ONE CLASS OF SMOOTH BOUNDED SOLUTIONS TO THE CAUCHY PROBLEM FOR A THREE-DIMENSIONAL FILTRATION MODEL WITH DARCY'S LAW
DOI:
https://doi.org/10.26577/JMMCS.2022.v114.i2.013Keywords:
continuity equation, Darcy’s law, four-dimensional function, Cauchy-Riemann conditionAbstract
In the filtration theory there are numerous approaches to solving three-dimensional problems of fluid motion in a porous medium. Basically, the solutions of such problems are obtained by numerical methods. The question of finding an analytical solution of precisely three-dimensional problems of fluid motion is open.
The first results on the use of the apparatus of four-dimensional mathematics for solving the three-dimensional model of the Navier-Stokes equations by the analytical method were obtained by the Kazakh mathematician Professor M.M. Abenov. After the author of this article with other researchers proved the theorem on the existence of a solution to the Cauchy problem for a three-dimensional model of filtration theory.
This paper is devoted to the study of a three-dimensional model of the filtration theory in one of the spaces of four-dimensional numbers. The purpose of this article is to obtain an analytical solution of the three-dimensional Cauchy problem for the mathematical model of linear filtration model by the method of four-dimensional regular functions.
In this study, a class of infinitely differentiable and bounded functions of the initial conditions of the Cauchy problem, satisfying the Cauchy-Riemann condition, with five degrees of freedom for a specific four-dimensional function is found, and also a class of infinitely differentiable and bounded solutions of this problem is found that satisfy the linear Darcy law.
References
[2] Frank, A. Coutelieris, J.M.P.Q Dlgado. “Transport Processes in Porous Media.” USA: Springer (2012).
[3] Oliver, Coussy. “Mechanics and Physics of Porous Solids.” UK: Wiley (2010).
[4] IHAR (International Association for Hydraulic Research). “Fundamentals of Transport Phenomena in Porous Media.”Israel: Elsevier (1972).
[5] Kambiz, Vafai. “Handbook of Porous Media.” USA: Taylor & Francis Group; Second Edition (2005).
[6] Entov, V.N. “Filtration theory.” Sorosov educational journal, no. 2. (1998): 121-128.
[7] Polubarinova-Kochina, P.Ya. “Теория движения грунтовых вод.” М.: Nauka (1977). 8. Dullien, F.A.L. “Porous Media: Fluid Transport and Pore Structure.” USA: Academic Press; Second Edition (1991).
[8] Whintaker, S. “The Forchheimer equation: A theoretical development.” Transport in Porous Media , no. 25 (1996): 27–61.
[9] Derek, B. Ingham, Ioan Pop. “Transport Phenomena in Porous Media.” USA: Great Britain (2005).
[10] Tamer, O.S., Toropov, E.S., Shevnina, T.E., Vorobieva, T.I. “Research of Reservoir Rock Properties in Violation of Darcy’s Linear Law. Transport and Storage of Hydrocarbons.” IOP Conf. Series: Materials Science and Engineering 154 (2016). https://doi:10.1088/1757-899X/154/1/012006
[11] Hassanizadeh, S.M., Gray, W.G. “High Velocity Flow in Porous Media.” Transport in Porous Media 2 (1987): 521-531.
[12] Bear, J. “Dynamics of fluids in porous media.” Elsevier, New York (1972).
[13] Alekseev, B.V. “Analytical solution of the Leibenson equation and filtration theory.” Fine chemical technologies 11, no. 1 (2016): 34-39.15.
[14] Abenov M.M. Chetirehmernaya matematika. Metody i prilozheniya. Nauchnaya monographia [Four-dimensional mathematics: Methods and applications. Scientific monograph]. Almaty.: Publishing House Kazakh University, 2019. -176.
[15] Abenov M.M., Gabbassov M.B. Anyzotropnie chetirehmernie prostranstva ili novie kvaternioni [Anisotropic fourdimensional spaces or new quaternions]. Preprint, Nur-Sultan. 2020
[16] Rakhymova A.T., Gabbassov M.B., Shapen K.M., “On one space of four-dimensional numbers,” Journal of Mathematics, Mechanics and Computer Science (Vol 4) (2020): 199-225.
[17] Rakhymova A.T., Gabbassov M.B., Shapen K.M., “Functions in one space of four-dimensional numbers,” Journal of Mathematics, Mechanics and Computer Science (Vol 2) (2021): 139-154.
[18] Rakhymova A.T., Gabbassov M.B., Ahmedov A.A. Analytical Solution of the Cauchy Problem for a Nonstationary Threedimensional Model of the Filtration Theory. Journal of Advanced Research in Fluid Mechanics and Thermal Sciences. Vol. 87 No. 1: November (2021): 118 – 133
