In J. Parisi, S. C. Muller, W. Zimmermann (Eds.), \A perspective look at. Experimental Study of Horizontally Shaken

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1 In J. Parisi, S. C. Muller, W. Zimmermann (Eds.), \A perspective look at nonlinear media in Physics, chemistry, and biology", Springer (Berlin, 1997) p Experimental Study of Horizontally Shaken Granular Matter { The Swelling Eect Thorsten Poschel and Dirk E. Rosenkranz Humboldt-Universitat zu Berlin, Institut fur Physik, Invalidenstrae 110, D Berlin, Germany Abstract. We report on a new eect observed in horizontally shaken granular material in a rectangular box. Within a certain region of parameters of the oscillation the granular material swells recurrently with a characteristic frequency (see This frequency is signicantly smaller than the frequency of forcing. We claim that this new eect is caused by the the interplay between Reynolds dilatancy due to convective motion and mechanical stability of the diluted granular material in the box. Measurements of the dissipated energy per time supports this explanation. 1 Introduction Shaken granular materials reveal dierent types of interesting macroscopic eects including surface uidization (Warr et al. (1995)), structure formation (Melo et al. (1995), Metcalfe et al. (1997)), localized excitations (Umbanhowar et al. (1996)), compaction, (e.g. Ayer and Soppet (1965/1966), Esipov et al. (1996)), segregation, (e.g. Williams (1976), Knight et al. (1993), Gallas et al. (1994)), convection and others. When granular material in a rectangular container is shaken in either direction, vertically or horizontally, one observes convective motion. Convection due to vertical oscillations has been subjected to many experimental (e.g. Ratkai (1976), Laroche et al. (1989), Fauve et al. (1989), Jaeger and Nagel (1992), Ehrichs et al. (1995), Jaeger et al. (1996)) and analytical (e.g. Bourzutschky and Miller (1995), Hayakawa et al. (1995), Salue~na et al. (1997), Esipov et al. (1996)) investigations. The typical convection cells could be reproduced by molecular dynamics simulations in two (Gallas et al. (1992), Taguchi (1992), Poschel and Herrmann (1995)) as well as in three dimensions (Gallas et al. (1994)).

2 2 Thorsten Poschel and Dirk E. Rosenkranz In contrast there are relatively few results on horizontally shaken granular material. Numerical molecular dynamics simulations and analytical investigations predicted that convection occurs in horizontally shaken containers too (Esipov et al. (1996), Salue~na et al. (1997), Liman and Metcalfe (1997)). Recently this eect has been investigated quantitatively (Tennakoon and Behringer (1997), Rosenkranz and Poschel (1997)). Despite of convective motion we know that horizontally shaken granular material reveals a rich variety of complex patterns (Iwashita et al. (1988), Strassburger et al. (1997), Ristow et al. (1997)). Up to now we are far from a real understanding of these complex phenomena. 2 The Swelling Eect In the present paper we want to report on a new eect in horizontally shaken granular matter, which we call the \swelling eect": Suppose a rectangular container lled with granular material up to a height z 0 f is shaken sinusoidal in horizontal direction x(t) = A cos(2ft) with amplitude A and frequency f. In a certain interval of parameters A, f and z 0 f we observe time dependent variation of the volume of the material and hence of its overall density. This eect can be measured by recording the lling height z f (t) > z 0 f. While there are several experimental investigations of exciting time dependent phenomena in vertically vibrated containers (Clement et al. (1996), Umbanhowar et al. (1996), Melo et al. (1995)) there is a main dierence between those eects and the swelling eect: All these eects act on time scales which are comparable with the period of forcing 1=f. The swelling eect, however, has a period which is up to several ten to a hundred times larger than the period of oscillation 1=f. Furthermore we do not observe a period doubling scenario as seen e.g. by Melo et al. (1995). Video sequences of the new eect are presented in the internet at URL: 3 Experimental Setup The experimental setup we have used for our investigations, is shown in Fig. 1. The probe carrier with the rectangular container was mounted to a precisely balanced horizontal linear bearing. A second linear bearing was driven by a stepping motor via a crankshaft. In between both bearings there was a piezo force sensor (Kistler, type 9203 with Kistler charge amplier, type 5001) which measured the force acting from the motor driven bearing on the bearing which holds the probe carrier. The force was registered by a computer with a sampling rate of 2 khz. The stepping motor was computer controlled, i.e. per precisely 2,000 impulses the motor axis revolves once. This high angular resolution provides a quasi steady motion, the nite step size per computer signal does not inuence the convective behavior. The amplitude of oscillation was adjustable by changing the eccentricity of the crankshaft with

3 Horizontally Shaken Granular Matter { The Swelling Eect 3 a precision of 0:05 mm. The entire mechanical device was xed on a oscillation damping table. For a detailed description of the equipment containing all technical details see (Rosenkranz (1997)). container with granular material force sensor connecting both linear bearings adjustable crankshaft linear bearings stepping motor Fig. 1. Experimental setup The container of size 6 cm10 cm consists of transparent plastic material. It was illuminated from the top and a camera was located in a distance of 4 m in a direction perpendicular to the axis of oscillation. The camera was levelled exactly with the upper surface of the granular material at rest. The convex shape of the upper surface of the shaken granular matter forms a bright region in the camera plane, because of the top lighting. Hence the position and width of this region give us measures of the height and the curvature of the upper material surface. A second camera was mounted about 1 m above the equipment to monitor the motion of the material at the upper surface. The output of both cameras has been digitized and recorded by standard digital equipment and has been processed afterwards. 4 Qualitative Description Figure 2 displays a time series of snapshots of the middle part of the surface region taken with time delay t = 0:2 sec. The container was horizontally shaken with amplitude A = 0:15 cm and frequency f = 25 sec?1. The dashed line is at xed height. The displayed series covers slightly more than one swelling oscillation period which is about 2:2 sec, i.e. about 55 times the period of the driving oscillation. One sees clearly the height oscillation the surface, which is in its dependence on the forcing parameters A and F and the lling height z 0 f the subject of our interest. The swelling oscillation shown in Fig. 2 could be observed for a wide range of frequency, amplitude and lling height and for a variety of materials and grain sizes (r = 20 m 100 m). The swelling was accompanied by a rapid

4 4 Thorsten Poschel and Dirk E. Rosenkranz Fig. 2. Time series of snapshots of the surface region (time grows from top to bottom). The dashed line leads the eye to a xed height above the container bottom.

5 Horizontally Shaken Granular Matter { The Swelling Eect 5 convective motion of the material at the surface. This ow was not uniform but its intensity varied with the same characteristic frequency as the swelling itself. 5 Quantitative Results To analyze the eect quantitatively we recorded time series of several observables, i.e. { the height of the material surface in the middle of the container { the motion of the material at the upper surface { the force acting on the probe carrier. 5.1 Height Oscillations Figure 3 displays the heights z over a period of 40 sec for an amplitude A = 0:15 cm and dierent driving frequencies f. The total height of the displayed region in each gure is 0:4 cm. The material lling height at rest was z 0 f = 2:5 cm, and the average grain size was 100 m. For all frequencies shown in Fig. 3 we observe, that the material height varies with time. For small driving frequencies (f. 23 sec?1 ) and large driving frequencies (f & 30 sec?1 ) the height changes irregularly with time, whereas for a characteristic region of frequency (23 sec?1. f. 29 sec?1 ) we observe regular, almost periodic oscillation of the material height. If there exists a characteristic frequency, it should be found in the Fourier spectra Figure 4 shows the Fourier spectra of the full serieses with total lengths 60 sec. For small driving frequencies f < 23 sec?1 no characteristic frequency of swelling can be observed, i.e. the Fourier spectrum reveals many frequencies. At f = 23 sec?1 there seems to be a sharp transition into another regime, where we nd a characteristic frequency of the swelling oscillation indicated by a peak in the Fourier spectrum. The peak vanishes at f 30 sec?1 due to vanishing amplitude of the height oscillation. A hysteresis of the transition point when increasing and decreasing the driving frequency has not be observed. In the region 23 sec?1. f. 29 sec?1 the frequency of swelling is not constant, but it is a function of the driving frequency f (Fig. 4). The function F (f) for dierent lling heights z 0 f is shown in Fig Surface Flow When granular material is shaken horizontally one observes convective motion. This eect has been described already theoretically as well as based on molecular dynamics simulations in two dimensions (Salue~na et al. (1997), Esipov et al. (1996), Liman and Metcalfe (1997)). Essentially one observes

6 6 Thorsten Poschel and Dirk E. Rosenkranz t in sec Fig. 3. Height of the granular material in the middle of the container over time for driving frequencies f = 17 sec?1 : : : 32 sec?1.

7 Horizontally Shaken Granular Matter { The Swelling Eect F in sec 1 Fig. 4. Fourier spectra of the time serieses shown in Fig. 3.

8 8 Thorsten Poschel and Dirk E. Rosenkranz 0.4 F in sec mm 25mm 30mm f in sec 1 Fig. 5. Swelling frequency F as a function of the frequency of forcing f for various lling heights z 0 f. pairs of symmetric convection cells: the material vanishes at the walls perpendicular to the shaking direction and reappears in the center of the container. However, the convection patterns in the experimental three dimensional system appears to be much more complex. Due to our knowledge this convection has never been described in detail and this work is in progress (Rosenkranz and Poschel (1997)). In the region of driving parameters A, f and z 0 f, where one observes the swelling eect, there is an intensive convective motion, however, the intensity of the convective motion oscillates as well. Viewing the container from the top (second camera) one nds that the oscillation of material height (Fig. 3) corresponds to a varying particle velocity at the upper surface. When the material is swelled there is an intensive ow, whereas the ow comes almost to rest instantly when the material is collapsed. We want to show now that the frequency of the character of surface ow coincides with the frequency of swelling. To this end we put tracer particles into the granular system and take a series of snapshots of the container viewing from top. Then we calculate a corresponding sequence of pictures by subtracting the gray scale values of the pixels of consecutive snapshots. The average value G of the gray scale of the pixels of the dierence pictures provides a measure of the material ow at the surface. If there would be no ow, the dierence pictures appear to be equal coloured, i.e G = 0. If there would be a steady ow, we would get G const. For the case of a time varying ow we nd a characteristic time series G(t) of which we expect to have the same characteristic frequencies as the swelling itself. Figure 6 shows the Fourier transform of G for a driving frequency of f = 25 sec?1. The Fourier spectrum of the average pixel dierences G of the gray scale of dierence pictures reveals a signicant peak. One notes

9 Horizontally Shaken Granular Matter { The Swelling Eect 9 that the variation of the surface ow has the same characteristic frequency as the swelling (c.f. Fig. 4). Hence, we claim that convective motion in the horizontally shaken container and swelling are closely related eects F in sec 1 Fig. 6. Fourier spectrum of the oscillation of the material ow at the surface over time for f = 25 sec? Dissipation of Mechanical Energy Using the force sensor, which measures the force F (t) acting on the probe carrier, we calculated the energy applied by the motor to the probe carrier during the nth period E n = 2fA Z (n+1)=f n=f F (t) cos (2ft) dt : (1) E n is the energy dissipated during the nth period by the entire system, i.e. by the linear bearing and by the motion of the granular material in the container. This time series E n can be analyzed by Fourier analysis similar to the surface ow intensity and the height of the material. Figure 7 shows the Fourier transform of the dissipated energy, again for amplitude A = 0:15 cm and a range of driving frequencies f. Comparing Figs. 4 and 7 we notice that both gures agree well, i.e. the characteristic frequencies of swelling oscillation and of energy dissipation coincide.

10 10 Thorsten Poschel and Dirk E. Rosenkranz F in sec 1 Fig. 7. Fourier spectra of the dissipated energy E for a range of frequencies.

11 Horizontally Shaken Granular Matter { The Swelling Eect The Inuence of Air on The Swelling Eect For the cases of other dynamical eects there has been a controversial discussion about the inuence of air. Whereas for the eect of irregular motion and clogging ow of granular material through a pipe there is experimental evidence that surrounding air has major inuence on the motion of the sand (Raafat et al. (1996), Horikawa et al. (1995)), the inuence of air on the spontaneous heap formation of vertically vibrated granular material seems to be unclear yet (Laroche et al. (1989), Evesque and Rajchenbach (1989), Evesque (1990), Savage (1988)). Hence, the inuence of air on the swelling eect has to be discussed. One could imagine that the convective motion of sand \pumps" air into the bulk of the material which leads to swelling. When the pressure of air exceeds a certain critical value a \bubble" escapes and the material collapses. If this explanation is valid, the swelling eect should vanish if the air pressure in the container is reduced to an amount so that the mean free path of air molecules comes into the region of the mean free path of the sand grains. In the experiment, however, we found that reducing the pressure to p = 50 Pa, which corresponds to a mean free path of air 130 m, the swelling eect does not disappear. Since air 130 m is smaller than the typical mean free path of sand grains, we conclude, that air does not play the major role for the eect of recurrent swelling. 6 Possible Explanation of the Swelling Eect Presently we do not have a complete explanation of the swelling eect. Nonetheless we want to suggest a possible explanation based on the measurement of dissipated mechanical energy and surface ow: In the horizontally shaken container we observe convection, i.e. motion of the particles with respect to each other. To allow for macroscopic motion in a granular material the material has to be diluted below a certain density R before. This process is called Reynolds dilatancy (Reynolds (1885), Wang and Campbell (1992)). When the local density of the material is below R the material can start to ow. The convection in the container implies shear ow, which causes further dilution and uidization. This mechanism is explained by Spahn et al. (1997). Hence we believe, that there are two contradicting eects. First, due to shear the material tends to dilute. Second, the material has to remain mechanically stable, i.e. the grains on top have to be supported by the grains below them and these have to be stabilized by the grains of the next layer etc. When the shaken material swells, the material becomes diluted and becomes less mechanically stable at the same time. At a certain moment the material becomes diluted to an extend that it loses mechanical stability and collapses. (The situation resembles somewhat the transition in a uid pipe ow above the critical Reynolds number.) Then the material starts again to swell.

12 12 Thorsten Poschel and Dirk E. Rosenkranz Our proposed explanation is supported by the measurements of the dissipation of mechanical energy in the system (Sect. 5.3) and of the surface material ow (Sect. 5.2). If the material is collapsed, i.e. if a large part of the granular material has a density above R, the grains in these regions cannot move with respect to each other due to the Reynolds dilatancy eect. Hence, in these regions there is no shear motion and therefore no or low dissipation of mechanical energy. From this consideration we conclude, that the collapsed material should dissipate less energy per time than the swelled material. This behavior coincides with what we found in the measurement of the dissipated energy in Sect Moreover the results in Sect. 5.2 show that the intensity of surface ow oscillates with the same frequency F. Therefore the experimental results on energy dissipation and surface ow support our hypothesis on the origin of the swelling eect. If this explanation is valid it explains the swelling eect itself and the saw teeth shape of the height over time curves (Fig. 3). However, it does not explain the transition from periodic to irregular behavior of swelling and the characteristic shift of swelling frequency with driving frequency in the periodic regime. 7 Conclusion We have described a new eect observed in a container lled with granular material which is subjected to sinusoidal horizontal oscillations. Within a certain range of parameters of driving the material swells recurrently. Due to our knowledge this eect has not been reported in the literature so far. For xed amplitude of driving A we nd for low as well as for high frequencies f, that the material swells irregularly, whereas in an intermediate frequency interval swelling occurs almost periodically with a certain frequency F. In this interval the swelling frequency F is about 50 to 100 times smaller than the driving frequency f, hence we claim that the eect reported in this paper is not comparable with periodic eects reported by other authors (e.g. Clement et al. (1996), Metcalfe et al. (1997), Umbanhowar et al. (1996), Melo et al. (1995)), where the frequency of the eect is the same as the frequency of driving or small multiples of it. In the periodic interval the swelling frequency F is a characteristic function of the driving frequency f. Similar as the height of the material, the intensity of the convective ow of the granular material in the container as well as the energy dissipated by the granular system vary in time too. Comparing the characteristic frequencies by Fourier analysis we noticed, that all three oscillating observables are closely related. Finally we proposed a possible explanation of the eect which is based on the eect of Reynolds dilatancy in owing granular matter and which is supported by the coincidence of the measured time serieses of material height, surface ow intensity and energy dissipation.

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15 Horizontally Shaken Granular Matter { The Swelling Eect 15 Tennakoon, S. G. K. and Behringer, R. P. (1997): Liquefaction of a horizontally vibrated granular bed. preprint. Umbanhowar, P. B., Melo, F. and Swinney, H. L. (1996): Localized excitations in a vertically vibrated granular layer. Nature 382, 793{796. Nature (1996). Wang, D. G. and Campbell, C. S. (1992): Reynolds' Analogy for a Shearing Granular Material. J. Fluid Mech. 224, 527{546. Warr, S., Huntley, J. M., and Jacques, G. T. H. (1995): Fluidization of a twodimensional granular system: Experimental study and scaling behavior. Phys. Rev. E, 52, 5583{5595. Williams, J. C. (1976): The Segregation of Particulate Materials. A Review. Powder Technology 15, 245{251.