Environmental Engineering Reference
In-Depth Information
nutrients) because the equilibrium heights attained through the xylem are unlimited
and it promotes photosynthesis and transpiration in the leaves.
Historically, for nearly three hundred years the dynamic problem, to the time
evolution liquid's free surface, towards the equilibrium, remained unresolved until
by 2008 our group published a paper with the solution to this problem (Higuera et al.
2008
).
In this work we report a series of experiments where the wedge's arista is tilted
with to the vertical. This slope changes the shape of the equilibrium and dynamical
profiles with respect to cases with vertical edges. Particularly, we are interested in
finding the equilibrium shapes of the free surfaces, theoretically and experimentally.
To reach such a goal, this work is divided as follows: in the next section we reviewed
the problem of the capillary rise in the Taylor-Hauksbee cell. Later on, in the same
section, we proposed a theoretical description of the equilibrium profiles in cells with
tilted edges. Experimental data yield that such description is a suitable way to show
such profiles. In Sect.
3
we show a set of dynamical profiles. Finally in Sect.
4
we
present the main conclusions for this work.
2 Capillary Rise in the Taylor-Hauksbee Cell
2.1 Cell with a Vertical Edge
The wedge-shaped gap between two vertical plates intersecting at an angle
1
is initially empty. At a certain moment, the lower edges of the plates get in touch
with a liquid of density
ˁ
, dynamic viscosity
ʱ
μ
, and surface tension
˃
. The liquid
wets the plates with a contact angle
2 and therefore rises between the plates
by capillary action as shown in Fig.
1
a. The ratio of the two principal curvatures
of the free surface of the liquid between the plates is small, on the order of
ʸ
<
ˀ
/
.The
normal section of maximum curvature, by a plane nearly normal to the plates, is
approximately a circular arc of radius
ʱ
, where x is the distance to the line
of intersection of the plates. The pressure jump across the surface is approximately
ʱ
x/2cos
ʸ
2
˃
cos
ʸ
p
=
.
(1)
ʱ
x
At equilibrium, the height
H
e
(
x
)
of the meniscus above the level of the outer
liquid is determined by the balance
p
=
ˁg
H
e
,
(2)
where
g
is the acceleration due to gravity. This balance gives the rectangular hyper-
bola (Concus and Finn
1969
)
2
˃
cos
ʸ
H
e
=
x
.
(3)
ˁgʱ
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