V. V. Privalova, E. Yu. Prosviryakov
EXACT SOLUTIONS FOR A COUETTE–HIEMENZ CREEPING CONVECTIVE FLOW WITH LINEAR TEMPERATURE DISTRIBUTION ON THE UPPER BOUNDARY
A new exact solution is found for the plane convection of a viscous incompressible fluid describable by the Oberbeck–Boussinesq equation system in an infinite thin layer. The solution of the boundary-value problem is obtained for the fluid flow arising in the case of inhomogeneous velocity distribution and a linear heat source at the upper boundary of an infinite layer of a viscous incompressible fluid. A creeping convective flow is studied by the generalization of the isothermal class of the Hiemenz exact solutions. The temperature and pressure fields are linear with respect to the horizontal coordinate in this class of solutions. The analysis of polynomial solutions describing natural fluid convection in the Stokes approximation is presented. The paper shows the existence of points where the velocity field vanishes inside the fluid layer. These points are termed stagnation points and indicate the presence of counterflows in the fluid. Similar investigations are carried out for the temperature and pressure fields. The isotherms and isobars are shown to be always locally parabolic or hyperbolic, i.e. to have an infinitely distant point, due to the structure of the discussed exact solution.
Keywords: counterflow, exact solution, Couette flow, Hiemenz flow, Oberbeck–Boussinesq equation, Stokes approximation, stagnation point, stratification
1.Couette M. Etudes sur le frottement des liquids. Ann. Chim. Phys., 1890, vol. 21, pp. 433–510.
2.Landau L.D., Lifshitz E.M. Course of Theoretical Physics. Vol. 6. Fluid mechanics. Pergamon Press, Oxford, 1987, 539 р.
3.Drazin P.G. The Navier–Stokes equations: A classification of flows and exact solutions. Cambridge, Cambridge Univ. Press, 2006, 196 p.
4.Boronin S.A. Stability of the plane Couette flow of a disperse medium with a finite volume fraction of the particles. Fluid Dynamics, 2011, vol. 46, pp. 64–71.
5.Georgiyevskii D.V. Generalized Joseph Estimates of Stability of Plane Shear Flows with Scalar Nonlinearity. Bulletin of the Russian Academy of Sciences: Physics, 2011, vol. 75, no. 1, pp. 149–152. DOI: 10.3103/S1062873810121044.
6.Zhuk V.I., Protsenko I.G. Asymptotic model for the evolution of perturbations in the plane Couette-Poiseuille flow. Doklady Mathematics, 2006, vol. 74, no. 3, pp. 896–900. DOI: 10.1134/S1064562406060287.
7.Rudyak V.Ya., Isakov E.B., Bord E.G. Instability of plane Couette flow of two-phase liquids. Technical Physics Letters, 1998, vol. 24, pp. 199–200. DOI: 10.1134/1.1262051.
8.Troshkin O.V. Nonlinear stability of Couette, Poiseuille and Kolmogorov plane channel flows. Doklady Mathematics, 2012, vol. 85, no. 2, pp. 181–185. DOI: 10.1134/S1064562412020068.
9.Shalybkov D.A. Hydrodynamic and hydromagnetic stability of the Couette flow. Physics-Uspekhi, 2009, vol. 52, no. 9, pp. 915–935. DOI: 10.3367/UFNe.0179.200909d.0971.
10.Abramyan A.K., Mirantsev L.V., Kuchmin A.Yu. Modeling of processes at Couette simple fluid flow in flat nano-scopic canal. Matematicheskoe Modelirovanie, 2012, vol. 24, no. 4, pp. 3–21 (In Russian).
11.Aristov S.N., Prosviryakov E.Yu. Nonuniform convective Couette flow. Fluid Dynamics, 2016, vol. 51, pp. 581–587. DOI: 10.1134/S001546281605001X.
12.Aristov S.N., Prosviryakov E.Yu. A new class of exact solutions for three-dimensional thermal diffusion equations. Theor. Found. Chem. Eng., 2016, vol. 50, no. 3, pp. 286–293. DOI: 10.1134/S0040579516030027.
13.Babkin V. A. Plane Turbulent Couette Flow. Journal of Engineering Physics and Thermophysics, vol. 76, iss. 6, pp. 1251–1254. DOI: 10.1023/B:JOEP.0000012026.19646.c6.
14.Belyaeva N.A., Kuznetsov K.P. Analysis of a nonlinear dynamic model of the Couette flow for structured liquid in a flat gap. Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2012, no. 2 (27), pp. 85–92. DOI: 10.14498/vsgtu1018. (In Russian).
15.Gavrilenko S.L., Shil'ko S.V., Vasin R.A. Characteristics of a viscoplastic material in the couette flow. Journal of Applied Mechanics and Technical Physics, 2002, vol. 43, iss. 3, pp 439–444. DOI: 10.1023/A:1015378622918.
16.Kudinov V.A., Kudinov I.V. Reception of exact analytical decisions of the hyperbolic equations of movement at a dispersed current of Kuetta. Izvestiya Rossiyskoy Akademii Nauk. Energetika, 2012, no. 1, pp. 119–133 (In Russian).
17.Malyshev V.A., Manita A.D. Stochastic Micromodel of the Couette Flow. Theory of Probability & its Applications, 2009, vol. 53, iss. 4, pp. 716–727. DOI: 10.1137/S0040585X97983924.
18.Pukhnachev V.V., Pukhnacheva T.P. Couette Problem for Kelvin-Voigt Medium. Journal of Mathematical Sciences, 2012, vol. 186, iss. 3, pp. 495–510. DOI: 10.1007/s10958-012-1003-0.
19.Skulsky O.I., Aristov S.N. Mekhanika anomalno vyazkikh zhidkostey [Mechanics of Quasi-Viscous Fluids]. Moscow–Izhevsk, NITs Regularnaya i Khaoticheskaya Dinamika Publ., 2003, 156 p. (In Russian).
20.Aristov S.N., Privalova V.V., Prosviryakov E.Yu. Stationary nonisothermal Couette ﬂow. Quadratic heating of the upper boundary of the ﬂuid layer. Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 2, pp. 167–178. DOI: 10.20537/nd1602001. (In Russian).
21.Aristov S.N., Prosviryakov E.Yu. On one class of analytic solutions of the stationary axisymmetric convection Bénard–Maragoni viscous incompreeible fluid. Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki, 2013, no. 3 (32), pp. 110–118. DOI: 10.14498/vsgtu1205. (In Russian).
22.Aristov S.N. Prosviryakov E.Yu. Exact Solutions of Thermocapillary Convection in a Localized Heating of a Plane Layer of a Viscous Incompressible Fluid. Vestnik KGTU im. A.N. Tupoleva, 2014, no. 3, pp. 7–12. (In Russian).
23.Aristov S.N., Vlasova S.S., Prosviryakov E.Yu. Linear Benard-Marangoni convection with quadratic heating on top of a plane layer of a viscous incompressible fluid. Polzunovskiy Vestnik, 2014, no. 4–2, pp. 95‒102. (In Russian).
24.Aristov S.N., Shvarts K.G. Convective heat transfer in a locally heated plane incompressible fluid layer. Fluid Dynamics, 2013, vol. 48, pp. 330–335. DOI: 10.1134/S001546281303006X.
25.Vlasova S.S., Prosviryakov E.Y. Parabolic convective motion of a fluid cooled from below with the heat exchange at the free boundary. Russian Aeronautics, 2016, vol. 59, no. 4, pp. 529‒535. DOI: 10.3103/S1068799816040140.
26.Vlasova S.S., Prosviryakov E.Yu. Two-dimensional convection of an incompressible viscous fluid with the heat exchange on the free border. Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki, 2016, vol. 20, no. 3, pp. 567–577. DOI: 10.14498/vsgtu1483.
27.Hiemenz K. Die Grenzschicht an einem in den gleichförmigen Flüssigkeit-sstrom eingetauchten geraden Kreiszylinder. Dingler’s Politech. J., 1911, vol. 326, pp. 321–324.
28.Andreev V.K., Cheremnykh E.N. 2D thermocapillary motion of three fluids in a flat channel. Journal of Siberian Federal University. Mathematics and physics, 2016, vol. 9, issue. 4, pp. 404–415. DOI: 10.17516/1997-1397-2016-9-4-404-415.
29.Andreev V.K., Cheremnykh E.N. The joint creeping motion of three viscid liquids in a plane layer: A priori estimates and convergence to steady flow. Journal of Applied and Industrial Mathematics, 2016, vol. 10, no. 1, pp. 7–20. DOI: 10.1134/S1990478916010026.
30.Andreev V.K., Sobachkina N.L. Dvizhenie binarnoy smesi v ploskikh i tsilindricheskikh oblastyakh [The Motion of a Binary Mixture in Planar and Cylindrical Regions]. Krasnoyarsk, SFU Publ., 2012, 187 p. (In Russian).
31.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. Theoretical Foundations of Chemical Engineering, 2009, vol. 43, no. 5, pp. 642–662.
32.Pukhnachev V.V. Group Properties of the Navier-Stokes Equations in a Plane Case. Prikl. Mekh. Tekh. Fiz., 1960, no. 1, pp. 83–90.
33.Ekman V.W. On the Influence of the Earth’s Rotation on Ocean-Currents. Ark. Mat. Astron. Fys., 1905, vol. 2, no. 11, pp. 1–52.
34.Gershuni G.Z., Zhukhovitskii E.M. Convective Stability of Incompressible Fluids. Israel Program for Scientific Translations. Jerusalem: Keter Publishing House, 1976, 330 p.
35.Andreev V.K., Gaponenko Ya.A., Goncharova O.N., Pukhnachev V.V. Mathematical Models of Convection. Berlin, Boston: De Gryuter Publ., 2012, 417 p. DOI: 10.1515/9783110258592.
36.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., 1989, vol. 30, no. 2, pp. 197–203. DOI: 10.1007/BF00852164.
37.Lin C.C. Note on a class of exact solutions in magneto-hydrodynamics. Arch. Rational Mech. Anal., 1958, vol. 1, pp. 391–395.
Privalova V. V., Prosviryakov E. Yu. Exact Solutions for a Couette–hiemenz Creeping Convective Flow with Linear Temperature Distribution on the Upper Boundary // Diagnostics, Resource and Mechanics of materials and structures. -
2018. - Iss. 2. - P. 92-109. -
DOI: 10.17804/2410-9908.2018.2.092-109. -