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Abstract

Knowledge about complex physical phenomena used in the casting process simulation requires continuous complementary research and improvement in mathematical modeling. The basic mathematical model taking into account only thermal phenomena often becomes insufficient to analyze the process of metal solidification, therefore more complex models are formulated, which include coupled heat-flow phenomena, mechanical or shrinkage phenomena. However, such models significantly complicate and lengthen numerical simulations; therefore the work is limited only to the analysis of coupled thermal and flow phenomena. The mathematical description consists then of a system of Navier-Stokes differential equations, flow continuity and energy. The finite element method was used to numerically modeling this problem. In computer simulations, the impact of liquid metal movements on the alloy solidification process in the casting-riser system was assessed, which was the purpose of this work, and the locations of possible shrinkage defects were pointed out, trying to ensure the right supply conditions for the casting to be free from these defects.
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Authors and Affiliations

L. Sowa
1
ORCID: ORCID
T. Skrzypczak
1
ORCID: ORCID
P. Kwiatoń
1
ORCID: ORCID

  1. Czestochowa University of Technology, Department of Mechanics and Machine Design Fundamentals, 73 Dąbrowskiego Str., 42-200 Częstochowa, Poland
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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.
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[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.
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[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

The article presents "-approximation of hydrodynamics equations’ stationary model along with the proof of a theorem about existence of a hydrodynamics equations’ strongly generalized solution. It was proved by a theorem on the existence of uniqueness of the hydrodynamics equations’ temperature model’s solution, taking into account energy dissipation. There was implemented the Galerkin method to study the Navier–Stokes equations, which provides the study of the boundary value problems correctness for an incompressible viscous flow both numerically and analytically. Approximations of stationary and non-stationary models of the hydrodynamics equations were constructed by a system of Cauchy–Kovalevsky equations with a small parameter ". There was developed an algorithm for numerical modelling of the Navier– Stokes equations by the finite difference method.
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Bibliography

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[5] S.Sh. Kazhikenova, S.N. Shaltakov, D. Belomestny, and G.S. Shai- hova: Finite difference method implementation for numerical integration hydrodynamic equations melts. Eurasian Physical Technical Journal, 17(1), (2020), 50–56.
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[15] 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), 1176–1191, http://umj.imath.kiev.ua/index.php/ umj/article/view/1508.
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Authors and Affiliations

Saule Sh. Kazhikenova
1
ORCID: ORCID

  1. Head of the Department of Higher Mathematics, Karaganda Technical University, Kazakhstan
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Abstract

We demonstrate in this study that a rotating magnetic field (RMF) and spinning magnetic particles using this kind of magnetic field give rise to a motion mechanism capable of triggering mixing effect in liquids. In this experimental work two mixing mechanisms were used, magnetohydrodynamics due to the Lorentz force and mixing due to magnetic particles under the action of RMF, acted upon by the Kelvin force. To evidence these mechanisms,we report mixing time measured during the neutralization process (weak acid-strong base) under the action of RMF with and without magnetic particles. The efficiency of the mixing process was enhanced by a maximum of 6.5% and 12.8% owing to the application of RMF and the synergistic effect of magnetic field and magnetic particles, respectively.
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Authors and Affiliations

Rafał Rakoczy
1
ORCID: ORCID
Marian Kordas
1
ORCID: ORCID
Agata Markowska-Szczupak
1
ORCID: ORCID
Maciej Konopacki
1
ORCID: ORCID
Adrian Augustyniak
1
ORCID: ORCID
Joanna Jabłońska
1
Oliwia Paszkiewicz
1
ORCID: ORCID
Kamila Dubrowska
1
Grzegorz Story
1
Anna Story
1
Katarzyna Ziętarska
1
Dawid Sołoducha
1
Tomasz Borowski
1
Marta Roszak
2
Bartłomiej Grygorcewicz
2
ORCID: ORCID
Barbara Dołęgowska
2
ORCID: ORCID

  1. West Pomeranian University of Technology in Szczecin, Faculty of Chemical Technology and Engineering, Department of Chemical and Process Engineering, al. Piastów 42,71-065 Szczecin, Poland
  2. Pomeranian Medical University in Szczecin, Chair of Microbiology, Immunology and Laboratory Medicine, Department of Laboratory Medicine, al. Powstańców Wielkopolskich 72, 70-111 Szczecin, Poland

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