Applied Mechanics and Materials
ISSN: 1662-7482, Vols. 110-116, pp 880-885
doi:10.4028/www.scientific.net/AMM.110-116.880
© 2012 Trans Tech Publications, Switzerland
Online: 2011-10-24
Super Accelerated Flow in Diverging Conical Pipes
G. J. Gutierrez1,a, A. Lopez Villa2,b, A. Torres3,c, S. Peralta4,d
and C. A. Vargas5,e
1,2,3,4
SEPI ESIME Azcapotzalco, Instituto PolitÉcnico Nacional, Mexico City, México
Departamento de Ciencias Básicas e Ingeniería, UAM Azcapotzalco, Mexico City, México
5
a,b,c,d
[email protected], [email protected]
Keywords: component; Free fall; inviscid liquids; nonlinear dynamics
Abstract. The motion of the upper free surface of a liquid column released from rest in a vertical,
conical container is analyzed theoretically and experimentally. An inviscid, one-dimensional model,
for a slightly expanding pipe's radius, describes how the recently reported super free fall of liquids
occurs in liquids of very low viscosity. Experiments agree with the theoretical results.
Introduction
Consider a chain of length L initially attached at both ends to a horizontal support. As one end is
suddenly released, the chain begins to fall in the gravitational field. Moreover, if the initial distance
between the ends of the chain is very close to L, that is, when the chain is initially stretched to its
maximum length, the vertical motion of the chain tip becomes identical with the motion of a freely
falling body [1]. However, when the horizontal separation ∆l between the ends of the chain is
shorter than L, that is, the chain is tightly folded, the falling chain tip will attains an acceleration
that is larger than the gravity acceleration, g, i.e., there is a super free fall. All these results have
been confirmed both in experiments and numerical simulations [1,2].
In this work the problem of the free fall of a mass of low viscosity liquid in a vertical, weakly
expanding (conical) pipe [3] is revisited. In this case, recently has been found that the motion of the
free surface of liquid has an effective acceleration that overcomes the acceleration due to the gravity
[3]. They have argued that this motion in a conical pipe super accelerates downward due to a force
originated by a positive pressure gradient at the upper interface. Thus, The pressure force added to
the pure gravitational body force induces this type of motion.
The theoretical model here developed, also based on the slender slope approximation, allows
showing that this condition is unnecessary. Instead, appears that the relative levels of filling are
crucial to get a super free fall in this geometry.
The division of this work is as follows. In next section the formulation of the governing
equations for the one-dimensional motion of an inviscid liquid in a vertical cone is given. Next, in
section 3 are discussed the numerical results concerning the super accelerated motion of the free
surface for several initial levels of filling. Finally, in section 4 the main conclusions and
perspectives of this work are presented.
Theory
When an ideal liquid is in a vertical, cylindrical pipe and suddenly the lower part of the cylinder
is opened, the gravity will accelerate the liquid. The main flow is one-dimensional and the
conservation equations of mass and momentum are, respectively,
∂u
= 0,
(1)
∂z
∂u
∂p
ρ
= − + ρg .
(2)
∂t
∂z
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans
Tech Publications, www.ttp.net. (#69828562, Pennsylvania State University, University Park, USA-19/09/16,06:11:25)
Applied Mechanics and Materials Vols. 110-116
881
In the previous equations u is the downward velocity, z is the vertical velocity pointing
downward, p is the pressure, ρ is the liquid density and t is the time. The solution of Eq. (1) allows
finding that
u = u (t ).
Now consider the same liquid in a vertical, conical pipe where the initial level of the free surface
measured from its apex is H2(0) and the position of the bottom exit is H1 (see Fig. 1). When the
bottom exit is suddenly opened the liquid falls due the gravity action. The continuity and
momentum equations, in the slender slope approximation, i.e., when the angle of aperture α is
small, which is valid for a smoothly expanding tube, are now
1 ∂ 2
(z u ) = 0,
(3)
z 2 ∂z
∂u 
∂p
 ∂u
(4)
ρ  + u  = − + ρg .
∂z 
∂z
 ∂t
In this case the velocity can be obtained from Eq. [3], in the form z 2 u = A(t ) and therefore
A(t )
u= 2 .
(5)
z
Using this last result in Eq. (4), it is obtained that
A2 
∂p
ρ  dA

− 2 3  = −
+ ρg .
2 
z  dt
z 
∂z
(6)
If Eq. (6) is integrated from H2 to H1 the resulting height-averaged momentum equation is
dA  H 1 − H 2  A 2  H 14 − H 24 
 = g (H 1 − H 2 ).


−
dt  H 1 H 2  2  H 14 H 24 
By the way, the liquid velocity is given also by
dH 2
u=
,
dt
and using the later equation in Eq. (5) it is easy to found that
dH 2 A(t )
= 2 ,
dt
H2
or
dH 23
= 3 A(t ).
dt
(7)
(8)
(9)
A second derivative of the previous equation gives
Fig. 1. Two-dimensional projection of a vertical conical pipe filled with an inviscid liquid. In the figure are shown the coordinate system and the
positions of the upper free surface (z=H₂(0)) and the lower free surface (z=H₁) which appears when the pipe is suddenly opened.
882
Mechanical and Aerospace Engineering, ICMAE2011
d 2 H 23
dA
=3 .
2
dt
dt
(10)
The substitution of Eq. (10) into Eq. (7) yields
1  d 2 H 23   H 1 − H 2  1  dH 23 



− 
3  dt 2   H 1 H 2  18  dt 
2
 H 14 − H 24

4
4
 H1 H 2

 =

(11)
g (H 1 − H 2 ).
Rearranging terms and using the identity
2
2
 d 2 H 23 
 dH 2 
2 d H2

 = 3H 2
+ 6H 2 
 ,
2 
dt 2
 dt 
 dt 
in Eq. [11] it is finally found that
2
2
d H 2 H1
H 2 H 22
1  dH 2   1
1 
=
g
+
+
+ 3 −



.
2
2
H2
2  dt   H 1 H 1 H 1 H 2 
dt
(12)
In this later equation it should be noted that the term on the left hand side is the acceleration of
the upper free surface of the liquid, H2. On the right hand side appears the term (H1/H2)g which can
be much larger than g if H2<<H1. The second term has acceleration units but involves to the square
of velocity of the free surface. This highly nonlinear equation should be solved numerically giving
the conditions of the free fall: at t=0 the upper surface is at H2(0) and there is not initial velocity,
(dH2/dt)t=0=0.
Numerical solutions
In the numerical computations was assumed that H1=1 m. The solution of Eq. (12) was obtained
by using a Runge-Kutta method of fourth-order. In order to show the effect of the level of filling,
H2(0), on the overall motion of the liquid, three values of this quantity were chosen:
Fig. 2. Plots of the evolution of the position of the upper free surface, H₂-H2(0), as a function of time. It is assumed that initially the upper free surface
was at the position H2(0). Three different initial positions are assumed, as can be seen in the box in the plot. In all cases the lower free surface was at
H₁=1 m
H2(0)=0.20 m, 0.55 m and 0.70 m. The first value corresponds to a high level of filling, whereas
H2(0)=0.70 m indicates that the level of filling is close to the bottom exit.
As a result, in Fig. 2 are plotted the spatial evolution of the free surfaces, H₂-H2(0), as a function
of time for the three different values of H2(0). There, also is plotted the free fall case. It is noted that
for these three cases the rate of change of the position of the free surface as a time function is faster
than that corresponding to the free fall. It is confirmed through the Figs. 3 and 4 where are plotted
the corresponding instantaneous velocities and the accelerations of the free surfaces.
Applied Mechanics and Materials Vols. 110-116
883
Fig. 3. Instantaneous velocities of the upper free surfaces, u= dH₂/dt, as functions of time. The different curves correspond to initial positions, H2(0) of
Fig. 2
Fig. 4. Plots of the non dimensional instantaneous accelerations of the upper free surfaces, g∗/g, as functions of time. Plots correspond to cases given in
Fig. 2
From Fig. 3 it is possible to conclude that, for H2(0)= 0.55 m and 0.70 m, the velocities behave
as linear functions of t. Thus, the motions occur following relations of the form u= dH₂/dt=g*t,
where g* is the effective acceleration and, in this case, it can be computed as g*= du/dt, which is the
slope of the straight lines in Fig. 3. This behavior is the mark of the uniformly accelerated motion
and the respective accelerations are: g*=13.92 m/s2 for H2(0)= 0.55 m and g*=15.15 m/s2 for
H2(0)= 0.70 m. For the value H2(0)= 0.20 m the behavior of the instantaneous velocity of the
interface is non linear and the motion is non uniformly accelerated.
In Fig. 4 are given the plots of the non dimensional accelerations, g*/g, as functions of time. In
Mexico City g=9.779 m/s2. There the plots show, in a more accurate way, that the kinematics of the
free surface when H2(0)=0.20 m is complex. In a first stage the motion is fast, it has a high
acceleration, after the acceleration of this motion decreases but even is super accelerated and finally
the acceleration increases again. In this case the description of the motion can be made through the
time-averaged acceleration which is defined as <g*>=∫∆tg*dt/∫∆tdt where ∆t is the time interval
during which the motion occurs. Computations allow found that for the case with H2(0)=0.20 m the
average acceleration of the overall motion is <g*>=28.83 m/s2 or <g*>/g=2.94. The estimation of
the average acceleration for the other two values of H2(0) yield essentially the same values obtained
through the estimation of the slope of the straight lines of Fig. 3.
In order to compare the goodness of the results here derived with the results given in [3] it is
convenient to remember that they give the initial acceleration in the form g*=g/(1-β) where, in the
present context, β=1-(H2(0)/H1). Villermaux and Pomeau [3] studied two cases. In case 1: β=0.33,
α=1.71o, H1=1.1 m and H2(0)=0.737 m. For case 2: β=0.52, α=4o, H1=0.52 m and H2(0)=0.25 m.
Using this values, in Fig. 5 are plotted the time-averaged accelerations by assuming the two
constant values of H1 and taken in to account several values of H2(0), i.e., several levels of filling.
884
Mechanical and Aerospace Engineering, ICMAE2011
In such a plot also were plotted the instantaneous accelerations computed by using the model
here developed. Both models give very similar results for levels of filling very close to the
respective value of H1. Moreover, the experiments discussed in [3] are two square points in the
plots.
Fig. 5. Plots of the average non dimensional acceleration of the upper free surface, <g∗>/g, as a function of the initial position of the upper free
surface, H2(0). The two square symbols correspond to the experimentally analyzed cases in [3]
Figure 5 shows that different size of conical containers and different levels of filling produce
different regimes of super free fall. Notice that when H2(0) is close to the apex, the ratio <g*>/g
increases its value and, logically, when H2(0)→H1, <g*>/g→1. It means that the greater level of
filling the larger value of the average acceleration. The substantial difference between the curves
given by the Villermaux et al model and the model developed here for small values of H2(0) may be
due to the acceleration in the first case corresponds to an instantaneous acceleration at short times
whereas the average acceleration in the model here discussed is for the overall flow.
Experiments
In this part an experimental realization of the super accelerated system it is discussed. A conical
glass pipe was used. Its dimensions were 48 cm length, 7 cm bottom diameter and 4 cm upper
diameter. In Fig. 6 it is shown a picture of the case where H2(0)=0.65 m and H1=0.79 m the pipe
was filled with ethanol, the picture corresponds to the initial position, when no flow is occurring.
Also, log-log plots are given on the left hand side of Fig. 6.
In the upper plot H2 (0) =0.36 m and <g*>/g=2. 3. For the lower plot H2(0)=0.65m and
*
<g >/g=1.2. In conclusion, both of these results show that effectively the super acceleration it is a
function of the level of filling.
Conclusions
It is apparent that the theoretical model here developed predicts the super free fall in the motion
of the upper interface of a liquid in a conical container when it falls only during the action of the
gravity field. Moreover, has been shown that the effective acceleration depends strongly on the
level of filling, H2(0) and on the length of the container, H1. Experiments given here show that the
theoretical predictions are corrects. Experiments by other authors [3] have shown that during the
free fall was observed the formation of nipples at the middle part of the free surface. The simple
model here presented does not allow quantifying the properties of such structure, but surely their
increase as the flow occurs, is due to the increase of cross-section. Theoretical and experimental
studies in this direction are now conducted.
.
Applied Mechanics and Materials Vols. 110-116
885
Fig. 6. Picture of the conical pipe used in experiments (rhs) and experimentalplots (lhs)of the position of as a function of time. There is indicated the
value of H2(0)
References
[1] Tomaszewski W, Pieranski P, Geminard JC, ( 2006) Am. J. Phys. 74:776-783.
[2] Calkin MG, March RH, Am. Jour. Phys. (1989) 57: 154-157.
[3] Villermaux E, Pomeau Y, Jour. Fluid Mech. (2010) 642: 147-157.