Advertisement

Microgravity Science and Technology

, Volume 30, Issue 4, pp 543–560 | Cite as

Influence of Gravity on the Stability of Evaporative Convection Regimes

  • V. B. Bekezhanova
  • I. A. Shefer
Original Article
  • 41 Downloads

Abstract

The characteristics of convective regimes in a two-layer system have been investigated in the framework of the Boussinesq approximation of the Navier–Stokes equations. An exact invariant solution of the convection equations is used to describe a joint stationary flow of an evaporating liquid and a gas-vapor mixture in a horizontal channel. Thermodiffusion effects in the gas-vapor phase are additionally taken into account in the governing equations and interface conditions. The influence of gravity and thickness of the liquid layer on the hydrodynamical, thermal and concentration characteristics of the regimes has been investigated. Flows of the pure thermocapillary, mixed and Poiseuille’s types are specified for different values of the problem parameters. The linear stability of the evaporative convection regimes has been studied. The types and properties of the arising perturbations have been investigated and the critical characteristics of the stability have been obtained. Disturbances can lead to the formation of deformed convective cells, vortex and thermocapillary structures. The change of the instability types and threshold thermal loads occurs with the increasing thickness of the liquid layer and gravity action.

Keywords

Evaporative convection Exact solution Characteristic perturbations Stability 

Notes

Compliance with Ethical Standards

Conflict of interests

The authors declare that they have no conflict of interest.

References

  1. Andreev, V.K., Kaptsov, O.V., Pukhnachov, V.V., Rodionov, A.A.: Applications of Group Theoretical Methods in Hydrodynamics, vol. 408. Kluwer Academic Publ., Dordrecht (1998)CrossRefMATHGoogle Scholar
  2. Andreev, V.K., Bublik, V.V., Bytev, V.O.: Symmetries of nonclassical models of hydrodynamics, vol. 352. Novosibirsk, Nauka (2003). [in Russian]Google Scholar
  3. Andreev, V.K., Gaponenko, Yu.A., Goncharova, O.N., Pukhnachov, V.V.: Mathematical models of convection (De Gruyter studies in Mathematical Physics), vol. 417. De Gruyter, Berlin/Boston (2012)Google Scholar
  4. Bar-Cohen, A., Wang, P.: Thermal managment of on-chip hot spot. J. Heat Transfer 134(5), 051017 (2012)CrossRefGoogle Scholar
  5. Bekezhanova, V.B.: Convective instability of Marangoni–Poiseuille flow under a longitudinal temperature gradient. J. Appl. Mech. Tech. Phys. 52(1), 74–81 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. Bekezhanova, V.B.: Three-dimensional disturbances of a plane-parallel two-layer flow of a viscous, heat-conducting fluid. Fluid Dyn. 47(6), 702–708 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. Bekezhanova, V.B., Goncharova, O.N.: Stability of the exact solutions describing the two-layer flows with evaporation at interface. Fluid Dyn. Res. 48(6), 061408 (2016)MathSciNetCrossRefGoogle Scholar
  8. Bekezhanova, V.B., Goncharova, O.N., Rezanova, E.V., Shefer, I.A.: Stability of two-layer fluid flows with evaporation at the interface. Fluid Dyn. 52(2), 189–200 (2017)MathSciNetCrossRefMATHGoogle Scholar
  9. Berg, J.C., Acrivos, A., Boudart, M.: Evaporative convection. Adv. Chem. Eng. 6, 61–123 (1966)CrossRefGoogle Scholar
  10. Birikh, R.V.: Thermocapillary convection in a horizontal layer of liquid. J. Appl. Mech. Tech. Phys. 3, 43–45 (1966)Google Scholar
  11. Burelbach, J.P., Banko, S.G., Davis, S.H.: Nonlinear stability of evaporating/condensing films. J. Fluid Mech. 195, 463–494 (1988)CrossRefMATHGoogle Scholar
  12. Colinet, P., Joannes, L., Iorio, C.S., Haute, B., Bestehorn, M., Lebon, G., Legros, J.-C.: Interfacial turbulence in evaporating liquids: Theory and preliminary results of the ITEL-master 9 sounding rocket experiment. Adv. Space Res. 32(2), 119–127 (2003)CrossRefGoogle Scholar
  13. Colinet, P., Legros, J.C., Velarde, M.G.: Nonlinear Dynamics of Surface-Tension-Driven Instabilities, vol. 512. Wiley-VCH, Berlin (2001)CrossRefMATHGoogle Scholar
  14. Das, K.S., Ward, C.A.: Surface thermal capacity and its effects on the boundary conditions at fluid-fluid interfaces. Phys. Rev. E 75, 1–4 (2007)CrossRefGoogle Scholar
  15. Frezzotti, A.: Boundary conditions at the vapor–liquid interface. Phys. Fluids 23, 030609 (2011)CrossRefMATHGoogle Scholar
  16. Godunov, S.: On the numerical solution of boundary value problems for systems of ordinary linear equations. Uspekhi Matem Nauk 16(3(99)), 171–174 (1961)Google Scholar
  17. Goncharova, O.N.: Modeling of flows under conditions of heat and mass transfer at the interface. Izvestiya of Altai State University Journal 73(1/2), 12–18 (2012)Google Scholar
  18. Goncharova, O.N., Hennenberg, M., Rezanova, E.V., Kabov, O.A.: Modeling of the convective fluid flows with evaporation in the two-layer systems. Interfacial Phenomena and Heat Transfer 1(4), 317–338 (2013)CrossRefGoogle Scholar
  19. Goncharova, O.N., Kabov, O.A.: Investigation of the two-layer fluid flows with evaporation at interface on the basis of the exact solutions of the 3D problems of convection. J. Phys.: Conf. Ser. 754, 032008 (2016)Google Scholar
  20. Goncharova, O.N., Rezanova, E.V.: Example of an exact solution of the stationary problem of two-layer flows with evaporation at the interface. J. Appl. Mech. Techn. Phys. 55(2), 247–257 (2014)CrossRefMATHGoogle Scholar
  21. Goncharova, O.N., Rezanova, E.V.: Construction of a mathematical model of flows in a thin liquid layer on the basis of the classical convection equations and generalized conditions on an interface. Izvestiya of Altai State University Journal 85(1/1), 70–74 (2015)Google Scholar
  22. Goncharova, O.N., Rezanova, E.V., Lyulin, Yu.V., Kabov, O.A.: Modeling of two-layer liquid-gas flow with account for evaporation. Thermophys. Aeromech. 22(5), 631–637 (2015)CrossRefGoogle Scholar
  23. Haut, B., Colinet, P.: Surface-tension-driven instability of a liquid layer evaporating into an inert gas. J. Colloid Interface Sci. 285, 296–305 (2005)CrossRefGoogle Scholar
  24. Hoke, B.C., Chen, J.C.: Mass transfer in evaporating falling liquid film mixtures. AIChE J. 38(5), 781–787 (1992)CrossRefGoogle Scholar
  25. Iorio, C.S., Goncharova, O.N., Kabov, O.A.: Study of evaporative convection in an open cavity under shear stress flow. Microgravity Sci. Technol. 21(1), 313–320 (2009)CrossRefGoogle Scholar
  26. Iorio, C.S., Kabov, O.A., Legros, J.-C.: Thermal Patterns in evaporating liquid. Microgravity Sci. Technol. XIX(3/4), 27–29 (2007)CrossRefGoogle Scholar
  27. Kabov, O.A., Kuznetsov, V.V., Kabova, Yu.O.: Evaporation, dynamics and interface deformations in thin liquid films sheared by gas in a microchannel (Chapter 2), encyclopedia of two-phase heat transfer and flow II: special topics and applications, volume 1: special topics in boiling in microchannels / micro-evaporator cooling systems. In: Thome, J.R., Kim, J. (eds.) , pp 57–108. World Scientific Publishing Company, Singapore (2015)Google Scholar
  28. Kabova, Yu., Kuznetsov, V.V., Kabov, O., Gambaryan-Roisman, T., Stephan, P.: Evaporation of a thin viscous liquid film sheared by gas in a microchannel. Int. J. Heat Mass Transf. 68, 527–541 (2014)CrossRefGoogle Scholar
  29. Kandlikar, S.G., Colin, S., Peles, Y., Garimella, S., Pease, R.F., Brandner, J.J., Tuckerman, D.B.: Heat transfer in microchannels –2012 status and research needs. J. Heat Transfer 135(9), 091001 (2013)CrossRefGoogle Scholar
  30. Kimball, J.T., Hermanson, J.C., Allen, J.S.: Experimental investigation of convective structure evolution and heat transfer in quasi-steady evaporating liquid films. Phys. Fluids 24, 052102 (2012)CrossRefGoogle Scholar
  31. Klentzman, J., Ajaev, V.S.: The effect of evaporation on fingering instabilities. Phys. Fluids 21(12), 122101 (2009)CrossRefMATHGoogle Scholar
  32. Kuznetsov, V.V.: Heat and mass transfer on a liquid– vapor interface. Fluid Dyn. 46(5), 754–763 (2011)MathSciNetCrossRefMATHGoogle Scholar
  33. Kuznetsov, V.V., Andreev, V.K.: Liquid film and gas flow motion in a microchannel with evaporation. Thermophys. Aeromech. 20(1), 17–28 (2013)CrossRefGoogle Scholar
  34. Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics, Volume 6: Fluid Mechanics, 2nd edn., vol. 554. Pergamon Press, Oxford (1987)Google Scholar
  35. Li, P., Chen, Z., Shi, J.: Numerical study on the effects of gravity and surface tension on condensation process in square minichannel. Microgravity Sci. Technol. 30, 19–24 (2018)CrossRefGoogle Scholar
  36. Liu, R., Kabov, O.A.: Instabilities in a horizontal liquid layer in co-current gas flow with an evaporating interface. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 85(6), 066305 (2012)CrossRefGoogle Scholar
  37. Lyulin, Y., Kabov, O.: Evaporative convection in a horizontal liquid layer under shear-stress gas flow. Int. J. Heat Mass Transf. 70, 599–609 (2014)CrossRefGoogle Scholar
  38. Lyulin, Y., Kabov, O.: Measurement of the evaporation mass flow rate in a horizontal liquid layer partly opened into flowing gas. Tech. Phys. Lett. 39, 795–797 (2013)CrossRefGoogle Scholar
  39. Mancini, H., Maza, D.: Pattern formation without heating in an evaporative convection experiment. Europhys. Lett. 66(6), 812–818 (2004)CrossRefGoogle Scholar
  40. Margerit, J., Colinet, P., Lebon G., Iorio, C.S., Legros, J.C.: Interfacial nonequilibrium and Benard-Marangoni instability of a liquid–vapor system. Phys. Rev. E 68, 1–14 (2003)CrossRefGoogle Scholar
  41. Merkt, D., Bestehorn, M.: Benard–marangoni convection in a strongly evaporating field. Phys. D 185, 196–208 (2003)MathSciNetCrossRefMATHGoogle Scholar
  42. Molenkamp, T.: Marangoni Convection Mass Transfer and Microgravity. 240 Ph.D. Dissertation, Rijksuniversiteit Groningen, Groningen (1998)Google Scholar
  43. Napolitano, L.G.: Plane Marangoni–Poiseuille flow two immiscible fluids. Acta Astronaut. 7, 461–478 (1980)CrossRefMATHGoogle Scholar
  44. Narendranath, A.D., Hermanson, J.C., Kolkka, R.W., Struthers, A.A., Allen, J.S.: The effect of gravity on the stability of an evaporating liquid film. Microgravity Sci. Technol. 26(3), 189–199 (2014)CrossRefGoogle Scholar
  45. Nie, Z.H., Kumacheva, E.: Patterning surfaces with functional polymers. Nat. Mater. 7, 277–290 (2008)CrossRefGoogle Scholar
  46. Nepomnyashchy, A.A., Velarde, M.G., Colinet, P.: Interfacial Phenomena and Convection, vol. 360. Chapman & Hall/CRC, Boca Raton (2002)MATHGoogle Scholar
  47. Oron, A.: Nonlinear dynamics of irradiated thin volatile liquid films. Phys. Fluids 12(1), 29 (2000)CrossRefMATHGoogle Scholar
  48. Ostroumov, G.A.: Free Convection under the Conditions of an Internal Problem, vol. 286. Gostekhizdat Press, Moscow–Leningrad (1952). [in Russian]Google Scholar
  49. Oron, A., Davis, S.H., Bankoff, S.C.: Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69(3), 931–980 (1997)CrossRefGoogle Scholar
  50. Ozen, O., Narayanan, R.: The physics of evaporative and convective instabilities in bilayer systems: linear theory. Phys. Fluids 16(12), 4644 (2004)CrossRefMATHGoogle Scholar
  51. Prosperetti, A.: Boundary conditions at a liquid–vapor interface. Mechanica 14(1), 34–47 (1979)CrossRefMATHGoogle Scholar
  52. Pukhnachov, V.V.: A plane steady-state free boundary problem for the Navier–Stokes equations. J. Appl. Mech. Techn. Phys. 13(3), 340–351 (1972)CrossRefGoogle Scholar
  53. Pukhnachov, V.V.: Group-theoretical nature of the Birikh’s solution and its generalizations. In: Book of Proc. Symmetry and Differential Equations, Krasnoyarsk. [in Russian], pp 180–183 (2000)Google Scholar
  54. Pukhnachov, V.V.: Symmetries in the Navier–Stokes equations. Uspekhi Mechaniki 4(1), 6–76 (2006). [in Russian]Google Scholar
  55. Puknachov, V.V.: Thermocapillary convection under low gravity. Fluid Dynamics Transactions 14, 140–204 (1989)Google Scholar
  56. Reutov, V.P., Ezersky, A.B., Rybushkina, G.V., Chernov, V.V.: Convective structures in a thin layer of an evaporating liquid under an airflow. J. Appl. Mech. Techn. Phys. 48(4), 469–478 (2007)CrossRefMATHGoogle Scholar
  57. Rezanova, E.V., Shefer, I.A.: Influence of thermal load on the characteristics of a flow with evaporation. J. Appl. Ind. Math. 11(2), 274–283 (2017)MathSciNetCrossRefMATHGoogle Scholar
  58. Rodionova, A.V., Rezanova, E.V.: Stability of two-layer fluid flow. J. Appl. Mech. Tech. Phys. 57(4), 588–595 (2016)MathSciNetCrossRefMATHGoogle Scholar
  59. Saenz, P.J., Valluri, P., Sefiane, K., Karapetsas, G., Matar, O.K.: Linear and nonlinear stability of hydrothermal waves in planar liquid layers driven by thermocapillarity. Phys. Fluids 25(9), 094101 (2013)CrossRefGoogle Scholar
  60. Saenz, P.J., Valluri, P., Sefiane, K., Karapetsas, G., Matar, O.K.: On phase change in Marangoni-driven flows and its effects on the hydrothermal-wave instabilities. Phys. Fluids 26(2), 024114 (2014)CrossRefGoogle Scholar
  61. Scheid, B., Margerit, J., Iorio, C.S., Joannes, L., Heraud, M., Dauby, P.C., Colinet, P.: Onset of thermal ripples at the interface of an evaporating liquid under a flow of inert gas. Exp. Fluids 52, 1107–1119 (2012)CrossRefGoogle Scholar
  62. Shi, W.-Y., Rong, S.-M., Feng, L.: Marangoni convection instabilities induced by evaporation of liquid layer in an open rectangular pool. Microgravity Sci. Technol. 29, 91–96 (2017)CrossRefGoogle Scholar
  63. Shklyaev, O.E., Fried, E.: Stability of an evaporating thin liquid film. J. Fluid Mech. 584, 157–183 (2007)MathSciNetCrossRefMATHGoogle Scholar
  64. Shliomis, M.I., Yakushin, V.I.: Convection in a two-layers binary system with an evaporation. Collected papers: Uchenye zapiski Permskogo Gosuniversiteta, seriya Gidrodinamika 4, 129–140 (1972). [in Russian]Google Scholar
  65. Sultan, E., Boudaoud, A., Amat, M.B.: Evaporation of a thin film: diffusion of the vapour and Marangoni instabilities. J. Fluid Mech. 543, 183–202 (2005)MathSciNetCrossRefMATHGoogle Scholar
  66. Voropai, P.I., Shlepov, A.A.: Enhancement of Reliability and Efficiency of Reciprocating Compressors, vol. 359. Nedra, Moscow (1980)Google Scholar
  67. Zeytounian, R.: The Benard–Marangoni thermocapillary-instability problem. Usp. Phys. Nauk 168(3), 259–286 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Differential Equations of MechanicsInstitute of Computational Modelling SB RASKrasnoyarskRussia
  2. 2.Institute of Mathematics and Computer ScienceSiberian Federal UniversityKrasnoyarskRussia

Personalised recommendations