Environmental Engineering Reference
In-Depth Information
3 Quasi-Two-Dimensional Numerical Model
Quasi-two-dimensional (Q2D) models for thin liquid layers have been successfully
applied in both hydrodynamic (Satijn et al. 2001 ; Clercx and Heijst 2003 ) and mag-
netohydrodynamic flows (Sommeria 1988 ; Figueroa et al. 2009 , 2014 ). In these
models, the governing equations are integrated in the vertical direction or along the
magnetic field lines in such a way that effects due to the boundary layer at the bot-
tom of the container are considered by means of a linear friction term. Since induced
currents are negligible in low-conductivity electrolytes, the Lorentz force is fully
known and the dimensionless governing equations of motion are
∇·
u
=
0
,
(1)
u
2 u
Re j
t + (
u
·∇ )
u
=−∇
p
+∇
+
×
B
,
(2)
U 0 , respectively.
where the velocity, u , and pressure, p , are normalized by U 0 and
ˁ
j 0 B max L 2
The characteristic velocity U 0
comes from a balance between
viscous and Lorentz forces, where the characteristic length L is the magnet side
length, j 0 the applied current density, B max the maximum magnetic field strength,
=
/ˁʽ
ˁ
the mass density and
the kinematic fluid viscosity. In turn, the current density
and the magnetic field are normalized by j 0 and B max . For numerical purposes
the Reynolds number in Eq. ( 2 ) is defined as Re
ʽ
, hence it depends
on the applied current. Due to the small layer thickness, the dominant magnetic
field component along the normal z -direction is the only one considered. For a
single permanent magnet, we assume that this component can be expressed as
(Figueroa et al. 2009 )
=
U 0 L
B z (
x
,
y
,
z
) =
B z (
x
,
y
)
g
(
z
),
(3)
where B z (
reproduces the variation of the magnetic field in the x - y plane
given by the superposition of two magnetized square surfaces, separated by a dis-
tance c , uniformly polarized in the normal direction (McCaig 1977 ). The term
g
x
,
y
)
(
) =
( ʳʵ
)
models the variation of the magnetic field in the normal direc-
tion, where z is normalized by the depth of the liquid layer h . In turn, the constant
ʳ =
z
exp
z
05 was obtained by fitting the experimental data of the magnetic field decay
(Figueroa et al. 2009 ), and
2
.
L is the aspect ratio. It has been shown that
Eq. ( 3 ) accurately reproduces the experimental magnetic field within the fluid layer
(Figueroa et al. 2009 ). An array of magnets ismodeled as the superposition of the field
generated by each magnet. In the Q2D approximation, we assume that the momen-
tum in the boundary layer at the bottom of the container is mainly transported by
diffusion in the normal direction to the wall. Therefore, the velocity components are
expressed in the form
ʵ =
h
/
u
(
x
,
y
,
z
,
t
) =[
u
(
x
,
y
,
t
)
f
(
x
,
y
,
z
),
v
(
x
,
y
,
t
)
f
(
x
,
y
,
z
),
0
] ,
(4)
 
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