Abstract
This paper deals with the evolution of a localized, constant-volume initial condition on an incline into a spreading descending thin-film solution. Clear fluids in this geometry are known to have a front position that moves according to a t 1/3 scaling law, based on similarity-solution analysis by Huppert (Nature 300:427–429, 1982). The same dynamics are investigated for particle-laden flow using a recently proposed lubrication model and physical experiments. The analysis includes the role of a precursor in the model. In the lubrication model, the height of the precursor significantly influences the position of the fluid front, independent of particles settling in the direction of flow. By comparing theory with experiments it is shown that the t 1/3 scaling law persists, to leading order, for particle-laden flows with particle settling. However, additional physics is needed in the existing lubrication models to quantitatively explain departures from clear-fluid self-similarity due to particle settling.
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Acknowledgement
Natalie Grunewald was supported by the DFG grant GR 3391/1-1. She thanks Andrea Bertozzi and UCLA for their kind hospitality during the last year. This work was supported by ONR grant N000140610059, and NSF grants ACI-0321917, DMS-0601395, and DMS-0502315, as well as UC Lab Fees Research Grant number 09-LR-04-116741-BERA. We thank Ben Cook and A. E. Hosoi for helpful discussions.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Grunewald, N., Levy, R., Mata, M. et al. Self-similarity in particle-laden flows at constant volume. J Eng Math 66, 53–63 (2010). https://doi.org/10.1007/s10665-009-9345-4
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DOI: https://doi.org/10.1007/s10665-009-9345-4