Scaling behavior in the convection-driven Brazil nut effect

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Scaling behavior in the convection-driven Brazil nut effect

Prakhyat Hejmady, Ranjini Bandyopadhyay, Sanjib Sabhapandit, and Abhishek Dhar

Phys. Rev. E 86, 050301(R) – Published 19 November 2012

Abstract

The Brazil nut effect is the phenomenon in which a large intruder particle immersed in a vertically shaken bed of smaller particles rises to the top, even when it is much denser. The usual practice while describing these experiments has been to use the dimensionless acceleration $Γ = a ω^{2} / g$ , where $a$ and $ω$ are, respectively, the amplitude and the angular frequency of vibration and $g$ is the acceleration due to gravity. Considering a vibrated quasi-two-dimensional bed of mustard seeds, we show here that the peak-to-peak velocity of shaking $v = a ω$ , rather than $Γ$ , is the relevant parameter in the regime where boundary-driven granular convection is the main driving mechanism. We find that the rise time $τ$ of an intruder is described by the scaling law $τ \sim {(v - v_{c})}^{- α}$ , where $v_{c}$ is identified as the critical vibration velocity for the onset of convective motion of the mustard seeds. This scaling form holds over a wide range of $(a, ω)$ , diameter, and density of the intruder.

Received 3 February 2012

DOI:https://doi.org/10.1103/PhysRevE.86.050301

Authors & Affiliations

Prakhyat Hejmady^*, Ranjini Bandyopadhyay^†, Sanjib Sabhapandit^‡, and Abhishek Dhar^§

Raman Research Institute, CV Raman Avenue, Sadashivanagar, Bangalore 560080, India

^*phejmady@rri.res.in
^†ranjini@rri.res.in
^‡sanjib@rri.res.in
^§dabhi@rri.res.in

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Vol. 86, Iss. 5 — November 2012

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Images

Figure 1
Experimental setup. (a) The initial position of an acrylic intruder with a diameter of $1.7$ cm embedded in a column of mustard seeds of sizes between $1.5$ and 2 mm. The bottom of the intruder is placed at a height $h_{1} = 5$ mm from the floor of the cell. The rise time $τ$ is the time taken by the center of the intruder to move from height $h_{2}$ to height $h_{3}$ . A typical trajectory of the rising intruder is shown by the vertical white line. (b) Convection pattern obtained by superposing many consecutive time-lapsed images of the vibrated bed of black mustard seeds in which yellow mustard seeds are distributed randomly to serve as tracer particles. The two loops [yellow (light gray)] with arrows indicate the flow direction of the mustard seeds. (c) Intruders used in the experiment with relative densities ranging between 0.30 and 2.34.Reuse & Permissions
Figure 2
A series of snapshots of the vibrated bed taken at regular intervals of 8 s. The top and bottom panels are for experiments done without and with an intruder, respectively. The shaking parameters were set at $f = 50$ Hz and $a = 1.6$ mm. In the bottom panel, the intruder diameter is $1.7$ cm and $ρ_{r} = 1.07$ . The alternate black and yellow mustard layers initially destabilize at the walls, and eventually, bulk convection rolls are formed. An inspection of these figures confirms that the granular convection patterns do not change significantly as a result of the presence of the intruder, except close to the intruder boundary.Reuse & Permissions
Figure 3
Intruder rise-time plots. The top inset shows plots of the rise time $τ$ vs shaking amplitude $a$ for different frequencies: (i) 20 Hz (squares), (ii) 35 Hz (triangles), and (iii) 50 Hz (circles) for an intruder with a diameter of $1.7$ cm and for four different relative densities $ρ_{r}$ $=$ 0.3 (blue), 0.8 (green), 1.07 (red), and 2.34 (magenta). The solid symbols correspond to a 1 cm diameter intruder with $ρ_{r} = 1.07$ . The bottom inset shows a plot of the same data vs the dimensionless acceleration $Γ$ . The main plot shows $τ$ vs peak-to-peak velocity of shaking $v = a ω$ . The data collapse for the entire range of shaking velocities, for all relative densities, and for the two intruder sizes. The solid line shows the functional form $τ \sim {(v - v_{c})}^{- α}$ with $α = 2.0$ and $v_{c} = 16$ cm/s.Reuse & Permissions
Figure 4
Convection velocities at the side walls. The convection velocities, in the absence of the intruder, are plotted against the peak-to-peak velocity of shaking $v = a ω$ for shaking frequencies (i) 20 Hz (squares), (ii) 35 Hz (triangles), and (iii) 50 Hz (circles). Each linear fit (denoted by the solid line) is extrapolated to estimate the peak-to-peak shaking velocity for the onset of convection. The $x$ intercepts lie in the range of 16–17 cm/s. The inset shows a plot of the convection velocities at the side wall as a function of $Γ$ .Reuse & Permissions
Figure 5
Phase diagram. Each data point (circles) corresponds to a set of (amplitude, frequency) parameter values for which BNE was observed in our experiments, while the points (triangles) represent data sets where no bulk convection, and therefore no BNE, was observed. The solid line corresponds to $v = a (2 π f) = v_{c}$ , while the dashed line corresponds to $Γ = a {(2 π f)}^{2} / g = 1$ . Bulk convection occurs only in the region where both the conditions $Γ > 1$ and $v > v_{c}$ are satisfied.Reuse & Permissions

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