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Evidence for a universal saturation profile for radial viscous fingers

Complex fingering patterns develop when a low viscosity fluid is injected from a point source into the narrow space between two parallel plates initially saturated with a more viscous, immiscible fluid. We combine historical and new experiments with (a) a constant injection rate; (b) a constant sour...

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Autores principales: Beeson-Jones, Tim H., Woods, Andrew W.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Nature Publishing Group UK 2019
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6533247/
https://www.ncbi.nlm.nih.gov/pubmed/31123288
http://dx.doi.org/10.1038/s41598-019-43728-z
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author Beeson-Jones, Tim H.
Woods, Andrew W.
author_facet Beeson-Jones, Tim H.
Woods, Andrew W.
author_sort Beeson-Jones, Tim H.
collection PubMed
description Complex fingering patterns develop when a low viscosity fluid is injected from a point source into the narrow space between two parallel plates initially saturated with a more viscous, immiscible fluid. We combine historical and new experiments with (a) a constant injection rate; (b) a constant source pressure; and (c) a linearly increasing injection rate, together with numerical simulations based on a model of diffusion limited aggregation (DLA), to show that for viscosity ratios in the range 300–10,000, (i) the finger pattern has a fractal dimension of approximately 1.7 and (ii) the azimuthally-averaged fraction of the area occupied by the fingers, S(r,t), is organised into three regions: an inner region of fixed radius, r < r(b), which is fully saturated with injection fluid, S = 1; a frozen finger region, r(b) < r < r(f) (t), in which the saturation is independent of time, S(r) = (r/r(b))(−0.3); and an outer growing finger region, r(f)(t) < r < 1.44 r(f)(t), in which the saturation decreases linearly to zero from the value (r(f)/r(b))(−0.3) at r(f)(t). For a given injected volume per unit thickness of the cell, V ≫ πr(b)(2), we find r(f) = 0.4r(b) (V/r(b)(2))(1/1.7). This apparent universality of the saturation profile of non-linear fingers in terms of the inner region radius, r(b), and the injected volume V, demonstrates extraordinary order in such a complex and fractal instability. Furthermore, control strategies designed to suppress viscous fingering through variations in the injection rate, based on linear stability theory, are less effective once the instability becomes fully nonlinear.
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spelling pubmed-65332472019-06-03 Evidence for a universal saturation profile for radial viscous fingers Beeson-Jones, Tim H. Woods, Andrew W. Sci Rep Article Complex fingering patterns develop when a low viscosity fluid is injected from a point source into the narrow space between two parallel plates initially saturated with a more viscous, immiscible fluid. We combine historical and new experiments with (a) a constant injection rate; (b) a constant source pressure; and (c) a linearly increasing injection rate, together with numerical simulations based on a model of diffusion limited aggregation (DLA), to show that for viscosity ratios in the range 300–10,000, (i) the finger pattern has a fractal dimension of approximately 1.7 and (ii) the azimuthally-averaged fraction of the area occupied by the fingers, S(r,t), is organised into three regions: an inner region of fixed radius, r < r(b), which is fully saturated with injection fluid, S = 1; a frozen finger region, r(b) < r < r(f) (t), in which the saturation is independent of time, S(r) = (r/r(b))(−0.3); and an outer growing finger region, r(f)(t) < r < 1.44 r(f)(t), in which the saturation decreases linearly to zero from the value (r(f)/r(b))(−0.3) at r(f)(t). For a given injected volume per unit thickness of the cell, V ≫ πr(b)(2), we find r(f) = 0.4r(b) (V/r(b)(2))(1/1.7). This apparent universality of the saturation profile of non-linear fingers in terms of the inner region radius, r(b), and the injected volume V, demonstrates extraordinary order in such a complex and fractal instability. Furthermore, control strategies designed to suppress viscous fingering through variations in the injection rate, based on linear stability theory, are less effective once the instability becomes fully nonlinear. Nature Publishing Group UK 2019-05-23 /pmc/articles/PMC6533247/ /pubmed/31123288 http://dx.doi.org/10.1038/s41598-019-43728-z Text en © The Author(s) 2019 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
spellingShingle Article
Beeson-Jones, Tim H.
Woods, Andrew W.
Evidence for a universal saturation profile for radial viscous fingers
title Evidence for a universal saturation profile for radial viscous fingers
title_full Evidence for a universal saturation profile for radial viscous fingers
title_fullStr Evidence for a universal saturation profile for radial viscous fingers
title_full_unstemmed Evidence for a universal saturation profile for radial viscous fingers
title_short Evidence for a universal saturation profile for radial viscous fingers
title_sort evidence for a universal saturation profile for radial viscous fingers
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6533247/
https://www.ncbi.nlm.nih.gov/pubmed/31123288
http://dx.doi.org/10.1038/s41598-019-43728-z
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