<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Publishing DTD v1.3 20210610//EN" "JATS-journalpublishing1-3.dtd">
<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="ru"><front><journal-meta><journal-id journal-id-type="publisher-id">jamt</journal-id><journal-title-group><journal-title xml:lang="ru">Journal of Advanced Materials and Technologies</journal-title><trans-title-group xml:lang="en"><trans-title>Journal of Advanced Materials and Technologies</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">2782-2192</issn><issn pub-type="epub">2782-2206</issn><publisher><publisher-name>ФГБОУ ВО «ТГТУ»</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.17277/jamt.2023.01.pp.049-059</article-id><article-id custom-type="elpub" pub-id-type="custom">jamt-36</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>Производственные процессы и системы</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="en"><subject>Manufacturing processes and systems</subject></subj-group></article-categories><title-group><article-title>Механизм и кинетика сепарации примеси частиц различной плотности в быстром гравитационном потоке зернистого материала</article-title><trans-title-group xml:lang="en"><trans-title>Mechanism and kinetics of separation of impurity particles with different densities in a rapid gravity flow of granular material</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-9868-5487</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Иванов</surname><given-names>О. О.</given-names></name><name name-style="western" xml:lang="en"><surname>Ivanov</surname><given-names>O. O.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Олег Олегович Иванов, кандидат технических наук, доцент, советник по региональному развитию</p><p>141071;  ул. Трудовая, д. 1; Московская область; Королев</p></bio><bio xml:lang="en"><p>Oleg O. Ivanov, Cand. Sc. (Eng.), Associate Professor, Regional Development Advisor</p><p>141071; 1, Trudovaya St.; Moscow region; Korolev</p></bio><email xlink:type="simple">iooc4@mail.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-6227-5224</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Долгунин</surname><given-names>В. Н.</given-names></name><name name-style="western" xml:lang="en"><surname>Dolgunin</surname><given-names>V. N.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Виктор Николаевич Долгунин, доктор технических наук, профессор</p><p>392000; ул. Ленинградская, д. 1; Тамбов</p></bio><bio xml:lang="en"><p>Viktor N. Dolgunin, D. Sc. (Eng.), Professor</p><p>392000; 1, Leningradskaya St.; Tambov</p></bio><email xlink:type="simple">dolgunin-vn@yandex.ru</email><xref ref-type="aff" rid="aff-2"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-4902-982X</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Тараканов</surname><given-names>А. Г.</given-names></name><name name-style="western" xml:lang="en"><surname>Tarakanov</surname><given-names>A. G.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Александр Геннадьевич Тараканов, аспирант</p><p>392000; ул. Ленинградская, д. 1; Тамбов</p></bio><bio xml:lang="en"><p>Alexander G. Tarakanov, Postgraduate</p><p>392000; 1, Leningradskaya St.; Tambov</p></bio><email xlink:type="simple">uazqaaz@gmail.com</email><xref ref-type="aff" rid="aff-2"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0009-0009-5068-2234</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Куди</surname><given-names>К. А.</given-names></name><name name-style="western" xml:lang="en"><surname>Kudi</surname><given-names>K. A.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Константин Андреевич Куди, главный финансовый аналитик</p><p>115114; ул. Кожевническая, д. 14; Москва</p></bio><bio xml:lang="en"><p>Konstantin A. Kudi, Chief Financial Analyst</p><p>115114; 14, Kozhevnicheskaya St.; Moscow</p></bio><email xlink:type="simple">kostya.kudi@mail.com</email><xref ref-type="aff" rid="aff-3"/></contrib></contrib-group><aff-alternatives id="aff-1"><aff xml:lang="ru"><institution>АНО «Вдохновители»</institution><country>Россия</country></aff><aff xml:lang="en"><institution>ANO “Vdokhnoviteli”</institution><country>Russian Federation</country></aff></aff-alternatives><aff-alternatives id="aff-2"><aff xml:lang="ru"><institution>Тамбовский государственный технический университет</institution><country>Россия</country></aff><aff xml:lang="en"><institution>Tambov State Technical University</institution><country>Russian Federation</country></aff></aff-alternatives><aff-alternatives id="aff-3"><aff xml:lang="ru"><institution>Коммерческий банк «Ренессанс кредит»</institution><country>Россия</country></aff><aff xml:lang="en"><institution>CB “Renaissance Credit” (LLC)</institution><country>Russian Federation</country></aff></aff-alternatives><pub-date pub-type="collection"><year>2023</year></pub-date><pub-date pub-type="epub"><day>14</day><month>02</month><year>2026</year></pub-date><volume>8</volume><issue>1</issue><fpage>49</fpage><lpage>59</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Иванов О.О., Долгунин В.Н., Тараканов А.Г., Куди К.А., 2026</copyright-statement><copyright-year>2026</copyright-year><copyright-holder xml:lang="ru">Иванов О.О., Долгунин В.Н., Тараканов А.Г., Куди К.А.</copyright-holder><copyright-holder xml:lang="en">Ivanov O.O., Dolgunin V.N., Tarakanov A.G., Kudi K.A.</copyright-holder><license xml:lang="ru" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>Данная работа распространяется под лицензией Creative Commons Attribution 4.0.</license-p></license><license xml:lang="en" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>This work is licensed under a Creative Commons Attribution 4.0 License.</license-p></license></permissions><self-uri xlink:href="https://jamt.elpub.ru/jour/article/view/36">https://jamt.elpub.ru/jour/article/view/36</self-uri><abstract><p>   Проведено исследование эффективности процесса сепарации примеси частиц различной плотности в быстром гравитационном потоке монодисперсного зернистого материала. Исследование выполнено методом математического моделирования динамики распределения концентрации примеси частиц на шероховатом скате с использованием математической модели, учитывающей кинетику эффектов конвекции, перемешивания, сегрегации и квазидиффузионной сепарации. Исследование, проведенное при варьировании относительной плотностью частиц примеси в интервале 0,2…4,0 по отношению к плотности частиц базового компонента, подтверждает определяющую роль эффекта квазидиффузионной сепарации. Установлено, что профили распределения концентрации примеси частиц с малой и высокой плотностью имеет форму, аналогичную форме профилей объемных долей пустот и твердой фазы, соответственно. При одинаковой степени отличия частиц легких и тяжелых примесей от частиц базового компонента более высокую склонность к сепарации имеют легкие частицы. Склонность к сепарации с увеличением степени отличия плотности частиц примеси от плотности частиц базового компонента повышается более интенсивно для примеси частиц с малой плотностью. Установлено, что с уменьшением концентрации частиц примеси в потоке интенсивность их сепарации повышается для частиц с малой плотностью и снижается для частиц с высокой плотностью. Наиболее высокой склонностью к сепарации частиц по плотности характеризуется бинарная смесь однородных по размеру частиц с минимальным объемным содержанием примеси частиц, имеющих малую плотность.</p></abstract><trans-abstract xml:lang="en"><p>   The efficiency of the separation process of impurity particles with different densities in a rapid gravity flow of monodisperse granular material has been studied. The study was carried out by mathematical modeling of particle concentration distribution dynamics on a rough chute using a mathematical model that takes into account the kinetics of convection, mixing, segregation and quasi-diffusion separation effects. During the research process, the relative density of impurity particles in the range of 0.2–4.0 was varied with respect to the base component particle density, and this confirms the defining role of the quasi-diffusive separation effect. It was found that the concentration distribution profiles of low- and high-density particles have a shape similar to the shape of the profiles of the volume fractions of voids and solids, respectively. With the same degree of difference between the light and heavy impurity particles from the base component particles, the light particles have a higher tendency to separate. The tendency to separation with increasing degree of difference between the density of the impurity particles and the density of the base component particles increases more intensively for the impurity particles with low density. It is found that with decreasing concentration of impurity particles in the flow, the intensity of their separation increases for particles with low density and decreases for particles with high density. A binary mixture of homogeneous particles with the minimum volume content of low density particles is characterized by the highest tendency to particle density separation.</p></trans-abstract><kwd-group xml:lang="ru"><kwd>зернистый материал</kwd><kwd>быстрое гравитационное течение</kwd><kwd>сепарация частиц по плотности</kwd><kwd>сегрегация</kwd><kwd>квазидиффузионная сепарация (миграция)</kwd><kwd>скорость сдвига</kwd><kwd>объемная доля пустот</kwd></kwd-group><kwd-group xml:lang="en"><kwd>granular material</kwd><kwd>rapid gravity flow</kwd><kwd>particle density separation</kwd><kwd>segregation</kwd><kwd>quasi-diffusion separation (migration)</kwd><kwd>shear rate</kwd><kwd>void volume fraction</kwd></kwd-group><funding-group><funding-statement xml:lang="ru">Исследование выполнено при поддержке гранта РФФИ № 09-08-97521</funding-statement><funding-statement xml:lang="en">The study was supported by the RFBR grant No. 09-08-97521</funding-statement></funding-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">Goldhirsch I. Rapid granular flows. Annual Review of Fluid Mechanics. 2003;35:267-293. DOI: 10.1146/annurev.fluid.35.101101.161114</mixed-citation><mixed-citation xml:lang="en">Goldhirsch I. Rapid granular flows. Annual Review of Fluid Mechanics. 2003;35:267-293. DOI: 10.1146/annurev.fluid.35.101101.161114</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">Forterre Y, Pouliquen O. Flows of dense granular media. Annual Review of Fluid Mechanics. 2008;40:1-24. DOI: 10.1146/annurev.fluid.40.111406.102142</mixed-citation><mixed-citation xml:lang="en">Forterre Y, Pouliquen O. Flows of dense granular media. Annual Review of Fluid Mechanics. 2008;40:1-24. DOI: 10.1146/annurev.fluid.40.111406.102142</mixed-citation></citation-alternatives></ref><ref id="cit3"><label>3</label><citation-alternatives><mixed-citation xml:lang="ru">Pudasaini SP. A general two-phase debris flow model. Journal of Geophysical Research: Earth Surface. 2012;117(3):F03010. DOI: 10.1029/2011JF002186</mixed-citation><mixed-citation xml:lang="en">Pudasaini SP. A general two-phase debris flow model. Journal of Geophysical Research: Earth Surface. 2012;117(3):F03010. DOI: 10.1029/2011JF002186</mixed-citation></citation-alternatives></ref><ref id="cit4"><label>4</label><citation-alternatives><mixed-citation xml:lang="ru">Savage SB. Granular flows down rough inclines - Review and extension. In: Jenkins JT, Satake M. (eds.) Studies in Applied Mechanics. 1983;7:261-282. DOI: 10.1016/B978-0-444-42192-0.50028-1</mixed-citation><mixed-citation xml:lang="en">Savage SB. Granular flows down rough inclines - Review and extension. In: Jenkins JT, Satake M. (eds.) Studies in Applied Mechanics. 1983;7:261-282. DOI: 10.1016/B978-0-444-42192-0.50028-1</mixed-citation></citation-alternatives></ref><ref id="cit5"><label>5</label><citation-alternatives><mixed-citation xml:lang="ru">Domnik B, Pudasaini SP, Katzenbach R, Miller SA. Coupling of full two-dimensional and depth-averaged models for granular flows. Journal of Non-Newtonian Fluid Mechanics. 2013;201:56-68. DOI: 10.1016/j.jnnfm.2013.07.005</mixed-citation><mixed-citation xml:lang="en">Domnik B, Pudasaini SP, Katzenbach R, Miller SA. Coupling of full two-dimensional and depth-averaged models for granular flows. Journal of Non-Newtonian Fluid Mechanics. 2013;201:56-68. DOI: 10.1016/j.jnnfm.2013.07.005</mixed-citation></citation-alternatives></ref><ref id="cit6"><label>6</label><citation-alternatives><mixed-citation xml:lang="ru">Brennen CE. Fundamentals of multiphase flows. Cambridge: Cambridge University Press; 2005. 410 p. DOI: 10.1017/CBO9780511807169</mixed-citation><mixed-citation xml:lang="en">Brennen CE. Fundamentals of multiphase flows. Cambridge: Cambridge University Press; 2005. 410 p. DOI: 10.1017/CBO9780511807169</mixed-citation></citation-alternatives></ref><ref id="cit7"><label>7</label><citation-alternatives><mixed-citation xml:lang="ru">Dolgunin VN, Kudi AN, Tuev MA. Mechanisms and kinetics of gravity separation of granular materials. Uspekhi fizicheskikh nauk = Advances in Physical Sciences. 2020;190(6):545-561. DOI: 10.3367/UFNe.2020.01.038729. (In Russ.).</mixed-citation><mixed-citation xml:lang="en">Dolgunin VN, Kudi AN, Tuev MA. Mechanisms and kinetics of gravity separation of granular materials. Uspekhi fizicheskikh nauk = Advances in Physical Sciences. 2020;190(6):545-561. DOI: 10.3367/UFNe.2020.01.038729. (In Russ.).</mixed-citation></citation-alternatives></ref><ref id="cit8"><label>8</label><citation-alternatives><mixed-citation xml:lang="ru">Jian FX, Cai ZY, Guo WI. Laboratory-scale investigation of the material distribution characteristics of landslide dams in U-shaped valleys. Journal of Mountain Science, 2023;20:688-704. DOI: 10.1007/s11629-022-7664-3</mixed-citation><mixed-citation xml:lang="en">Jian FX, Cai ZY, Guo WI. Laboratory-scale investigation of the material distribution characteristics of landslide dams in U-shaped valleys. Journal of Mountain Science, 2023;20:688-704. DOI: 10.1007/s11629-022-7664-3</mixed-citation></citation-alternatives></ref><ref id="cit9"><label>9</label><citation-alternatives><mixed-citation xml:lang="ru">Hutter K, Sheiwiller T. Rapid plane flow of granular materials down a chute. In: Jenkins JT, Satake M. (eds.) Mechanics of granular materials. Amsterdam: Elsevier Science Publ.; 1983. pp. 283-293.</mixed-citation><mixed-citation xml:lang="en">Hutter K, Sheiwiller T. Rapid plane flow of granular materials down a chute. In: Jenkins JT, Satake M. (eds.) Mechanics of granular materials. Amsterdam: Elsevier Science Publ.; 1983. pp. 283-293.</mixed-citation></citation-alternatives></ref><ref id="cit10"><label>10</label><citation-alternatives><mixed-citation xml:lang="ru">Bates L. User guide to segregation. United Kingdom: Elsinore House; 1997. 134 p.</mixed-citation><mixed-citation xml:lang="en">Bates L. User guide to segregation. United Kingdom: Elsinore House; 1997. 134 p.</mixed-citation></citation-alternatives></ref><ref id="cit11"><label>11</label><citation-alternatives><mixed-citation xml:lang="ru">Gray JMNT, Thornton AR. A theory for particle size segregation in shallow granular free-surface flows. Proceedings of the Royal Society of London. 2005;461(2057):1447-1473. DOI: 10.1098/rspa.2004.1420</mixed-citation><mixed-citation xml:lang="en">Gray JMNT, Thornton AR. A theory for particle size segregation in shallow granular free-surface flows. Proceedings of the Royal Society of London. 2005;461(2057):1447-1473. DOI: 10.1098/rspa.2004.1420</mixed-citation></citation-alternatives></ref><ref id="cit12"><label>12</label><citation-alternatives><mixed-citation xml:lang="ru">Gray JMNT, Chugunov VA. Particle-size segregation and diffusive remixing in shallow granular avalanches. Journal of Fluid Mechanics. 2006;569:365-398. DOI: 10.1017/S0022112006002977</mixed-citation><mixed-citation xml:lang="en">Gray JMNT, Chugunov VA. Particle-size segregation and diffusive remixing in shallow granular avalanches. Journal of Fluid Mechanics. 2006;569:365-398. DOI: 10.1017/S0022112006002977</mixed-citation></citation-alternatives></ref><ref id="cit13"><label>13</label><citation-alternatives><mixed-citation xml:lang="ru">Gray JMNT, Shearer M, Thornton AR. Time-dependent solutions for particle-size segregation in shallow granular avalanches. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2006;462:947-972. DOI: 10.1098/rspa.2005.1580</mixed-citation><mixed-citation xml:lang="en">Gray JMNT, Shearer M, Thornton AR. Time-dependent solutions for particle-size segregation in shallow granular avalanches. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2006;462:947-972. DOI: 10.1098/rspa.2005.1580</mixed-citation></citation-alternatives></ref><ref id="cit14"><label>14</label><citation-alternatives><mixed-citation xml:lang="ru">Dolgunin VN, Kudi AN, Tarakanov AG. Structural inhomogeneity and effects of separation by size and density in gravity flow of granular materials. Journal of Engineering Physics and Thermophysics. 2022;95(2): 464-494. DOI: 10.1007/s10891-022-02505</mixed-citation><mixed-citation xml:lang="en">Dolgunin VN, Kudi AN, Tarakanov AG. Structural inhomogeneity and effects of separation by size and density in gravity flow of granular materials. Journal of Engineering Physics and Thermophysics. 2022;95(2): 464-494. DOI: 10.1007/s10891-022-02505</mixed-citation></citation-alternatives></ref><ref id="cit15"><label>15</label><citation-alternatives><mixed-citation xml:lang="ru">Hill KM, Fan Y. Granular temperature and segregation in dense sheared particulate mixtures. KONA Powder and Particle Journal. 2016;33:150-168. DOI: 10.14356/kona.2016022</mixed-citation><mixed-citation xml:lang="en">Hill KM, Fan Y. Granular temperature and segregation in dense sheared particulate mixtures. KONA Powder and Particle Journal. 2016;33:150-168. DOI: 10.14356/kona.2016022</mixed-citation></citation-alternatives></ref><ref id="cit16"><label>16</label><citation-alternatives><mixed-citation xml:lang="ru">Dolgunin VN, Kudi AN, Ukolov AA, Tuev MA. Rapid granular flows on a vibrated rough chute: behavior patterns and interaction effects of particles. Chemical Engineering Research and Design. 2017;122:22-32. DOI: 10.1016/j.cherd.2017.03.038</mixed-citation><mixed-citation xml:lang="en">Dolgunin VN, Kudi AN, Ukolov AA, Tuev MA. Rapid granular flows on a vibrated rough chute: behavior patterns and interaction effects of particles. Chemical Engineering Research and Design. 2017;122:22-32. DOI: 10.1016/j.cherd.2017.03.038</mixed-citation></citation-alternatives></ref><ref id="cit17"><label>17</label><citation-alternatives><mixed-citation xml:lang="ru">Widerseiner SN, Andreini N, Epely-Chauvin G, et al. Experimental investigation into segregating granular flows down chutes. Physics of Fluids. 2011;23:013301-013310. DOI: 10.1063/1.3536658</mixed-citation><mixed-citation xml:lang="en">Widerseiner SN, Andreini N, Epely-Chauvin G, et al. Experimental investigation into segregating granular flows down chutes. Physics of Fluids. 2011;23:013301-013310. DOI: 10.1063/1.3536658</mixed-citation></citation-alternatives></ref><ref id="cit18"><label>18</label><citation-alternatives><mixed-citation xml:lang="ru">Khakhar DV, McCarthy JJ, Ottino JM. Radial segregation of granular mixtures in rotating cylinders. Physics of Fluids. 1997;9:3600. DOI :10.1063/1.869498</mixed-citation><mixed-citation xml:lang="en">Khakhar DV, McCarthy JJ, Ottino JM. Radial segregation of granular mixtures in rotating cylinders. Physics of Fluids. 1997;9:3600. DOI :10.1063/1.869498</mixed-citation></citation-alternatives></ref><ref id="cit19"><label>19</label><citation-alternatives><mixed-citation xml:lang="ru">Hill KM, Tan DS. Segregation in dense sheared flows: gravity, temperature gradients, and stress partitioning. Journal of Fluid Mechanics. 2014;756:54-88. DOI: 10.1017/jfm.2014.271</mixed-citation><mixed-citation xml:lang="en">Hill KM, Tan DS. Segregation in dense sheared flows: gravity, temperature gradients, and stress partitioning. Journal of Fluid Mechanics. 2014;756:54-88. DOI: 10.1017/jfm.2014.271</mixed-citation></citation-alternatives></ref><ref id="cit20"><label>20</label><citation-alternatives><mixed-citation xml:lang="ru">Duan Y, Peckham J, Umbanhowar PB, Ottino JM, Lueptow RM. Designing minimally segregating granular mixtures for gravity-driven surface flows. AIChE Journal. 2023;e18032. DOI: 10.1002/aic.18032</mixed-citation><mixed-citation xml:lang="en">Duan Y, Peckham J, Umbanhowar PB, Ottino JM, Lueptow RM. Designing minimally segregating granular mixtures for gravity-driven surface flows. AIChE Journal. 2023;e18032. DOI: 10.1002/aic.18032</mixed-citation></citation-alternatives></ref><ref id="cit21"><label>21</label><citation-alternatives><mixed-citation xml:lang="ru">Ivanov OO, Dolgunin VN, Tarakanov AG, Zhilo AA, Pronin VA. Evolution of separation effects under variable conditions of gravity flows of granular materials. Journal of Advanced Materials and Technologies 2022;7(3):181-191. DOI: 10.17277/jamt.2022.03.pp.181-191.</mixed-citation><mixed-citation xml:lang="en">Ivanov OO, Dolgunin VN, Tarakanov AG, Zhilo AA, Pronin VA. Evolution of separation effects under variable conditions of gravity flows of granular materials. Journal of Advanced Materials and Technologies 2022;7(3):181-191. DOI: 10.17277/jamt.2022.03.pp.181-191.</mixed-citation></citation-alternatives></ref><ref id="cit22"><label>22</label><citation-alternatives><mixed-citation xml:lang="ru">Kudi AN, Fedosov NA, Sergeev VV, Tarakanov AG, Pronin VA, Goncharova AE. Multifractional separation of polydisperse particles varying in shape and density. Vestnik Tambovskogo Gosudarstvennogo Tehnicheskogo Universiteta. 2021;27(2):285-293. DOI: 10.17277/vestnik.2021.02.pp.285-293</mixed-citation><mixed-citation xml:lang="en">Kudi AN, Fedosov NA, Sergeev VV, Tarakanov AG, Pronin VA, Goncharova AE. Multifractional separation of polydisperse particles varying in shape and density. Vestnik Tambovskogo Gosudarstvennogo Tehnicheskogo Universiteta. 2021;27(2):285-293. DOI: 10.17277/vestnik.2021.02.pp.285-293</mixed-citation></citation-alternatives></ref><ref id="cit23"><label>23</label><citation-alternatives><mixed-citation xml:lang="ru">Dolgunin VN, Ukolov AA, Ivanov OO. Segregation Kinetics in the rapid gravity flow of granular materials. Theoretical Foundations of Chemical Engineering. 2006;40(4):393-404. DOI: 10.1134/S0040579506040099</mixed-citation><mixed-citation xml:lang="en">Dolgunin VN, Ukolov AA, Ivanov OO. Segregation Kinetics in the rapid gravity flow of granular materials. Theoretical Foundations of Chemical Engineering. 2006;40(4):393-404. DOI: 10.1134/S0040579506040099</mixed-citation></citation-alternatives></ref><ref id="cit24"><label>24</label><citation-alternatives><mixed-citation xml:lang="ru">Ferziger JH, Kaper HG. Mathematical theory of transport processes in gases. Amsterdam: North-Holland Publ.; 1972. 568 p.</mixed-citation><mixed-citation xml:lang="en">Ferziger JH, Kaper HG. Mathematical theory of transport processes in gases. Amsterdam: North-Holland Publ.; 1972. 568 p.</mixed-citation></citation-alternatives></ref><ref id="cit25"><label>25</label><citation-alternatives><mixed-citation xml:lang="ru">Shen HH, Ackermann NL. Constitutive relationships for fluid-solid mixture. Journal of the Engineering Mechanics Division, ASCE. 1982;108:748-763. DOI: 10.1061/jmcea3.0002868</mixed-citation><mixed-citation xml:lang="en">Shen HH, Ackermann NL. Constitutive relationships for fluid-solid mixture. Journal of the Engineering Mechanics Division, ASCE. 1982;108:748-763. DOI: 10.1061/jmcea3.0002868</mixed-citation></citation-alternatives></ref><ref id="cit26"><label>26</label><citation-alternatives><mixed-citation xml:lang="ru">Marchuk GI. Methods of numerical mathematics. Applications of mathematics. New York: Springer-Verlag; vol. 2, 1975. 316 p.</mixed-citation><mixed-citation xml:lang="en">Marchuk GI. Methods of numerical mathematics. Applications of mathematics. New York: Springer-Verlag; vol. 2, 1975. 316 p.</mixed-citation></citation-alternatives></ref><ref id="cit27"><label>27</label><citation-alternatives><mixed-citation xml:lang="ru">Chuyev AS. On the inconsistency of definitions of physical quantities dynamic and kinematic viscosity. Zakonodatel'naya i prikladnaya metrologiya. 2012;1:54-60. (In Russ.)</mixed-citation><mixed-citation xml:lang="en">Chuyev AS. On the inconsistency of definitions of physical quantities dynamic and kinematic viscosity. Zakonodatel'naya i prikladnaya metrologiya. 2012;1:54-60. (In Russ.)</mixed-citation></citation-alternatives></ref><ref id="cit28"><label>28</label><citation-alternatives><mixed-citation xml:lang="ru">Nagel SR. Experimental soft-matter science. Reviews of Modern Physics. 2017;89(2):025002 DOI: 10.1103/RevModPhys.89.025002</mixed-citation><mixed-citation xml:lang="en">Nagel SR. Experimental soft-matter science. Reviews of Modern Physics. 2017;89(2):025002 DOI: 10.1103/RevModPhys.89.025002</mixed-citation></citation-alternatives></ref><ref id="cit29"><label>29</label><citation-alternatives><mixed-citation xml:lang="ru">Gidaspow D. Multiphase flow and fluidization: Continuum and kinetic theory description. San Diego: Academic press; 1994. 467 p.</mixed-citation><mixed-citation xml:lang="en">Gidaspow D. Multiphase flow and fluidization: Continuum and kinetic theory description. San Diego: Academic press; 1994. 467 p.</mixed-citation></citation-alternatives></ref><ref id="cit30"><label>30</label><citation-alternatives><mixed-citation xml:lang="ru">Jenkins JT, Savage SB. A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. Journal of Fluid Mechanics. 1983;130:187-202. DOI: 10.1017/S0022112083001044</mixed-citation><mixed-citation xml:lang="en">Jenkins JT, Savage SB. A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. Journal of Fluid Mechanics. 1983;130:187-202. DOI: 10.1017/S0022112083001044</mixed-citation></citation-alternatives></ref><ref id="cit31"><label>31</label><citation-alternatives><mixed-citation xml:lang="ru">Jenkins J, Berzi D. Kinetic theory applied to inclined flows. Granular Matter. 2012;14(2):79-84. DOI: 10.1007/s10035-011-0308-x</mixed-citation><mixed-citation xml:lang="en">Jenkins J, Berzi D. Kinetic theory applied to inclined flows. Granular Matter. 2012;14(2):79-84. DOI: 10.1007/s10035-011-0308-x</mixed-citation></citation-alternatives></ref><ref id="cit32"><label>32</label><citation-alternatives><mixed-citation xml:lang="ru">Umbanhowar PB, Lueptow RM, Ottino JM. Modeling segregation in granular flows. Annual Review of Chemical and Biomolecular Engineering. 2019;10:129-153 DOI: 10.1146/annurev-chembioeng-060718-030122</mixed-citation><mixed-citation xml:lang="en">Umbanhowar PB, Lueptow RM, Ottino JM. Modeling segregation in granular flows. Annual Review of Chemical and Biomolecular Engineering. 2019;10:129-153 DOI: 10.1146/annurev-chembioeng-060718-030122</mixed-citation></citation-alternatives></ref></ref-list><fn-group><fn fn-type="conflict"><p>The authors declare that there are no conflicts of interest present.</p></fn></fn-group></back></article>
