An Explicit Method for Numerical Solution of the Equation Governing the Motion of a Particle Under Arbitrary Force Fields

Subject Areas : International Journal of Mathematical Modelling & Computations

Ghiyam Eslami ^{1 *} , Masoumeh Zeinali ²

1 - Department of Mechanical engineering, Islamic Azad University, Ahar Branch, Ahar, Iran
2 - Faculty of mathematics, statistics and computer sciences, University of Tabriz,Tabriz, Iran

Received: 2023-06-23 Accepted : 2023-08-10 Published : 2023-09-17

Keywords: Numerical Solution, Integro-differential equation, Particle motion, Basset history force, Creeping flow,

Abstract :

In this paper, an implicit second order integro-differential equation governing unsteady motion of a solid particle submerged in a fluid medium and, affected by an arbitrary force field is solved numerically. It is assumed that the particle Reynolds number is quite small to use the well-known Basset kernel for the history force. The implicitness and singularity of the equation are removed by using a hybrid quadrature rule (HQR) and a generalized quadrature rule (GQR), respectively. A recursive plan is used to reduce the required CPU time. Two schemes along with the associated numerical solution algorithms are presented. It is described how the accuracy of the method can be increased in a systematic way. The results obtained by several examples show the effectiveness of the method.

References:

[1] M. Aghakhani, G. Eslami, Thermal comfort Assessment of Underfloor vs. Overhead AirDistribution
System, Journal of Applied Sciences 12 (2012) 473-479.
[2] AJ. Dorgan, E. Loth, Efficient calculation of the history force at finite Reynolds numbers, International
journal of multiphase flow, 33(8) (2007) 833-848.
[3] HA. Elghannay, DK. Tafti, Development and validation of a reduced order history force model,
International Journal of Multiphase Flow, 85 (2016) 284-297.
[4] G. Eslami, E. Esmaeilzadeh, P. Garca Snchez, A. Behzadmehr, S. Baheri, Heat transfer enhancement
in a stagnant dielectric liquid by the up and down motion of conductive particles induced by coulomb
forces, Journal Of Applied Fluid Mechanics, 10(1) (2017) 169-182.
[5] G. Eslami, E. Esmaeilzadeh, AT. Prez, Modeling of conductive particle motion in viscous medium affected by an electric field considering particle-electrode interactions and microdischarge phenomenon,
Physics of Fluids, 28(10) (2016).
[6] YH. Feng, YH. Zhangx, ZQ. Zhao, XD. Guo, Dissipative particle dynamics aided design of drug
delivery systems: a review, Molecular Pharmaceutics, 17(6) (2020) 1778-1799.
[7] W. Hackbusch, Integral Equations: Theory and Numerical Treatment, Birkhuser Verlag, Basel, 1995.
[8] C. Henry, JP. Minier, S. Brambilla, Particle resuspension: challenges and perspectives for future
models, Physics Reports, 1007 (2023) 1-98.
[9] C. Iliescu, GL. Xu, PL. Ong, KJ. Leck, Dielectrophoretic separation of biological samples in a 3D
filtering chip, Journal of Micromechanics and Microengineering, (2007) 17(7):S128.
[10] D. Jaganathan, SG. Prasath, R. Govindarajan, V. Vasan, The BassetBoussinesq history force: its
neglect, validity, and recent numerical developments. Frontiers in Physics, (2023) May 22;11:370.
[11] M. Kim, H. Shim, YG. Na, HK. Lee, JS. Baek, CW. Cho, Practical Manipulation for the Preparation
of Mesoporous Silica Nanoparticles for Drug Delivery Vehicles, Current Drug Delivery, 20(8) (2023)
1206-1215.
[12] H. Ma, L. Zhou, Z. Liu, M. Chen, X. Xia, Y. Zhao, A review of recent development for the CFD-DEM
investigations of non-spherical particles. Powder Technology. (2022) Oct 2:117972.
[13] E. Moeendarbary, TY. Ng, M. Zangeneh, Dissipative particle dynamics: introduction, methodology and complex fluid applicationsa review, International Journal of Applied Mechanics, (2009)
Dec;1(04):737-63.
[14] PA. Moreno-Casas, FA. Bombardelli, Computation of the Basset force: recent advances and environmental flow applications, Environmental Fluid Mechanics, 16 (2016) 193-208.
[15] M. Parmar, S. Annamalai, S. Balachandar, A. Prosperetti, Differential formulation of the viscous
history force on a particle for efficient and accurate computation, Journal of fluid mechanics, 844
(2018) 970-993.
[16] A. Schrder, D. Schanz, 3D Lagrangian particle tracking in fluid mechanics, Annual Review of Fluid
Mechanics, 55 (2023) 511-540.
[17] V. Shelukhin, Rotational particle separation in solutions: Micropolar fluid theory approach. Polymers,
3(7) (2021) 1072.
[18] L. Shi, W. Tao, N. Zheng, T. Zhou, Z. Sun, Numerical study of convective heat transfer and particle distribution subject to magneto-static field in a square cavity, International Journal of Thermal
Sciences, 185 (2023) 108081.
[19] V. Sizikov, D. Sidorov, Generalized quadrature for solving singular integral equations of Abel type in
application to infrared tomography, Applied Numerical Mathematics, 106 (2016) 69-78.
[20] Ye.V. Visitskii, A.G. Petrov and M.M. Shunderyuk, The motion of a particle in a viscous fluid under
gravity, vibration and Basset’s force, Journal of Applied Mathematics and Mechanics, 73 (2009) 548-
557.
[21] D. J. VOJIR and E. E. MICHAELIDES, Effect of the history term on the motion of rigid spears ina
viscous fluid, Int. J. Multiphase Flow Vol., 20 (1994) 547-556.
[22] M. Wang, J. Zou, H. Zhang, Y. Wei, S. Liu, Numerical modeling of particles separation method based
on compound electric field, Applied Sciences, 10(17) (2020) 5999.
[23] S. Yokojima, Y. Shimada, K. Mukaiyama, A further study of history force effect on particle clustering
in turbulence, European Journal of Mechanics-B/Fluids, 103 (2024) 11-24.
14 Gh. Eslami et al./ IJM2C, 13 - 03 (2023) 1-14.
[24] M. Zeinali, The existence result of a fuzzy implicit integro-differential equation in semilinear Banach
space, Computational Methods for Differential Equations, 5(3) (2017) 232-245.
[25] M. Zeinali, S. Shahmorad, An equivalence lemma for a class of fuzzy implicit integro-differential
equations, Journal of Computational and Applied Mathematics, 327 (2018) 388-399.
[26] M. Zeinali, S. Shahmorad, K. Mirnia, Fuzzy integro-differential equations: Discrete solution and error
estimation, Iranian Journal of Fuzzy Systems, 10(1) (2013) 107-122.
[27] Z. Zhang, V. Bertin, M. Essink, H. Zhang, N. Fares, Z. Shen, T. Bickel, T. Salez, A. Maali, Unsteady
drag force on an immersed sphere oscillating near a wall, arXiv preprint arXiv. 2307.05991. 2023 Jul
12