Construction of mathematical models of the statics of grain media considering the Reynolds effect
Keywords:granular materials, equilibrium thermodynamics, Reynolds effect, horizontal grain layer, boundary-value problem, boundary conditions
This study addresses the construction of a mathematical model, the statement of boundary-value problems on the statics of a grainy material related to the technological processes of agricultural production. A working apparatus employed to construct the model of a grainy material is the methods of equilibrium thermodynamics. We have stated the main thermodynamic equality, which makes it possible to derive a rheological ratio that establishes the connection between stresses and deformations of the granular material. The chosen grainy material is a granular medium that manifests a Reynolds effect. This effect occurs in the case of small deformations and indicates the presence of a dependence of dilation on the stress tensor deviator. In contrast to the classical methods that consider a model of continuous medium with the non-deformed and smooth grain’s particles, the present work takes into consideration both a Reynolds effect and the presence of elastic deformations. The resulting rheological ratio produces the dependence for a stress tensor on the deformation tensor corresponding to ratios from the linear theory of elasticity.
For the case of an isothermal process of deformation, a boundary-value problem on the grain material’s statics in the field of gravity has been stated. This paper shows the statement and solution to two particular tasks on the balance of a granular layer along the horizontal plane: in the absence of surface forces and under the action of tangent surface forces on a free surface.
The boundary problems on the equilibrium of a granular material are nonlinear in character, and the resulting solution represents a complex mathematical apparatus involving numerical methods.The obtained models for the statics of a continuous environment precede the consideration of dynamic problems, in particular, the study of equilibrium stability
- Aranson, I. S., Tsimring, L. S. (2006). Patterns and collective behavior in granular media: Theoretical concepts. Reviews of Modern Physics, 78 (2), 641–692. doi: https://doi.org/10.1103/revmodphys.78.641
- Börzsönyi, T., Halsey, T. C., Ecke, R. E. (2008). Avalanche dynamics on a rough inclined plane. Physical Review E, 78 (1). doi: https://doi.org/10.1103/physreve.78.011306
- Gujjula, R., Mangadoddy, N. (2015). Hydrodynamic Study of Gas–Solid Internally Circulating Fluidized Bed Using Multiphase CFD Model. Particulate Science and Technology, 33 (6), 593–609. doi: https://doi.org/10.1080/02726351.2015.1013590
- Das, P., Puri, S., Schwartz, M. (2018). Granular fluids with solid friction and heating. Granular Matter, 20 (1). doi: https://doi.org/10.1007/s10035-018-0789-y
- Schwedes, J. (2003). Review on testers for measuring flow properties of bulk solids. Granular Matter, 5 (1), 1–43. doi: https://doi.org/10.1007/s10035-002-0124-4
- Nanka, A., Ievlev, I. (2017). About separation of the impurity in the carrying stream of the grain environment. Visnyk Kharkivskoho natsionalnoho tekhnichnoho universytetu silskoho hospodarstva imeni Petra Vasylenka, 181, 215–222.
- Tishchenko, L., Kharchenko, S., Kharchenko, F., Bredykhin, V., Tsurkan, O. (2016). Identification of a mixture of grain particle velocity through the holes of the vibrating sieves grain separators. Eastern-European Journal of Enterprise Technologies, 2 (7 (80)), 63–69. doi: https://doi.org/10.15587/1729-4061.2016.65920
- Nesterenko, O. V., Leshchenko, S. M., Vasylkovskyi, O. M., Petrenko, D. I. (2017). Analytical assessment of the pneumatic separation quality in the process of grain multilayer feeding. INMATEH. Agricultural Engineering, 53 (3), 65–70.
- Minh, N. H., Cheng, Y. P. (2016). On the contact force distributions of granular mixtures under 1D-compression. Granular Matter, 18 (2). doi: https://doi.org/10.1007/s10035-016-0625-1
- Swisher, N. C., Utter, B. C. (2014). Flow profile of granular avalanches with imposed vertical vibration. Granular Matter, 16 (2), 175–183. doi: https://doi.org/10.1007/s10035-014-0488-2
- Kaviani Rad, H., Nejat Pishkenari, H. (2018). Frictional viscoelastic based model for spherical particles collision. Granular Matter, 20 (4). doi: https://doi.org/10.1007/s10035-018-0835-9
- Gnoli, A., Lasanta, A., Sarracino, A., Puglisi, A. (2016). Unified rheology of vibro-fluidized dry granular media: From slow dense flows to fast gas-like regimes. Scientific Reports, 6 (1). doi: https://doi.org/10.1038/srep38604
- Dijksman, J. A., Wortel, G. H., van Dellen, L. T. H., Dauchot, O., van Hecke, M. (2011). Jamming, Yielding, and Rheology of Weakly Vibrated Granular Media. Physical Review Letters, 107 (10). doi: https://doi.org/10.1103/physrevlett.107.108303
- Gnoli, A., Puglisi, A., Sarracino, A., Vulpiani, A. (2014). Nonequilibrium Brownian Motion beyond the Effective Temperature. PLoS ONE, 9 (4), e93720. doi: https://doi.org/10.1371/journal.pone.0093720
- Liu, H., Yoon, S., Li, M. (2016). Three-dimensional computational fluid dynamics (CFD) study of the gas–particle circulation pattern within a fluidized bed granulator: By full factorial design of fluidization velocity and particle size. Drying Technology, 35 (9), 1043–1058. doi: https://doi.org/10.1080/07373937.2016.1230628
- Ford, K. J., Gilchrist, J. F., Caram, H. S. (2009). Transitions to vibro-fluidization in a deep granular bed. Powder Technology, 192 (1), 33–39. doi: https://doi.org/10.1016/j.powtec.2008.11.017
- Jaeger, H. M., Nagel, S. R., Behringer, R. P. (1996). The Physics of Granular Materials. Physics Today, 49 (4), 32–38. doi: https://doi.org/10.1063/1.881494
- Schreck, C. F., O’Hern, C. S., Shattuck, M. D. (2013). Vibrations of jammed disk packings with Hertzian interactions. Granular Matter, 16 (2), 209–216. doi: https://doi.org/10.1007/s10035-013-0458-0
- Chou, C. (2004). The kinematic model for granular flow in a two‐dimensional symmetrical louvered moving granular filter bed. Journal of the Chinese Institute of Engineers, 27 (2), 299–304. doi: https://doi.org/10.1080/02533839.2004.9670876
- Dahl, S. R., Hrenya, C. M., Garzó, V., Dufty, J. W. (2002). Kinetic temperatures for a granular mixture. Physical Review E, 66 (4). doi: https://doi.org/10.1103/physreve.66.041301
- Pouliquen, O., Forterre, Y. (2009). A non-local rheology for dense granular flows. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 367 (1909), 5091–5107. doi: https://doi.org/10.1098/rsta.2009.0171
- Das, M. K., Mukherjee, P. P., Muralidhar, K. (2018). Modeling Transport Phenomena in Porous Media with Applications. Springer. doi: https://doi.org/10.1007/978-3-319-69866-3
- Bezmaternykh, A. V., Ofrikhter, V. G. (2017). The phenomenon of dilatancy and its impact оn the nature of deformation of soil under load. MASTER`S JOURNAL, 2, 85–90.
- Bileush, A. I., Krivonog, A. I., Krivonog, V. V., Filimonov,V. Yu. (2011). Strength of granular soil having dilatancy. Prykladna hidromekhanika, 13 (3), 23–32.
- Liu, H., Zhang, S.-H., Cheng, M., Song, H.-W., Trentadue, F. (2015). A minimum principle for contact forces in random packings of elastic frictionless particles. Granular Matter, 17 (4), 475–482. doi: https://doi.org/10.1007/s10035-015-0567-z
- Nicolis, G. (1970). Thermodynamic theory of stability, structure and fluctuations. Pure and Applied Chemistry, 22 (3-4), 379–392. doi: https://doi.org/10.1351/pac197022030379
- Marin, V. I., Didenko, B. A. (2002). Modelirovanie akusticheskogo trakta ustroystva izmereniya protsentnogo soderzhaniya svyazuyushchego. Matematicheskie metody v tehnike i tehnologiyah: sb. tr. XV Mezhdunar. nauch. konf. Tambov, 59–62.
- Langston, P. A., Nikitidis, M. S., Tüzün, V., Heyes, D. M. (1998). Tomographic measurements and distinct element simulations of binary granular flow voidage. World Congress on particle Technology 3. Brighton, UK, 333.
- Millen, M. J., Sowerby, B. D., Abemethy, D. A., Kingsiey, R., Grima, C. (1997). On-line measurement of pulverised coal mass flow using an ultrasonic technique. Powder technology, 92, 105–113.
- Schlaberg, H. I., Podd, F. J. W., Hoyle, B. S. (2000). Ultrasound process tomography system for hydrocyclones. Ultrasonics, 38 (1-8), 813–816. doi: https://doi.org/10.1016/s0041-624x(99)00189-4
- Dolgunin, V. N., Ivanov, O. O., Borshchev, V. Ya. (2016). Sdvigovye techeniya zernistyh sred: zakonomernosti i tehnologicheskie aspekty. Tambov: Izd-vo FGBOU VO «TGTU», 168.
- Jaeger, H. M., Nagel, S. R., Behringer, R. P. (1996). Granular solids, liquids, and gases. Reviews of Modern Physics, 68 (4), 1259–1273. doi: https://doi.org/10.1103/revmodphys.68.1259
How to Cite
Copyright (c) 2019 Alexander Nanka, Ivan Iyevlev, Vitaliy Sementsov, Denis Boiko, Viktor Duhanets
This work is licensed under a Creative Commons Attribution 4.0 International License.
The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.