Environmental Engineering Reference
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nonlinear effects determine the flow field that results from vortex interaction. One of
the classical problems of fluid dynamics consists in finding the motion of a set of
point vortices, and particularly to determine the conditions under which a certain
configuration of point vortices is stable. Since the pioneering work of Helmholtz
(
1858
) and Kelvin (
1867
), many important contributions have been published on
this problem, see for example Aref (
2007
,
2009
).Themainissueistoestablishthe
relative equilibrium of identical point vortices that rotate uniformly without change
in shape or size. The term relative equilibrium is used to distinguish it from absolute
equilibrium of the system at rest. In these systems, some points in the fluid named
co-rotating points, are in equilibrium with respect to the pattern of vortices. A com-
pilation of different flow patterns that show relative equilibrium between co-rotating
points and ideal point vortices has been presented by Dirksen (
2012
).
From the experimental side, to produce flow patterns that behave closely as a
configuration of ideal point vortices is not an easy task. However, we can explore
in a simple way the interaction of vortices created by electromagnetic forces in
a thin layer of electrolyte. This is a common non-intrusive method that has been
successfully applied to investigate the vortex dynamics and transport processes in
quasi-two-dimensional systems (Figueroa et al.
2009
,
2011
,
2014
; Durán-Matute
et al.
2010
; Rossi and Lardeau
2011
). This paper aims to study the stability and
spatio-temporal behavior of electromagnetically driven sets of vortices in a thin
electrolytic layer. Lorentz forces are generated by the interaction of a uniform D.C.
current with localized magnetic fields produced by arrays of three to ten permanent
magnets placed equidistantly on the perimeter of a circle whose diameter is much
larger than the diameter of the magnets. When a single magnet is considered, the
Lorentz force originates a dipolar vortex. If we place more than one magnet at a
close enough distance, dipolar vortices produced by each magnet interact with each
other and lead to more complex flow patterns. Therefore, the question is whether the
resulting flow is stable or unstable and presents a steady or time-dependent behavior.
Evidently, the resulting flow not only depends on the number and separation of the
magnets but also on the intensity of the applied current. In addition to the experimental
flow visualization, we obtain numerically the velocity fields in different magnet
configurations which are used to perform a Lagrangian tracking.
2 Experimental Setup and Flow Visualization
The experimental setup consists of an open rectangular frame with interior dimen-
sions of 28 cm
1.3 cm; three sides are made of Plexiglas and one is made
of glass. The container is filled up with an electrolyte solution of sodium bicarbonate
(NaHCO
3
) at 8.6% in weight. The depth of the electrolyte layer is 0.4 cm with a total
volumen of 400 cm
3
. The mass density, kinematic viscosity, and electrical conduc-
tivity of the electrolyte to ambient temperature are, respectively,
×
36 cm
×
086kg/m
3
,
ˁ
=
1
,
10
−
6
m
2
/s, and
ʽ
=
36S/m. Cooper electrodes are placed along the farther
sides of the cell connected to an adjustable D.C. voltage power supply so that a uni-
˃
=
6
.
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