Skip to main content
SpringerLink
Log in

Bénard–Marangoni convection in a small circular container: influence of the Biot and Prandtl numbers on pattern dynamics and free surface deformation

  • Research Article
  • Published:
Experiments in Fluids Aims and scope Submit manuscript

Abstract

We studied, experimentally, the pattern dynamics and free surface deformation in Bénard–Marangoni convection, in a circular container (aspect ratio = 6). The free surface deformation fields were visualized by interferometry and temperature fields by infrared thermography. We considered the influence of the Marangoni (up to 2,623), Biot and Prandtl numbers. More dynamics are induced by increasing the Biot number and transition to a time-dependent flow has been observed. Conversely, increasing the Prandtl number reduces the dynamics. The deformation increases as a function of the Marangoni number until it reaches asymptotic values, which are functions of the Biot and Prandtl numbers.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Abbreviations

d :

layer depth (m)

g :

acceleration due to gravity (m/s2)

L :

Biot number

Ma :

Marangoni number

N c :

number of convective cells

Pr :

Prandtl number

R :

radius of the container (m)

Ra :

Rayleigh number

ΔT :

temperature difference applied to the liquid layer (°C)

ΔT p :

temperature difference between the sapphire and the copper plate (°C)

σ :

surface tension (kg/m2)

ρ :

density (kg/m3)

ν :

kinematic viscosity (m2/s)

κ :

thermal diffusivity (m2/s)

α :

expansion coefficient (K−1)

λ :

thermal conductivity (W/m K)

Γ :

aspect ratio

l:

liquid

g:

gas

References

  • Bénard H (1900) “Les tourbillons cellulaires dans une nappe liquide. Première partie: description”. Rev Gen Sci Pure Appl 11:1261–1271

    Google Scholar 

  • Boeck T (2005) Bénard–Marangoni convection at large Marangoni numbers: results of numerical simulations. Adv Space Res 36:4–10

    Article  Google Scholar 

  • Bok-Cheol S, Zebib A (2002) Effect of free surface heat loss and rotation on transition to oscillatory thermocapillary convection. Phys Fluids 14(1):225–231

    Article  Google Scholar 

  • Carpenter BM, Homsy GM (1989) Combined buoyant-thermocapillary flow in a cavity. J Fluid Mech 207:121–132

    Article  MATH  Google Scholar 

  • Cerisier P, Jamond C, Pantaloni J, Charmet JC (1984) Déformation de la surface libre en convection de Bénard–Marangoni. J Phys 45:405–411

    Google Scholar 

  • Cerisier P, Perez-Garcia C, Jamond C, Pantaloni J (1987) Wavelength selection in Bénard Marangoni convection. Phys Rev A 35:1949–1952

    Article  Google Scholar 

  • Cerisier P, Billia B, Rahal S (2001) Evolution and stability limits of patterns in surface-tension-driven Bénard convection. J Non-Equilib Thermodyn 26(02):99–118

    Article  MATH  Google Scholar 

  • Chiang KT (2005) Effect of a non-uniform basic temperature gradient on the onset of Bénard–Marangoni convection: stationary and oscillatory analyses. Int Commun Heat Mass Transf 32:192–203

    Article  Google Scholar 

  • Colinet P, Legros JC, Velarde MG (2001) Nonlinear dynamics of surface-tension-driven instabilities. Wiley-VCH, Berlin

    MATH  Google Scholar 

  • Colinet P, Nepomnyashchy AA, Legros JC (2002) Multiplication of defects in hexagonal patterns. Europhys Lett 57:480–486

    Article  Google Scholar 

  • Davis SH, Homsy GM (1980) Energy stability theory for free-surface problems: buoyancy–thermocapillary layers. J Fluid Mech 98:527–553

    Article  MATH  MathSciNet  Google Scholar 

  • Dijkstra HA (1995a) Surface tension driven cellular patterns in three-dimensional boxes—part II: a bifurcation study. Microgravity Sci Technol 8(2):70–76

    Google Scholar 

  • Dijkstra HA (1995b) Surface tension driven cellular patterns in three-dimensional boxes—linear stability. Microgravity Sci Technol 7(4):307–312

    Google Scholar 

  • Dijkstra HA (1995c) Surface tension driven cellular patterns in three-dimensional boxes—part III: the formation of hexagonal patterns. Microgravity Sci Technol 8(3):155–162

    MathSciNet  Google Scholar 

  • Hadji L (1996) Nonlinear analysis of the coupling between interface deflection and hexagonal patterns in Rayleigh–Benard–Marangoni convection. Phys Rev E 53:5982–5992

    Article  Google Scholar 

  • Johnson D, Narayanan R (1996) Experimental observation of dynamic modes switching in interfacial-tension-driven-convection near a codimension-two point. Phys Rev E 54:3102–3104

    Article  Google Scholar 

  • Koschmieder EL, Prahl SA (1990) Surface tension-driven Bénard convection in small containers. J Fluid Mech 215:571–583

    Article  Google Scholar 

  • Koschmieder EL, Switzer DW (1992) The wavenumbers of supercritical surface-tension-driven Bénard convection J Fluid Mech 240:533–548

    Article  Google Scholar 

  • Kraska JR, Sani RL (1979) Finite amplitude Bénard–Rayleigh convection. Int J Heat Mass Transf 22:535–546

    Article  MATH  Google Scholar 

  • Merkt D, Bestehorn M (2003) Bénard–Marangoni convection in a strongly evaporating fluid. Phys D 185:196–208

    Article  MATH  MathSciNet  Google Scholar 

  • Narayanan V, Page R, Seyed-Yagoobi J (2003) Visualization of air flow using infrared thermography. Exp Fluids 34(2):275–284

    Article  Google Scholar 

  • Nepomnyashchy AA, Velarde MG, Colinet P (2002) Interfacial phenomena and convection. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  • Ondarçuhu T, Mindlin G, Mancini HL, Pérez-Garcia C (1993) Dynamical patterns in Bénard–Marangoni convection in a square container. Phys Rev Lett 70:3892–3895

    Article  Google Scholar 

  • Ondarçuhu T, Mindlin G, Mancini HL, Pérez- Garcia C (1994) The chaotic evolution of patterns in Bénard–Marangoni convection with square symmetry. J Phys Condens Matter 6:A427–A432

    Article  Google Scholar 

  • Parmentier PM, Regnier VC, Lebon G, Legros JC (1996) Nonlinear analysis of coupled gravitational and capillary thermoconvection in thin fluid layers. Phys Rev E 54(1):411–423

    Article  Google Scholar 

  • Perez-Garcia C, Pantaloni J, Occelli R, Cerisier P (1985) Linear analysis of surface deflection in Bénard–Marangoni instability. J Phys 46:2047–2051

    Google Scholar 

  • Regnier VC, Dauby PC, Parmentier PM, Lebon G (1997) Square cells in gravitational and capillary thermoconvection. Phys Rev E 55(6):6860–6865

    Article  Google Scholar 

  • Regnier VC, Dauby PC, Lebon G (2000) Linear and nonlinear Rayleigh–Bénard–Marangoni instability with surface deformations. Phys Fluids 12(11):2787–2799

    Article  Google Scholar 

  • Schatz MF, Neitzel GP (2001) Experiments on thermocapillary instabilities. Ann Rev Fluid Mech 33(6):93–127

    Article  Google Scholar 

  • Schwabe D (2006) Marangoni instabilities in small circular containers under microgravity. Exp Fluids 40(6):942–950

    Article  Google Scholar 

  • Scriven LE, Sternling CV (1964) On cellular convection driven by surface-tension gradients: effects of mean surface tension and surface viscosity. J Fluid Mech 19:321–340

    Article  MATH  MathSciNet  Google Scholar 

  • Young Y, Riecke H (2002) Mean flow in hexagonal convection: stability and nonlinear dynamics. Phys D 163:166–183

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Batna, Rue Boukhlouf Mohamed el Hadi, 05000, Batna, Algeria

    S. Rahal

  2. IUSTI, CNRS UMR 6595, Polytech’Marseille, Technopôle de Château-Gombert, 5 rue Enrico Fermi, 13453, Marseille, France

    P. Cerisier

  3. Department of Aerospace Engineering, Osaka Prefecture University, Gakuen-cho 1-1, Sakai, Osaka, 599-8531, Japan

    H. Azuma

Authors
  1. S. Rahal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Cerisier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Azuma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Rahal.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rahal, S., Cerisier, P. & Azuma, H. Bénard–Marangoni convection in a small circular container: influence of the Biot and Prandtl numbers on pattern dynamics and free surface deformation. Exp Fluids 43, 547–554 (2007). https://doi.org/10.1007/s00348-007-0323-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00348-007-0323-1

Keywords

Search

Navigation