7 results
Granular surface flow on an asymmetric conical heap
- Sandip Mandal, D. V. Khakhar
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- Journal:
- Journal of Fluid Mechanics / Volume 865 / 25 April 2019
- Published online by Cambridge University Press:
- 18 February 2019, pp. 41-59
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We carry out an experimental study of the granular surface flow of nearly monodisperse glass beads on a conical heap formed on a rough circular disc by a narrow stream of the particles from a hopper, with the pouring point displaced from the centre of the disc. During the growth phase, an axisymmetric heap is formed, which grows either by periodic avalanches or by non-periodic avalanches that occur randomly over the azimuthal location of the heap, depending on the operating conditions and system properties. The dynamics of heap growth is characterized by the variation of the heap height, angle of repose and the angular velocity of the periodic avalanche with time, for different mass flow rates from the hopper. When the base of the heap reaches the edge of the disc closest to the pouring point, the heap stops growing and a steady surface flow of particles is developed on the heap surface, with particles flowing over the edge of the disc into a collection tray. The geometry is a unique example of a granular flow on an erodible bed without any bounding side walls. The corresponding steady state geometry of the asymmetric heap is characterized by means of surface contours and angles of repose. The streamwise and transverse surface velocities are measured using high-speed video photography and image analysis for different mass flow rates. The flowing layer thickness is measured by immersing a coated needle in the flow at different positions on the mid-line of the flow. The surface angle of the flowing layer is found to be significantly smaller than the angle of repose and to be independent of the mass flow rate. The velocity profiles at different streamwise positions for different mass flow rates are found to be geometrically similar and are well described by Gaussian functions. The flowing layer thickness is calculated from a model using the measured surface velocities. The predicted values match the measured values quite well.
Density difference-driven segregation in a dense granular flow
- Anurag Tripathi, D. V. Khakhar
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- Journal:
- Journal of Fluid Mechanics / Volume 717 / 25 February 2013
- Published online by Cambridge University Press:
- 01 February 2013, pp. 643-669
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We consider the segregation of spheres of equal size and different density flowing over an inclined plane, theoretically and computationally by means of distinct element method (DEM) simulations. In the first part of the work, we study the settling of a single higher-density particle in the flow of otherwise identical particles. We show that the motion of the high-density tracer particle can be understood in terms of the buoyancy and drag forces acting on it. The buoyancy force is given by Archimedes principle, with an effective volume associated with the particle, which depends upon the local packing fraction, $\phi $. The buoyancy arises primarily from normal forces acting on the particle, and tangential forces have a negligible contribution. The drag force on a sphere of diameter $d$ sinking with a velocity $v$ in a granular medium of apparent viscosity $\eta $ is given by a modified Stokes law, ${F}_{d} = c\pi \eta dv$. The coefficient ($c$) is found to decrease with packing fraction. In the second part of the work, we consider the case of binary granular mixtures of particles of the same size but differing in density. A continuum model for segregation is presented, based on the single-particle results. The number fraction profile for the heavy particles at equilibrium is obtained in terms of the effective temperature, defined by a fluctuation–dissipation relation. The model predicts the equilibrium number fraction profiles at different inclination angles and for different mass ratios of the particles, which match the DEM results very well. Finally, a complete model for the theoretical prediction of the flow and number fraction profiles for a mixture of particles of different density is presented, which combines the segregation model with a model for the rheology of mixtures. The model predictions agree quite well with the simulation results.
Rheology of surface granular flows
- ASHISH V. ORPE, D. V. KHAKHAR
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- Journal:
- Journal of Fluid Mechanics / Volume 571 / 25 January 2007
- Published online by Cambridge University Press:
- 04 January 2007, pp. 1-32
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Surface granular flow, comprising granular material flowing on the surface of a heap of the same material, occurs in several industrial and natural systems. The rheology of such a flow was investigated by means of measurements of velocity and number-density profiles in a quasi-two-dimensional rotating cylinder, half-filled with a model granular material – monosize spherical stainless-steel particles. The measurements were made at the centre of the cylinder, where the flow is fully developed, using streakline photography and image analysis. The stress profile was computed from the number-density profile using a force balance which takes into account wall friction. Mean-velocity and root-mean-square (r.m.s.)-velocity profiles are reported for different particle sizes and cylinder rotation speeds. The profiles for the mean velocity superimpose when distance is scaled by the particle diameter d and velocity by a characteristic shear rate and the particle diameter, where βm is the maximum dynamic angle of repose and βs is the static angle of repose. The maximum dynamic angle of repose is found to vary with the local flow rate. The scaling is also found to work for the r.m.s. velocity profiles. The mean velocity is found to decay exponentially with depth in the bed, with decay length λ = 1.1d. The r.m.s. velocity shows similar behaviour but with λ = 1.7d. The r.m.s. velocity profile shows two regimes: near the free surface the r.m.s. velocity is nearly constant and below a transition point it decays linearly with depth. The shear rate, obtained by numerical differentiation of the velocity profile, is not constant anywhere in the layer and has a maximum which occurs at the same depth as the transition in the r.m.s. velocity profile. Above the transition point the velocity distributions are Gaussian and below the transition point the velocity distributions gradually approach a Poisson distribution. The shear stress increases roughly linearly with depth. The variation in the apparent viscosity η with r.m.s. velocity u shows a relatively sharp transition at the shear-rate maximum, and in the region below this point the apparent viscosity η ∼ u−1.5. The measurements indicate that the flow comprises two layers: an upper low-viscosity layer with a nearly constant r.m.s. velocity and a lower layer of increasing viscosity with a decreasing r.m.s. velocity. The thickness of the upper layer depends on the local flow rate and is independent of particle diameter while the reverse is found to hold for the lower-layer thickness. The experimental data is compared with the predictions of three models for granular flow.
Deformation and breakup of slender drops in linear flows
- D. V. Khakhar, J. M. Ottino
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- Journal:
- Journal of Fluid Mechanics / Volume 166 / May 1986
- Published online by Cambridge University Press:
- 21 April 2006, pp. 265-285
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We study the deformation and breakup of a low-viscosity slender drop in a linear flow, $\overline{\boldmath v}^{\infty} \overline{\boldmath L}\cdot\overline{\boldmath x} $, assuming that the drop remains axisymmetric. We find that the drop stretches as if it were immersed in an axisymmetric extensional flow with a strength $\overline{\boldmath D}:\overline{\boldmath m}\overline{\boldmath m} $, where $\overline{\boldmath D} = \frac{1}{2}(\overline{\boldmath L}+\overline{\boldmath L}^T)$, and $\overline{\boldmath m} $ is the orientation of the drop, and rotates as if it were a material element in a hypothetical flow $\overline{\boldmath M}=G\overline{\boldmath D}+\overline{\Omega} $, where $\overline{\Omega} = \frac{1}{2}(\overline{\boldmath L}^T - \overline{\boldmath L})$, and G is a known function of the drop length. The approximations involved in the model are quite good when $\overline{\boldmath M}$ has only one eigenvalue with a positive real part, and somewhat less precise when $\overline{\boldmath M}$ has two eigenvalues with positive real parts. In the suitable limits the model reduces to Buckmaster's (1973) model for axisymmetric extensional flow and to the linear-axis version of the more general model proposed by Hinch & Acrivos (1980) for simple shear flow. In establishing a criterion for breakup for all linear flows, we find that the relevant quantity that specifies the flow is the largest positive real part of the eigenvalues of $\overline{\boldmath M}$, which depends on the drop length and the imposed flow. Our predictions are in reasonable agreement with the recent experimental data of Bentley (1985) for general two-dimensional linear flows and those of Grace (1971) for simple shear and hyperbolic extensional flow. We also present calculations for a class of three-dimensional flows as an illustration of the behaviour of three-dimensional flows in general.
Analysis of chaotic mixing in two model systems
- D. V. Khakhar, H. Rising, J. M. Ottino
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- Journal:
- Journal of Fluid Mechanics / Volume 172 / November 1986
- Published online by Cambridge University Press:
- 21 April 2006, pp. 419-451
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We study the chaotic mixing in two periodic model flows, the ‘tendril–whorl’ flow and the ‘Aref-blinking-vortex’ flow, with the objective of supplying evidence for the primary mechanisms responsible for mixing in two-dimensional deterministic flows. The analysis is based on tools of dynamical systems theory but it is clear that the mixing problem generates several questions of its own: low periodic points and horseshoes dominate the picture, since we want to achieve mixing quickly; Poincaré sections, popular in dynamical systems analyses, might give misleading information with regard to dispersion at short times. Our analysis shows that both flows are able to stretch and fold material lines well below the lengthscale of the flows themselves. The inner workings of the two systems are revealed by studying the local and global bifurcations. Computations for the blinking-vortex system indicate the existence of an optimum period at which the average efficiency is maximized, whereas the intensity of segregation – a classical parameter in mixing studies – decays rapidly to an asymptotic value in the globally chaotic region. Even though our flows are not turbulent the results might have some implications for pointing to the limits of similar studies in actual turbulent flows (e.g. line stretching).
Surface flow of granular materials: model and experiments in heap formation
- D. V. KHAKHAR, ASHISH V. ORPE, PETER ANDRESÉN, J. M. OTTINO
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- Journal:
- Journal of Fluid Mechanics / Volume 441 / 25 August 2001
- Published online by Cambridge University Press:
- 15 August 2001, pp. 255-264
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Granular surface flows are important in industrial practice and natural systems, but the understanding of such flows is at present incomplete. We present a combined theoretical and experimental study of quasi-two-dimensional heap formation by pouring particles continuously at a point. Two cases are considered: open systems and closed systems. Experimental results show that the shear rate in the flowing layer is nearly independent of the mass flow rate, and the angle of static friction at the bed–layer interface increases with flow rate. Predictions of the model for the flowing layer thickness and interface angles are in good agreement with experiments.
Chaotic mixing in a bounded three-dimensional flow
- G. O. FOUNTAIN, D. V. KHAKHAR, I. MEZIĆ, J. M. OTTINO
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- Journal:
- Journal of Fluid Mechanics / Volume 417 / 25 August 2000
- Published online by Cambridge University Press:
- 25 August 2000, pp. 265-301
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Even though the first theoretical example of chaotic advection was a three-dimensional flow (Hénon 1966), the number of theoretical studies addressing chaos and mixing in three-dimensional flows is small. One problem is that an experimentally tractable three-dimensional system that allows detailed experimental and computational investigation had not been available. A prototypical, bounded, three-dimensional, moderate-Reynolds-number flow is presented; this system lends itself to detailed experimental observation and allows high-precision computational inspection of geometrical and dynamical effects. The flow structure, captured by means of cuts with a laser sheet (experimental Poincaré section), is visualized via continuously injected fluorescent dye streams, and reveals detailed chaotic structures and chains of high-period islands. Numerical experiments are performed and compared with particle image velocimetry (PIV) and flow visualization results. Predictions of existing theories for chaotic advection in three-dimensional volume-preserving flows are tested. The ratio of two frequencies of particle motion – the frequency of motion around the vertical axis and the frequency of recirculation in the plane containing the axis – is identified as the crucial parameter. Using this parameter, the number of islands in the chain can be predicted. The same parameter – using as a base-case the integrable motion – allows the identification of operating conditions where small perturbations lead to nearly complete mixing.