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Exact solutions to generalized plane Beltrami-Trkal and Ballabh flows. (English) Zbl 1463.76032

Summary: Nonstationary plane flows of a viscous incompressible fluid in a potential field of external forces are considered. An elliptic partial differential equation is obtained, with each solution being a vortex flow stream function described by an exact solution to the Navier-Stokes equations. The obtained solutions generalize the Beltrami-Trkal and Ballabh flows. Examples of such new solutions are given. They are intended to verify numerical algorithms and computer programs.

MSC:

76F02 Fundamentals of turbulence
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76F45 Stratification effects in turbulence
76R05 Forced convection
76U05 General theory of rotating fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Loitsyanskii L. G., Mechanics of Liquids and Gases, Pergamon Press, Oxford, 1966 · Zbl 0247.76001
[2] Lamb H., Hydrodynamics, Cambridge Univ., Cambridge, 1924 · JFM 50.0567.01
[3] Zhuravlev V. M., “A new representation of the two-dimensional equations of the dynamics of an incompressible fluid”, J. Appl. Math. Mech., 58:6 (1994), 1003-1009 · Zbl 0881.76103
[4] Chernyi G. G., “Plane steady self-similar vortex flows of an ideal fluid (Keplerian motions)”, Dokl. Math., 42:1 (1997), 52-55 · Zbl 0926.76017
[5] Ladyzhenskaya O. A., “Sixth problem of the millennium: Navier-Stokes equations, existence and smoothness”, Russ. Math. Surv., 58:2 (2003), 251-286 · Zbl 1062.35067
[6] Aristov S. N., Pukhnachev V. V., “On the equations of axisymmetric motion of a viscous incompressible fluid”, Dokl. Phys., 49:2 (2004), 112-115
[7] Pukhnachev V. V., “Integrals of motion of an incompressible fluid occupying the entire space”, J. Appl. Mech. Tech. Phys., 45:2 (2004), 167-171 · Zbl 1050.35075
[8] Moshkin N. P., Poochinapan K., Christov C. I., “Numerical implementation of Aristov-Pukhnachev”s formulation for axisymmetric viscous incompressible flows”, Int. J. Numer. Meth. Fluids, 62:10 (2010), 1063-1080 · Zbl 1425.76173
[9] Moshkin N. P., Poochipan K., “Novel finite difference scheme for the numerical solution of two-dimensional incompressible Navier-Stokes equations”, Int. J. Numer. Anal. Mod., 7:2 (2010), 321-329
[10] Golubkin V. N., Markov V. V., Sizykh G. B., “The integral invariant of the equations of motion of a viscous gas”, J. Appl. Math. Mech., 79:6 (2015), 566-571 · Zbl 1432.76216
[11] Aristov S. N., Knyazev D. V., Polyanin A. D., “Exact solutions of the Navier-Stokes equations with the linear dependence of velocity components on two space variables”, Theor. Found. Chem. Eng., 43:5 (2009), 642-662
[12] Lin C. C., “Note on a class of exact solutions in magneto-hydrodynamics”, Arch. Rational Mech. Anal., 1:1 (1958), 391-395 · Zbl 0083.42103
[13] Neményi P. F., “Recent developments in inverse and semi-inverse methods in the mechanics of continua”, R. von Mises, Th. von Kármán (eds.), Advances in Applied Mechanics, v. 2, Academic Press, New York, 1951, 123-151 · Zbl 0044.20403
[14] Sidorov A. F., “Two classes of solutions of the fluid and gas mechanics equations and their connection to traveling wave theory”, J. Appl. Mech. Tech. Phys., 30:2 (1989), 197-203
[15] Meleshko S. V., Pukhnachev V. V., “One class of partially invariant solutions of the Navier-Stokes equations”, J. Appl. Mech. Tech. Phys., 40:2 (1999), 208-216 · Zbl 0934.35116
[16] Ludlow D. K., Clarkson P. A., Bassom A. P., “Similarity reductions and exact solutions for the two‐dimensional incompressible Navier-Stokes equations”, Stud. Appl. Math., 103:3 (1999), 183-240 · Zbl 1136.76342
[17] Polyanin A. D., “Exact solutions to the Navier-Stokes equations with generalized separation of variables”, Dokl. Phys., 46:10 (2001), 726-731
[18] Meleshko S. V., “A particular class of partially invariant solutions of the Navier-Stokes equations”, Nonlinear Dynam., 36:1 (2004), 47-68 · Zbl 1098.76059
[19] Pukhnachev V. V., “Symmetries in Navier-Stokes equations”, Usp. Mekh., 4:1 (2006), 6-76 (In Russian)
[20] Drazin P. G., Riley N., The Navier-Stokes Equations: A Classification of Flows and Exact Solutions, Cambridge Univ., Cambridge, 2006 · Zbl 1154.76019
[21] Polyanin A. D., Aristov S. N., “A new method for constructing exact solutions to three-dimensional Navier-Stokes and Euler equations”, Theor. Found. Chem. Eng., 45:6 (2011), 885-890
[22] Aristov S. N., Polyanin A. D., “New classes of exact solutions and some transformations of the Navier-Stokes equations”, Russ. J. Math. Phys., 17:1 (2010), 1-18 · Zbl 1192.35128
[23] Maslov V. P., Shafarevich A. I., “Asymptotic solutions of Navier-Stokes equations and topological invariants of vector fields and Liouville foliations”, Theor. Math. Phys., 180:2 (2014), 967-982 · Zbl 1308.76071
[24] Allilueva A. I., Shafarevich A. I., “Asymptotic solutions of linearized Navier-Stokes equations localized in small neighborhoods of curves and surfaces”, Russ. J. Math. Phys., 22:4 (2015), 421-436 · Zbl 1333.76033
[25] Broman G. I., Rudenko O. V., “Submerged Landau jet: exact solutions, their meaning and application”, Physics-Uspekhi, 53:1 (2010), 91-98
[26] Aristov S. N., Polyanin A. D., “New classes of exact solutions of Euler equations”, Dokl. Phys., 53:3 (2008), 166-171 · Zbl 1257.35142
[27] Couette M., “Études sur le frottement des liquids”, Ann. de Chim. et Phys. (6), 21 (1890), 433-510 (In French) · JFM 22.0964.01
[28] Poiseuille J., “Récherches expérimentales sur le mouvement des liquides dans les tubes de très petits diamètres”, C. R. Acad. Sci., 11 (1840), 961-967, 1041-1048
[29] Poiseuille J., “Récherches expérimentales sur le mouvement des liquides dans les tubes de très petits diamètres”, C. R. Acad. Sci., 12 (1841), 112-115
[30] Stokes G. G., “On the effect of the internal friction of fluid on the motion of pendulums”, Trans. Cambridge Philos. Soc., 9 (1851), 8-106
[31] v. Kármán Th., “Über laminare und turbulente Reibung”, ZAMM, 1:4 (1921), 233-252 (In German) · JFM 48.0968.01
[32] Hiemenz K., “Die Grenzschicht an einem inden gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder”, Dinglers Polytech. J., 326 (1911), 321-324
[33] Aristov S. N., Prosviryakov E. Yu., “Inhomogeneous Couette flow”, Nelin. Dinam., 10:2 (2014), 177-182 (In Russian) · Zbl 1310.76074
[34] Aristov S. N., Prosviryakov E. Yu., “Stokes waves in vortical fluid”, Nelin. Dinam., 10:3 (2014), 309-318 (In Russian) · Zbl 1310.76075
[35] Aristov S. N., Prosviryakov E. Yu., “Unsteady layered vortical fluid flows”, Fluid Dyn., 51:2 (2016), 148-154 · Zbl 1342.76048
[36] Aristov S. N., Shvarts K. G., Vortical Flows of the Advective Nature in a Rotating Fluid Layer, Perm State Univ., Perm, 2006 (In Russian)
[37] Aristov S. N., Shvarts K. G., Vortical Flows in Thin Fluid Layers, Vyatka State Univ., Kirov, 2011, 207 pp. (In Russian)
[38] Andreev V. K., Bekezhanova V. B., “Stability of nonisothermal fluids (Review)”, J. Appl. Mech. Tech. Phys., 54:2 (2013), 171-184 · Zbl 1298.76083
[39] Ryzhkov I. I., Thermal Diffusion in Mixtures: Equations, Symmetries, Solutions and their Stability, Sib. Otd. Ross. Akad. Nauk, Novosibirsk, 2013 (In Russian)
[40] Aristov S. N., Prosviryakov E. Yu., “A new class of exact solutions for three dimensional thermal diffusion equations”, Theor. Found. Chem. Eng., 50:3 (2016), 286-293
[41] Beltrami E., “Considerazioni idrodinamiche”, Rend. Inst. Lombardo Acad. Sci. Lett., 22 (1889), 122-131 · JFM 21.0948.04
[42] Trkal V., “Poznámka k hydrodynamice vazkých tekutin”, Časopis, 48 (1919), 302-311 (In Czech) · JFM 47.0768.02
[43] Gromeka I. S., Collected Works, Akad. Nauk SSSR, Moscow, 1952 (In Russian)
[44] Batchelor G. K., An Introduction to Fluid Dynamics, Cambridge Univ., Cambridge, 2000 · Zbl 0958.76001
[45] Lavrent’ev M. A., Shabat B. V., Methodsof the Theory of Functions of a Complex Variable, Nauka, Moscow, 1987 (In Russian)
[46] Markov V. V., Sizykh G. B., “Exact solutions of the Euler equations for some two-dimensional incompressible flows”, Proc. Steklov Inst. Math., 294:1 (2016), 283-290 · Zbl 1365.35114
[47] Ballabh R., “Self superposable motions of the type \(\xi=\lambda u\), etc.”, Proc. Benares Math. Soc., n. Ser., 2 (1940), 85-89
[48] Ballabh R., “Superposable motions in a heterogeneous incompressible fluid”, Proc. Benares Math. Soc., n. Ser., 3 (1941), 1-9 · Zbl 0061.43403
[49] Ballabh R., “On coincidence of vortex and stream lines in ideal liquids”, Ganita, 1 (1950), 1-4 · Zbl 0041.11104
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