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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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[5] Y. Seokwan and K. Dochan: Three-dimensional incompressible Navier– Stokes solver using lower-upper symmetric Gauss–Seidel algorithm. AIAA Journal, 29(6), (1991), 874–875, DOI: 10.2514/3.10671.
[6] P.M. Gresho: Incompressible fluid dynamics: some fundamental formulation issues. Annual Review of Fluid Mechanics, 23, (1991), 413–453, DOI: 10.1146/annurev.fl.23.010191.002213.
[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

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.
[6] O.A. Ladijenskaya: Boundary Value Problems of Mathematical Physics. Nauka, Moscow, 1973.
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[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.kiev.ua/index.php/umj/article/view/1768.
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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.
[16] S.Sh. Kazhikenova, M.I. Ramazanov, and A.A. Khairkulova: epsilon- Approximation of the temperatures model of inhomogeneous melts with allowance for energy dissipation. Bulletin of the Karaganda University- Mathematics, 90(2), (2018), 93–100, DOI: 10.31489/2018M2/93-100.
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[22] M. Rosenfeld and M. Israeli: Numerical solution of incompressible flows by a marching multigrid nonlinear method. AIAA 7th Comput. Fluid Dyn. Conf.: Collect. Techn. Pap., New-York, (1985), 108–116.92.


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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

The paper presents the results of the computer simulations of solidification with consideration of the liquid phase movement. Simulations were conducted in a real, complex cast. There is a multi-stage resolution to the problem of convection in solidification simulations. The most important resolution concerns the development of the numerical model with the momentum and continuity equations, as well as conditions which are determined by the convection. Simulations were carried out with the use of our authorial software based on stabilized finite elements method (Petroy-Galerkin). In order to solve Navier-Stokes equation (with the convection element), Boussinesq’s approximation were used. Finite Elements Method (FEM) was responsible for the solidification. FEM is close to the heat conduction equation solution (with the internal heat source responsible for the heat released during phase transformation). Convection causes movement in the liquid phase in the solidifying cast and can significantly influence the process of heat transfer from the cast. It may change the distribution of the defects. Results of this article make it possible to assess the conditions in which the influence of the convection on solidification is significant.
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Authors and Affiliations

E. Gawrońska
R. Dyja
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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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Abstract

In the presented work Egorov’s approach (adding a source term to the ω-equation in the k-! model, which mimics the damping of turbulence close to a solid wall) was implemented in on the subclass of shear stress transport models. Hence, turbulence damping is available for all shear stress transport type models, including hybrid models that are based on the ω-equation. It is shown that turbulence damping improves the prediction of the axial velocity profile not only for Reynolds-averaged Navier–Stokes simulation but also for detached eddy simulation and delayed detached eddy simulation models. Furthermore, it leads to a more realistic estimation of the pressure drop and, hence, to a more correct prediction of the liquid level. In this paper, simulation results for four different turbulence models are presented and validated by comparison with experimental data. Furthermore, the influence of the magnitude of the damping factor on the pressure drop in the channel is investigated for a variety of different gas-to-liquid flow rate ratios. These investigations show that higher gas-to-liquid flow rate ratios require higher damping factors to correctly predict the pressure drop. In the end, advice is formulated on how an appropriate damping factor can be determined for a specific test case.
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Authors and Affiliations

Jiri Polansky
1
Sonja Schmelter
2

  1. Czech Technical University in Prague, Jugoslávských partyzánu 1580/3, 160 00 Prague 6 – Dejvice, Czech Republic
  2. Physikalisch-Technische Bundesanstalt (PTB), Abbestraße 2-12, D-10587 Berlin-Charlottenburg, Germany

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