
μ
T
0
)
, and the dimensionless
viscosity
μ/μ
0
=
exp(
−
B
ʸ
), where
B
=−
ln(
μ/μ
0
)
. Here,
g
is the acceleration due
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