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RAS Chemistry & Material ScienceРасплавы Melts

  • ISSN (Print) 0235-0106
  • ISSN (Online) 3034-5715

Linear stability analysis of a solidification process with convection in a bounded region of space

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PII
10.31857/S0235010624020048-1
DOI
10.31857/S0235010624020048
Publication type
Article
Status
Published
Authors
Александров Дмитрий Валерьевич
  • Affiliation: Ural Federal University named after the First President of Russia B.N. Yeltsin
Volume/ Edition
Volume / Issue number 2
Pages
189-210
Abstract
The morphological/dynamic instability of crystallization process in a bounded region in the presence of intense convection in liquid is studied. The paper considers a linear theory of morphological instability with a flat solid-liquid interface on the example of molten metal and magma. The mathematical model includes heat transfer equations and convective type boundary conditions at the interface. The equations for perturbations of the temperature field and interfacial boundary are found, allowing to obtain the dispersion relation. Its analysis has shown the existence of morphological instability of the flat interfacial boundary for a wide range of wavenumbers. Dynamic perturbations (perturbations of the quasi-stationary crystallization velocity) were also analyzed and two solutions for the perturbation frequency were obtained. One of them is stable and the other one is unstable. The system selects one of them depending on the action of convection. The result of morphological and dynamic instability is the appearance of a two-phase region in front of a flat solid-liquid interface. Therefore, the paper also considers the dynamic instability of stationary crystallization with a two-phase region replaced by a discontinuity surface. In this case, the dynamic instability was also found for a wide range of crystallization velocities.
Keywords
направленная кристаллизация конвекция линейный анализ устойчивости аналитические решения
Date of publication
17.09.2025
Year of publication
2025
Number of purchasers
0
Views
6

References

  1. 1. Dantzig J.A., Rappaz M. Solidification: 2nd Edition – Revised & Expanded. EPFL Press, Lausanne. 2017.
  2. 2. Fisher D., Kurz W. Fundamentals of solidification // Fundamentals of Solidification. 1998. P. 1–316.
  3. 3. Alexandrov D.V., Zubarev A.Y. Patterns in soft and biological matters // Philosophical Transactions of the Royal Society A. 2020. 378. № 2171. P. 20200002.
  4. 4. Kurz W., Fisher D.J., Trivedi R. Progress in modelling solidification microstructures in metals and alloys: dendrites and cells from 1700 to 2000 // International Materials Reviews. 2019. 64. № 6. P. 311–354.
  5. 5. Kurz W., Rappaz M., Trivedi R. Progress in modelling solidification microstructures in metals and alloys. Part II: dendrites from 2001 to 2018 // International Materials Reviews. 2021. 66. № 1. P. 30–76.
  6. 6. Galenko P.K., Toropova L.V., Alexandrov D.V., Phanikumar G., Assadi H., Reinartz M., Paul P., Fang Y., Lippmann S. Anomalous kinetics, patterns formation in recalescence, and final microstructure of rapidly solidified Al-rich Al-Ni alloys // Acta Materialia. 2022. 241. P. 118384.
  7. 7. Alexandrov D.V., Galenko P.K. The shape of dendritic tips // Philosophical Transactions of the Royal Society A. 2020. 378. № 2171. P. 20190243.
  8. 8. Александров Д.В., Дубовой Г.Ю., Малыгин А.П., Низовцева И.Г., Торопова Л.В. Затвердевание трехкомпонентных систем с нелинейной фазовой диаграммой // Расплавы. 2017. № 1. С. 3–17.
  9. 9. Александров Д.В., Низовцева И.Г., Александрова И.В., Иванов А.А., Стародумов И.О., Торопова Л.В., Гусакова О.В., Шепелевич В.Г. К теории направленного затвердевания при наличии области фазового превращения // Расплавы. 2020. № 5. С. 469–479.
  10. 10. Ivanov A.A., Alexandrova I.V., Alexandrov D.V. Towards the theory of phase transformations in metastable liquids. Analytical solutions and stability analysis // European Physical Journal Special Topics. 2020. 229. P. 365–373.
  11. 11. Wang H., Liu F., Wang K., Zhai H. Oscillatory morphological stability for rapid directional solidification: effect of non-linear liquidus and solidus // Acta Materialia. 2011. 59. № 14. P. 5859–5867.
  12. 12. Makoveeva E.V., Ivanov A.A., Alexandrova I.V., Alexandrov D.V. Directional crystallization with a mushy region. Part 1: linear analysis of dynamic stability // European Physical Journal Special Topics. 2023. 232. P. 1119–1127.
  13. 13. Makoveeva E.V., Ivanov A.A., Alexandrova I.V., Alexandrov D.V. Directional crystallization with a mushy region. Part 2: nonlinear analysis of dynamic stability // European Physical Journal Special Topics. 2023. 232. P. 1129–1139.
  14. 14. Mullins W.W., Sekerka R.F. Stability of a planar interface during solidification of a dilute binary alloy // Journal of Applied Physics. 1964. 35. № 2. P. 444–451.
  15. 15. Sekerka R.F. A stability function for explicit evaluation of the Mullins‐Sekerka interface stability criterion // Journal of Applied Physics. 1965. 36. № 1. P. 264–268.
  16. 16. Sekerka R.F. Morphological stability // Journal of Crystal Growth. 1968. 3. P. 71–81.
  17. 17. Cserti J., Tichy G. Stability of anisotropic liquid-solid interfaces // Acta Metallurgica. 1986. 34. № 6. P. 1029–1034.
  18. 18. Wilson L.O. The effect of fluctuating growth rates on segregation in crystals grown from the melt // Journal of Crystal Growth. 1980. 48. № 3. P. 435–458.
  19. 19. Wheeler A.A. The effect of a periodic growth rate on the morphological stability of a freezing binary alloy //Journal of Crystal Growth. 1984. 67. № 1. P. 8–26.
  20. 20. Wollhöver K., Scheiwe M.W., Hartmann U., Körber C. On morphological stability of planar phase boundaries during unidirectional transient solidification of binary aqueous solutions // International Journal of Heat and Mass Transfer. 1985. 28. № 5. P. 897–902.
  21. 21. Alexandrov D.V. Self-similar solidification: morphological stability of the regime // International Journal of Heat and Mass Transfer. 2004. 47. № 6–7. P. 1383–1389.
  22. 22. Laxmanan V. Morphological transitions in the rapid solidification regime: a re-examination of the fundamental validity of the absolute stability concept of Mullins and Sekerka //Acta Metallurgica. 1989. 37. № 4. P. 1109–1119.
  23. 23. Alexandrov D.V., Ivanov A.O. Dynamic stability analysis of the solidification of binary melts in the presence of a mushy region: changeover of instability // Journal of Crystal Growth. 2000. 210. № 4. P. 797–810.
  24. 24. Alexandrov D.V., Malygin A.P. Convective instability of directional crystallization in a forced flow: the role of brine channels in a mushy layer on nonlinear dynamics of binary systems // International Journal of Heat and Mass Transfer. 2011. 54. № . 5–6. P. 1144–1149.
  25. 25. Solomon T.H., Hartley R.R., Lee A.T. Aggregation and chimney formation during the solidification of ammonium chloride // Physical Review E. 1999. 60. № 3. P. 3063.
  26. 26. Katz R.F., Worster M.G. Simulation of directional solidification, thermochemical convection, and chimney formation in a Hele-Shaw cell // Journal of Computational Physics. 2008. 227. № 23. P. 9823–9840.
  27. 27. Delves R.T. Theory of stability of a solid-liquid interface during growth from stirred melts // Journal of Crystal Growth. 1968. 3. P. 562–568.
  28. 28. Coriell S.R., Hurle D.T. J., Sekerka R.F. Interface stability during crystal growth: the effect of stirring // Journal of Crystal Growth. 1976. 32. № 1. P. 1–7.
  29. 29. Favier J.J., Rouzaud A. Morphological stability of the solidification interface under convective conditions // Journal of Crystal Growth. 1983. 64. № 2. P. 367–379.
  30. 30. Forth S.A., Wheeler A.A. Coupled convective and morphological instability in a simple model of the solidification of a binary alloy, including a shear flow // Journal of Fluid Mechanics. 1992. 236. P. 61–94.
  31. 31. Chen Y.J., Davis S.H. Directional solidification of a binary alloy into a cellular convective flow: localized morphologies // Journal of Fluid Mechanics. 1999. 395. P. 253–270.
  32. 32. Alexandrov D.V., Malygin A.P. Flow-induced morphological instability and solidification with the slurry and mushy layers in the presence of convection // International Journal of Heat and Mass Transfer. 2012. 55. № 11–12. P. 3196–3204.
  33. 33. Makoveeva E.V., Alexandrov D.V. Morphological stability analysis of a planar crystallization front with convection // European Physical Journal Special Topics. 2023. 232. P. 1109–1117.
  34. 34. Kerr R.C., Woods A.W., Worster M.G., Huppert H.E. Solidification of an alloy cooled from above Part 1. Equilibrium growth // Journal of Fluid Mechanics. 1990. 216. P. 323–342.
  35. 35. Turner J.S., Huppert H.E., Sparks R.S.J. Komatiites II: Experimental and theoretical investigations of post-emplacement cooling and crystallization // Journal of Petrology. 1986. 27. № 2. P. 397–437.
  36. 36. Peppin S.S. L., Aussillous P., Huppert H.E., Worster M.G. Steady-state mushy layers: experiments and theory //Journal of Fluid Mechanics. 2007. 570. P. 69–77.
  37. 37. Peppin S.S. L., Huppert H.E., Worster M.G. Steady-state solidification of aqueous ammonium chloride // Journal of Fluid Mechanics. 2008. 599. P. 465–476.
  38. 38. Huguet L., Alboussiere T., Bergman M.I., Deguen R., Labrosse S., Lesœur G. Structure of a mushy layer under hypergravity with implications for Earth’s inner core // Geophysical Journal International. 2016. 204. № 3. P. 1729–1755.
  39. 39. Alexandrov D.V., Netreba A.V., Malygin A.P. Time-dependent crystallization in magma chambers and lava lakes cooled from above: the role of convection and kinetics on nonlinear dynamics of binary systems // International Journal of Heat and Mass Transfer. 2012. 55. № 4. P. 1189–1196.
  40. 40. Nizovtseva I.G., Alexandrov D.V. The effect of density changes on crystallization with a mushy layer // Philosophical Transactions of the Royal Society A. 2020. 378. № 2171. P. 20190248.
  41. 41. Makoveeva E.V. Steady-state crystallization with a mushy layer: a test of theory with experiments // European Physical Journal Special Topics. 2023. P. 1165 –1169.
  42. 42. Galenko P.K., Danilov D.A. Selection of the dynamically stable regime of rapid solidification front motion in an isothermal binary alloy // Journal of Crystal Growth. – 2000. 216. № 1–4. P. 512–526.
  43. 43. Alexandrov D.V., Mansurov V.V. Dynamic stability of a solidification process of a binary melt in the presence of a broad quasiequilibrium mushy region //Scripta Materialia. 1996. 35. № 7. P. 787–790.
  44. 44. Buevich Y.A., Iskakova L.Y., Mansurov V.V. Design of processes involving directional solidification with an equilibrium two-phase zone // Teplofizika Vysokikh Temperatur. 1991. 29. № 2. P. 286–293.
  45. 45. Alexandrov D., Buyevich Y., Mansurov V., Vabischevich P., Churbanov A. Directional solidification of binary melts in the presence of a mushy region // Numerical Methods in Engineering’96 (Paris, 9–13 September 1996). 1996. С. 697–706.
  46. 46. Чернов А.А. Об одной модели затвердевания магмы в процессе эксплозивного вулканического извержения // Прикладная механика и техническая физика. ٢٠٠٣. 44. № 5. С. 89–90.
  47. 47. Долбак А.Е., Жачук Р.А., Ольшанецкий Б.З. Диффузия Cu по чистой поверхности Si (١١١) // Физика и техника полупроводников. ٢٠٠١. 35. № 9. С. 1063–1066.
  48. 48. Pelcé P. Dynamics of Curved Fronts. Academic Press, Boston, MA. 1988.
  49. 49. Bouissou P., Pelcé P. Effect of a forced flow on dendritic growth // Physical Review A. 1989. 40. № 11. P. 6673.
  50. 50. Alexandrov D.V., Galenko P.K. Dendritic growth with the six-fold symmetry: theoretical predictions and experimental verification // Journal of Physics and Chemistry of Solids. 2017. 108. P. 98–103.
  51. 51. Kessler D.A., Koplik J., Levine H. Pattern selection in fingered growth phenomena // Advances in Physics. 1988. 37. № 3. P. 255–339.
  52. 52. Торопова Л.В., Александров Д.В., Галенко П.К. К вопросу об устойчивом росте анизотропного дендрита при конвективном теплопереносе в жидкой фазе у поверхности дендрита // Расплавы. ٢٠١٨. № 3. С. 320–329.
  53. 53. Feltham D.L., Worster M.G. Flow-induced morphological instability of a mushy layer // Journal of Fluid Mechanics. 1999. 391. P. 337–357.
  54. 54. Feltham D.L., Worster M.G., Wettlaufer J.S. The influence of ocean flow on newly forming sea ice //Journal of Geophysical Research: Oceans. 2002. 107. – № C2. P. 1119.
  55. 55. Alexandrov D.V., Malygin A.P. Convective instability of solidification with a phase transition zone // Journal of Experimental and Theoretical Physics. 2011. 112. P. 596–601.
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