Tank Drainage for an Electrically Conducting Newtonian Fluid with the use of the Bessel Function
Abstract
In this study, an unsteady flow for drainage through a circular tank of an isothermal and incompressible Newtonian magnetohydrodynamic (MHD) fluid has been investigated. The series solution method is employed, and an analytical solution is obtained. Expressions for the velocity field, average velocity, flow rate, fluid depth at different times in the tank and time required for the wide-ranging drainage of the fluid (time of efflux) have been obtained. The Newtonian solution is attained by assuming σΒ02=0. The effects of various developing parameters on velocity field υz and depth of fluid H(t) are presented graphically. The time needed to drain the entire fluid and its depth are related and such relations are obtained in closed form. The effect of electromagnetic forces is analyzed. The fluid in the tank will drain gradually and it will take supplementary time for complete drainage.
Keywords:
tank drainage, Newtonian MHD fluid, analytical solution, series solutionDownloads
References
D. D. Joye, B. C. Barrett, “The tank drainage problem revisited: Do these equations actually work?”, The Canadian Journal of Chemical Engineering, Vol. 81, No. 5, pp. 1052–1057, 2003 DOI: https://doi.org/10.1002/cjce.5450810516
A. N. Hrymak, “Applied fluid mechanics by T. C. Papanastasiou, Prentice-Hall, Englewood Cliffs, NJ, 1994, 520 pp.”, AIChE Journal, Vol. 41, No. 10, pp. 2342–2342, 1995 DOI: https://doi.org/10.1002/aic.690411022
R. B. Bird, W. E. Stewart, E. N. Lightfoot, Transport phenomena, John Wiley and Sons, 1961 DOI: https://doi.org/10.1149/1.2428074
E. J. Crosby, Experiments in transport phenomena, Wiley, 1961
D. Hanesian, A. Perna, A laboratory manual for fundamentals of engineering design, chemical engineering module: Measurements laboratory, New Jersey Institute of Technology, 1995
N. D. Nevers, Fluid mechanics for chemical engineers, McGraw-Hill Education, 2005
J. O. Wilkes, Fluid mechanics for chemical engineers with microfluidics and CFD, Prentice Hall, 2006
S. Channer, K. N. Memon, A. A. Ghoto, A. M. Siddiqui, S. F. Shah, “Analytical solutionof lift for thin film flow for phan thien tanner fluid”, Sindh University Research Journal, Vol. 51, No. 2, pp. 215–222, 2019 DOI: https://doi.org/10.26692/sujo/2019.6.37
L. K. Forbes, G. C. Hocking, “Unsteady draining flows from a rectangular tank”, Physics of Fluids, Vol. 19, No. 8, Article ID 082104, 2007 DOI: https://doi.org/10.1063/1.2759891
L. K. Forbes, G. C. Hocking, “Unsteady draining of a fluid from a circular tank”, Applied Mathematical Modelling, Vol. 34, No. 12, pp. 3958–3975, 2010 DOI: https://doi.org/10.1016/j.apm.2010.03.032
G. V. S. K. Reddy, C. V. Subbarao, “Comparison of efflux times between cylindrical and spherical tank through an exit pipe”, International Journal of Engineering and Applied Sciences, Vol. 3, No. 2, pp. 61–68, 2011
C. V. Subbarao, P. S. Rao, G. M. J. Raju, V. S. R. K. Prasad, “Slow draining of large spherical tank under gravity”, Elixir International Journal, Vol. 50, pp. 10346-10348, 2012
C. V. Subbarao, Y. P. Yadav, P. King, “Drag reduction by surfactant solutions in gravity driven flow systems”, Iranian Journal of Chemistry and Chemical Engineering, Vol. 32, No. 2, pp. 119–123, 2013
C. Y. Wang, “On a class of exact solutions of the navier-stokes equations”, Journal of Applied Mechanics, Vol. 33, No. 3, pp. 696–698, 1966 DOI: https://doi.org/10.1115/1.3625151
A. A. Mahessar, A. L. Qureshi, A. N. Laghari, S. Qureshi, S. F. Shah, F. A. Shaikh, “Impact of Hairdin, Miro Khan and Shahdad Kot drainage on Hamal dhand, Sindh”, Engineering, Technology & Applied Science Research, Vol. 8, No. 6, pp. 3652–3656, 2018 DOI: https://doi.org/10.48084/etasr.2389
K. N. Memon, A. M. Siddiqui, S. F. Shah, “Exact solution of tank drainage for Newtonian fluid with slip condition”, Sindh University Research Journal, Vol. 49, No. 2, pp. 283–288, 2017
S. Islam, K. N. Memon, A. M. Siddiqui, S. F. Shah, “Analytical solution of tank drainage for electrically conducting power law fluid”, available at: https://www.preprints.org/manuscript/201802.0033/v1, 2018 DOI: https://doi.org/10.20944/preprints201802.0033.v1
K. Afanasiev, A. Munch, B. Wagner, “Landau-Levich problem for non-Newtonian liquids”, Physical Review E, Vol. 76, No. 3, Article ID 036307, 2007 DOI: https://doi.org/10.1103/PhysRevE.76.036307
N. Ali, A. Abbasi, I. Ahmad, “Channel flow of Ellis fluid due to peristalsis”, AIP Advances, Vol. 5, No. 9, Article ID 097214, 2015 DOI: https://doi.org/10.1063/1.4932042
K. N. Memon, S. F. Shah, A. M. Siddiqui, “Exact solution of unsteady tank drainage for Ellis Fluid”, Journal of Applied Fluid Mechanics, Vol. 11, No. 6, pp. 1629–1636, 2018 DOI: https://doi.org/10.29252/jafm.11.06.28890
M. R. Mohyuddin, T. Gotz, “Resonance behaviour of viscoelastic fluid in Poiseuille flow in the presence of a transversal magnetic field”, International Journal for Numerical Methods in Fluids, Vol. 49, No. 8, pp. 837–847, 2005 DOI: https://doi.org/10.1002/fld.1026
J. J. V. Rossum, “Viscous lifting and drainage of liquids”, Applied Scientific Research, Section A, Vol. 7, pp. 121–144, 1958 DOI: https://doi.org/10.1007/BF03184644
I. Pop, M. Kumari, G. Nath, “Conjugate MHD flow past a flat plate”, Acta Mechanica, Vol. 106, pp. 215–220, 1994 DOI: https://doi.org/10.1007/BF01213563
S. Abel, P. H. Veena, K. Rajgopal, V. K. Pravin, “Non-Newtonian magnetohydrodynamic flow over a stretching surface with heat and mass transfer”, International Journal of Non-Linear Mechanics, Vol. 39, No. 7, pp. 1067–1078, 2004 DOI: https://doi.org/10.1016/S0020-7462(03)00097-0
M. Abdullah, N. Saada, “Free convection MHD couette flow with application of periodic temperature and constant heat flux on walls”, Engineering, Technology & Applied Science Research, Vol. 9, No. 2, pp. 4007–4011, 2019 DOI: https://doi.org/10.48084/etasr.2611
G. F. Round, V. K. Garg, Applications of fluid dynamics, E. Arnold, 1986
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