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Abstract

The basic objective of the research is to construct a difference model of the melt motion. The existence of a solution to the problem is proven in the paper. It is also proven the convergence of the difference problem solution to the original problem solution of the melt motion. The Rothe method is implemented to study the Navier–Stokes equations, which provides the study of the boundary value problems correctness for a viscous incompressible flow both numerically and analytically.
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Bibliography

[1] R. Lakshminarayana, K. Dadzie, R. Ocone, M. Borg, and J. Reese: Recasting Navier–Stokes equations. Journal of Physics Communications, 3(10), (2019), 13–18, DOI: 10.1088/2399-6528/ab4b86.
[2] S.Sh. Kazhikenova, S.N. Shaltaqov, D. Belomestny, and G.S. Shai- hova: Finite difference method implementation for Numerical integration hydrodynamic equations melts. Eurasian Physical Technical Journal, 17(33), (2020), 50–56.
[3] C. Bardos: A basic example of non linear equations: The Navier– Stokes equations. Mathematics: Concepts and Foundations, III (2002), http://www.eolss.net/sample-chapters/c02/e6-01-06-02.pdf.
[4] J.XuandW.Yu:ReducedNavier–Stokes equations with streamwise viscous diffusion and heat conduction terms. AIAA Pap., 1441 (1990), 1–6, DOI: 10.2514/6.1990-1441.
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[7] S.E. Rogers, K. Dochan, and K. Cetin: Steady and unsteady solutions of the incompressible Navier–Stokes equations. AIAA Journal, 29(4), (1991), 603–610, DOI: 10.2514/3.10627.
[8] S. Masayoshi, T. Hiroshi, S. Nobuyuki, and N. Hidetoshi: Numerical simulation of three-dimensional viscous flows using the vector potential method. JSME International Journal, 34(2), (1991), 109–114, DOI: 10.1299/jsmeb1988.34.2_109.
[9] E. Sciubba: A variational derivation of the Navier–Stokes equations based on the exergy destruction of the flow. Journal of Mathematical and Physical Sciences, 25(1), (1991), 61–68.
[10] A. Bouziani and R. Mechri: The Rothe’s method to a parabolic integrodifferential equation with a nonclassical boundary conditions. International Journal of Stochastic Analysis, Article ID 519684, (2010), DOI: 10.1155/2010/519684.
[11] N. Merazga and A. Bouziani: Rothe time-discretization method for a nonlocal problem arising in thermoelasticity. Journal of Applied Mathematics and Stochastic Analysis, 2005(1), (2005), 13–28, DOI: 10.1080/00036818908839869.
[12] T.A. Barannyk, A.F. Barannyk, and I.I. Yuryk: Exact solutions of the nonliear equation. Ukrains’kyi Matematychnyi Zhurnal, 69(9), (2017), 1180–1186, http://umj.imath.[K]iev.ua/index.php/umj/article/view/1768.
[13] N.B. Iskakova, A.T. Assanova, and E.A. Bakirova: Numerical method for the solution of linear boundary-value problem for integrodifferential equations based on spline approximations. Ukrains’kyi Matematychnyi Zhurnal, 71(9), (2019), 1176–91, http://umj.imath.[K]iev.ua/index.php/ umj/article/view/1508.
[14] S.L. Skorokhodov and N.P. Kuzmina: Analytical-numerical method for solving an Orr-Sommerfeld type problem for analysis of instability of ocean currents. Zh. Vychisl. Mat. Mat. Fiz., 58(6), (2018), 1022–1039, DOI: 10.7868/S0044466918060133.
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Authors and Affiliations

Saule Sh. Kazhikenova
1
ORCID: ORCID
Sagyndyk N. Shaltakov
1
ORCID: ORCID
Bekbolat R. Nussupbekov
2
ORCID: ORCID

  1. Karaganda Technical University, Kazakhstan
  2. Karaganda University E.A. Buketov, Kazakhstan
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Abstract

This article proposes a method for grinding coal based on the use of the energy of a pulsed shock wave resulting from a spark electric discharge in a liquid. The main purpose of the scientific work is the development of an electric pulse device for producing coal powder, the main component of coal-water fuel. The diameter of the initial coal fraction averaged 3 mm, and the size of the resulting product was 250 μm. To achieve this goal, the dependence of the length of a metal rod electrode (positive electrode) on the length and diameter of its insulation is investigated. Various variants of the shape of the base (bottom) of the device acting as a negative electrode are considered, and an effective variant based on the results of coal grinding is proposed. An experimental electric pulse installation is described, the degree of coal grinding is determined depending on the geometric parameters. The optimal characteristics of the obtained coal powder have been established.
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Bibliography

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[6] A. Atal and Y.A. Levendis. Observations on the combustion behavior of coal water fuels and coal water fuels impregnated with calcium magnesium acetate. Combustion and Flame, 93(1-2):61–89. 1993. doi: 10.1016/0010-2180(93)90084-G.
[7] S. Yavuzkurt and M.Y Ha. A model of the enhancement of combustion of coal-water slurry fuels using high-intensity acoustic fields. Journal of Energy Resources Technology, 113(4):268–276, 1991. doi: 10.1115/1.2905911.
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[13] J. Robak, K. Ignasiak, and M. Rejdak. Coal micronization studies in vibrating mill in terms of coal water slurry (CWS) fuel preparation. Journal of Ecological Engineering, 18(2):111–118. 2017. doi: 10.12911/22998993/68214.
[14] A.R. Rizun, T.D. Denisyuk, Y.V. Golen, V.Y. Kononov, and A.N. Rachkov. Electric discharge disintegration and coal desulphurization in the manufacture of water-coal fuel. Surface Engineering and Applied Electrochemistry, 47(1):100–102. 2011. doi: 10.3103/S1068375511010170.
[15] I. Kuritnik, B.R. Nussupbekov A.K. Khassenov, D.Zh. Karabekova. Disintegration of copper ores by electric pulses. Archives of Metallurgy and Materials, 60(4):2449–2551. 2015. doi: 10.1515/amm-2015-0412.
[16] L.A. Yutkin. Electrohydraulic effect and its application in industry. Mechanical Engineering, 1986. (in Russian).
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Authors and Affiliations

Igor P. Kurytnik
1
ORCID: ORCID
Ayanbergen K. Khassenov
2
ORCID: ORCID
Ulan B. Nussupbekov
2
ORCID: ORCID
Dana Z. Karabekova
2
ORCID: ORCID
Bekbolat R. Nussupbekov
2
ORCID: ORCID
Madina Bolatbekova
2
ORCID: ORCID

  1. The Witold Pilecki State Higher School, Oświęcim, Poland
  2. E.A.Buketov University of Karaganda, Kazakhstan

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