Skip to main content
SpringerLink
Log in

Stability of an Advective Flow in a Horizontal Fluid Layer Heat-Insulated from Below with Rigid Boundaries

  • Published:
Fluid Dynamics Aims and scope Submit manuscript

Abstract—

In this paper, we study the stability of an advective flow in a flat horizontal layer of an incompressible fluid with rigid boundaries. A linear temperature distribution is set on the upper boundary of the layer while the lower boundary is thermally insulated. The plane-parallel flow due to the action of horizontal convection is described analytically as an exact solution of the Navier–Stokes equations in the Boussinesq approximation. In the linear theory, the stability of an advective flow to normal perturbations is studied at various values of the Prandtl number. The most dangerous modes are determined, and neutral curves are plotted. In the nonlinear formulation of the problem, the structure of finite-amplitude perturbations in the supercritical region near the minima of the neutral curves is studied.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. Gershuni, G.Z., Zhukhovitskii, E.M., and Nepomnyashchii, A.A., Ustoichivost’ konvektivnykh techenii (Stability of Convective Flows), Moscow: Nauka, 1989.

  2. Ostroumov, G.A., Free Convection under the Conditions of the Internal Problem, NASA TM, 1958.

    Google Scholar 

  3. Andreev, V.K., Birikh solutions for the convection equations and its certain generalizations, Preprint of Institute of Numerical Mathematics, Siberian Branch RAS, 2010, nos. 1–10.

  4. Birikh, R.V., Thermocapillary convection in a horizontal layer of liquid, J. Appl. Mech. Techn. Phys., 1966, vol. 7, no. 3, pp. 43–44.

    Article  ADS  Google Scholar 

  5. Gershuni, G.Z., Laure, P., Myznikov, V.M., Roux, B., and Zhukhovitsky, E.M., On the stability of plane-parallel advective flows in long horizontal layers, Microgravity Q., 1992, vol. 2, no. 3, pp. 141–151.

    Google Scholar 

  6. Andreev, V.K. and Bekezhanova, V.B., Stability of non-isothermal fluids (review), J. Appl. Mech. Techn. Phys., 2013, no. 2, pp. 171–184.

  7. Schwarz, K.G., Stability of thermocapillary advective flow in a slowly rotating liquid layer under microgravity conditions, Fluid Dyn., 2012, vol. 47, no. 1, pp. 37–49.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Aristov, S.N. and Shvarts, K.G., Advective flow in a rotating liquid film, J. Appl. Mech. Techn. Phys., 2016, vol. 57, no. 1, pp. 188–194. https://doi.org/10.1134/S0021894416010211

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Schwarz, K.G., Plane-parallel advective flow in a horizontal incompressible fluid layer with rigid boundaries, Fluid Dyn., 2014, vol. 49, no. 4, pp. 438–442. https://doi.org/10.1134/S0015462814040036

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Shvarts, K.G., Plane-parallel advective flow in a horizontal layer of an incompressible fluid with an internal linear heat source, Fluid Dyn., 2018, vol. 53, suppl. 1, pp. S24–S28. https://doi.org/10.1134/S0015462818040237

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Schwarz, K.G. and Schwarz, Yu.A., Stability of advective flow in a horizontal incompressible fluid layer in the presence of the Navier slip condition, Fluid Dyn., 2020, vol. 55, no. 1, pp. 31–42.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Lyubimov, D.V. and Shklyaev, S.V., Thermal convection in an acoustic field, Fluid Dyn., 2000, vol. 35, no. 3, pp. 321–330.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Lyubimova, T.P., Nikitin, D.A., and Skuridin, R.V., Acoustic wave effect onto the stability of advective flow in the plane layer, Vestn. Perm. Univ., Ser.: Mat., Mekh. Inform., 2011, no. 5(9), pp. 143–147.

  14. Ivantsov, A.O., Weakly non-linear analysis of thermoacoustic advective flow stability, Vestn. Perm. Univ., Fiz., 2019, no. 3, pp. 28–44.

  15. Slavchev, S., Hennenberg, M., Valhev, G., et al., Stability of ferrofluid flows in a horizontal channel subjected to a longitudinal temperature gradient and an oblique magnetic field, Microgravity Sci. Technol., 2008, vol. 20, no. 1, pp. 199–203.

    Article  ADS  Google Scholar 

  16. Aristov, S.N. and Shvarts, K.G., Convective heat transfer in a locally heated plane incompressible fluid layer, Fluid Dyn., 2013, vol. 48, no. 3, pp. 330–335.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Aristov, S.N. and 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.

    Article  Google Scholar 

  18. Burmasheva, N.V., Larina, E.A., and Prosviryakov, E.Yu., A Couette-type flow with a perfect slip condition on a solid surface, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2021, no. 74, pp. 79–94. https://doi.org/10.17223/19988621/74/9

  19. Demin, V.A., Convective separators, Prikl. Fiz., 2013, no. 4, pp. 60–67. https://applphys.orion-ir.ru/appl-13/13-4/PF-13-4-60.pdf.

  20. Hart, J., A note on the stability of low-Prandtle-number Hadley circulations, J. Fluid Mech., 1983, vol. 132, pp. 271–281.

    Article  ADS  MATH  Google Scholar 

  21. Laure, P., Etude des mouvements de convection dans une cavite rectangulaire soumise a un gradient de temperature horizontal, J. Mec. Theor., 1987, vol. 6, pp. 351–382.

    MATH  Google Scholar 

  22. Kuo, H.P. and Korpela, S.A., Stability and finite amplitude natural convection in a shallow cavity with insulated top and bottom and heated from a side, Phys. Fluids, 1988, vol. 31, no. 1, pp. 33–42.

    Article  ADS  MATH  Google Scholar 

  23. Wang, P. and Daniels, P.G., Numerical solutions for the flow near the end of a shallow laterally heated cavity, J. Eng. Math., 1994, vol. 28, pp. 211–226.

    Article  MathSciNet  MATH  Google Scholar 

  24. Lyubimov, D.V., Lyubimova, T.P., Nikitin, D.A., et al., Stability of a binary-mixture advective flow in a plane horizontal layer with perfectly heat conducting boundaries, Fluid Dyn., 2010, vol. 45, no. 3, pp. 458–467.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Lybimova, T.P., Lybymov, D.V., Morozov, V.A., et al., Stability of convection in a horizontal channel subjected to a longitudinal temperature gradient. Pt. 1. Effect of aspect ratio and prandtl number, J. Fluid Mech., 2009, vol. 635, pp. 275–295.

    Article  ADS  MathSciNet  Google Scholar 

  26. Lyubimova, T.P. and Nikitin, D.A., Stability of the advective flow in a horizontal rectangular channel with adiabatic boundaries, Fluid Dyn., 2011, vol. 46, no. 2, pp. 240–249.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Mizev, A., Mosheva, E., Kostarev, K., et al., Stability of solutal advective flow in a horizontal shallow layer, Phys. Rev. Fluids, 2017, vol. 2, no. 10, p. 103903.

  28. Demin, V.A., Kostarev, K.G., Mizev, A.I., et al., On convective instability of the counter propagating fluxes of inter-soluble liquids, Russ. J. Nonlin. Dyn., 2014, vol. 10, no. 2, pp. 195–208.

    MATH  Google Scholar 

  29. Schwarz, K.G., Stability of advective flow in a rotating horizontal incompressible fluid layer heat-insulated from below with rigid boundaries at low Prandtl number, Fluid Dyn., 2022, vol. 57, no. 2, pp. 146–157.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Aristov, S.N. and Shvarts, K.G., Vikhrevye techeniya advektivnoi prirody vo vrashchayushchemsya sloe zhidkosti (Vortex Flows of Advective Nature in a Rotating Fluid Layer), Perm: Perm Univ., 2006.

  31. Tarunin, E.L. and Shvarts, K.G., Investigation of the linear stability of advective flow by the grid method, Vychisl. Tekhnol., 2001, vol. 6, no. 6, pp. 108–117.

    MathSciNet  MATH  Google Scholar 

  32. Shvarts, K.G., Finite-amplitude spatial perturbations of advective flow in the rotating horizontal fluid layer, Vychisl. Tekhnol., 2001, vol. 6, special issue, part 2: Proc. Int. Conf. RDAMM, Moscow, 2001, pp. 702–707.

  33. Tarunin, E.L., Vychislitel’nyi eksperiment v zadachakh svobodnoi konvektsii (Computational Experiment in Problems of Free Convection), Irkutsk: Irkutsk Univ., 1990.

Download references

Author information

Authors and Affiliations

  1. Perm State National Research University, 614068, Perm, Russia

    K. G. Shvarts & Yu. A. Shvarts

Authors
  1. K. G. Shvarts
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu. A. Shvarts
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to K. G. Shvarts or Yu. A. Shvarts.

Ethics declarations

We declare that we have no conflicts of interest.

Additional information

Translated by A. Ivanov

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shvarts, K.G., Shvarts, Y.A. Stability of an Advective Flow in a Horizontal Fluid Layer Heat-Insulated from Below with Rigid Boundaries. Fluid Dyn 57, 973–981 (2022). https://doi.org/10.1134/S0015462822080055

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0015462822080055

Keywords:

Search

Navigation