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International Journal of Physics. 2022, 10(2), 93-101
DOI: 10.12691/ijp-10-2-2
Open AccessArticle

On the Incompressible Fluid Flow over the Prismatic Bodies

Khatiashvili Nino1,

1I. Vekua Institute of Applied Mathematics, Iv. Javakhishvili Tbilisi State University, Tbilisi, Georgia

Pub. Date: March 20, 2022
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Cite this paper:
Khatiashvili Nino. On the Incompressible Fluid Flow over the Prismatic Bodies. International Journal of Physics. 2022; 10(2):93-101. doi: 10.12691/ijp-10-2-2

Abstract

In the paper the unsteady incompressible fluid flow over the infinite and finite prismatic bodies is studied. Mathematically this problem is modeled as 3D Navier-Stokes equations (NSE) for the fluid velocity components with the appropriate initial-boundary conditions. The study of the fluid flow over the bodies with the sharp edges is the important problem of Aerodynamics and Hydrodynamics. We admit that near the sharp edges the velocity components are non-smooth. By the methods of mathematical physics the bounded novel exact solutions are obtained for the specific pressure. The profile of the velocity is plotted by means of “Maplesoft”.

Keywords:
incompressible fluid flows Navier-stokes equations prismatic bodies

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References:

[1]  Batchelor, G.K, An Introduction to Fluid Dynamics, Cambridge Univ. Press, Cambridge, 2000.
 
[2]  Cermak, J.E., Cochran, L.S., “Physical modeling of the atmosphe-ric surface layer”, J. Wind Eng. Ind. Aerodyn., 41, 935-946, 1992.
 
[3]  Chacon-Rebollo, T., Lewandowski R., Mathematical and numerical foundations of turbulence models and applications, in: Modeling and Simulation in Science, Engineering and Technology, Springer, New-York, 2014.
 
[4]  Chorin, A.J., Marsden, J.E., A Mathematical Introduction to Fluid Mechanics, 2-nd ed., Springer, NY, 1984.
 
[5]  Constantin, P., Foias, C., Navier-Stokes equations, Chicago Lectures in Mathematics. University of Chicago Press, 1988.
 
[6]  Kim, S., Karrila, S., Microhydrodynamics: Principles and Selected Applications, Dover, 2005.
 
[7]  Kirby, B.J., Micro and Nanoscale Fluid Mechanics in Microfluidic Devices, Cambridge University Press, 2010.
 
[8]  Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow, Mathematics and its Applications. 2nd Ed., Gordon and Breach, 1969.
 
[9]  Ladyzhenskaya, O. A. “The Sixth Millennium Problem: Navier-Stokes Equations, Existence and Smoothness”, Russ. Math. Surv., 58 (2), 251-286. 2003.
 
[10]  Lamb Horace, Hydrodynamics (Sixth ed.), Dover, NY, 1945.
 
[11]  Landay, L.D., Lifshitz, E.M., Fluid Mechanics, Course of Theoretical Physics. 6, Pergamon Press, 1987.
 
[12]  Lautrup, B., Physics of Continuous Matter, Second Edition: Exotic and Everyday Phenomena in the Macroscopic World, Second Ed., CRC Press, 2011.
 
[13]  Lavrentiev, M.A., Shabat, B.V., Problems of hydrodynamics and their mathematical Models (Russian), Nauka, Moskow, 1977.
 
[14]  Lions, P.L., Mathematical Topics in Fluid Mechanics, The Clarendon Press, Vols. 1,3 Oxford University Press, NY, 1996.
 
[15]  Majda, A.J., Bertozzi, A.L., Vorticity and Incompressible Flow, Vol.27, Cambridge University Press, 2002.
 
[16]  Milne-Thompson, L.M., Theoretical Hydrodynamics, 5-th ed, Macmillan, 1968.
 
[17]  Ockendon, H and Ockendon J., Viscous Flow, Cambridge University Press, 1995.
 
[18]  Prandtl, L., Essentials of Fluid Mechanics, third ed. in: Applied Mathematical Sciences, Vol. 158, Springer, NY, 2010.
 
[19]  Rosenhead, L., (ed.), Laminar boundary layers, Clarendon Press, 1963.
 
[20]  Temam, R., Navier-Stokes Equations, Theory and numerical Analysis, AMS Chelsea, 2001.
 
[21]  Tychonov, A.N., Samarski, A.A., Partial Differential Equations of Mathematical Physics, vol. I. Translated by S. Radding, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964.
 
[22]  Whitham, G. B., Linear and Nonlinear Waves. Reprint of the 1974 original. Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1999.
 
[23]  Eringen, A. C., “Theory of micropolar fluids”, International Journal of Mathematics and Mechanics 16. 1-18. 1966.
 
[24]  Nabok, A., Organic and Inorganic Nanostructures, Artech House, MEMS series, Boston London, 2005.
 
[25]  Hsiao, K. L., “Micropolar nanofluid flow with MHD and viscous dissipation effects towards a stretching sheet with multimedia feature”. Int. J. Heat Mass Transfer. 12. 983-90.2013.
 
[26]  Mishra, S. R., Khan, I., Al-mdallal Q.M., Asifa, T., “Free convective micropolar fluid flow and heat transfer over a shrinking sheet with heat source”. Case Studies in Thermal Engineering, 11. 113-119. 2018.
 
[27]  Sun, X., Animasaun, I. L., Swain, K., Shah, N. A., Wakif, A., Olanrewaju, P. O., “Significance of nanoparticle radius, inter‐particle spacing, inclined magnetic field, and space‐dependent internal heating: The case of chemically reactive water conveying copper nanoparticles”, ZAMM - Journal of Applied Mathematics and Mechanics. 202100094. 2021.
 
[28]  Saleem, S., Animasaun, I. L., Yook, S. J., Al-Mdallal, Q. M., Shah, N. A., Faisal, M., “Insight into the motion of water conveying three kinds of nanoparticles shapes on a horizontal surface: Significance of thermo-migration and Brownian motion”, Surfaces and Interfaces, 30. 101854.2022.
 
[29]  Animasaun, I. L., Shah, N. A., Wakif, A., Mahanthesh, B., Sivaraj, R., Koriko, O. K., Ratio of Momentum Diffusivity to Thermal Diffusivity: Introduction, Meta-analysis, and Scrutinization. Chapman and Hall/CRC. New York, 2022.
 
[30]  Crane, L.J., “Flow past a stretching plate”, ZAMP, 21. 645-647. 1970.
 
[31]  Ahmadi, G., “Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite plate”, Int. J. Eng. Sci., 14. 639-646. 1976.
 
[32]  Miklavcˇicˇ, M., Wang, C.Y., “Viscous flow due a shrinking sheet”, Q. Appl. Math., 64. 283-290. 2006.
 
[33]  Hayat, T., Javed, T., Sajid, M., “Analytic solution for MHD rotating flow of a second grade fluid over a shrinking surface”, Phys. Lett. A, 372. 3264-3273. 2008.
 
[34]  Fang, T., J. Zhang J., “Closed-form exact solution of MHD viscous flow over a shrinking sheet”. Commun. Nonlinear Sci. Numer. Simul., 14. 2853-2857. 2009.
 
[35]  Noor, N.F., Kechil, S.A., I. Hashim, I., “Simple non-perturbative solution for MHD viscous flow due to a shrinking sheet”, Commun. Nonlinear Sci. Numer. Simul., 15. 144-148. 2010.
 
[36]  Khan, Z. H., Qasim, M., Haq, R. U., Al-Mdallal, Q. M., “Closed form dual nature solutions of fluid flow and heat transfer over a stretching/shrinking sheet in a porous medium’, Chinese Journal of Physics, 55(4). 1284-1293. 2017.
 
[37]  Berker, R., “An exact solution of the Navier-Stokes equation, the vortex with curvilinear axis”, International Journal of Engineering Science, 20. 217-230. 1982.
 
[38]  Boussinesque J., “Mémoire sur l'influence des Frottements dans les des Fluids”. J. Math. Pures Appl. 13 (2): 377-424, 1868.
 
[39]  Burgers, J. M., “A mathematical model illustrating the theory of turbulence”, Adv. Appl.Mech., 1. 171-199. 1948.
 
[40]  Chwang, A., and Wu, T., “Hydromechanics of low-Reynolds-number flow. Singularity method for Stokes flows”, J. Fluid Mech., 62. 65-70. 1974.
 
[41]  Clayton, B. R., Massey, B. S., “Exact similar solution for an axisymmetric laminar boundary layer on a circular cone” AIAA J., 17(7). 785-786. 1979.
 
[42]  Couette, M. “Études sur le frottement des liquids”, Ann. Chim. Phys. 21, 433-510.1890.
 
[43]  Dorrepaal, J. M., “An exact solution of the Navier-Stokes equations which describes non-orthogonal stagnation-point flow in two dimensions”, J. Fluid Mech. 163. 141-147. 1986.
 
[44]  Drazin, P.G., Riley, N., The Navier-Stokes equations: a classification of flows and exact solutions, 334, Cambridge University Press, 2006.
 
[45]  Ethier, C. R., Steinman, D. A., “Exact fully 3D Navier-Stokes solutions for benchmarking”, International Journal for Numerical Methods in Fluids, 19 (5). 369-375. 1994.
 
[46]  Fraenkel, L. E., “Laminar flow in symmetrical channels with slightly curved walls, I. On the Jeffery-Hamel solutions for flow between plane walls”, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 267. 119-138. 1962.
 
[47]  Jeffery, G. B., “The two-dimensional steady motion of a viscous fluid”, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 29 (172). 455-465. 1915.
 
[48]  Hiemenz, K., “Die Grenschicht an einem in den gleichfoermigen Fluessigkeitsstrom eingetauchten geraden Kreiszy-linder”, Dingler’s Ploytech. J. 326, 321-324.1911.
 
[49]  Karman, T., “Über laminare und turbulente Reibung”, Zeitschrift für Angewandte Mathematik und Mechanik, 1 (4). 233-252. 1921.
 
[50]  Kerr, O. S., Dold, J. W., “Periodic steady vortices in a stagnation-point flow”, Journal of Fluid Mechanics, 276. 307-325. 1994.
 
[51]  Khatiashvili, N., Pirumova, K., Janjgava, D., “On the Stokes flow over ellipsoidal type bodies”, Lecture Notes in Engineering and Computer Science, Proceedings of The World Congress on Engineering, 1.IAENG, London, 148-151. 2013.
 
[52]  Khatiashvili, N.,Pirumova, K., Janjgava, D., „On some Effective Solutions of Stokes Axisymmetric Equation for a Viscous Fluid”, Proceedings of World Academy of Science, Engineering and Technology, 79, London, 690-694. 2013.
 
[53]  Khatiashvili, N., “On the Non-Smooth Solutions of 3D Navier-Stokes Equations for the Incompressible Fluid Flows”, International Journal of Physics, 9 (3). 178-185. 2021.
 
[54]  Khatiashvili, N., Shanidze, R., Janjgava, D., “On effective solutions of the nonlinear Schrödinger equation”, J. Phys.: Conf. Ser., 482(1). 012020. 2014.
 
[55]  Khatiashvili, N., “On one Class of Elliptic Equations Connected with the nonlinear Waves”, Trans. A.Razmadze Math.Inst., 173.61-70.2019.
 
[56]  Landau, L. D., “New exact solution of the Navier-Stokes equations”, Doklady Akademii Nauk SSSR, 44. 311-314. 1944.
 
[57]  Lewandowski, R., “Navier-Stokes equations in the whole space with an eddy viscosity”, J. Math. Anal. Appl., 478 (2). 698-742. 2019.
 
[58]  Neuber, H,. “Einneuer Ansatzzur Losungtaumlicher Probleme der Elastizitats theorie”, Jornal of Applied Mathematics and Mechanics Academy, 14. 203-212. 1934.
 
[59]  Papkovich, P.F., “Solution Generale des equations differentials fondamentales delasticite exprimee par trois fonctions harmoniques”, Compt. Rend. Acad. Sci. Paris, 195, 513-515. 1932.
 
[60]  Poiseuille, J., “Recherches expérimentales sur le mouvement des liquides dans les tubes de très petits diamètres”. Mémoires présentés par Divers Savants à l'Académie Royale des sciences de l'Institut de France. 9, Paris. 433-544. 1846.
 
[61]  Polyanin, A. D., “Exact solutions to the Navier-Stokes equations with generalized separation of variables”, Dokl. Phys. 46. 726-731. 2001.
 
[62]  Proudman, J., “Notes on the motion of viscous liquids in channels”, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 28(163). 30-36. 1914.
 
[63]  Prosviryakov E.Yu., “New Class of Exact Solutions of Navier-Stokes Equations with Exponential Dependence of Velocity on Two Spatial Coordinates”, Theor. Found. Chemic. Eng., 53, 107-114. 2019.
 
[64]  Proudman, J., “On the motion of solids in a liquid possessing vorticity”, Proc. R. Soc. Lond. A. 92. 408-424. 1916.
 
[65]  Rajagopal, K. R., “A class of exact solutions to the Navier-Stokes equations”, International Journal of Engineering Science. 22. 451-455. 1984.
 
[66]  Rao, A. R., Kasiviswanathan, S. R., “On exact solutions of the unsteady Navier-Stokes equation the vortex with instantaneous curvilinear axis”, International Journal of Engineering Science. 25. 337-349. 1987.
 
[67]  Rayleigh, Lord,. “On the motion of solid bodies through viscous liquid”, Phil. Mag. 21, 697-711. 1911.
 
[68]  Riabouchinsky, D., Mémoires sur la Mécanique des Fluides. Offerts à M. Dimitri, P. Riabouchinsky. Publications Scientifiques et Techniques du Ministère de l'Air, Paris, 1954.
 
[69]  Rosenhead, L., “The steady two-dimensional radial flow of viscous fluid between two inclined plane walls,” Proceedings of the Royal Society of London A: Mathematical Physical and Engineering Sciences, The Royal Society, 963- 175. 1940.
 
[70]  Rott, N., “Unsteady viscous flow in the vicinity of a stagnation point”, Q. Appl. Maths. 444-451. 1956.
 
[71]  Sheng Yin Cheng, Falin Chen, “On the stagnation point position of the flow impinging obliquely on a moving flat plate”, J.Fluid Mech, 889. 2020.
 
[72]  Siddiqui, A.M., M. R. Mohyuddin, M.R., Hayat, T, and Asghar, S, “Some more inverse solutions for steady flows of a second-grade fluid,” Archives of Mechanics, 55(4). 373-387. 2003.
 
[73]  Siddiqui, A. M., Iftikhar, S., “Unsteady flow of a viscous fluid between two oscillating cylinders”, Appl. Math. Inf. Sci. 6 (3). 483-489. 2012.
 
[74]  Squire, H. B., “The round laminar jet”, The Quarterly J. of Mechanics and Applied Mathematics, 4(3). 321-329. 1951.
 
[75]  Stokes, G. G., “On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids”, Transactions of the Cambridge Philosophical Society, 8. 287-341. 1845.
 
[76]  Stokes, G.G., “On the steady motion of incompressible fluids”. Transactions of the Cambridge Philosophical Society. 7, Mathematical and Physical Papers, Cambridge University Press. 75-129. 1880.
 
[77]  Taylor, G.I., “Motion of solids in fluids when the flow is not irrotational”, Proc. R. Soc. Lond. A. 93. 92-113. 1917.
 
[78]  Taylor, G.I., “Stability of a viscous liquid containing between two rotating cylindres”, Philosophical Transactions of the Royal Society of London. Series A, 223, 605-615. 289-343. 1923.
 
[79]  Taylor, G.I., “On the decay of vortices in a viscous fluid,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 46.274, 671-674. 1923.
 
[80]  Taylor, G.I. and Green, A.E., “Mechanism of the Production of Small Eddies from Large Ones”, Proc. R. Soc. Lond. A, 158. 499-521. 1937.
 
[81]  Tsangaris, S., Kondaxakis, D., Vlachakis, N. W., “Exact solution for flow in a porous pipe with unsteady wall suction and/or injection”, Commun. Nonlinear Sci. Numer. Simul., 12. 1181-1189. 2007.
 
[82]  Tsien Hsue-Shen., “Symmetrical Joukowsky Airfoils in shear flow”, Quarterly of Applied Mathematics, 1: 130-48.1943.
 
[83]  Yatseyev, V. I., “About a class of exact solutions of viscous fluid equations of motion”, Zhurnal Tekhnicheskoj Fiziki, 20 (11). 1031-1034. 1950.
 
[84]  Yih, C.S., Wu, F., Garg, A. K., and Leibovich, S., “Conical vortices: a class of exact solutions of the Navier-Stokes equations”, Phys. Fluids, 25(12). 2147-2158. 1982.
 
[85]  Waleed, S. K., “Classical Fundamental Unique Solution for the Incompressible Navier-Stokes Equation in RN”, J. Appl. Math. and Physics, 5(4). 30-40. 2017.
 
[86]  Wang, C. Y., “Exact solutions of the steady-state Navier-Stokes equations”, Annual Review of Fluid Mechanics, 23. 159-177. 1991.
 
[87]  Wang, C. Y., “Flow due to a stretching boundary with partial slip an exact solution of the Navier-Stokes equations”, Chem. Eng. Sci., 57. 3745-3747. 2002.
 
[88]  Zeb, A., Siddiqui, A.M., Ahmed, M., “An Analysis of the Flow of a Newtonian Fluid Between Two Moving Parallel Plates”, International Scholary Research Notices, 2013.
 
[89]  Zubarev, N.M., Prosviryakov, E.Yu., “Exact Solutions for Layered Three Dimensional Nonstationary Isobaric Flows of a Viscous Incompressible Fluid”, J. Appl.Mech. and Thechnical Physics, 60(6), 1031-1037. 2019.
 
[90]  Khatiashvili, N., Komurjishvili, O., Papukashvili, A., Shanidze, R. et, al., “On Some Mathematical Models of Growth of Solid Crystals and Nanowires”, Bulletin of TICMI, 17(1). 28-48, 2013.
 
[91]  Sieden, L.S., Buckminster Fuller’s Universe: His Life and Work. Perseus Books Group, New York, 2000.
 
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