From streamline jumping to strange eigenmodes: Bridging the Lagrangian and Eulerian pictures of the kinematics of mixing in granular flows

Ivan C. Christov; Julio M. Ottino; Richard M. Lueptow

doi:10.1063/1.3653280

From streamline jumping to strange eigenmodes: Bridging the Lagrangian and Eulerian pictures of the kinematics of mixing in granular flows

Ivan C. Christov, Julio M. Ottino, Richard M. Lueptow^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

27 Scopus citations

Abstract

Through a combined computational-experimental study of flow in a slowly rotating quasi-two-dimensional container, we show several new aspects related to the kinematics of granular mixing. In the Lagrangian frame, for small numbers of revolutions, the mixing pattern is captured by a model termed "streamline jumping." This minimal model, arising at the limit of a vanishingly thin surface flowing layer, possesses no intrinsic stretching or streamline crossing in the usual sense, yet it can lead to complex particle trajectories. Meanwhile, for intermediate numbers of revolutions, we show the presence of naturally persistent granular mixing patterns, i.e., "strange" eigenmodes of the advection-diffusion operator governing the mixing process in Eulerian frame. Through a comparative analysis of the structure of eigenmodes and the corresponding Poincaré section and finite-time Lyapunov exponent field of the flow, the relationship between the Eulerian and Lagrangian descriptions of mixing is highlighted. Finally, we show how the mapping method for scalar transport can be modified to include diffusion. This allows us to examine (for the first time in a granular flow) the change in shape, lifespan, and eventual decay of eigenmodes due to diffusive effects at larger numbers of revolutions.

Original language	English (US)
Article number	103302
Journal	Physics of Fluids
Volume	23
Issue number	10
DOIs	https://doi.org/10.1063/1.3653280
State	Published - Oct 10 2011

Funding

We would like to thank Gabriel Juarez for helpful discussions on the material in Section . The images of the monodisperse and bidisperse experiments were kindly provided by Emre Yildiz and Florent Pignatel, respectively. The careful reading and insightful comments by the referees are much appreciated. This work was supported, in part, by NSF Grant CMMI-1000469. ICC was also supported, in part, by a Walter P. Murphy Fellowship from the Robert R. McCormick School of Engineering and Applied Science at Northwestern University.

ASJC Scopus subject areas

Computational Mechanics
Condensed Matter Physics
Mechanics of Materials
Mechanical Engineering
Fluid Flow and Transfer Processes

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abstract = "Through a combined computational-experimental study of flow in a slowly rotating quasi-two-dimensional container, we show several new aspects related to the kinematics of granular mixing. In the Lagrangian frame, for small numbers of revolutions, the mixing pattern is captured by a model termed {"}streamline jumping.{"} This minimal model, arising at the limit of a vanishingly thin surface flowing layer, possesses no intrinsic stretching or streamline crossing in the usual sense, yet it can lead to complex particle trajectories. Meanwhile, for intermediate numbers of revolutions, we show the presence of naturally persistent granular mixing patterns, i.e., {"}strange{"} eigenmodes of the advection-diffusion operator governing the mixing process in Eulerian frame. Through a comparative analysis of the structure of eigenmodes and the corresponding Poincar{\'e} section and finite-time Lyapunov exponent field of the flow, the relationship between the Eulerian and Lagrangian descriptions of mixing is highlighted. Finally, we show how the mapping method for scalar transport can be modified to include diffusion. This allows us to examine (for the first time in a granular flow) the change in shape, lifespan, and eventual decay of eigenmodes due to diffusive effects at larger numbers of revolutions.",

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note = "Funding Information: We would like to thank Gabriel Juarez for helpful discussions on the material in Section . The images of the monodisperse and bidisperse experiments were kindly provided by Emre Yildiz and Florent Pignatel, respectively. The careful reading and insightful comments by the referees are much appreciated. This work was supported, in part, by NSF Grant CMMI-1000469. ICC was also supported, in part, by a Walter P. Murphy Fellowship from the Robert R. McCormick School of Engineering and Applied Science at Northwestern University.",

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N2 - Through a combined computational-experimental study of flow in a slowly rotating quasi-two-dimensional container, we show several new aspects related to the kinematics of granular mixing. In the Lagrangian frame, for small numbers of revolutions, the mixing pattern is captured by a model termed "streamline jumping." This minimal model, arising at the limit of a vanishingly thin surface flowing layer, possesses no intrinsic stretching or streamline crossing in the usual sense, yet it can lead to complex particle trajectories. Meanwhile, for intermediate numbers of revolutions, we show the presence of naturally persistent granular mixing patterns, i.e., "strange" eigenmodes of the advection-diffusion operator governing the mixing process in Eulerian frame. Through a comparative analysis of the structure of eigenmodes and the corresponding Poincaré section and finite-time Lyapunov exponent field of the flow, the relationship between the Eulerian and Lagrangian descriptions of mixing is highlighted. Finally, we show how the mapping method for scalar transport can be modified to include diffusion. This allows us to examine (for the first time in a granular flow) the change in shape, lifespan, and eventual decay of eigenmodes due to diffusive effects at larger numbers of revolutions.

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From streamline jumping to strange eigenmodes: Bridging the Lagrangian and Eulerian pictures of the kinematics of mixing in granular flows

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