Electronic Scientific Journal
 
Diagnostics, Resource and Mechanics 
         of materials and structures
Рус/Eng  

 

advanced search

IssuesAbout the JournalAuthorContactsNewsRegistration

2019 Issue 2

2021 Issue 2
 
2021 Issue 1
 
2020 Issue 6
 
2020 Issue 5
 
2020 Issue 4
 
2020 Issue 3
 
2020 Issue 2
 
2020 Issue 1
 
2019 Issue 6
 
2019 Issue 5
 
2019 Issue 4
 
2019 Issue 3
 
2019 Issue 2
 
2019 Issue 1
 
2018 Issue 6
 
2018 Issue 5
 
2018 Issue 4
 
2018 Issue 3
 
2018 Issue 2
 
2018 Issue 1
 
2017 Issue 6
 
2017 Issue 5
 
2017 Issue 4
 
2017 Issue 3
 
2017 Issue 2
 
2017 Issue 1
 
2016 Issue 6
 
2016 Issue 5
 
2016 Issue 4
 
2016 Issue 3
 
2016 Issue 2
 
2016 Issue 1
 
2015 Issue 6
 
2015 Issue 5
 
2015 Issue 4
 
2015 Issue 3
 
2015 Issue 2
 
2015 Issue 1

 

 

 

 

 

V. V. Privalova, E. Yu. Prosviryakov

VORTEX FLOWS OF A VISCOUS INCOMPRESSIBLE FLUID AT CONSTANT VERTICAL VELOCITY UNDER PERFECT SLIP CONDITIONS

In this paper, we study exact solutions of shear flows for a viscous incompressible fluid. The proposed solutions for the velocity components are linear functions of the longitudinal coordinates. Such solutions belong to the class of Lin-Sidorov-Aristov solutions for isobaric and isothermal processes. The obtained exact solution of the Navier-Stokes equation describes a new mechanism of momentum transfer in a medium and the flow of a vertically whirling fluid. A vertical twist in a fluid arises due to the allowance for inertial forces and a nonuniform distribution of velocities at the free boundary of the fluid layer. This solution allows us to describe the counterflow of incompressible fluid in a thin layer. The condition of perfect slip on the lower solid surface of the fluid layer is considered for the obtained exact general solution. The existence of points at which the velocity field vanishes inside the fluid layer is shown. It determines the existence of stagnant points and counterflows in the fluid.

Keywords: exact solution, vertical vortex, counterflow, stagnation point, perfect slip

Bibliography:

  1. Ovsyannikov L.V. Gruppovoy analiz differentsialnykh uravneniy [Group Analysis of Differential Equations]. Moscow, Nauka Publ., 1978, 400 p. (In Russian).
  2. Andreev V.K., Khapzov O.V., Pukhnachev V.V., Rodionov A.A. Primenenie teoretiko-gruppovykh metodov v gidrodinamike [Application of Group-Theoretical Methods in Hydrodynamics]. Novosibirsk, Nauka Publ., 1994, 320 p. (In Russian).
  3. 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., 2009, vol. 43, no. 5, pp. 642‒662. DOI: 10.1134/S0040579509050066.
  4. Aristov S.N., Prosviryakov E.Yu. Inhomogeneous Couette Flows. Nelin. Dyn., 2014, vol. 10, no. 2, pp. 177‒182. DOI: 10.20537/nd1402004. (In Russian).
  5. Aristov S.N., Prosviryakov E.Yu. Stokes waves in vortical fluid. Nelin. Dyn., 2014, vol. 10, no. 3, pp. 309‒318. DOI: 10.20537/nd1403005. (In Russian).
  6. 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.
  7. Prosviryakov E.Yu. New Class of Exact Solutions of Navier-Stokes Equations with Exponential Dependence of Velocity on Two Spatial Coordinates. Theor. Found. Chem. Eng., 2019, vol. 53, no. 1, pp. 107‒114. DOI: 10.1134/S0040579518060088.
  8. Lin C.C. Note on a class of exact solutions in magneto-hydrodynamics. Arch. Rational Mech. Anal., 1957, vol. 1, pp. 391‒395. DOI: 10.1007/BF00298016.
  9. Sidorov A.F. One class of solutions of equations of gas dynamics and natural convection. In: Numerical and Analytical Methods of Solving Problems of Mechanics of Continuous Media (collected scientific papers), eds. A. F. Sidorov, Yu. N. Kondyurin, UNTs, AN SSSR Publ., Sverdlovsk, 1981, pp. 101–117. (In Russian).
  10. 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.
  11. Aristov S.N., Shvarts K.G. Vikhrevye techeniya advektivnoy prirody vo vrashchayushchemsya sloe zhidkosti [Vortex Flows of Advective Nature in Rotating Liquid Layer]. Perm, Perm State Univ. Publ., 2006. (In Russian).
  12. Aristov S.N., Shvarts K.G. Vikhrevye Techeniya v Tonkikh Sloyakh Zhidkosti [Vortical Flows in Thin Fluid Layers]. Kirov, Vyatka State Univ. Publ., 2011, 207 p. (In Russian).
  13. Andreev V.K. Resheniya Birikha uravneniy konvektsii i nekotorye ego obobshcheniya: Preprint no. 1–10 [Birikh Solutions to Convection Equations and Some of its Extensions]. IVM SO RAN Publ., Krasnoyarsk, 2010, 68 p. (In Russian).
  14. Andreev V.K., Bekezhanova V.B. Stability of nonisothermal fluids (Review). J. Appl. Mech. Tech. Phys., 2013, vol. 54, no. 2, pp. 171–184. DOI: 10.1134/S0021894413020016.
  15. Andreev V.K., Stepanova I.V. Unidirectional flows of binary mixtures within the framework of the Oberbeck–Boussinesq model. Fluid Dyn., 2016, vol. 51, no. 2, pp. 136–147. DOI: 10.1134/S0015462816020022.
  16. Goncharova O.N., Hennenberg M., Rezanova E.V., Kabov O.A. Modening of the convective fluid flows with evaporation in the two-layer system. Interfacial Phenomena and Heat Transfer, 2013, vol. 1, no. 4, pp. 317‒338. DOI: 10.1615/InterfacPhenomHeatTransfer.v1.i4.20.
  17. 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., 2009, vol. 43, no. 5, pp. 642–662. DOI: 10.1134/S0040579509050066.
  18. Betyaev V.K. Asimptoticheskie metody klassicheskoy dinamiki zhidkosti [Asymptotic Methods of Classical Fluid Dynamics]. Izhevsk, Institut Kompyuternykh Issledovaniy Publ., 2014, 516 p. (In Russian).
  19. Andreev V.K. Influence of the interfacial internal energy on the thermocapillary steady flow. Journal of Siberian Federal University. Mathematics & Physics, 2017, vol. 10, no. 4, pp. 537‒547. DOI: 10.17516/1997-1397-2017-10-4-537-547.
  20. Andreev V.K., Bekezhanova V.B., Efimova M.V., Ryzhkov I.I., Stepanova I.V. Non-classical convection models: exact solutions and their stability. Computational Technologies, 2009, vol. 14, no. 6, pp. 5‒18.
  21. Bekezhanova V.B., Shefer I.A., Goncharova O.N., Rezanova E.B. Stability of two-layer fluid flows with evaporation at the interface. Fluid Dynamics, 2017, vol. 52, no. 2, pp. 189–200. DOI: 10.1134/S001546281702003X.
  22. Goncharova O.N., Rezanova E.V., Lyulin Y.V., Kabov O.A. Analysis of a convective fluid flow with a concurrent gas flow with allowance for evaporation. High Temperature, 2017, vol. 55, no. 6, pp. 887–897. DOI: 10.1134/S0018151X17060074.
  23. Aristov S.N., Prosviryakov E.Y. Large-scale flows of viscous incompressible vortical fluid. Russian Aeronautics (Iz. VUZ), 2015, vol. 58, no. 4, pp. 413‒418. DOI: 10.3103/S1068799815040091.
  24. Aristov S.N., Prosviryakov E.Y. Unsteady layered vertical fluid flows. Fluid Dynamics, 2016, vol. 51, no. 2, pp. 148‒154. DOI: 10.1134/S0040579516030027.
  25. Aristov S.N., Privalova V.V., Prosviryakov E.Yu. Stationary nonisothermal Couette flow. Quadratic heating of the upper boundary of the fluid layer. Nonlin. Dyn., 2016, vol. 12, no. 2, pp. 167–178. DOI: 10.20537/nd1602001. (In Russian).
  26. Privalova V.V., Prosviryakov E.Yu. Steady convective Coutte flow for quadratic heating of the lower boundary fluid layer. Nonlin. Dyn., 2018, vol. 14, no. 1, pp. 69–79. DOI: 10.20537/nd1801007. (In Russian).
  27. Privalova V.V., Prosviryakov E.Yu. Couette–Hiemenz exact solutions for the steady creeping convective flow of a viscous incompressible fluid, with allowance made for heat recovery. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 2018, vol. 22, no. 3, pp. 532–548. DOI: https://doi.org/10.14498/vsgtu1638. (In Russian).
  28. Vinogradova O.I., Belyaev A.V. Wetting, roughness and flow boundary conditions. J. Phys.: Condens. Matter, 2011, vol. 23, p. 184104. DOI: 10.1088/0953-8984/23/18/184104.
  29. Stone H., Stroock A., Ajdari A. Engineering flows in small devices: Microfluidics toward a lab-on-a-chip. Annual Review of Fluid Mechanics, 2004, vol. 36, pp. 381–411. DOI: 10.1146/annurev.fluid.36.050802.122124.
  30. Quere D. Wetting and roughness. Annual Review of Materials Research, 2008, vol. 38, pp. 71–99. DOI: 10.1146/annurev.matsci.38.060407.132434.
  31. Whitesides G.M. The origins and the future of microfluidics. Nature, 2006, vol. 442, pp. 368–373. DOI: 10.1038/nature05058.
  32. Belyaev A.V., Vinogradova O.I. Effective slip in pressure-driven flow past superhydrophobic stripes. J. Fluid Mech., 2010, vol. 652, pp. 489–499. DOI: 10.1017/S0022112010000741.
  33. Landau L.D., Lifshitz E.M. Fluid Mechanics. Pergamon Press, Oxford, 1987.
  34. Pedlosky J. Geophysical Fluid Dynamics. Springer-Verlag, New York Inc., 1982.
  35. Vinogradova O.I. Drainage of a thin liquid film confined between hydrophobic surfaces. Langmuir, 1995, vol. 11, iss. 6, pp. 2213–2220. DOI: 10.1021/la00006a059.
  36. Mehdizadeh A., Oberlack M. Analytical and numerical investigations of laminar and turbulent Poiseuille–Ekman flow at different rotation rates. Physics of Fluids, vol. 22, no. 10, pp. 105104 -11. DOI: 10.1063/1.3488039.
  37. Korotaev G.K., Mikhailova E.N., Shapiro N.B. Teoriya ekvatorialnykh protivotecheniy v Mirovom okeane [Theory for the Equatorial Countercurrents in the World's Ocean]. Kiev, Naukova Dumka Publ., 1986. (In Russian). 


PDF        

 

impact factor
RSCI 0.42

 

MRDMS 2021
Google Scholar


NLR

 

Founder:  Institute of Engineering Science, Russian Academy of Sciences (Ural Branch)
Chief Editor:  S.V. Smirnov
When citing, it is obligatory that you refer to the Journal. Reproduction in electronic or other periodicals without permission of the Editorial Board is prohibited. The materials published in the Journal may be used only for non-profit purposes.
Contacts  
 
Home E-mail 0+
 

ISSN 2410-9908 Registration SMI Эл № ФС77-57355 dated March 24, 2014 © IMACH of RAS (UB) 2014-2021, www.imach.uran.ru