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71Конвективные течения..., 2003
GRANULAR MATTER SEDIMENTATION IN TILTED CHANNEL FILLED WITH LIQUID AND SUBJECT TO LONGITUDINAL VIBRATION
V.G. Kozlov1, A.V. Chigrakov2, A.A. Ivanova2, P. Evesque3
'institute of Continuous Media Mechanics RAS,
1, Acad. Korolev str., 614013, Perm, Russia
2Perm State Teacher Training University,
24, Sibirskaya str., 614990, Perm, Russia
3Ecole Centrale Paris, France
Abstract. New experimental results of mean behaviour of granular matter in viscous liquid in tilted and vertical channels subject to longitudinal vibration are presented. The effect of compaction of liquefied sand into blocks is found. The blocks are extended along the channel and fill all the cross section that reduces the sedimentation speed by few orders. The qualitatively new effect is the vibrational stabilisation of the lower boundaries of the blocks due to mean interaction (attraction) of particles in viscous liquid. The demarcations are characterised with sharp discontinuity of density (the concentration of granular particles below the boundary is few orders less than above it). In the case of vertical channel it results in the formation of liquefied granular matter into a single block with stable lower and upper demarcations, slowly falling down.
The study of granular matter behaviour under vibrations is an actual problem from a practical point of view; a large amount of articles are devoted to the behaviour of dry non-cohesive granular matter under vibrations. Great number of effects have been observed and reviewed [1]. However, the examination of vibrational dynamics of "wet" sand in fluid demonstrates qualitatively new effects. One of them - the formation of a quasi-stationary spatial relief at the demarcation between liquefied sand and fluid in a cavity making high-frequency horizontal vibrations has been found in [2] and studied [3, 4]. This relief, which is "static" in
© Kozlov V.G., Chigrakov A.V., Ivanova A.A., Evesque P., 2003
mean in the cavity frame, is made of a series of hills and valleys arranged periodically in the direction of the vibration. This relief formation requires the sand near the interface to be in a completely liquefied state. The relief presents on some parts oblique quasi-steady interface, whose stability in the gravity field is not ensured by dry friction but is determined by mean hydrodynamic interaction of particles with each other and with an oscillating fluid flow.
The quasi-steady relief formation takes place due to Kelvin - Helmholtz mechanism of instability averaged upon the time, the mean dynamics of the interface between the liquefied sand and the fluid is similar to the one between two immiscible fluids. The mean action of vibration on the interface is determined by the vibrational parameter W = b2W2 /(gh), where b and W are respectively the amplitude and radian frequency of the vibration, h and g are the cavity height and the gravity acceleration. The wavelength and the height of the relief grow with W . At large W the gradual transport of liquefied granular matter along the vibration axis takes place: after some time one can find all the granular matter, except a very small amount transported and compacted in one of the cavity ends. In the cylindrical cavity [3] under intensive horizontal vibration the quasi-steady demarcation of liquefied sand-liquid becomes nearly vertical. More complete description of the mean behaviour of a "liquid - sand" interface can be found in [4] which includes other vibration polarisations.
Another mean effect of spontaneous organisation of sedimenting granular matter into the periodical system of horizontal thin clouds [5] was observed in a flat, i.e. nearly 2d (d - grain diameter), hourglass filled with liquid and subject to intensive vertical vibration. The density of granular matter was lower than one of the liquid. The formation of vibrational patterns strongly modifies the sedimentation process.
The present article presents the results of experimental study of the average effect of intensive longitudinal vibrations on the sedimentation process of granular matter in inclined and vertical channels filled with viscous liquid.
1. EXPERIMENTAL TECHNIQUE AND RESULTS
Experiments have been carried out on a mechanical vibrator, which provides the harmonic translational oscillation of the cavity. The frequency f = W /(2p) of vibration could be smoothly adjusted within a relative accuracy Sf / f = 0.5 % in the interval f = 1 - 50 Hz, and
measured within a 0.1 % accuracy using either an optoelectronic tachometer or a stroboscope. The amplitude of vibration varied in the range b = 0 -1.5 cm, that allowed to generate accelerations as large as 70 g , and has been measured within a 5 % relative accuracy from direct measurement on a video screen and using a camera.
The sedimentation process in tilted and vertical channels was studied in relatively long rectangular metal channel with plexiglass cover (L = 280 mm long, 20 mm wide and 3.0 mm thick), the sand volume in this case was about 1/10 part of the cavity volume. The internal part of the back wall has been painted in black to improve the image contrast of the sand particles.
Experiments were carried out with water-glycerine mixtures at different concentration and kinematic viscosity v; silicate glass beads with density pS = 2.5 g/cm3 of different size (d = 0.36 ± 0.03 mm,
0.09 ± 0.01 mm, and 0.06 ± 0.01 mm) were used as the granular matter. The dynamics of sand was observed using stroboscopic illumination and recorded on video.
Experimental method was the next. Before each experiment the sand has been moved to one side of the channel. At the beginning the vibrator was settled in the horizontal position and the model was fixed to the vibrator. Then a definite frequency and amplitude of vibration has been imposed to the model and the angle of vibration has been varied gradually up to a = 45o or 90o. The grains start sedimenting as soon as one begins changing the orientation of the channel; however the speed of sedimentation of the lower boundary of sand is considerably reduced by vibration, it allows to fix the model in the vertical or inclined position before the sedimentation has occurred significantly.
Different technique was used in experiments with small particles (d = 0.09 mm). In this case, all the granular matter was preliminary well packed under vibration in one side of the channel. When the cavity is turned over the process of particles sedimentation from the lower boundary of packed granular matter goes so slowly, that the experimenter has enough time to fasten the cavity on the vertical or inclined vibrator and impose the vibration after it.
1.1. Tilted channel
Experiments are performed in thin layer h = 3 mm. It is worth mentioning that the value of the vibrational parameter in the experiments is large enough, W ~10 - 60, when the effect of gradual sand transport along the channel and collapsing of all the hills into blocks [1] takes
place just at the beginning of experiment when the channel has horizontal orientation. One can see that in the inclined channel under intensive vibration the sand distribution looks like a succession of sand blocks expanding over the whole channel cross section (Fig. 1). The longitudinal size and the number of blocks depend on the initial distribution of sand in the channel. The length of the block is much larger than the channel thickness. These blocks move slowly downwards with a speed, which depends on the block height and intensity of vibration.
Fig. 1. Sketch of the cavity
One can see parametric oscillations at sand-liquid interface (sides of the blocks perpendicular to the vibration axis). For some layers the amplitude of parametric oscillations is so large that it is comparable the length of the block itself, in this case the boundaries are not very sharp. It is worth mentioning that in spite of it the boundaries of the layers remain stable and definite.
The transport of sand along the channel is not only due to the motion of layers as a whole but also due to the motion of separate particles. The relatively slow transport of particles from the upper layers to the lower ones which modifies the amount of granular matter in the layers is observed. The parametric oscillations of boundaries intensify this process. The height of the most upper layer changes rather fast in time, whereas the size of the lower layers is more stable.
The typical sand distribution in tilted (45o with the horizontal) channel under the longitudinal vibration and its variation with time is presented in Fig. 2. The shaded areas correspond to the blocks consisting of diluted sand. The height of the upper layer monotonically descends at the expense of shedding sand from its lower boundary. The lower layers sometimes have rather stable sizes due to constant process of particle
inflow and particle shedding by the upper and the lower boundaries respectively. As the thickness of the top layer decreases with time, this layer disappears and the second layer becomes the top one after some time. One can see also the relatively slow motion of the layers downward itself.
100
x,mm
50
III, %
2 %
0
80
t,s
160
Fig. 2. Sand distribution in tilted channel versus the time (f = 20.5 Hz, b = 11 mm)
0
The velocity of the upper interface of the top block is determined by filtration of liquid through the porous granular matter. One can see that this upper boundary moves down with constant speed, it mean that the permeability of suspended granular matter in blocks remains constant.
The decrease of the height of the top block with time is determined with the speed of granular matter shedding through the lower boundary. One can see (Fig. 4) that it depends on the intensity of vibration, the higher the intensity, the lower the speed of block mass decreasing.
After the vibration is switched off suddenly, the system of stable layers disappears, the sand falls down the lower plane of the inclined channel and slips to the bottom few orders faster than in case of sedimentation under vibration.
Fig. 3. The coordinate of upper sand boundary versus the time t (1-3 correspond to f (Hz) = 15.3, 17.2, 20.5 at b = 11 mm)
Fig. 4. Dependence of the upper layer height L on time t; f (Hz) = 15.3, 17.2, 18.5, 20.5 at b = 11 mm (1-4)
The low speed of sedimentation under vibration is determined with liquid filtration through the liquefied granular matter filling the whole channel cross section. At the same time the vibration provides the stability of lower boundary of each sand layer.
1.2. Vertical channel
The experiments are started with the sand compacted in the top part of the cavity (Fig. 5). In the absence of vibration the spreading of the sand depends on the grains diameter and on liquid viscosity, in the case of d = 0.09 mm in water it runs relatively slowly at a speed much lower than the normal speed of sedimentation of a single particle; this is due to the back liquid flow which is generated due to the grain motion.
However the speed of spreading of sand increases considerably with vibration because of the fluidisation of the sand at the lower boundary, which allows also the forcing of parametrical oscillations and local organisation of flows with local inhomogeneities. With further increase of the vibration the locally dense zones are stretched horizontally reducing the mean vertical speed.
However such flat formations are not steady and collapse fast.
At some threshold intensity of vibration the flat cloud of sand becomes able to block the whole cross-section of the cavity, forming then a stable layer. It results from the formation of a lower stable boundary between the liquefied sand and the liquid below it. This layer is slowly moving down. The spreading of sand from the lower boundary of this layer is decelerated and takes place essentially at one of the walls of the cavity. However the intensity of shedding decreases with further increase of the frequency of vibration.
At the top of the cavity the sand remains packed quite long. Thus, the sand dropping from the surface of the packed layer can create jumps of concentration, at which the complete blocking of the channel is possible.
a
w
O
o
L1
Fig. 5. Sketch of the vertical cavity
g
L
2
So in the experiments the formation at once of several steady layers was observed. At a constant vibration rate all the layers slowly move downwards.
0 -----------------------------------------------
0 40 t,s 80
Fig. 6. Longitudinal distribution of granular matter in vertical channel under vibration versus the time ( b = 11 mm, f = 30.4 Hz)
At decreasing the frequency of vibration the intensity of shedding of separate particles from a layer increases. At some threshold frequency the layer with definite concentration, or with a definite jump of concentration on the interface, becomes unstable and falls off rapidly. The threshold frequency at which a layer becomes unstable is notably lower than the frequency demanded for a creation of a stable layer.
The downfall of one of the layers has no considerable effect upon the stability of the rest irrespective of their position. In the experiments the cases were observed, when the underlayer remained steady after a downfall of the upper layer, and when the disturbances caused by a downfall of underlayer did not influence the stability of the upper one.
The destruction of all the layers with decreasing of the vibration intensity results in rather homogeneous suspension of sand slowly subsiding. It is worth noting that the increase of vibration rate does not result in the formation of bedded structure from homogeneous suspension.
If the vibration is intensive enough the spreading of the sand from the lower boundary of liquefied matter (cloud of suspended granular matter) is absent and the speed of sedimentation is very low, so during the gradual liquefaction of packed sand a single layer of diluted granular matter appears. In Fig. 6 the diagram reflecting the dynamics of such a layer of sand at vertical vibration for a combination water - glass beads
d = 0.09 mm is plotted. The shaded area between curves corresponds to diluted sand, falling downwards as a whole.
Fig. 7. The co-ordinate of the upper block demarcation versus the time at different vibration intensity: b = 11 mm, f = 30.4 Hz (1); 12, 36.5 (2) and 6.5, 39.1 (3)
Fig. 8. Block velocity versus the dimensionless acceleration
The speed of downward motion, as well as for angle 45o , is rather small. In Fig. 7 the longitudinal co-ordinate of the upper boundary of falling stable blocks in vertical channel under different vibration inten-
sity is presented. One can see that the velocity of blocks does not change with time and strongly depends on intensity of vibration. Taking into account, that the block speed is determined with mean velocity of filtration through the porous granular matter, one comes to a conclusion about the increase of particles concentration on the lower boundary of the block with vibration intensity. The dependence of velocity of granular matter falling down on the dimensionless acceleration of vibration (Fig. 8) demonstrates its gradual decrease with intensity of vibration.
2. DISCUSSION
A sand-liquid interface is not stable at translational vibration directed parallel to the interface. This effect [1] takes place due Kelvin - Helmholtz instability. At large enough W the tops of hills formed by vibration reach the ceiling and block all the cross-section of the cavity (Fig. 1). Further sand dynamics is determined by some mean vibrational interaction of particles inside the sand blocks and on the interface. It is worth mentioning that the hills consist of diluted granular matter but have quite definite and sharp slopes nearly perpendicular to the vibration axis. It is well observed in the case of fine particles in viscous liquid, when the particle-liquid interaction is determined mainly by viscosity. One can suppose that it takes place due to some mean vibrational attraction of the particles. This conclusion is in agreement with the observation of the behaviour of sand-liquid system in gravity field under vertical vibrations [6], where the effect of vibrational solidification of the interface, oriented perpendicular to the axis of vibration was observed. At the same time the present experiments with vertical orientation of the channel demonstrates that it is not possible to form horizontal layer structure from an initially homogeneous suspension of relatively low concentration. It leads to conclusion that particles concentration and perhaps the jump of density at the interface play the determining role in mean attraction of particles.
The conclusion about mean vibrational mutual attraction of particles in viscous liquid in the direction of vibration is proved by the effect of spontaneous formation of granular matter jump in inclined and vertical channels.
The present observations are in agreement with the results of [7], where the particles segregation from the initially homogeneous distribution under translational vibration takes place only above some critical concentration of particles.
It is possible to speak about the concept of efficient "surface tension" on the interface of the liquefied sand and pure liquid (on jump of con-
centration of granular matter in a liquid) under intensive vibrations along the density gradient.
Conclusion. New mean vibrational effects are found in the experimental study of suspension in tilted and vertical channels subjected to intensive longitudinal oscillations. First is the effect of stabilisation of sharp potentially unstable horizontal demarcation of heavy suspended matter - pure liquid (liquid is below). It results in formation of clouds of concentrated suspended matter with sharp and stable boundaries perpendicular to the vibration axis. The effect demonstrates the mean vibrational hydrodynamic interaction (attraction) of particles in oscillating force field. The clouds completely fill the cross section of the channel and slowly fall down in gravity field. The effect takes place at large vibrational acceleration and essentially depends on particle concentration and jump of density on demarcation. Another effect consists in strong reduction of granular matter sedimentation speed. The speed of sedimentation of diluted granular matter is few orders lower than without vibration, decreases with dimensionless acceleration and is determined by non-linear interaction of oscillating porous media and liquid.
The new phenomena could be useful for practical purpose for granular matter control in ordinary and microgravity conditions.
ACKNOWLEDGEMENT
The work is partly supported by RFBR (03-01-00552) and the joint program CNRS-RFBR (PICS № 1170 / RFBR 01-01-22005 NCSI-a).
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3. Ivanova A., Kozlov V., Evesque P. Patterning of liquefied sand surface in a cylinder filled with liquid and subjected to horizontal vibrations // Europhys. Letters. 1996. V. 35. № 3. P. 159-164.
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5. Evesque P. Sablier inverse II Pour la Science. 1997. № 239. P. 94-9б.
6. Ivanova AA., Kozlov V.G., Evesque P. Fluidization of a granular medium in a viscous fluid under vertical vibration II Izv. RAS. Mech. Zhidk. i Gaza. 2000. № 3. P. 113-122.
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ОСЕДАНИЕ СЫПУЧЕЙ СРЕДЫ В НАКЛОННОМ КАНАЛЕ С ЖИДКОСТЬЮ, СОВЕРШАЮЩЕМ ПРОДОЛЬНЫЕ ВИБРАЦИИ
В.Г. Козлов, А.В. Чиграков, А. А. Иванова, П. Эвеск
Представлены новые экспериментальные результаты исследования осредненной динамики сыпучей среды в вязкой жидкости в наклонном канале, совершающем продольные вибрации. обнаружен эффект образования устойчивых блоков из ожиженного песка. Блоки вытянуты вдоль канала и занимают все поперечное сечение, что приводит к снижению скорости оседания сыпучей среды на несколько порядков по сравнению с отсутствием вибраций. Качественно новым является эффект вибрационной стабилизации нижней границы блоков, связанный с осредненным взаимодействием частиц в вязкой жидкости. Граница раздела характеризуется резким скачком плотности (концентрация частиц под границей на несколько порядков ниже, чем над ней). в вертикальном канале вибрации приводят к формированию одиночного блока сыпучей среды с устойчивыми нижней и верхней границами, медленно перемещающегося вниз.
