Analytical Solutions of the Riccati Differential Equation: Particle Deposition in a Viscous Stagnant Fluid

In this communication, the solution of the differential Riccati equation is shown to provide a closed analytical expression for the transient settling velocity of arbitrary non-spherical particles in a still, unbounded viscous fluid. Such a solution is verified against the numerical results of the i...

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Autores:
Laín Beatove, Santiago
García González, Diego Fernando
Gandini Ayerbe, Mario Andrés
Tipo de recurso:
Article of investigation
Fecha de publicación:
2023
Institución:
Universidad Autónoma de Occidente
Repositorio:
RED: Repositorio Educativo Digital UAO
Idioma:
eng
OAI Identifier:
oai:red.uao.edu.co:10614/15846
Acceso en línea:
https://hdl.handle.net/10614/15846
https://doi.org/10.3390/math11153262
https://red.uao.edu.co/
Palabra clave:
Riccati differential equation
Closed analytical solution
Non-spherical particle
Unbounded viscous fluid
Settling velocity
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openAccess
License
Derechos reservados - MDPI, 2023
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oai_identifier_str oai:red.uao.edu.co:10614/15846
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repository_id_str
dc.title.eng.fl_str_mv Analytical Solutions of the Riccati Differential Equation: Particle Deposition in a Viscous Stagnant Fluid
title Analytical Solutions of the Riccati Differential Equation: Particle Deposition in a Viscous Stagnant Fluid
spellingShingle Analytical Solutions of the Riccati Differential Equation: Particle Deposition in a Viscous Stagnant Fluid
Riccati differential equation
Closed analytical solution
Non-spherical particle
Unbounded viscous fluid
Settling velocity
title_short Analytical Solutions of the Riccati Differential Equation: Particle Deposition in a Viscous Stagnant Fluid
title_full Analytical Solutions of the Riccati Differential Equation: Particle Deposition in a Viscous Stagnant Fluid
title_fullStr Analytical Solutions of the Riccati Differential Equation: Particle Deposition in a Viscous Stagnant Fluid
title_full_unstemmed Analytical Solutions of the Riccati Differential Equation: Particle Deposition in a Viscous Stagnant Fluid
title_sort Analytical Solutions of the Riccati Differential Equation: Particle Deposition in a Viscous Stagnant Fluid
dc.creator.fl_str_mv Laín Beatove, Santiago
García González, Diego Fernando
Gandini Ayerbe, Mario Andrés
dc.contributor.author.none.fl_str_mv Laín Beatove, Santiago
García González, Diego Fernando
Gandini Ayerbe, Mario Andrés
dc.subject.proposal.eng.fl_str_mv Riccati differential equation
Closed analytical solution
Non-spherical particle
Unbounded viscous fluid
Settling velocity
topic Riccati differential equation
Closed analytical solution
Non-spherical particle
Unbounded viscous fluid
Settling velocity
description In this communication, the solution of the differential Riccati equation is shown to provide a closed analytical expression for the transient settling velocity of arbitrary non-spherical particles in a still, unbounded viscous fluid. Such a solution is verified against the numerical results of the integrated differential equation, establishing its accuracy, and validated against previous experimental, theoretical and numerical studies, illustrating the effect of particle sphericity. The developed closed analytical formulae are simple and applicable to general initial velocity conditions in the Stokes, transitional and Newtonian regimes, extending the range of application of former published analytical approximate solutions on this subject
publishDate 2023
dc.date.issued.none.fl_str_mv 2023-07
dc.date.accessioned.none.fl_str_mv 2024-10-04T18:43:45Z
dc.date.available.none.fl_str_mv 2024-10-04T18:43:45Z
dc.type.spa.fl_str_mv Artículo de revista
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dc.type.content.eng.fl_str_mv Text
dc.type.driver.eng.fl_str_mv info:eu-repo/semantics/article
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dc.identifier.citation.spa.fl_str_mv Lain Beatove, S.; García González, D. F. y Gandini Ayerbe, M. A. (2023). "Analytical Solutions of the Riccati Differential Equation: Particle Deposition in a Viscous Stagnant Fluid". Mathematics 11(15), 13 p. ISSN: 2227-7390. https://doi.org/10.3390/math11153262
dc.identifier.issn.spa.fl_str_mv 22277390
dc.identifier.uri.none.fl_str_mv https://hdl.handle.net/10614/15846
dc.identifier.doi.spa.fl_str_mv https://doi.org/10.3390/math11153262
dc.identifier.instname.spa.fl_str_mv Universidad Autónoma de Occidente
dc.identifier.reponame.spa.fl_str_mv Respositorio Educativo Digital UAO
dc.identifier.repourl.none.fl_str_mv https://red.uao.edu.co/
identifier_str_mv Lain Beatove, S.; García González, D. F. y Gandini Ayerbe, M. A. (2023). "Analytical Solutions of the Riccati Differential Equation: Particle Deposition in a Viscous Stagnant Fluid". Mathematics 11(15), 13 p. ISSN: 2227-7390. https://doi.org/10.3390/math11153262
22277390
Universidad Autónoma de Occidente
Respositorio Educativo Digital UAO
url https://hdl.handle.net/10614/15846
https://doi.org/10.3390/math11153262
https://red.uao.edu.co/
dc.language.iso.eng.fl_str_mv eng
language eng
dc.relation.citationendpage.spa.fl_str_mv 13
dc.relation.citationissue.spa.fl_str_mv 15
dc.relation.citationstartpage.spa.fl_str_mv 1
dc.relation.citationvolume.spa.fl_str_mv 11
dc.relation.ispartofjournal.eng.fl_str_mv Mathematics
dc.relation.references.none.fl_str_mv 1. Anderson, B.D.; Moore, J.B. Optimal Control-Linear Quadratic Methods; Prentice-Hall: Hoboken, NJ, USA, 1999.
2. Nowakowski, M.; Rosu, H.C. Newton’s laws of motion in form of Riccati equation. Phys. Rev. E 2002, 65, 047602. [CrossRef]
3. Fraga, E.S. The Schrodinger and Riccati Equations; Lecture Notes in Chemistry; Springer: Berlin, Germany, 1999; Volume 70.
4. Dieter, S. Nonlinear Riccati Equations as a Unifying Link between Linear Quantum Mechanics and Other Fields of Physics. J. Phys. Conf. Ser. 2014, 538, 012019. [CrossRef]
5. Lain, S.; Gandini,M.A. Ideal reactors as an illustration of solving transport phenomena problems in Engineering. Fluids 2023, 8, 58. [CrossRef]
6. Faraoni, V. Solving for the dynamics of the universe. Am. J. Phys. 1999, 67, 732–734. [CrossRef]
7. Boyle, P.P.; Tian, W.; Guan, F. The Riccati Equation in Mathematical Finance. J. Symb. Comput. 2002, 33, 343–355. [CrossRef]
8. Kamke, E. Differentialgleichungen Lösungsmethoden und Lösungen; Vieweg+Teubner Verlag: Leipzig, Germany, 1977.
9. Murphy, G.M. Ordinary Differential Equations and Their Solutions; Van Nostrand: New York, NY, USA, 1960.
10. Polyanin, A.D.; Zaitsev, V.F. Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed.; Chapman & Hall/CRC Press: Boca Raton, FL, USA; London, UK, 2003.
11. Polyanin, A.D.; Zaitsev, V.F. Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems; CRC Press: Boca Raton, FL, USA; London, UK, 2018.
12. Reid, W.T. Riccati Differential Equations; Academic Press: New York, NY, USA, 1980.
13. Ndiaye, M. The Riccati equation, differential transform, rational solutions and applications. Appl. Math. 2022, 13, 774–792. [CrossRef]
14. Mordant, N.; Pinton, J.F. Velocity measurement of a settling sphere. Eur. Phys. J. B 2000, 18, 343–352. [CrossRef]
15. Thompson, M.; Hourigan, K.; Cheung, A.; Leweke, T. Hydrodynamics of a particle impact on a wall. Appl. Math. Model. 2006, 30, 1356–1369. [CrossRef]
16. Lyotard, N.; Shew,W.; Bocquet, L.; Pinton, J.F. Polymer and surface roughness effects on the drag crisis for falling spheres. Eur. Phys. J. B 2007, 60, 469–476. [CrossRef]
17. Ganji, D. A semi-analytical technique for non-linear settling particle equation of motion. J. Hydro-Environ. Res. 2012, 6, 323–327. [CrossRef]
18. Nouri, R.; Ganji, D.; Hatami, M. Unsteady sedimentation analysis of spherical particles in newtonian fluid media using analytical methods. Propul. Power Res. 2014, 3, 96–105. [CrossRef]
19. Habte, M.;Wu, C. Particle sedimentation using hybrid Lattice Boltzmann-immersed boundary method scheme. Powder Technol. 2017, 315, 486–498. [CrossRef]
20. Guo, J.K. Motion of spheres falling through fluids. J. Hydraul. Res. 2011, 49, 32–41. [CrossRef]
21. Engelund, F.; Hansen, E. A Monograph on Sediment Transport in Alluvial Streams; TEKNISKFORLAG Skelbrekgade 4: Copenhagen, Denmark, 1967.
22. Chang, T.J.; Yen, B.C. Gravitational fall velocity of sphere in viscous fluid. J. Eng. Mech. 1998, 124, 1193–1199. [CrossRef]
23. Allen, H.S. The motion of a sphere in a viscous fluid. Philos. Mag. 1900, 50, 519–534. [CrossRef]
24. Moorman, R.W. Motion of a Spherical Particle in the Acceleration Portion of Free Fall. Ph.D. Dissertation, University of Iowa, Iowa City, IA, USA, 1955.
25. Mann, H.; Mueller, P.; Hagemeier, T.; Roloff, C.; Thevenin, D.; Tomas, J. Analytical description of the unsteady settling of spherical particles in Stokes and Newton regimes. Granul. Matter 2015, 17, 629–644. [CrossRef]
26. Hagemeier, T.; Thevenin, D.; Richter, T. Settling of spherical particles in the transitional regime. Int. J. Multiph. Flow 2021, 138, 103589. [CrossRef]
27. Kalman, H.; Portnikov, D. New model to predict the velocity and acceleration of accelerating spherical particles. Powder Technol. 2023, 415, 118197. [CrossRef]
28. Kalman, H.; Matana, E. Terminal velocity and drag coefficient for spherical particles. Powder Technol. 2022, 396, 181–190. [CrossRef]
29. Chien, S.F. Settling Velocity of Irregularly Shaped Particles. SPE Drill. Complet. 1994, 9, 281–289. [CrossRef]
30. Jalaal, M.; Ganji, D.D.; Ahmadi, G. Analytical investigation on acceleration motion of a vertically falling spherical particle in incompressible Newtonian media. Adv. Powder Technol. 2010, 21, 298–304. [CrossRef]
31. Yaghoobi, H.; Torabi, M. Analytical solution for settling of non-spherical particles in incompressible Newtonian media. Powder Technol. 2012, 221, 453–463. [CrossRef]
32. Malvandi, A.; Ganji, D.D.; Malvandi, A. Analytical study on accelerating falling of non-spherical particle in viscous fluid. Int. J. Sediment Res. 2014, 29, 423–430. [CrossRef]
33. Zolfagharian, A.; Darzi, M.; Ghasemi, S.E. Analysis of nano droplet dynamics with various sphericities using efficient computational techniques. J. Cent. South Univ. 2017, 24, 2353–2359. [CrossRef]
34. Wadell, H. The coefficient of resistance as a function of Reynolds number for solids of various shapes. J. Frankl. Inst. 1934, 217, 459–490. [CrossRef]
35. Yin, Z.; Wang, Z.; Liang, B.; Zhang, L. Initial Velocity Effect on Acceleration Fall of a Spherical Particle through Still Fluid. Math. Probl. Eng. 2017, 2017, 9795286. [CrossRef]
36. Mandø, M.; Rosendahl, L. On the motion of non-spherical particles at high Reynolds number. Powder Technol. 2010, 202, 1–13. [CrossRef]
37. Castang, C.; Lain, S.; Sommerfeld, M. Pressure center determination for regularly shaped non-spherical particles at intermediate Reynolds number range. Int. J. Multiph. Flow 2021, 137, 103565. [CrossRef]
38. Chen, H.; Ding, W.; Wei, H.; Saxen, H.; Yu, Y. A Coupled CFD-DEM Study on the Effect of Basset Force Aimed at the Motion of a Single Bubble. Materials 2022, 15, 5461. [CrossRef]
39. Lain, S.; Sommerfeld, M. A study of the pneumatic conveying of non-spherical particles in a turbulent horizontal channel flow. Braz. J. Chem. Eng. 2007, 24, 535–546. [CrossRef]
40. Haider, A.; Levenspiel, O. Drag coefficient and terminal velocity of spherical and nonspherical particles. Powder Technol. 1989, 58, 63–70.
41. Cheng, N.S. Comparison of formulas for drag coefficient and settling velocity of spherical particles. Powder Technol. 2009, 189, 395–398. [CrossRef]
42. Sommerfeld, M.; Lain, S. Stochastic modelling for capturing the behaviour of irregular-shaped non-spherical particles in confined turbulent flows. Powder Technol. 2018, 332, 253–264. [CrossRef]
43. Rubey,W.W. Settling velocity of gravel, sand, and silt particles. Am. J. Sci. 1933, 225, 325–338.
44. Michaelides, E.E. Hydrodynamic force and heat/mass transfer from particles, bubbles and drops—The Freeman Scholar Lecture. J. Fluids Eng. 2003, 125, 209–238.
45. Daitche, A. On the role of the history force for inertial particles in turbulence. J. Fluid Mech. 2015, 782, 567–593.
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spelling Laín Beatove, Santiagovirtual::5704-1García González, Diego FernandoGandini Ayerbe, Mario Andrésvirtual::5705-12024-10-04T18:43:45Z2024-10-04T18:43:45Z2023-07Lain Beatove, S.; García González, D. F. y Gandini Ayerbe, M. A. (2023). "Analytical Solutions of the Riccati Differential Equation: Particle Deposition in a Viscous Stagnant Fluid". Mathematics 11(15), 13 p. ISSN: 2227-7390. https://doi.org/10.3390/math1115326222277390https://hdl.handle.net/10614/15846https://doi.org/10.3390/math11153262Universidad Autónoma de OccidenteRespositorio Educativo Digital UAOhttps://red.uao.edu.co/In this communication, the solution of the differential Riccati equation is shown to provide a closed analytical expression for the transient settling velocity of arbitrary non-spherical particles in a still, unbounded viscous fluid. Such a solution is verified against the numerical results of the integrated differential equation, establishing its accuracy, and validated against previous experimental, theoretical and numerical studies, illustrating the effect of particle sphericity. The developed closed analytical formulae are simple and applicable to general initial velocity conditions in the Stokes, transitional and Newtonian regimes, extending the range of application of former published analytical approximate solutions on this subject13 páginasapplication/pdfengMDPIBasel, SwitzerlandDerechos reservados - MDPI, 2023https://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessAtribución-NoComercial-SinDerivadas 4.0 Internacional (CC BY-NC-ND 4.0)http://purl.org/coar/access_right/c_abf2Analytical Solutions of the Riccati Differential Equation: Particle Deposition in a Viscous Stagnant FluidArtículo de revistahttp://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/articlehttp://purl.org/redcol/resource_type/ARTinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/version/c_970fb48d4fbd8a851315111Mathematics1. Anderson, B.D.; Moore, J.B. Optimal Control-Linear Quadratic Methods; Prentice-Hall: Hoboken, NJ, USA, 1999.2. Nowakowski, M.; Rosu, H.C. Newton’s laws of motion in form of Riccati equation. Phys. Rev. E 2002, 65, 047602. [CrossRef]3. Fraga, E.S. The Schrodinger and Riccati Equations; Lecture Notes in Chemistry; Springer: Berlin, Germany, 1999; Volume 70.4. Dieter, S. Nonlinear Riccati Equations as a Unifying Link between Linear Quantum Mechanics and Other Fields of Physics. J. Phys. Conf. Ser. 2014, 538, 012019. [CrossRef]5. Lain, S.; Gandini,M.A. Ideal reactors as an illustration of solving transport phenomena problems in Engineering. Fluids 2023, 8, 58. [CrossRef]6. Faraoni, V. Solving for the dynamics of the universe. Am. J. Phys. 1999, 67, 732–734. [CrossRef]7. Boyle, P.P.; Tian, W.; Guan, F. The Riccati Equation in Mathematical Finance. J. Symb. Comput. 2002, 33, 343–355. [CrossRef]8. Kamke, E. Differentialgleichungen Lösungsmethoden und Lösungen; Vieweg+Teubner Verlag: Leipzig, Germany, 1977.9. Murphy, G.M. Ordinary Differential Equations and Their Solutions; Van Nostrand: New York, NY, USA, 1960.10. Polyanin, A.D.; Zaitsev, V.F. Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed.; Chapman & Hall/CRC Press: Boca Raton, FL, USA; London, UK, 2003.11. Polyanin, A.D.; Zaitsev, V.F. Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems; CRC Press: Boca Raton, FL, USA; London, UK, 2018.12. Reid, W.T. Riccati Differential Equations; Academic Press: New York, NY, USA, 1980.13. Ndiaye, M. The Riccati equation, differential transform, rational solutions and applications. Appl. Math. 2022, 13, 774–792. [CrossRef]14. Mordant, N.; Pinton, J.F. Velocity measurement of a settling sphere. Eur. Phys. J. B 2000, 18, 343–352. [CrossRef]15. Thompson, M.; Hourigan, K.; Cheung, A.; Leweke, T. Hydrodynamics of a particle impact on a wall. Appl. Math. Model. 2006, 30, 1356–1369. [CrossRef]16. Lyotard, N.; Shew,W.; Bocquet, L.; Pinton, J.F. Polymer and surface roughness effects on the drag crisis for falling spheres. Eur. Phys. J. B 2007, 60, 469–476. [CrossRef]17. Ganji, D. A semi-analytical technique for non-linear settling particle equation of motion. J. Hydro-Environ. Res. 2012, 6, 323–327. [CrossRef]18. Nouri, R.; Ganji, D.; Hatami, M. Unsteady sedimentation analysis of spherical particles in newtonian fluid media using analytical methods. Propul. Power Res. 2014, 3, 96–105. [CrossRef]19. Habte, M.;Wu, C. Particle sedimentation using hybrid Lattice Boltzmann-immersed boundary method scheme. Powder Technol. 2017, 315, 486–498. [CrossRef]20. Guo, J.K. Motion of spheres falling through fluids. J. Hydraul. Res. 2011, 49, 32–41. [CrossRef]21. Engelund, F.; Hansen, E. A Monograph on Sediment Transport in Alluvial Streams; TEKNISKFORLAG Skelbrekgade 4: Copenhagen, Denmark, 1967.22. Chang, T.J.; Yen, B.C. Gravitational fall velocity of sphere in viscous fluid. J. Eng. Mech. 1998, 124, 1193–1199. [CrossRef]23. Allen, H.S. The motion of a sphere in a viscous fluid. Philos. Mag. 1900, 50, 519–534. [CrossRef]24. Moorman, R.W. Motion of a Spherical Particle in the Acceleration Portion of Free Fall. Ph.D. Dissertation, University of Iowa, Iowa City, IA, USA, 1955.25. Mann, H.; Mueller, P.; Hagemeier, T.; Roloff, C.; Thevenin, D.; Tomas, J. Analytical description of the unsteady settling of spherical particles in Stokes and Newton regimes. Granul. Matter 2015, 17, 629–644. [CrossRef]26. Hagemeier, T.; Thevenin, D.; Richter, T. Settling of spherical particles in the transitional regime. Int. J. Multiph. Flow 2021, 138, 103589. [CrossRef]27. Kalman, H.; Portnikov, D. New model to predict the velocity and acceleration of accelerating spherical particles. Powder Technol. 2023, 415, 118197. [CrossRef]28. Kalman, H.; Matana, E. Terminal velocity and drag coefficient for spherical particles. Powder Technol. 2022, 396, 181–190. [CrossRef]29. Chien, S.F. Settling Velocity of Irregularly Shaped Particles. SPE Drill. Complet. 1994, 9, 281–289. [CrossRef]30. Jalaal, M.; Ganji, D.D.; Ahmadi, G. Analytical investigation on acceleration motion of a vertically falling spherical particle in incompressible Newtonian media. Adv. Powder Technol. 2010, 21, 298–304. [CrossRef]31. Yaghoobi, H.; Torabi, M. Analytical solution for settling of non-spherical particles in incompressible Newtonian media. Powder Technol. 2012, 221, 453–463. [CrossRef]32. Malvandi, A.; Ganji, D.D.; Malvandi, A. Analytical study on accelerating falling of non-spherical particle in viscous fluid. Int. J. Sediment Res. 2014, 29, 423–430. [CrossRef]33. Zolfagharian, A.; Darzi, M.; Ghasemi, S.E. Analysis of nano droplet dynamics with various sphericities using efficient computational techniques. J. Cent. South Univ. 2017, 24, 2353–2359. [CrossRef]34. Wadell, H. The coefficient of resistance as a function of Reynolds number for solids of various shapes. J. Frankl. Inst. 1934, 217, 459–490. [CrossRef]35. Yin, Z.; Wang, Z.; Liang, B.; Zhang, L. Initial Velocity Effect on Acceleration Fall of a Spherical Particle through Still Fluid. Math. Probl. Eng. 2017, 2017, 9795286. [CrossRef]36. Mandø, M.; Rosendahl, L. On the motion of non-spherical particles at high Reynolds number. Powder Technol. 2010, 202, 1–13. [CrossRef]37. Castang, C.; Lain, S.; Sommerfeld, M. Pressure center determination for regularly shaped non-spherical particles at intermediate Reynolds number range. Int. J. Multiph. Flow 2021, 137, 103565. [CrossRef]38. Chen, H.; Ding, W.; Wei, H.; Saxen, H.; Yu, Y. A Coupled CFD-DEM Study on the Effect of Basset Force Aimed at the Motion of a Single Bubble. Materials 2022, 15, 5461. [CrossRef]39. Lain, S.; Sommerfeld, M. A study of the pneumatic conveying of non-spherical particles in a turbulent horizontal channel flow. Braz. J. Chem. Eng. 2007, 24, 535–546. [CrossRef]40. Haider, A.; Levenspiel, O. Drag coefficient and terminal velocity of spherical and nonspherical particles. Powder Technol. 1989, 58, 63–70.41. Cheng, N.S. Comparison of formulas for drag coefficient and settling velocity of spherical particles. Powder Technol. 2009, 189, 395–398. [CrossRef]42. Sommerfeld, M.; Lain, S. Stochastic modelling for capturing the behaviour of irregular-shaped non-spherical particles in confined turbulent flows. Powder Technol. 2018, 332, 253–264. [CrossRef]43. Rubey,W.W. Settling velocity of gravel, sand, and silt particles. Am. J. Sci. 1933, 225, 325–338.44. Michaelides, E.E. Hydrodynamic force and heat/mass transfer from particles, bubbles and drops—The Freeman Scholar Lecture. J. Fluids Eng. 2003, 125, 209–238.45. Daitche, A. On the role of the history force for inertial particles in turbulence. J. Fluid Mech. 2015, 782, 567–593.Riccati differential equationClosed analytical solutionNon-spherical particleUnbounded viscous fluidSettling velocityComunidad generalPublication082b0926-3385-4188-9c6a-bbbed7484a95virtual::5704-11b7ae0bb-d40b-4d15-94ee-5d8949aad3c5virtual::5705-1082b0926-3385-4188-9c6a-bbbed7484a95virtual::5704-11b7ae0bb-d40b-4d15-94ee-5d8949aad3c5virtual::5705-1https://scholar.google.com/citations?user=g-iBdUkAAAAJ&hl=esvirtual::5704-10000-0002-0269-2608virtual::5704-10000-0002-6430-2601virtual::5705-1https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0000262129virtual::5704-1https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0000952028virtual::5705-1ORIGINALAnalytical_Solutions_of_the_Riccati_Differential_Equation_Particle_Deposition_in_a_Viscous_Stagnant_Fluid.pdfAnalytical_Solutions_of_the_Riccati_Differential_Equation_Particle_Deposition_in_a_Viscous_Stagnant_Fluid.pdfArchivo texto completo del artículo de revista, PDFapplication/pdf1616486https://red.uao.edu.co/bitstreams/e66abbcc-090e-4c67-b718-6fe30db337ab/download079231fb13d89267de40778469d3992bMD51LICENSElicense.txtlicense.txttext/plain; charset=utf-81672https://red.uao.edu.co/bitstreams/b000b43b-3a72-4eee-95c1-c1a90042f29b/download6987b791264a2b5525252450f99b10d1MD52TEXTAnalytical_Solutions_of_the_Riccati_Differential_Equation_Particle_Deposition_in_a_Viscous_Stagnant_Fluid.pdf.txtAnalytical_Solutions_of_the_Riccati_Differential_Equation_Particle_Deposition_in_a_Viscous_Stagnant_Fluid.pdf.txtExtracted texttext/plain57715https://red.uao.edu.co/bitstreams/5a07d0c8-4226-493a-9684-d721e0526053/download0145df39983ca43840c68fb4d3f83321MD53THUMBNAILAnalytical_Solutions_of_the_Riccati_Differential_Equation_Particle_Deposition_in_a_Viscous_Stagnant_Fluid.pdf.jpgAnalytical_Solutions_of_the_Riccati_Differential_Equation_Particle_Deposition_in_a_Viscous_Stagnant_Fluid.pdf.jpgGenerated Thumbnailimage/jpeg14838https://red.uao.edu.co/bitstreams/51f803c9-531c-48e7-ab41-9b8119794e18/downloadd25aed9fe73a2bf129bf60c16cd3519cMD5410614/15846oai:red.uao.edu.co:10614/158462024-10-05 03:00:24.431https://creativecommons.org/licenses/by-nc-nd/4.0/Derechos reservados - MDPI, 2023open.accesshttps://red.uao.edu.coRepositorio Digital Universidad Autonoma de Occidenterepositorio@uao.edu.coPHA+RUwgQVVUT1IgYXV0b3JpemEgYSBsYSBVbml2ZXJzaWRhZCBBdXTDs25vbWEgZGUgT2NjaWRlbnRlLCBkZSBmb3JtYSBpbmRlZmluaWRhLCBwYXJhIHF1ZSBlbiBsb3MgdMOpcm1pbm9zIGVzdGFibGVjaWRvcyBlbiBsYSBMZXkgMjMgZGUgMTk4MiwgbGEgTGV5IDQ0IGRlIDE5OTMsIGxhIERlY2lzacOzbiBhbmRpbmEgMzUxIGRlIDE5OTMsIGVsIERlY3JldG8gNDYwIGRlIDE5OTUgeSBkZW3DoXMgbGV5ZXMgeSBqdXJpc3BydWRlbmNpYSB2aWdlbnRlIGFsIHJlc3BlY3RvLCBoYWdhIHB1YmxpY2FjacOzbiBkZSBlc3RlIGNvbiBmaW5lcyBlZHVjYXRpdm9zLiBQQVJBR1JBRk86IEVzdGEgYXV0b3JpemFjacOzbiBhZGVtw6FzIGRlIHNlciB2w6FsaWRhIHBhcmEgbGFzIGZhY3VsdGFkZXMgeSBkZXJlY2hvcyBkZSB1c28gc29icmUgbGEgb2JyYSBlbiBmb3JtYXRvIG8gc29wb3J0ZSBtYXRlcmlhbCwgdGFtYmnDqW4gcGFyYSBmb3JtYXRvIGRpZ2l0YWwsIGVsZWN0csOzbmljbywgdmlydHVhbCwgcGFyYSB1c29zIGVuIHJlZCwgSW50ZXJuZXQsIGV4dHJhbmV0LCBpbnRyYW5ldCwgYmlibGlvdGVjYSBkaWdpdGFsIHkgZGVtw6FzIHBhcmEgY3VhbHF1aWVyIGZvcm1hdG8gY29ub2NpZG8gbyBwb3IgY29ub2Nlci4gRUwgQVVUT1IsIGV4cHJlc2EgcXVlIGVsIGRvY3VtZW50byAodHJhYmFqbyBkZSBncmFkbywgcGFzYW50w61hLCBjYXNvcyBvIHRlc2lzKSBvYmpldG8gZGUgbGEgcHJlc2VudGUgYXV0b3JpemFjacOzbiBlcyBvcmlnaW5hbCB5IGxhIGVsYWJvcsOzIHNpbiBxdWVicmFudGFyIG5pIHN1cGxhbnRhciBsb3MgZGVyZWNob3MgZGUgYXV0b3IgZGUgdGVyY2Vyb3MsIHkgZGUgdGFsIGZvcm1hLCBlbCBkb2N1bWVudG8gKHRyYWJham8gZGUgZ3JhZG8sIHBhc2FudMOtYSwgY2Fzb3MgbyB0ZXNpcykgZXMgZGUgc3UgZXhjbHVzaXZhIGF1dG9yw61hIHkgdGllbmUgbGEgdGl0dWxhcmlkYWQgc29icmUgw6lzdGUuIFBBUkFHUkFGTzogZW4gY2FzbyBkZSBwcmVzZW50YXJzZSBhbGd1bmEgcmVjbGFtYWNpw7NuIG8gYWNjacOzbiBwb3IgcGFydGUgZGUgdW4gdGVyY2VybywgcmVmZXJlbnRlIGEgbG9zIGRlcmVjaG9zIGRlIGF1dG9yIHNvYnJlIGVsIGRvY3VtZW50byAoVHJhYmFqbyBkZSBncmFkbywgUGFzYW50w61hLCBjYXNvcyBvIHRlc2lzKSBlbiBjdWVzdGnDs24sIEVMIEFVVE9SLCBhc3VtaXLDoSBsYSByZXNwb25zYWJpbGlkYWQgdG90YWwsIHkgc2FsZHLDoSBlbiBkZWZlbnNhIGRlIGxvcyBkZXJlY2hvcyBhcXXDrSBhdXRvcml6YWRvczsgcGFyYSB0b2RvcyBsb3MgZWZlY3RvcywgbGEgVW5pdmVyc2lkYWQgIEF1dMOzbm9tYSBkZSBPY2NpZGVudGUgYWN0w7phIGNvbW8gdW4gdGVyY2VybyBkZSBidWVuYSBmZS4gVG9kYSBwZXJzb25hIHF1ZSBjb25zdWx0ZSB5YSBzZWEgZW4gbGEgYmlibGlvdGVjYSBvIGVuIG1lZGlvIGVsZWN0csOzbmljbyBwb2Ryw6EgY29waWFyIGFwYXJ0ZXMgZGVsIHRleHRvIGNpdGFuZG8gc2llbXByZSBsYSBmdWVudGUsIGVzIGRlY2lyIGVsIHTDrXR1bG8gZGVsIHRyYWJham8geSBlbCBhdXRvci4gRXN0YSBhdXRvcml6YWNpw7NuIG5vIGltcGxpY2EgcmVudW5jaWEgYSBsYSBmYWN1bHRhZCBxdWUgdGllbmUgRUwgQVVUT1IgZGUgcHVibGljYXIgdG90YWwgbyBwYXJjaWFsbWVudGUgbGEgb2JyYS48L3A+Cg==

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