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Microgravity Science and Technology

, Volume 31, Issue 3, pp 293–304 | Cite as

Linear-Stability Analysis of Thermocapillary-Buoyancy Convection in an Annular Two-Layer System with Upper Rigid Wall Subjected to a Radial Temperature Gradient

  • Dong-Ming Mo
  • Deng-Fang RuanEmail author
Original Article
  • 26 Downloads

Abstract

This paper presented a linear-stability analysis on thermocapillary-buoyancy convection in radially heated annular pool filled in 5cSt silicone oil / HT-70 two-layer fluids. The influences of the aspect ratio, radius ratio and thickness ratio on the flow stability were discussed. Results indicate that the system is most unstable when the thickness ratio is close to 0.375. Flow bifurcation appears when the thickness ratio varies from 0.625 to 0.875. Three corresponding destabilization mechanisms were revealed. Effect of buoyancy convection and the upper boundary condition on the flow stability were also analyzed. The upper rigid wall can enhance the flow stability in the annular two-layer system, especially when the thickness ratio is near 0.75. Furthermore, the buoyancy can greatly reduce the flow stability comparing with that under the microgravity condition.

Keywords

Thermocapillary-buoyancy convection Two-layer system Linear-stability analysis Annular pool 

Nomenclature

A

Complex matrix

B

Complex matrix

e

Unit vector

h

Depth, m

H

Dimensionless depth

m

Wave number

Ma

Marangoni number

P

Dimensionless pressure

Pr

Prandtl number

r

Radius, m

R

Dimensionless radius

T

Temperature, K

U

Dimensionless radial velocity

v

dimensionless azimuthal velocity

V

Dimensionless velocity vector

W

Dimensionless vertical velocity

z

Axial coordinate, m

Z

Dimensionless axial coordinate

Greek Symbols

α

Thermal diffusivity, m2/s

β

Thermal expansion coefficient, 1/K

γT

Surface tension coefficient, N/(m.K)

ε

Thickness ratio, ε = h1/h

η

Aspect ratio, η = h/(ro-ri)

Θ

Dimensionless temperature, Θ = (T-Tc)/(Th-Tc)

θ

Azimuthal coordinate, rad

κ

Thermal conductivity, W/(m.K)

λ

Eigenvalues

μ

Dynamic viscosity, kg/(m.s)

ν

Kinematic viscosity, m2/s

ρ

Density, kg/m3

τ

Dimensionless time

Γ

Radius ratio, Γ = ri/ro

ϕ

Temperature perturbation

φ

Propagation angle of the hydrothermal wave, °

ψ

Dimensionless stream function

ω

Phase velocity

Subscripts

c

Cold

cri

Critical point

h

Hot

i

Inner

j

Liquid layer, the lower layer (j = 1) and upper layer(j = 2)

o

Outer

Notes

Acknowledgments

This work is supported by Chongqing Research Program of Basic Research and Frontier Technology (Grant No. cstc2018jcyjAX0597) and Science and Technology Research Program of Chongqing Municipal Education Commission of China (Grant No. KJQN201803201).

References

  1. Andereck, C.D., Colovas, P.W., Degen, M.M., Renardy, Y.Y.: Instabilities in two layer Rayleigh-Bénard convection: overview and outlook. Int. J. Eng. Sci. 36(12–14), 1451–1470 (1998)CrossRefzbMATHGoogle Scholar
  2. Boeck, T., Nepomnyashchy, A.A., Simanovskii, I.B., Golovin, A., Braverman, L.M., Thess, A.: Three-dimensional convection in a two-layer system with anomalous thermocapillary effect. Phys. Fluids. 14(11), 3899–3911 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bratsun, D.A.: On Rayleigh-Benard mechanism of alignment of salt fingers in reactive immiscible two-layer systems. Microgravity Sci. Technol. 26(5), 293–303 (2014)CrossRefGoogle Scholar
  4. Davalos-Orozco, L.A.: Longwave stability of two liquid layers coating both sides of a thick wall in the absence of gravity. Microgravity Sci. Technol. 30, 209–228 (2018)CrossRefGoogle Scholar
  5. Doi, T., Koster, J.N.: Thermocapillary convection in two immiscible liquid layers with free surface. Phys. Fluids. 5(8), 1914–1927 (1993)CrossRefzbMATHGoogle Scholar
  6. Gong, X.W., Mo, D.M., Wu, C.M., Li, Y.R.: Linear-stability analysis of thermocapillary-buoyancy convection in annular two-layer system with a radial temperature gradient. Int. Commun. Heat Mass Transfer. 66, 58–62 (2015)CrossRefGoogle Scholar
  7. Guo, W., Narayanan, R.: Onset of Rayleigh-Marangoni convection in a cylindrical annulus heated from below. J. Colloid Interface Sci. 314(2), 727–732 (2007)CrossRefGoogle Scholar
  8. Gupta, N.R., Hossein, H., Borhan, A.: Thermocapillary convection in double-layer fluid structures: an effective single-layer model. J. Colloid Interface Sci. 293(1), 158–171 (2006)CrossRefGoogle Scholar
  9. Gupta, N.R., Hossein, H., Borhan, A.: Thermocapillary convection in double-layer fluid structures within a two-dimensional open cavity. J. Colloid Interface Sci. 315(1), 237–247 (2007)CrossRefGoogle Scholar
  10. Hurle, D.T.J.: Thermo-hydrodynamic oscillation in liquid metals: the cause of impurities striations in melt-grown crystals. J. Phys. Chem. Solids. 1, 659–669 (1967)Google Scholar
  11. Juel, A., Burgess, J.M., McCormick, W.D., Swift, J.B., Swinney, H.L.: Surface tension- driven convection patterns in two liquid layers. Physica D Nonlinear Phenomena. 143(1–4), 169–186 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Lehoucq, R.B.: Implicitly restarted Arnoldi methods and subspace iteration. J. Soc. Ind. Appl. Math. 23(2), 551–562 (2001)MathSciNetzbMATHGoogle Scholar
  13. Li, Y.R., Wang, S.C., Wu, S.Y., Peng, L.: Asymptotic solution of thermocapillary convection in thin annular two-layer system with upper free surface. Int. J. Heat Mass Transf. 52, 4769–4777 (2009)CrossRefzbMATHGoogle Scholar
  14. Li, Y.R., Wang, S.C., Wu, S.Y., Shi, W.Y.: Asymptotic solution of thermocapillary convection in two immiscible liquid layers in a shallow annular cavity. SCIENCE CHINA Technol. Sci. 53(6), 1655–1665 (2010a)CrossRefzbMATHGoogle Scholar
  15. Li, Y.R., Zhang, W.J., Peng, L.: Thermal convection in an annular two-layer system under combined action of buoyancy and thermocapillary forces. J. Supercond. Nov. Magn. 23(6), 1219–1223 (2010b)CrossRefGoogle Scholar
  16. Li, Y.R., Zhang, W.J., Wang, S.C.: Two-dimensional numerical simulation of thermocapillary convection in annular two-layer system. Microgravity Sci. Technol. 20(3–4), 313–317 (2008)CrossRefGoogle Scholar
  17. Liu, Q.S., Chen, G., Roux, B.: Thermogravitational and thermocapillary convection in a cavity containing two superposed immiscible liquid layers. Int. J. Heat Mass Transf. 36(1), 101–117 (1993)CrossRefzbMATHGoogle Scholar
  18. Liu, Q.S., Zhou, B.H., Liu, R., Nguyen-Thi, H., Billia, B.: Oscillatory instabilities of two-layer Rayleigh-Marangoni-Bénard convection. Acta Astronaut. 59(1–5), 40–45 (2006)CrossRefGoogle Scholar
  19. Madruga, S., Pérez-García, C., Lebon, G.: Convective instabilities in two superposed horizontal liquid layers heated laterally. Phys. Rev. E. 68(1), 041607 (2003)CrossRefGoogle Scholar
  20. Mikishev, A.B., Nepomnyashchy, A.A.: Parametric excitation of Marangoni instability in a heated thin layer covered by insoluble surfactant. Microgravity Sci. Technol. 30, 173–181 (2018)CrossRefGoogle Scholar
  21. Mo, D.M., Li, Y.R., Shi, W.Y.: Linear-stability analysis of thermocapillary flow in an annular two-layer system with upper rigid wall. Microgravity Sci. Technol. 23(S1), 43–48 (2011)CrossRefGoogle Scholar
  22. Mo, D.M., Ruan, D.F.: Linear-stability analysis of thermocapillary convection in an annular two-layer system with free surface subjected to a radial temperature gradient. J. Mech. Sci. Technol. 32(7), 3437–3444 (2018)CrossRefGoogle Scholar
  23. Nepomnyashchy, A.A., Simanovskii, I.B.: Convective flows in a two-layer system with a temperature gradient along the interface. Phys. Fluids. 18, 032015 (2006)Google Scholar
  24. Nepomnyashchy, A.A., Simanovskii, I.B.: Convective Instabilities in Systems with Interface. Gordon and Breach, London (1993)zbMATHGoogle Scholar
  25. Nepomnyashchy, A.A., Simanovskii, I.B., Legros, J.C.: Interfacial Convection in Multilayer Systems, Second edn. Springer, New York (2012)Google Scholar
  26. Peng, L., Li, Y.R., Shi, W.Y.: Three-dimensional thermocapillary-buoyancy flow of silicone oil in a differentially heated annular pool. Int. J. Heat Mass Transf. 50(5), 872–880 (2007)CrossRefzbMATHGoogle Scholar
  27. Simanovskii, I.B., Kabov, O.A.: Nonlinear convective oscillations in two-layer systems with different aspect ratios. Microgravity Sci. Technol. 24(2), 127–137 (2012)CrossRefGoogle Scholar
  28. Simanovskii, I.B., Nepomnyashchy, A.A., Viviani, A., Dubois, F.: Nonlinear waves in two-layer systems with a temperature-dependent interfacial heat release. Microgravity Sci. Technol. 28(4), 381–393 (2016)CrossRefzbMATHGoogle Scholar
  29. Simanovskii, I.B., Viviani, A., Dubois, F.: Convective flows in a two-layer system with an interfacial heat release under the action of an imposed temperature gradient. Int. J. Therm. Sci. 113, 51–64 (2017)CrossRefGoogle Scholar
  30. Simanovskii, I.B., Viviani, A., Dubois, F., Queeckers, P.: Nonlinear convective flows in a laterally heated two-layer system with a temperature-dependent heat release/consumption at the interface. Microgravity Sci. Technol. 30, 243–256 (2018)CrossRefGoogle Scholar
  31. Someya, S., Munakata, T., Nishio, M., Okamoto, K., Madarame, H.: Flow observation in two immiscible liquid layers subject to a horizontal temperature gradient. J. Cryst. Growth. 235(1), 626–632 (2002)CrossRefGoogle Scholar
  32. Tavener, S.J., Cliffe, K.A.: Two-fluid Marangoni-Benard convection with a deformable interface. J. Comput. Phys. 182(1), 277–300 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Villers, D., Platten, J.K.: Thermal convection in superposed immiscible liquid layers. Appl. Sci. Res. 45(2), 145–152 (1988)CrossRefGoogle Scholar
  34. Wang, P., Kahawita, R., Nguyen, D.L.: Numerical simulation of buoyancy-Marangoni convection in two superposed immiscible liquid layers with a free surface. Int. J. Heat Mass Transf. 37(7), 1111–1112 (1994)CrossRefzbMATHGoogle Scholar
  35. Zhou, B.H., Liu, Q.S., Hu, L., Yao, Y.L., Hu, W.: R.: space experiments of thermocapillary convection in two-liquid layers. Science in China series E: technological. Sciences. 45(5), 552–560 (2002)Google Scholar
  36. Zhou, B.H., Liu, Q.S., Tang, Z.M.: Rayleigh-Marangoni-Benard instability in two-layer fluid system. Acta Mech. Sinica. 20(4), 366–373 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Mechanical TransmissionChongqing UniversityChongqingChina
  2. 2.Department of Mechanical EngineeringChongqing Industry Polytechnic CollegeChongqingChina
  3. 3.College of Automotive EngineeringChongqing UniversityChongqingChina

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