Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-12T20:21:26.565Z Has data issue: false hasContentIssue false
  • English
  • Français

Dissolution of a cylindrical disk in Hele-Shaw flow: a conformal-mapping approach

Published online by Cambridge University Press:  06 October 2020

Anthony J. C. Ladd Open the ORCID record for Anthony J. C. Ladd [Opens in a new window] ,
Liang Yu  and
Piotr Szymczak Open the ORCID record for Piotr Szymczak [Opens in a new window]
Anthony J. C. Ladd*
Affiliation:
Department of Chemical Engineering, University of Florida, Gainesville, FL32611-6005, USA
Liang Yu
Affiliation:
Department of Chemical Engineering, University of Florida, Gainesville, FL32611-6005, USA
Piotr Szymczak
Affiliation:
Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093Warsaw, Poland
*
†Email address for correspondence: tladd@che.ufl.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We apply conformal mapping to find the evolving shapes of a dissolving cylinder in a potential flow. Similar equations can be used to describe melting in a flowing liquid phase. Results are compared with microfluidic experiments and numerical simulations. Shapes predicted by conformal mapping agree almost perfectly with experimental observations, after a modest (20 %) rescaling of the time. Finite-volume simulations show that the differences with experiment are connected to the underlying assumptions of the analytical model: potential flow and diffusion-limited dissolution. Approximate solutions of the equations describing the evolution of the shape of the undissolved solid can be derived from a Laurent expansion of the mapping function from the unit circle. Asymptotic expressions for the evolution of the area of the disk and the shift in its centre of mass have been derived at low and high Péclet number. Analytic approximations to the leading-order Laurent coefficients provide additional insight into the mechanisms underlying pore-scale dissolution.

JFM classification

Low-Reynolds-number flows: Hele-Shaw flows Mathematical Foundations: General fluid mechanics Micro-/Nano-fluid dynamics: Microfluidics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A46
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Adda Bedia, M. & Amar, M. B. 1994 Investigations on the dendrite problem at zero surface tension in 2D and 3D geometries. Nonlinearity 7 (3), 765–776.CrossRefGoogle Scholar
Andersson, C., Führer, C. & Åkesson, J. 2015 Assimulo: a unified framework for ODE solvers. Maths Comput. Simul. 116, 26–43.CrossRefGoogle Scholar
Bauer, M. & Bernard, D. 2006 2D growth processes: SLE and Loewner chains. Phys. Rep. 432 (3–4), 115–221.CrossRefGoogle Scholar
Bazant, M. Z. 2004 Conformal mapping of some non-harmonic functions in transport theory. Proc. R. Soc. Lond. A 460 (2045), 1433–1452.CrossRefGoogle Scholar
Bazant, M. Z. 2006 Interfacial dynamics in transport-limited dissolution. Phys. Rev. E 73 (6), 060601.CrossRefGoogle ScholarPubMed
Bazant, M. Z., Choi, J. & Davidovitch, B. 2003 Dynamics of conformal maps for a class of non-Laplacian growth phenomena. Phys. Rev. Lett. 91 (4), 045503.CrossRefGoogle ScholarPubMed
Bensimon, D., Kadanoff, L. P., Liang, S., Shraiman, B. I. & Tang, C. 1986 Viscous flows in two dimensions. Rev. Mod. Phys. 58, 977–1002.CrossRefGoogle Scholar
Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 2001 Transport Phenomena. John Wiley & Sons.Google Scholar
Chen, L., Kang, Q., Tang, Q., Robinson, B. A., He, Y.-L. & Tao, W.-Q. 2015 Pore-scale simulation of multicomponent multiphase reactive transport with dissolution and precipitation. Intl J. Heat Mass Transfer 85, 935–949.CrossRefGoogle Scholar
Choi, J., Margetis, D., Squires, T. M. & Bazant, M. Z. 2005 Steady advection–diffusion around finite absorbers in two-dimensional potential flows. J. Fluid Mech. 536, 155–184.CrossRefGoogle Scholar
Christoffersen, J. & Christoffersen, M. R. 1976 The kinetics of dissolution of calcium sulphate dihydrate in water. J. Cryst. Growth 35 (1), 79–88.CrossRefGoogle Scholar
Colombani, J. 2008 Measurement of the pure dissolution rate constant of a mineral in water. Geochim. Cosmochim. Acta 72, 5634–5640.CrossRefGoogle Scholar
Cummings, L. M., Hohlov, Y. E., Howison, S. D. & Kornev, K. 1999 Two-dimensional solidification and melting in potential flows. J. Fluid Mech. 378, 1–18.CrossRefGoogle Scholar
Davidovitch, B., Choi, J. & Bazant, M. Z. 2005 Average shape of transport-limited aggregates. Phys. Rev. Lett. 95 (7), 075504.CrossRefGoogle ScholarPubMed
De Baere, B., Molins, S., Mayer, K. U. & François, R. 2016 Determination of mineral dissolution regimes using flow-through time-resolved analysis (FT-TRA) and numerical simulation. Chem. Geol. 430, 1–12.CrossRefGoogle Scholar
Dutka, F., Starchenko, V., Osselin, F., Magni, S., Szymczak, P. & Ladd, A. J. C. 2020 Time-dependent shapes of a dissolving mineral grain: comparisons of simulations with microfluidic experiments. Chem. Geol. 540, 119459.CrossRefGoogle Scholar
Feigenbaum, M. J., Procaccia, I. & Davidovich, B. 2001 Dynamics of finger formation in Laplacian growth without surface tension. J. Stat. Phys. 103, 973–1007.CrossRefGoogle Scholar
Galin, L. A. 1945 Unsteady filtration with a free surface. Dokl. Akad. Nauk SSSR 47, 246–249.Google Scholar
Goldstein, M. E. & Reid, R. L. 1978 Effect of fluid flow on freezing and thawing of saturated porous media. Proc. R. Soc. Lond. A 364 (1716), 45–73.Google Scholar
Gubiec, T. & Szymczak, P. 2008 Fingered growth in channel geometry: a Loewner-equation approach. Phys. Rev. E 77, 041602.CrossRefGoogle ScholarPubMed
Gustafsson, B. & Vasil'ev, A. 2006 Conformal and Potential Analysis in Hele-Shaw Cells. Springer Science & Business Media.Google Scholar
Howison, S. D. 1992 Complex variable methods in Hele-Shaw moving boundary problems. Eur. J. Appl. Maths 3 (3), 209–224.CrossRefGoogle Scholar
Huang, J. M., Moore, M. N. J. & Ristroph, L. 2015 Shape dynamics and scaling laws for a body dissolving in fluid flow. J. Fluid Mech. 765, R3.CrossRefGoogle Scholar
Ivantsov, G. P. 1947 Temperature field around a spheroidal, cylindrical and acicular crystal growing in a supercooled melt. Dokl. Akad. Nauk SSSR 58, 567–569.Google Scholar
Kang, Q., Tsimpanogiannis, I. N., Zhang, D. & Lichtner, P. C. 2005 Numerical modeling of pore-scale phenomena during CO$_2$ sequestration in oceanic sediments. Fuel Process. Technol. 86, 1351–1370.CrossRefGoogle Scholar
Kondratiuk, P. & Szymczak, P. 2015 Steadily translating parabolic dissolution fingers. SIAM J. Appl. Maths 75, 2193–2213.CrossRefGoogle Scholar
Kornev, K. & Mukhamadullina, G. 1994 Mathematical theory of freezing for flow in porous media. Proc. R. Soc. Lond. A 447 (1930), 281–297.Google Scholar
Langer, J. S. 1980 Instabilities and pattern formation in crystal growth. Rev. Mod. Phys. 52, 1–30.CrossRefGoogle Scholar
Leshchiner, A., Thrasher, M., Mineev-Weinstein, M. B. & Swinney, H. L. 2010 Harmonic moment dynamics in Laplacian growth. Phys. Rev. E 81 (1), 016206.CrossRefGoogle ScholarPubMed
Li, L., Steefel, C. I. & Yang, L. 2008 Scale dependence of mineral dissolution kinetics within single pores and fractures. Geochim. Cosmochim. Acta 72, 360–377.CrossRefGoogle Scholar
Lichtner, P. C. & Kang, Q. 2007 Upscaling pore-scale reactive transport equations using a multiscale continuum formulation. Water Resour. Res. 43 (12), W12S15.CrossRefGoogle Scholar
Löwner, K. 1923 Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann. 89, 103.CrossRefGoogle Scholar
Mbogoro, M. M., Snowden, M. E., Edwards, M. A., Peruffo, M. & Unwin, P. R. 2011 Intrinsic kinetics of gypsum and calcium sulfate anhydrite dissolution: surface selective studies under hydrodynamic control and the effect of additives. J. Phys. Chem. C 115 (20), 10147–10154.CrossRefGoogle Scholar
Mineev-Weinstein, M., Wiegmann, P. B. & Zabrodin, A. 2000 Integrable structure of interface dynamics. Phys. Rev. Lett. 84 (22), 5106.CrossRefGoogle ScholarPubMed
Molins, S., Trebotich, D., Arora, B., Steefel, C. I. & Deng, H. 2019 Multi-scale model of reactive transport in fractured media: diffusion limitations on rates. Transp. Porous Med. 128 (2), 701–721.CrossRefGoogle Scholar
Molins, S., Trebotich, D., Steefel, C. I. & Shen, C. 2012 An investigation of the effect of pore scale flow on average geochemical reaction rates using direct numerical simulation. Water Resour. Res. 48 (3), W03527.CrossRefGoogle Scholar
Molins, S., Trebotich, D., Yang, L., Ajo-Franklin, J. B., Ligocki, T. J., Shen, C. & Steefel, C. I. 2014 Pore-scale controls on calcite dissolution rates from flow-through laboratory and numerical experiments. Environ. Sci. Technol. 48 (13), 7453–7460.CrossRefGoogle ScholarPubMed
Moore, M. N. J., Ristroph, L., Childress, S., Zhang, J. & Shelley, M. J. 2013 Self-similar evolution of a body eroding in a fluid flow. Phys. Fluids 25 (11), 116602.CrossRefGoogle Scholar
Needham, T. 2000 Visual Complex Analysis. Clarendon.Google Scholar
Nilson, R. H. & Griffiths, S. K. 1990 Wormhole growth in soluble porous materials. Phys. Rev. Lett. 65, 1583–1586.CrossRefGoogle ScholarPubMed
Oliveira, T. D. S., Blunt, M. J. & Bijeljic, B. 2019 Modelling of multispecies reactive transport on pore-space images. Adv. Water Resour. 127, 192–208.CrossRefGoogle Scholar
Pelcé, P. 2004 New Visions on Form and Growth: Fingered Growth, Dendrites, and Flames. Oxford University Press.Google Scholar
Pereira Nunes, J. P., Blunt, M. J. & Bijeljic, B. 2016 Pore-scale simulation of carbonate dissolution in micro-CT images. J. Geophys. Res. Solid 121 (2), 558–576.CrossRefGoogle Scholar
Polubarinova-Kochina, P. Y. 1945 On the motion of the oil contour. Dokl. Akad. Nauk SSSR 47, 254–257.Google Scholar
Pommerenke, C. 1975 Univalent Functions. Vanderhoeck und Ruprecht.Google Scholar
Quaife, B. D. & Moore, M. N. J. 2018 A boundary-integral framework to simulate viscous erosion of a porous medium. J. Comput. Phys. 375, 1–21.CrossRefGoogle Scholar
Richardson, S. 1972 Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56 (4), 609–618.CrossRefGoogle Scholar
Ristroph, L., Moore, M. N. J., Childress, S., Shelley, M. J. & Zhang, J. 2012 Sculpting of an erodible body by flowing water. Proc. Natl Acad. Sci. USA 109 (48), 19606–19609.CrossRefGoogle ScholarPubMed
Rycroft, C. H. & Bazant, M. Z. 2016 Asymmetric collapse by dissolution or melting in a uniform flow. Proc. R. Soc. Lond. A 472 (2185), 20150531.Google ScholarPubMed
Saffman, P. G. & Taylor, G. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245 (1242), 312–329.Google Scholar
Shraiman, B. & Bensimon, D. 1984 Singularities in nonlocal interface dynamics. Phys. Rev. A 30 (5), 2840–2842.CrossRefGoogle Scholar
Soulaine, C., Roman, S., Kovscek, A. & Tchelepi, H. A. 2017 Mineral dissolution and wormholing from a pore-scale perspective. J. Fluid Mech. 827, 457–483.CrossRefGoogle Scholar
Starchenko, V. & Ladd, A. J. C. 2018 The development of wormholes in laboratory scale fractures: perspectives from three-dimensional simulations. Water Resour. Res. 54 (10), 7946–7959.CrossRefGoogle Scholar
Starchenko, V., Marra, C. J. & Ladd, A. J. C. 2016 Three-dimensional simulations of fracture dissolution. J. Geophys. Res. Solid Earth 121, 6421–6444.CrossRefGoogle Scholar
Szymczak, P. & Ladd, A. J. C. 2012 Reactive infiltration instabilities in rocks. Fracture dissolution. J. Fluid Mech. 702, 239–264.CrossRefGoogle Scholar
Wiegmann, P. B. & Zabrodin, A. 2000 Conformal maps and integrable hierarchies. Commun. Math. Phys. 213 (3), 523–538.CrossRefGoogle Scholar
Supplementary material: File

Ladd et al. supplementary material

Supplementary figs and data

Download Ladd et al. supplementary material(File)
File 83.8 MB