Environmental Engineering Reference
In-Depth Information
)
reproduces the velocity profile in the layer thickness (Beltrán
2010
). Since the elec-
trical conductivity of the electrolyte is low compared with that of liquid metals, and
the magnetic field intensity of permanent magnets is weak, induced currents in the
fluid are negligible. Therefore, it becomes unnecessary to solve the induction equa-
tion to determine the induced magnetic field. Only the applied current is relevant for
calculating the Lorentz forces (Figueroa et al.
2009
).
Substituting Eqs. (
1
) and (
2
) in the three-dimensional equations of motion and
averaging along the height of the fluid layer, we obtain the Q2D equations. A detailed
description of the averaging procedure can be found in (Beltrán
2010
; Figueroa et al.
2009
). In dimensionless terms, the equations of motion in the Q2D approximation
take the form
(
,
,
where
u
and
v
are the mean velocity components in the
x
-
y
plane, while
f
x
y
z
∂
u
x
+
∂
v
y
=
0
,
(3)
∂
∂
u
∂
∂
u
u
v
∂
u
=−
∂
P
1
Re
∇
u
˄
2
↥
QB
z
,
t
+
x
+
x
+
u
+
−
(4)
∂
∂
∂
y
∂
u
∂
∂
v
v
v
∂
v
=−
∂
P
1
Re
∇
v
˄
,
2
↥
t
+
x
+
y
+
v
+
(5)
∂
∂
∂
y
∂
where the overline in the velocity components was dropped and subindex
↥
denotes
the projection of the
operator on the
x
-
y
plane. The velocity components,
u
and
v
,
the pressure,
P
, the applied current density,
j
, and the applied magnetic field,
B
z
,are
normalized by
U
,
∇
U
2
,
J
0
and
B
max
, respectively. Here,
U
is the uniform entrance
ˁ
velocity,
is the mass density,
B
max
is the maximum intensity of the magnetic field,
and
J
0
is the magnitude of the applied current density. Dimensionless coordinates
x
and
y
are normalized by
L
, while time,
t
, is normalized by
L
ˁ
/
U
. Dimensionless
parameters
Re
and
Q
stand for the Reynolds number
Re
=
UL
/ʽ
, where
ʽ
is the
U
2
which
is the ratio of a magnetic pressure drop caused by the applied Lorentz force and
the free-stream dynamic pressure. Essentially,
Q
characterizes the strength of the
Lorentz forces. The third term on the right-hand-side of Eqs. (
4
) and (
5
) represents
the Rayleigh friction between the fluid and the bottom wall. It involves a characteristic
dimensionless timescale,
kinematic viscosity, and the Lorentz force parameter
Q
=
J
0
B
max
L
/ˁ
˄
, for the decay of vorticity due to dissipation in the viscous
layers and is given by (Beltrán
2010
)
e
−
ʳ
)
ʳ(
1
−
˄
−
1
=
1
.
(6)
e
−
ʳ
)
+
2
e
−
ʳ
−
1
ʳ
(
1
−
The considered boundary conditions are the following. At the entrance, a uniform
flow is imposed in the
x
-direction, therefore
u
=
1
,
v
=
0
,
at
x
=
0
,
0
≤
y
≤
H
.
(7)
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